Nonlinear sigma models for monitored dynamics of free fermions

We derive field theory descriptions for measurement-induced phase transitions in free fermion systems. We focus on a multi-flavor Majorana chain, undergoing Hamiltonian evolution with continuous monitoring of local fermion parity operators. Using the replica trick, we map the dynamics to the imaginary time evolution of an effective spin chain, and use the number of flavors as a large parameter for a controlled derivation of the effective field theory. This is a nonlinear sigma model for an orthogonal $N\times N$ matrix, in the replica limit $N\to 1$. (On a boundary of the phase diagram, another sigma model with higher symmetry applies.) Together with known results for the renormalization-group beta function, this derivation establishes the existence of stable phases -- nontrivially entangled and disentangled respectively -- in the physically-relevant replica limit $N\to 1$. In the nontrivial phase, an asymptotically exact calculation shows that the bipartite entanglement entropy for a system of size $L$ scales as $(\log L)^2$, in contrast to findings in previously-studied models. Varying the relative strength of Hamiltonian evolution and monitoring, as well as a dimerization parameter, the model's phase diagram contains transitions out of the nontrivial phase, which we map to vortex-unbinding transitions in the sigma model, and also contains separate critical points on the measurement-only axis. We highlight the close analogies as well as the differences with the replica approach to Anderson transitions in disordered systems.


I. INTRODUCTION
This paper develops field theory descriptions for systems of free fermions that are continuously monitored.We may imagine a chain of fermions evolving under a quadratic hopping Hamiltonian (perhaps timedependent), and an experimentalist who makes repeated measurements of local fermion bilinears [1][2][3] at all positions throughout the chain.The evolving state of the fermions then depends on the specific random outcomes of these measurements, but we can ask about the statistical ensemble of evolving states (quantum trajectories).To answer the question "what are the universal properties of this ensemble?"we need appropriate long-wavelength descriptions.
It is clear that there is a landscape of phases and crit-ical points to explore here (and these quadratic models are much more accessible numerically than generic interacting systems [3,26]).There are also intriguing connections with other low-dimensional critical phenomena.Here we will construct effective field theories that allow universal results.As we will show, these effective field theories are close cousins of familiar models from magnetism and disordered systems, but with differences reflecting the state-dependent randomness of quantum measurements.
So far, however, free fermion measurement transitions (FFMTs) in generic models have been established only numerically.Majorana circuits with "swap" operations and projective measurements [6,27] are solvable by a classical mapping, but that mapping does not extend to general circuits or Hamiltonians.(Many qualitative features of this model are nevertheless shared with generic models [22], as we will discuss later.)Models in which the randomness of quantum measurements is eliminated by complete "postselection" of outcomes can also be solvable [9,18,21,28], but are in a different regime.
A more general approach is to derive an effective model by using the replica trick to average over the randomness, in analogy to monitored interacting circuits [29,30] (cf.also unitary circuits [31] and tensor networks [32] and related mappings [33][34][35]).This approach works with an effective dynamics for N identical replicas of the system.Since the dynamics is anyway random due to the random measurement outcomes, a natural simplification is to take the Hamiltonian also random in space and time, so that the averaging is over both types of randomness.
The replica trick introduces a symmetry: in the inter-arXiv:2302.12820v2[cond-mat.stat-mech]8 Dec 2023 acting case this involves discrete permutations of replicas, but for free fermions it is possible to make continuous rotations between replicas [8,10,25,35].As we will discuss, this is in close analogy to other disordered free fermion systems [8,36], but a distinctive feature of the measurement problems is that the required replica limit is N → 1, rather than N → 0 as in more familiar disordered systems.The "additional" replica of the system is used to express the Born-rule probability of a quantum trajectory, which must be included in averages.
Ref. [25] examined a simple quantum circuit for a 1D Majorana chain involving weak measurements of fermion parity iγ i γ i+1 for adjacent sites, and by averaging over randomness obtained an effective model that was analyzed for the case N = 2. (See also Ref. [10], where a different kind of effective model was derived.)However, obtaining an effective theory for the physically relevant case N → 1 requires the replica limit to be addressed, and this has proved challenging.
Building on the models of Refs.[6,25] (cf.also [22,37,38]), our starting point will be a generalized Majorana chain with continuous-time evolution and with an arbitrary number N F of "flavors" at each site (not to be confused with the number of replicas).In this context it will be possible to obtain a continuum description in a controlled way, using 1/N F as the control parameter.On symmetry grounds, we expect the resulting field theories to extend to all N F (including the case N F = 1, which is dual to a monitored Ising chain).The actions we obtain have the schematic form: where the matrix Q lives on an appropriate manifold.
In the most generic case that we discuss, Q ab (x, t) is an N × N orthogonal matrix, where N → 1 is the number of replicas.(The ". .." in Eq. 1 stands for a topological term that appears for the distinct theory that applies on a boundary of the model's phase diagram.)These NLσMs yield universal results for classes of monitored free fermion systems, as we will discuss.The monitored Majorana chain (and other models with the same symmetry) has a stable phase in which the nth Rényi entanglement entropy across a cut, in a system of size L, has the universal form Note that, unusually, this model allows an exact result for the von Neumann entropy (n = 1).We will also check this scaling numerically.The model shows phase transitions out of the stable phase whose exponents are not known exactly, but which can be understood qualitatively using the RG.The model also has disentangled phases, which are disordered phases for the sigma model field, driven by proliferation of topological defects (though unlike the transitions in the N = 2 model [25] these are not Kosterlitz-Thouless transitions).In Fig. 1 ∆ J 2 /Γ Figure 1.Structure of the phase diagram for the model with NF = 1, superimposed with a schematic illustration of the RG flows.Here, J 2 is the strength of the stochastic nearestneighbor hopping term.Weak measurements are performed on odd/even bonds with strength Γ(1 ± ∆) respectively.The pink region is a stable phase with nontrivial entanglement scaling (Eq.2).It is separated by continuous phase transitions (green lines) from two area-law phases (white regions), where the steady states are dressed versions of the fully dimerized states depicted at the bottom left and right of the diagram.The nontrivial phase is governed by marginal flow to the g = 0 fixed point of the SO(N ) NLσM (Eq. 1) and the green lines represent disordering transitions of this model.The J = 0 axis (blue) has a higher symmetry and has an unstable critical point at ∆ = 0 (blue dot).
we indicate schematically the topology of the phase diagram in the simplest case N F = 1.
In a notable step, Ref [8] made a symmetry classification of Gaussian random tensor networks (these can also be interpreted as evolution operators, though they do not correspond to monitored dynamics with Born's rule).The authors showed that the most generic Gaussian networks have symmetry matching that of network models [52] in symmetry class DIII [53], and proposed that the DIII NLσM at N → 0 should apply to such networks on grounds of symmetry.This is consistent with the symmetry we find in our explicit derivation of a field theory for a generic continuous time model.The universal behavior we find is however different, most importantly because of the change in N arising from Born's rule (this agrees with the claim in [35] and answers a question raised in [8]).Analyzing the field theory also shows that the scaling of entanglement in "metallic" phases differs from the scale-invariant form assumed previously in the tensor network context [8].
The different value of N in the NLσM leads to different amplitudes and exponents and in some cases to a different topology for phase diagram.However, key ideas from localization carry over to the N → 1 theories.(The localization field theory involves two spatial coordinates, rather than a space and a time coordinate, so the analogy is with "Anderson localization in spacetime".)Very loosely, entanglement in the measurement problem plays a similar role to conductance in the localization problem, because in the measurement problem the inverse of the running sigma model coupling, g(L) −1 (cf.Eq. 1) is a measure of the strength of entanglement at a certain lengthscale L, whereas in the localization context g(L) −1 is proportional to a conductivity.
A well-known phenomenon in localization is that the coupling can flow the "wrong way" in the replica limit, with g(L) −1 ∝ ln L; so that the flow approaches the "ordered" fixed point at g = 0, 1 contrary to conventional 2D models with continuous symmetry.In the present context, this flow leads to the bipartite entanglement scaling as in Eq. 2. As a result of the flow of 1/g (the increase in the "strength of entanglement" with scale) this entanglement is larger than the ln L that holds in scale-invariant theories [55].The more standard scale-invariant behavior holds at critical points (as opposed to the stable phase) that we find in the phase diagram of the measurement model.
It is interesting that the structure of the phase diagram in Fig. 1 for the fairly generic chain studied here is similar to that in the much simpler quantum circuit that maps to a classical loop model [6].This robustness of that phase diagram was previously found numerically in Ref. [22], which explored a different (but also relatively generic) model.We explain this similarity by close structural similarities between the sigma models describing the loop model and those describing more generic Majorana chains.However, the two kinds of problems are in different universality classes.
The recipe outlined in this paper, for deriving continuum nonlinear sigma model descriptions, can be applied to models with other symmetries or with other dimensionalities, and opens the way to a more general classification of monitored free fermion systems.(It could also be used to address other phenomena such as boundary decoherence [56,57].)A panoply of critical behaviors are shown by NLσMs for Anderson localization [42]: it will be very interesting to explore the corresponding landscape at the "N → 1" level that describes monitored dynamics.
This paper is organized as follows.Sec.II introduces our model and its hybrid dynamics, and derives the replica Hamiltonian.In Secs.III and IV we carry out the mapping to an effective spin chain and derive the effective SO(N ) NLσM description in the limit of large flavor index.In Sec.V we discuss the qualitative features of the phase diagram, while in Sec.VI we use this theory to compute the basic universal properties of the stable critical phase, which we verify in Sec.VI D. In Sec.VII we discuss the transition out of this phase via proliferation of Z 2 instantons.In Sec.VIII we discuss the different NLσM, on the manifold SO(2N )/U(N ), that we propose for the measurement-only version of the model, and the RG flows between models with different symmetries.Finally, we conclude in Sec.IX with an outlook.We consider a chain of Majorana fermions subject to continuous monitoring -in other words to repeated weak measurement in the limit where the measurements are very frequent and very weak.We expect the universal results also to apply to a larger family of monitored quantum circuit models [22,25], but the continuous time formulation simplifies the subsequent mappings.
Let us start with the Hamiltonian that generates the unitary part of the dynamics: Here γ j,µ are standard Majorana operators satisfying The fermions carry a site index j = 1, . . ., L, and we have also given them a flavor index µ = 1, . . ., N F .Flavors are a tool to aid the derivation of the NLσM, which will be controlled at large N F .However we also expect this continuum theory to apply for N F = 1, when there is just a single Majorana at each site.This case is also dual to a monitored Ising chain.We consider the limit in which the couplings J µν j (t) are white noise, i.e. are random variables with vanishing mean and with variance E G [•] denotes the Gaussian average.Note that J 2 has the units of a rate.We now include monitoring of all the fermion-parity operators iγ j,µ γ j+1,ν for adjacent sites.Physically, continuous-time monitoring can be thought of as the limit ∆t → 0 of a discrete-time process, with the strength of the measurements simultaneously approaching zero.A nice feature is that the resulting dynamics in the limit ∆t → 0 is independent of essentially all the details of the discrete-time measurement protocol, and is characterized solely by a measurement rate for each measured operator iγ j,µ γ j+1,ν .We will take this rate to be independent of the flavor indices, but we allow for a staggered dependence on the spatial position [6]: We include the dimerization ∆ in order to be able to drive a phase transition into a disentangled phase.In the regime where Γ ≫ J 2 , and where ∆ is close to either +1 or −1, the dynamics consists almost entirely of measurements of a single sublattice of bonds.When N F = 1, this manifestly leads at long times to area-law states of the forms shown in Fig 1 .In Fig 1 we anticipate the schematic phase diagram for N F = 1 that will result from our analysis (we will discuss the N F -dependence in later sections).When ∆ = 0 the model has translation symmetry by one site: we will find that when N F is odd this symmetry guarantees that the model is nontrivially entangled, consistent with Fig 2.
The evolution of the density matrix that results from the combined dynamics can be written down in many ways.In this section it will be convenient for us to define it directly in continuous time, using a loose notation 2 analogous to Eq. 3, in terms of a non-Hermitian extension of the Hamiltonian.This formulation is equivalent to the (perhaps more familiar) stochastic Schrödinger equation formalism [58][59][60] (we will use the latter for simulations in Sec.VI D).In Appendix A we give a more careful definition of the evolution as the ∆t → 0 limit of a discrete time process, and explain how the non-Hermitian evolution below arises.
(7) Schematically, the mapping to non-Hermitian Hamiltonian evolution arises because a discrete weak measurement of iγ j,µ γ j+1,ν involves conjugating the density matrix with a Kraus operator that is proportional to exp(−dM µν j γ j,µ γ j+1,ν ).Here dM µν j is proportional to the measurement outcome multiplied by an infinitesimal measurement strength: see Appendix A. In Eq. 7, 2 A standard fact about white-noise (such as J µν j (t)) is that it is not a well-defined function of t, due to divergences in the limit of vanishing correlation time.Therefore to give a precise meaning to the notion of a "realization" of J µν j (t) we should discretize time.But ultimately all we will need are averages such as Eq. 5, which are well-defined in the ∆t → 0 limit.The quantity M µν j (t) below (the measurement record) has a similar status: ultimately we will only need its expectation values.Appendix A gives Eqs. 8, 9 a precise meaning by discretizing time.
M µν j (t) is the "continuum limit" of the list of measurement outcomes (indexed by t) of the associated observable iγ j,µ γ j+1,ν .We will define M µν j (t) in the continuum limit through its statistics.While J µν j (t) is simple white noise, the statistics of M µν j (t) are nontrivial as a result of Born's rule.
First, however, consider the evolution of the state ρ J,M (t) conditioned on a given realization of the couplings J and given measurement outcomes M (i.e. on a given quantum trajectory), and starting in some initial state ρ(0).It is convenient first to define the evolution for an un-normalized version of the density matrix, which we denote by ρ(t): This has the same structure as for unitary evolution, but the time-evolution operator K J,M (t) is nonunitary: (T is time-ordering.)Note that ρJ,M (t) depends on J(t ′ ) and M (t ′ ) over the full time interval [0, t], i.e. on the complete trajectory.The physical density matrix is obtained by normalizing ρ(t), i.e. as ρ J,M (t) = ρJ,M (t)/tr ρJ,M (t).
To complete the definition of the continuous time dynamics we must specify how to average over M µν j (t).To be concrete, let us consider the physical average E[•] of some quantity that depends on the state ρ J,M (t), and therefore on J(t ′ ) and M (t ′ ) for t ′ ∈ [0, t].We may define these expectation values in two steps.
First, we define Gaussian averages, denoted E G [•], in which both J and M are treated as white noise: the variance of J is given in Eq. 5, and the variance of M is set by the measurement rate: (10) The correct measure for M is related to this white-noise measure by a factor that comes from Born's rule.In outline, the probability for a given measurement record {M µν k (t ′ )} is proportional to tr ρJ,M (t).(See Eq. 8 for the definition of ρ.)This modifies expectation values by the same factor of tr ρJ,M (t): The denominator here is a trivial constant ensuring that the measure on trajectories is normalized.For notational simplicity we can set this denominator to 1 simply by absorbing an additive constant into H, so that: For details of the mappings above, see Appendix A, where we make the formulas more precise by starting with a discrete-time measurement process defined in terms of Kraus operators.Eq. 12 is convenient because it expresses averages of physical quantities in terms of simple Gaussian averages.In particular, the basic object in the replica approach is the following Gaussian average of N copies of the unnormalized density matrix: Formally, this can be viewed as a density matrix for N copies of the Majorana chain.We will use two properties of this object.First, it has a simple time evolution, specified by an effective Hamiltonian that we will derive shortly.Second, by taking the replica limit N → 1, all the physical averages of interest can be expressed as traces of ρ (N ) (t) [29].

B. Review of replica formalism
In this section, we recall how the replica approach works for an illustrative class of observables.We simplify the discussion with respect to standard expositions by focusing on generating functions for the entropy, instead of averages involving logarithms.
The full probability distribution of the nth Rényi entropy is encoded in the generating function In particular, the behavior near k = 0 gives the mean of S n .Here it will be sufficient to treat k and n as positive integers, since we can analytically continue to real values at the very end.We have suppressed the subscripts on ρ(t) = ρ J,M (t) to avoid clutter.Using Eq. 12, the expectation value above becomes where the factors of Tr ρ(t) arise both from the normalization of ρ(t) and from the nontrivial factor in Eq. 12.
We now write this as lim We then study the limit by analytically continuing from integer values of N with N ≥ nk. 3 The utility of this is that the density matrices then appear only to positive powers, so that the above average can be written in terms of ρ(t) ⊗N : The trace is now taken in the Hilbert space of N copies of the system.C n is an operator that cyclically permutes n out of the N copies of the system: see Appendix C for an explanation of this notation.While the notation on the right hand side of (17) may appear formal, it is just a way of expressing the pattern of index contractions needed to give the traces in Eq. 16.
Finally, we can push the expectation value in (17) inside the trace, so that the generating function for the entropies is written in terms of a trace of the object ρ (N ) (t) defined in Eq. 13.

C. Effective replica Hamiltonian
In order to write the evolution of ρ (N ) (t) in a concise way, it will be convenient to implement a standard operator-to-state mapping.Instead of thinking of ρ (N ) (t) as a density matrix for N copies of the Majorana chain, we will think of it as a wavefunction (a ket) for 2N copies of the Majorana chain: This is a standard transformation for bosonic Hilbert spaces, and can be adapted to fermionic ones by mapping them to a bosonic one first.Here we follow the convention of Ref. [61], which we summarize in Appendix B. Under this operator-state mapping, traces of the replicated density matrix (which we will need to compute entanglement entropies) are mapped to state overlaps: schematically, where C is any operator on the N -copy Hilbert space and |C⟩ a corresponding state in the 2N -copy space.From Eq. 17, we then see that the generating function for S n (t) becomes such a transition amplitude, ⟨C k,n |ρ (N ) ⟩, for a state |C k,n ⟩ that we define precisely later.After this mapping, the evolution of the state |ρ (N ) ⟩ (for any fixed natural number N ) has the form of conventional imaginary time evolution with a Hamiltonian H: More precisely, there is a separate Hamiltonian H for each value of N , acting on the Hilbert space of 2N copies of the Majorana chain.Since the random quantities J and M have been averaged out, H is non-random, and independent of time (so depends only on N ).The evolution (20) of the replicated state follows from Eq. 8, and is of the form4 The term inside the average involves the exponential of 2N copies of the original Hamiltonian (7): Here, for each physical Majorana operator γ j,µ , we now have 2N "replicated" Majoranas.We have labelled these replicas by an index σ = ± which distinguishes the first N copies in Eq.21 from the last N copies, together with an index a = 1, . . ., N .
The Gaussian average (21) yields: We have separated terms coming from the unitary and non-unitary parts of the dynamics: This interacting fermion Hamiltonian is much more conveniently written as a spin model.We describe this next.

III. MAPPING TO A SPIN CHAIN
This Hamiltonian takes a much simpler form when reinterpreted in terms of generators of SO(2N ) rotations.Above we have used normalization conventions so that N F appears explicitly in (20): this is convenient because 1/N F will play the role of ℏ in the semi-classical treatment.
There is a natural action of so(2N ) at each site of the chain [25].For a moment, let us write the replica multiindex (σa) as a single index α = 1, . . ., 2N .Then the so(2N ) rotations are generated by the quantum operators The generator S αβ j corresponds to an infinitesimal rotation between Majoranas γ α and γ β from distinct replicas.They have commutation relations (26) Apart from the factor of 1/N F arising from our normalization convention, these are the standard commutation relations of rotation generators in 2N dimensions.
The set of generators S αβ j at a site j make up an antisymmetric 2N × 2N matrix of quantum operators, which we denote by S j (since it will play a role analogous to the spin operator S j in a Heisenberg antiferromagnet).Returning to the notation (σa) that distinguishes forward and backward replicas, we order the replicas as (+1), . . ., (+N ), (−1), . . ., (−N ).Then we denote the blocks of S j as: so that L = −L T , R = −R T , Q are N × N matrices whose entries are operators.
In terms of these operators, the Hamiltonian is (up to an additive constant that we neglect) We use "tr" to denotes traces for 2N × 2N (and later N × N ) matrices, reserving "Tr" for the many-body Hilbert space.The couplings in H are For now we have specialized to the undimerized case, Γ j = Γ; we will return to the effect of dimerization later.For generic J, Γ the symmetry of this Hamiltonian is only a subgroup of SO(2N ) [30], because the Q generators and the L, R generators appear with different coefficients in (28).This is analogous to the fact that the Hamiltonian of the XXZ spin chain, which is written in terms of the so(3) generators S x , S y and S z , is invariant only under a subgroup of SO (3).The symmetry of H is enlarged when either J 2 = 0, or Γ = 0, as we will discuss in Sec.VIII.
To complete the definition of the spin chain, we must specify which representation of SO(2N ) the generators in H act on: we do this in the next section.In the XXZ chain analogy, this would correspond to specifying the magnitude of the spin at each site.
In Sec.IV we will show that the low-energy dynamics of the spin chain are described by the SO(N ) non-linear sigma model (NLσM).The limit of large N F will allow a quantitatively controlled derivation (but, on grounds of symmetry, this continuum theory may be extended to small values of N F ).We see from Eq. 26 that the commutators of the S operators vanish when "ℏ" := 1/N F → 0, so that at large N F there is a regime where semiclassics is accurate.This is analogous to large spin in an su(2) spin chain.

A. Symmetries and local conserved quantities
The effective Hamiltonian H above possesses a large set of symmetries.We can split them into global symmetries, which will be present in the NLσM, and an extensive number of local integrals of motion, which will be completely fixed by the temporal boundary conditions for the evolution.For concreteness, we take the physical initial state to be the maximally mixed state, though, as we will discuss, the specific choice is not crucial.
H is invariant under an [O(N ) × O(N )] ⋊ Z 2 global replica symmetry.The two orthogonal groups correspond to rotations among Majorana operators within the same sector (i.e.within the σ = + or σ = − sector), and the corresponding symmetry generators are j L j and j R j respectively (replica indices omitted).The Z 2 operation is an exchange of forward and backward replicas: γ +a j,µ ↔ γ −a j,µ .In addition, each site possesses local integrals of motion that label choices of symmetry representation.The on-site Hilbert space splits into different representations of so(2N ), and the choice of representation is conserved in time.This conservation is due to the fact that H is written entirely in terms of local so(2N ) generators S αβ j .In fact, since S αβ j is itself a sum of generators acting on different flavors (see Eq. 25), the choice of representation is separately conserved for each site j and for each flavor µ.
Fortunately, boundary conditions greatly simplify the Hilbert space, by isolating a unique choice of representation at each site.The states that are in other representations can be discarded, since they have no overlap with the initial state that we have chosen.In fact, the same choice of representation is also fixed by the boundary states that we impose at the final time in order to compute entanglement entropies (Sec.VI), which is why the specific choice made for the initial state is not crucial.
It suffices to consider a single site j.First, for a given flavor index µ, the matrices 1  2 [γ α j,µ , γ β j,µ ] form a representation of so(2N ), isomorphic to the spin representation or its complex conjugate, with highest weight ω s or ω s respectively, depending on the value of the fermion parity number R j,µ = ±1.In our case the initial state fixes R j,µ = +1 for every site and flavor index, so that we are dealing with spin representations, as we explain in Appendix D.
Next, the operators (25) act on the tensor product of those representations, so that the N F flavors combine into various possible irreducible representations of so(2N ).Among those, the relevant irreducible representation is that with maximal highest weight, namely with weight N F ω s , cf.Appendix D. This is analogous to combining N F spin-1/2s into a state with maximum possible spin N F /2.This choice of representation is an invariant of the dynamics.
This representation is the one that maximizes the value of the quadratic Casimir on each site By a direct computation this is (see Appendix D) with C j ≃ 2N at large N F .

IV. MAPPING TO NONLINEAR σ-MODEL
Next we show that the low-energy dynamics of the spin chain ( 28) is captured by a NLσM for a field Q(x, t) that lives on the compact manifold SO(N ).More precisely, this applies for the generic case with nonzero J 2 and Γ: the case J = 0 has a higher symmetry and will be discussed separately in Sec.VIII.
An intuitive way to obtain the continuum theory is via the equations of motion that arise in the semiclassical (large N F or small ℏ) limit [62].These equations of motion allow us to deduce the Lagrangian, which may then be used to "requantize" the theory by writing the path integral.This is the route we will follow in this section.(An alternative route would be to start with a coherent states path integral on the lattice, cf.Appendix E.) We expect that our derivation of the continuum field theory, including the values of nonuniversal constants, is quantitatively controlled at large N F .
Let us outline the steps in a little more detail.First, in Sec.IV A we identify the "ultraviolet" degrees of freedom as antisymmetric matrices in SO(2N ).These are the fields (either in a path integral or in the semiclassical equations of motion) corresponding to the spin S αβ that appears in H.
Next, by analyzing the classical ground-state manifold (i.e. at large N F ), we see that only the field "Q" in the block decomposition (27) of the matrix S is a massless degree of freedom.This suggests that we should eliminate the modes L and R to obtain an effective Lagrangian L(Q).This can be done at the level of the equations of motion.
We obtain the semiclassical equations of motion from H in the usual way in Sec.IV B (the Heisenberg equation of motion for the operator S becomes a classical differential equation at large N F ).The large N F limit also allows us to take a controlled spatial continuum limit, and then to eliminate L, R.This gives a simple equation of motion for Q, from which we can identify the Lagrangian.

A. Degrees of freedom and classical ground states
In the limit N F → ∞ the matrix of quantum operators S αβ on a site becomes an antisymmetric matrix of classical phase space coordinates.In this limit we also have the constraint or in terms of the N × N block decomposition (27), Eq. 32 follows from applying the quantum operator (S T S) αβ to an arbitrary state in the representation determined in the previous section, and taking the large N F limit.We provide further details in Appendix D.
Eq. 32 shows that the appropriate semiclassical degree of freedom S lives on the space of antisymmetric orthogonal matrices.More precisely, S lives on the part of this space which is continuously connected to the point Q = 1 1, L = R = 0, as can be seen by noting that the Pfaffian of S -fixed by the so(2N ) representation -is Pf S = (−1) N (N −1)/2 for N F → ∞.Such matrices form a compact symmetric space isomorphic to SO(2N )/U(N ), cf.Appendix E.
We note that it is possible to formulate a coherent-state path integral for the spin chain as a functional integral over the local degrees of freedom S living in the same symmetric space, as we discuss in Appendix E. 5 This is reassuring as it confirms that the field theory degrees of freedom that we identify in the semiclassical limit also make sense at finite N F .
Next, consider the classical ground states that are obtained by minimizing our Hamiltonian H (28) with the relevant on-site representation.If the rates J 2 and Γ for the unitary and nonunitary parts of the dynamics are both nonzero, then J ⊥ > |J ∥ | in Eq. 28.In this case the energy is minimized by taking 6 L, R = 0, and Q j independent of j.By the constraints described above, the "order parameter" Q is then a (proper) rotation matrix: In passing, we note that there are more general ground states, with the same energy in the semiclassical limit, but which do not satisfy the constraint on the on-site representation imposed by the boundary conditions. 7

B. Equations of motion and continuum limit
The Heisenberg evolution of an operator O is 8 The factor of N F plays the role of an effective "1/ℏ", and is consistent with the normalization of the Hamiltonian in Eq. 20.In the N F → ∞ limit, Eq. 35 yields classical equations of motion for the variables L ab j , R ab j , and Q ab j . 5The procedure is standard [63]: we define coherent states |S⟩ that are labelled by the expectation value of S, and construct the path integral using resolutions of the identity of the schematic form dµ(S) |S⟩ ⟨S|. 6Recall that the Casimir s 2 a = N .These states have the same value of the quadratic Casimir, but in general have different values of the higher Casimirs.Our case (34) corresponds to taking sa = 1 for all a. 8 To avoid imaginary units in the equations of motion, we derive them for real time evolution, and Wick rotate back to imaginary time (as required for Eq.20) after obtaining the Lagrangian.
Using the commutation relations in Eq. 26 yields where "transp."denotes the transpose of the previous terms (ensuring L j and R j remain antisymmetric matrices).The equation for dR j /dt is obtained from that for dL j /dt by the exchanges L → R, Q → −Q T , as required by the Z 2 symmetry exchanging forward and backward replicas.These equations of motion simplify in the continuum limit, i.e. after keeping only lowest orders in a derivative expansion.Taking this limit amounts to isolating low-momentum fluctuations of the fields.This is quantitatively accurate at large N F . 9  If we denote the lattice spacing by a, then the continuum limit is formally equivalent to an expansion in a, so that a lattice operator O j±1 is expanded as Retaining the leading terms in a in the equations of motion, and making the rescaling L → L/a and R → R/a, we find: We used the fact that 2(J ⊥ − J ∥ ) = Γ is the measurement rate.The rescaling of L and R is possible because their value is zero in the ground-state manifold.In the following, we return to units where a = 1.
In passing we note that the continuum equations of motion could have been equivalently obtained, following Ref.[62], by first taking the continuum limit at the level of the Hamiltonian 9 In more detail: The justification for dropping higher derivative terms can be understood in field theory language (anticipating slightly).Dropping these terms amounts to modifying the Lagrangian for modes with momentum comparable to 1/a.We could worry that this will affect the IR Lagrangian, since integrating out high-momentum modes can renormalize the Lagrangian for the low-momentum modes.However, these renormalizations are of order "ℏ" ∼ 1/N F relative to the bare Lagrangian.This guarantees that there is a momentum scale Λ(N F ), which is parametrically small for large N F , above which the renormalizations are negligible.This small momentum scale justifies the derivative expansion.At still smaller scales, the nontrivial RG flow does become important: see Eq. 50.
up to a constant, and noting that only the commutation relations which involve L or R remain nontrivial in the continuum limit, while The equations above should be supplemented with the kinematic condition arising from the S T S = 1 constraint, which in terms of the rescaled variables gives Using these constraints, we can eliminate L and R from the equation of motion for ∂ 2 t Q that is obtained by differentiating (37).The terms with dL/dt and dR/dt can be eliminated using the relevant equations of motion (38), (39), and the terms L and R can be eliminating using the identities (42) which follow from (37) together with the constraint (41).Finally, Equivalently, we can rewrite this equation in the more suggestive form As we will see in the next section this is the equation of motion of the NLσM.

C. SO(N ) NLσM Lagrangian
The equation of motion above arises from a nonlinear sigma model action for Q, which, after rotating back to imaginary time, is The equations of motion can be recovered by noting that the allowed variations of Q are of the form Q → Q(1 + δq), with δq antisymmetric, and requiring δS/δq = 0.
The coupling constant g B does not appear in the equations of motion, but it is fixed by noting that the coefficient of the Tr(∂ x Q T ∂ x Q) term in the action is inherited directly from the Hamiltonian in Eq. 40 (see e.g.Eq.E5).Together with the result of the previous section for the velocity, this gives: This is the bare value of the coupling g: below we will consider its RG flow.This almost completes our derivation of the effective Langrangian for smooth Q configurations.The final point is that π 2 (SO(N )) = 0, which means that there is no topological Θ-term that can be added to L [64,65].(Topological Θ-terms do not affect the equations of motion, so could not be detected from them.)In Sec.VIII we will need another NLσM, where a Θ-term does play a role.
From now on we rescale time units so that v = 1, so where µ = x, t is summed over, and again Q ∈ SO(N ).
One final point will be crucial for understanding the phase diagram once we turn on the dimerization parameter.Above we discussed the regime of small g B , where Q can be treated as smooth at the lattice scale.However, since the fundamental group of the target space is we can in principle have pointlike Z 2 vortices where the order parameter Q is singular [66,67].These vortices are irrelevant at small g -this is analogous to the XY model at low temperature -so they do not affect the universal physics of the stable nontrivial phase, which we discuss in Sec.VI.However, vortices are responsible for the transition into a disentangled phase.We discuss this in Sec.VII.Remarkably, the Lagrangian in Eq. 47 gives access to exact universal results for the entanglement structure of the physical state.This is due to a special feature of the replica limit: the coupling g flows to smaller and smaller values at large scales, so that semiclassical calculations become better and better controlled.This flow gives rise to nontrivial scaling forms that can be computed exactly by combining the perturbative beta function with a saddle-point analysis of the field theory.
The two-dimensional SO(N ) NLσM appears in the context of Anderson localization, where the two dimensions are two spatial directions (i.e.there is no time coordinate).There, it describes Bogoliubov-de Gennes excitations of disordered superconductors in symmetry class DIII [53].The main difference, apart from the physical interpretation, is that in the Anderson localization problem one is interested in the limit N → 0, whereas here we need N → 1, cf.Sec.II A.
The perturbative beta function has been computed for arbitrary N [68][69][70]: The result is available up to O(g 5 R ), but the form above will suffice for now.The key point is the sign change at N = 2, which means that for each N < 2 there is a stable phase governed by the g R = 0 fixed point. 10ithin this phase for N = 1, solving (49) The universal constant of (8π) −1 in front of the logarithm will appear in the scaling of entanglement entropies.The bare value g B is nonuniversal, but we have fixed its value for large N F (Eq. 46).The explicit derivation of the sigma model in the previous section guarantees that for large N F we are in the basin of attraction of the g R = 0 fixed point.But in fact, we can argue on symmetry grounds that if N F is odd, and if the measurement rates are not dimerized, it is impossible for the NLσM to be in a disordered phase.The simplest scenario is that (in the absence of dimerization) the models with odd N F flow to the g R = 0 fixed point for any nonzero values of J 2 and Γ > 0. This is consistent with the picture for the RG flows that is proposed in the following sections.For N F even we expect flow to this fixed point for large enough J 2 /Γ , as we discuss briefly in Sec.V.
The fixed point is also completely stable (so long as replica symmetry is retained) since the only possible perturbations have larger numbers of derivatives.Therefore g R = 0 governs a stable phase of the monitored dynamics, whose properties we will discuss in Sec.VI.In particular, this phase is stable to perturbation by sufficiently weak dimerization ∆.
Of course it is possible to drive the dynamics into a disentangled phase with an appropriate perturbation that is large enough.In our model this can be done, by sufficiently strong and sufficiently strongly dimerized measurement (at least for N F = 1, and according to our proposed RG flows, for all N F ).The corresponding critical fixed point, driven by proliferation of vortices, is outside the range of validity of the perturbative beta function in Eq. 49, and will be discussed in Sec.VII.
We also emphasize that the measurement-only line J 2 = 0 is described by a different field theory, as a result of the enhanced replica symmetry [25] there: we defer discussing this theory to Sec.VIII.
Next -before describing the physical consequences of the above NLσM for the stable phase, or phase transitions out of this phase -let us preview the broader phase diagram of the monitored model.

V. PHASE DIAGRAM FOR NF = 1
In Fig. 2 we sketched our proposed phase diagram for the model with N F = 1.In this section we give a schematic overview of the basic features.The full justification for various features of this phase diagram will be given in later sections: the phase transition lines at J > 0 are discussed in terms of the SO(N → 1) NLσM in Sec.VII, while the line J = 0 is related to the SO(2N )/U(N )| N →1 NLσM in Sec.VIII, which also contains a conjecture about RG flows in that model.Fig. 2  ∆ Schematic phase diagram for NF = 1 (see also Fig. 1) The RG fixed points governing the phases and transitions are indicated by red, green and blue points, with a description of the associated field theory.Furthermore, two special points in the phase diagram are at (∆, J) = (±1, 0), denoted by black dots.Here unitary dynamics is absent and the measurements are fully dimerized.For NF = 1, the wavefunction then tends to an eigenstate of iγjγj+1 for odd (even) is also consistent with symmetry considerations for the lattice model.
For most of this Section we focus on the simplest case N F = 1, which corresponds to a chain with a single physical Majorana at each site.When N F > 1, so that each site hosts multiple Majoranas, we expect that the stable (pink) phase "touches" the J = 0 axis at a larger number of points: we comment on this briefly in Sec.V D and Sec.VIII.But while the phase diagram for N F > 1 is slightly more intricate, all the phase transition lines (at J > 0) remain in the same universality class (described by the same bulk CFT) as the phase transition lines in Fig. 2, and all critical points (at J = 0) remain in the same universality class as the J = 0 critical point in Fig. 2. The phase boundaries are symmetric under ∆ → −∆ for any value of N F .For N F odd, symmetry ensures that the model is nontrivially entangled everywhere on the vertical axis ∆ = 0.

A. Disentangled phases
A basic feature of the phase diagram is the existence of two distinct -stable -disentangled phases at large Γ, one for each sign of the dimerization ∆.These phases also appear in a Majorana model with discrete measurements [6] (or its Ising dual [11,27,71,72]) and in Refs.[22,25] where more general circuits are studied.
The extreme limits of these phases at J = 0, ∆ = ±1 have Majoranas dimerized in pairs, as shown in Fig. 2. In other words, at late times the state is a (random) eigenstate of the operators iγ j γ j+1 for either even j or odd j, depending on whether ∆ = 1 or ∆ = −1.On a finite chain, the former case has unpaired boundary Majoranas, as shown in Fig. 2. The fermion parity iγ 1 γ L of these two modes remains "hidden" from measurements, giving one "bit" that is not purified.These disentangled phases are stable (we checked this numerically using the techique described in Sec.VI D).
In terms of the NLσM, the disentangled phases are disordered phases, obtained by proliferating vortices [66,67,73].Anticipating Sec.VII, we find that the two disordered phases are distinguished by the sign of the vortex fugacity (and the line ∆ = 0 is forced to be nontrivially entangled because the vortex fugacity vanishes there).
The picture of paired Majoranas has a close analog in the effective spin chain H [8]: the spins are dimerized, and at ∆ = 1 there are gapless boundary spins that are not involved in any dimer. 11

B. Stable nontrivial phase and transition lines
Next, the phase diagram in Fig. 2 features a stable phase (shaded in pink) which, in the sigma model language, flows to g R = 0. Scaling properties of the entanglement inside this phase are discussed below in Sec.VI.The logarithmic flow of the NLσM coupling implies that entanglement entropies are larger at large scales than they would be at a conformal fixed point (Sec.VI).
Since the line ∆ = 0 is necessarily gapless, the simplest hypothesis is that this entire vertical axis, for J > 0, is in this phase.We expect the boundary of the gapless phase to meet the lines ∆ = ±1 at a finite value of J: that is, we expect that the dynamics is entangling whenever the measurements are sufficiently weak, regardless of whether they are dimerized.(There is nothing special about the lines ∆ = −1, 1 from the point of view of replica symmetry.) The phase boundary lines between the nontrivially entangled phase and the disentangled phases are governed by an RG fixed point that we discuss in Sec.VII.Fig. 1 above showed the schematic RG flow, which involves two copies of this fixed point, one for positive and one for negative dimerization (green dots).This fixed point is straightforwardly scale-invariant, so that for example the bipartite entanglement entropy is expected to scale as ln L on the phase boundary lines, as opposed to the (ln L) 2 in the stable nontrivial phase (see discussion just above Sec.VI A). 11 Nonrigorous extensions of standard ideas from spin chains may be used to argue that these two gapped phases of the spin chain cannot be connected without a phase transition (using the fact that the boundary spins transform projectively under global symmetry [74]), and that the line ∆ = 0 is gapless (in the spirit of the Lieb-Schultz-Mattis theorem [75]).The final aspect of the phase diagram is what happens to the transition lines as J 2 /Γ → 0. To answer this it is necessary to consider the distinct NLσM, with target space SO(2N )/U(N ), that applies on the J = 0 axis, as a result of higher replica symmetry there.This is discussed in Sec.VIII.There, we conjecture that the SO(2N )/U(N )| N →1 sigma model (which admits a Θ term) has an unstable fixed point with Θ = π that gives rise to a critical point at the origin of the phase diagram.We note that this critical point was already studied numerically in Ref. [12].The above conjecture for the sigma model gives the RG flow topology shown in Fig. 1.This fixed point is again scale-invariant, so that bipartite entanglement scales as ln L.

D. Phase diagram for general NF
The discussion of the Θ angle in Sec.VIII leads to the conjecture that, for a general number of flavors, N F critical points appear on the J = 0 axis.These critical points are due to Θ cycling N F times through π (mod 2π) as ∆ varies from −1 to 1.This is in close analogy to the sequence of phase transitions between dimerized and Haldane-like ground states in an antiferromagnetic chain with spin S = 2N F [76][77][78][79].
Each of the N F critical points is in the same (bulk) universality class as the J = 0 critical point of Fig. 2, so that the local topology of the phase boundaries is also the same near each critical point.As in the N F = 1 case, we expect that the chain is in the nontrivial phase when J 2 /Γ is sufficiently large, regardless of the value of ∆.These considerations give a simple conjecture for the phase diagram, shown in Fig. 3 for the case N F = 2.
Figure 4. Boundary conditions for the computation of the entropy of the whole state at finite time (left) and the bipartite entanglement entropy at asymptotically late times (right).Dashed lines denotes the direction along which ∇Q cl is non-zero, with Q cl denoting the classical configuration minimizing the energy.

E. Aside: Majorana loop model
The topology of the phase diagram is very similar to that of a Majorana quantum circuit model with discrete measurements and discrete unitary "swap" operations which maps to a classical loop model [6,67] (let us call this the "Majorana loop model").Like Fig. 2, the Majorana loop model shows a stable nontrivial phase where the entanglement scales as ln 2 L [6] (see Ref. [27] for an in-depth discussion of this phase).The similarity of the two phase diagrams is consistent with the numerical results of Ref. [22], who found previously that the phase diagram topology of the Majorana loop model was robust when more general unitary gates were allowed (and we expect that the gapless phase found here for J > 0 is the same as that found in Ref. [22] when generic unitaries are allowed).
But despite the similar phase diagram topology, the relevant universality classes are different in the loop model and in the "generic" models studied here.The similarity of the phase diagrams may be understood in terms of close similarities between the relevant field theories (which we discuss briefly in Sec.VIII).The RG flows have a similar topology, and a similar role is played by both vortex defects in the bulk of the phase diagram and a Θ term on the lower axis. 12However, there are also significant differences between the Majorana loop model and more general models.In the former, Majoranas are only ever entangled in pairs: this is an additional structure that is more constraining than Gaussianity.This results in a larger replica symmetry and a different NLσM.

VI. ENTANGLEMENT IN THE STABLE NON-TRIVIAL PHASE
We now turn to the universal properties of the stable nontrivial phase (shaded pink in Fig. 2), where entanglement entropies have non-trivial scaling with time and system size L. Asymptotically, the scaling depends only on the RG flow near the g R = 0 fixed point discussed in Sec.V, so is independent of the value of N F .In the following subsections we discuss both the purification of the state over time in a given quantum trajectory [82], and the entanglement of the pure states that arise at very long times.We first summarize the main results of this analysis.
Our initial state ρ(0) ∝ 1 has maximal entropy, for all Rényis.However, measurements tend to purify the state as t increases.We find that when L and t are both asymptotically large, and if t ≲ L, with the von Neumann entropy obtained as the n → 1 limit.The velocity v appearing here is nonuniversal: in Sec.IV we have calculated it for large N F and ∆ = 0.The above formula holds for S n , and not only for its mean S n : we find that the entropies and entanglement entropies are self-averaging (fluctuations are subleading compared to the mean).Note that the numerical prefactors in Eq. 52 and in Eq. 54 below are universal (ultimately arising from the RG flow in Eq. 50) and so apply to a larger class of monitored systems with the same symmetries.
At times t ≳ L, we find a crossover after which S n decays exponentially.The typical time of decay gives us a "purification timescale", after which the entropy becomes of order 1.The logarithms in this formula and the previous one are due to the marginal flow of the coupling g.In a model where the flow is to a fixed point at finite coupling they are absent, cf.Sec.VIII (the two kinds of flow can also be found in the Majorana loop model [6,22,27,35]).When t/τ P → ∞ we have a pure state, which we may characterize by its bipartite entanglement.For a region A including the leftmost L/2 sites (with non-periodic boundary conditions) we find that Again the additional logarithm, compared to the single power of ln L that would be dictated by scale invariance [32,55], is due to the marginal flow.Fluctuations are again subleading.
The above formulas are asymptotic formulas that hold for any fixed N F in the limit that min(t, L) becomes large.A factor (8π) −1 ln min(t, L) that contributes to these formulas arises from the asymptotic form of the inverse coupling 1/g R (L).If the number of flavors N F is large, we see from Eq. 50 and Eq.46 that there is a very large lengthscale below which the "bare" term in 1/g R (L) is larger than the logarithmic term.Consequently for large N F a much better formula is given by replacing (8π) −1 ln L by the flowing inverse coupling 1/g R (L)for example for the purification protocol with g −1 R given by Eq.50 (subleading terms in the beta function could also be taken into account to obtain more precise fits to data).
As mentioned in the introduction, the flow of 1/g to larger values as a function of lengthscale means that the entanglement structure of the state is not scale invariant (at least in the usual sense 13 ).In a scale or conformally invariant theory [32,55], the bipartite entanglement scales as ln L, which may be thought of heuristically (in the spirit of real space RG) as a sum of equal contributions from all logarithmically-spaced scales up to ln L.
Here, the contribution instead increases with scale.
While here we discuss the NLσM with N → 1 relevant to measurements, the considerations in this section carry over to the metallic phase of the theory with N → 0 which was proposed in Ref. [8] to describe Gaussian random tensor networks.Applied to this limit, our calculation gives the same (ln L) 2 entanglement scaling as in Eq. 54, but with a universal coefficient larger by a factor 2 as a result of Eq. 49.This result differs from the form proposed in Ref. [8], where scale invariance was assumed (cf. the discussion above about scale invariance).
We now describe how the results above may be obtained by minimizing the effective action S[Q] with the appropriate boundary conditions.This approach could be straightforwardly generalized to many other settings to study for example the entanglement of multiple intervals, or, say, other spatial boundary conditions. 14

A. Boundary conditions in the effective theory
We sketched in Secs.II B and Sec.II C how the generating function for any Rényi entropy S n (t) may be written as a transition amplitude in the spin chain: (56) Recall that in the replica approach, as we have formulated it, we initially take N > kn and then continue to N = 1.We have left the limit N → 1 implicit in Eq. 56: since we will always be interested in the limit N → 1 we will often simplify the notation by leaving the N dependence implict, for example in the state |C k,n ⟩ above.
The initial and final states appearing above are discussed in Appendix C. They are precisely (products of) coherent states |S⟩ of the form mentioned in Sec.IV A, which are labelled by a choice of expectation value for the matrix S. Since the states of interest have vanishing expectation values for L and R, they are parameterized by a value for the Q matrix: We have now labelled the initial and final states by their expectation values of Q (which are translation invariant in the present setting).In Appendix C we show that the initial maximally mixed state corresponds (after replication) to Q = 1 N (the N × N identity matrix), as indicated above, and that the final state |C k,n ⟩ = |Q k,n ⟩ corresponds to an N × N matrix with a block structure: The nontrivial n × n block has a cyclic form, with −1s on the subleading diagonal and +1 in the top right entry.
These Q matrices are closely related to the permutation matrices that can be used to express the pattern of index contractions at the boundary (in the language of a bosonic tensor network).However they are not equivalent to these objects due to their sign structure (see Appendix C).The signs (which would be absent in a permutation matrix) are crucial to ensure that det Q = 1 and Q ∈ SO(N ).
When we pass to the field theory, coarse-graining over microscopic scales, the initial and final states above set the boundary conditions for the field: These boundary conditions (BCs) are shown in Fig. 4 (Left).We have omitted a normalization constant in (60) since it will drop out automatically in the saddle-point calculation.
In the simplest case, we will be able to approximate the right-hand-side of (60) simply by the exponential of minus the action for the saddle-point solution Q cl(x,t) .In cases where the saddle-point configuration has variation on many lengthscales, we must separate out contributions to the free energy from different lengthscales since the value of g R renormalizes nontrivially as a function of scale.Crucially, these calculations are asymptotically exact since the theory flows to weak coupling (g R → 0).
Note that if the resulting action is simply proportional to k, then the generating function becomes that of a deterministic (nonrandom) variable.This is what we find in our leading-order calculation, indicating that fluctuations in the entanglement entropy are parametrically smaller than the mean.
The BCs above apply for the entropy of the entire system.A simple generalization gives the entropy of a subsystem A. Schematically, we must trace out region Ā from the physical density matrix: this means that the needed "index contractions" in region Ā at the final time are the same as at the initial time (here we are using the language of the equivalent bosonic system).As a result, the final time boundary condition becomes Q = 1 N in region Ā, and Q = Q k,n in region A. These BCs are shown in Fig. 4 (Right).We will focus on the case where t → ∞, so that we are computing the entanglement entropy of a pure state.
Next we apply the above to compute the entropy of the full state: due to the uniform boundary condition at time t this is the simplest case.Then we build on these results to compute the bipartite entanglement in Subsec.VI C.

B. Purification of mixed state
To compute the entropy of the final state we need (by Eq. 60) the minimal action configuration for the boundary conditions shown in Fig. 4 (Left), with Q = 1 N at the initial time and Q = Q k,n at the final time.Let us assume to begin with that t ≲ L, and for simplicity let us choose units so that the nonuniversal velocity is v = 1.
We first run the RG up to a scale that is comparable with (but somewhat less than) t, integrating out modes with shorter wavelengths.We expect the RG up to this scale to be approximately insensitive to the boundary conditions, and to produce a renormalized stiffness g −1 R (t) described by Eq. 50. 15We now have a system 15 For general N , integrating out high-frequency modes also contributes an additive constant to the free energy that is independent of boundary conditions.This is subleading at small g, but it also vanishes when N → 1, because the number N (N − 1)/2 of Goldstone modes vanishes.This is consistent with the fact that the transition amplitude with equal initial and final boundary conditions, ⟨1 N | e −tN F H |1 N ⟩, is trivial when N → 1.Up to which, measured in units of the new UV cutoff, is of length ∼ L/t in the spatial direction and ∼ 1 in the time direction.Since the stiffness is large and the cutoff is comparable with the temporal extent, we can neglect further renormalization effects and simply compute the path integral using saddle point.We need the lowest-action configuration Q cl k,n which interpolates between Q = 1 N and Q = Q k,n .Denoting the rescaled coordinates by x, t (so that t ∈ [0, 1]), we assume that the optimal Q cl k,n (x, t) is x-independent.The equation of motion then implies that Q cl k,n has the form where the antisymmetric matrix A k,n (the logarithm 16 of Q k,n ) also has a block structure, where, as usual, we leave the N -dependence of A k,n implicit.The gradient term in the renormalized action is then where the second equality is explained in Appendix I. Therefore, the minimal action in the renormalized theory is Comparing with Eq. 60 for the generating function then gives us the result for purification in Eq. 52 (where we have restored the nonuniversal velocity).Finally, for t > L we can proceed in a similar fashion.We first run the RG up to scale L, which renormalizes the coupling constant to g −1 R (L) ∼ ln L. At this stage it is convenient to use coordinates that are rescaled by a factor of L, so that the system has size 1 in the spatial direction and size t/L in the time direction (and a UV cutoff on the frequencies of order 1 in the rescaled units).The RG from this point onward is very different, as the system has become effectively one-dimensional.In 1D, the stiffness g −1 R is not dimensionless anymore, but rather it defines a correlation time t * ∝ g −1 R .Returning to the original units, this gives us the purification timescale in Eq. 53, which is the normalized sum of trajectory probabilities and is equal to 1 (Eq.12). 16While ln Q k,n is not unique, here we intend the branch of the ln that minimizes the Frobenius norm of A k,n , as this is the branch minimizing the energy: see Appendix I.
The action cost S, imposed by boundary conditions, scales as exp(−t/τ P ) (to leading order in the exponential), implying exponential decay of the entropies at very long times.Note that once t is larger than τ P , so that fluctuations become important in the effective 1D model, the (small) value of S n in a realization will no longer be close to the mean.
C. Bipartite entanglement in the pure state At times t ≫ τ P the dynamics generates an ensemble of random pure states that is independent of the initial state (pure or mixed).As discussed in Sec.VI A, the bipartite entanglement in this ensemble can be computed using the BCs shown in Fig. 4 (Right).We take t → ∞, so only the final-time BCs are important.
In contrast to the previous section, these BCs have a discontinuity.Taking A to be the right half of the chain and Ā to be the left, It is convenient to employ polar coordinates (r, θ) centered at the discontinuity, with θ ∈ [0, π].It is possible to find a vortex-like solution of the saddle-point equations that depends only on θ: where A k,n has been described above.We have neglected the spatial boundary conditions at large |x − L/2|, which will affect the solution on the largest scales but will not affect the leading term in the free energy.
The above solution has variations on all scales up to L. Since the effective coupling varies (albeit slowly) with scale, it would not be correct simply to approximate the path integral by the saddle-point action.However, this is simply remedied.
We split the free energy F into contributions from nested annuli at sequentially larger radius [83,84].Consider the additional contribution F [l,le s ] from an annulus r ∈ [l, le s ].Within this region, the typical lengthscale for variation of the expectation value of Q is l.A simple coarse-graining argument shows that F [l,le s ] may be approximated by the part of the saddle-point action for this annulus, but weighted by the coupling for this scale, g R (l). 17aking a small enough s, we obtain a differential form that may be integrated: (We may also think of e −F as a corrrelation function involving a boundary-condition-changing operator [85] at r = 0.In this language, Eq. 67 comes from the Callan-Symanzik equation for this correlator).
Using the form of the coupling (Eq.50) as L → ∞, the relation in Eq. 60 between the partition function Z = e −F and the entanglement entropy, and the value of tr a T n a n in Appendix I, gives the (ln L) 2 scaling stated in Eq. 54.However if N F is large, then there will be a very large range of scales where the bare term in 1/g R dominates, and in this intermediate range the entanglement entropies are proportional to N F ln L.

D. Numerical test of the entanglement scaling
We now proceed to numerically test the theory and in particular Eq.54 for the bipartite entanglement.For numerical convenience we restrict ourselves to N F = 1, so that larger system sizes can be reached.For the purpose of the simulation it is convenient to re-express the dynamics as a quantum state-diffusion equation [58][59][60] where H(t) is the Hamiltonian generating the unitary part of the evolution, (we use open boundary conditions for our numerics) and B j (t) depends on |ψ(t)⟩ itself: The big advantage of this approach is that dξ j is now simply the differential of a Brownian motion, i.e. dξ j = 0 and (dξ j ) 2 = Γ j dt, while higher-order cumulants vanish in the dt → 0 limit.Note that therefore we do not need to sample measurement outcomes explicitly -computing B j (t) at each time-step automatically takes Born's rule into account.We discuss how Eq. 68 can be efficiently implemented for Gaussian states in Appendix F. An important aspect is that Eq. 68 reproduces Born-rule sampling only in the dt → 0 limit, whereas the numerical simulations are necessarily performed using a finite time step dt.However, the deviation will tend to zero as dt → 0. In the following we will show the convergence of our results w.r.t.dt.
Initializing the state in the vacuum state |0⟩ associated with the fermions c j = (γ 2j−1 − iγ 2j )/2, we consider the evolution of the bipartite Rényi entropies S n,A of the evolved state as a function of time -see Appendix G for a plot as a function of time.After S n,A plateaus at longenough times, the prediction Eq. 54 is expected to hold.Averaging over a time window after the plateau, and over quantum trajectories, we can study the dependence of S n,A on the system size L.
Fig. 5 shows the result for S n,A at Γ = 1 and ∆ = 0 (we set J = 2).In the left-hand panel we see that S n,A appears to have a positive curvature as a function of ln L. Eq. 54 on its own is not a good fit, because subleading terms (neglected there) are still large at the system sizes we have access to.However, taking the discrete logarithmic derivative suppresses some of the contributions from short scales (for example a constant term would cancel out).Eq. 54 gives the scaling Plotting the LHS against ln L (Fig. 5 right), we indeed see a linear scaling with a universal slope compatible with that by predicted by the analysis in the previous sections.
In Fig. 6 we check that for a much smaller measurement rate, Γ = 0.1, the slope of ∆S n,A versus ln L remains compatible with the predictions, though with larger finite-size effects, presumably due to crossover from the unitary behavior [86].Finally, in Appendix G we show data testing the univeral dependence on the Rényi index, S n,A ∝ (n + 1)/n, finding (at these sizes) agreement at the level of a few percent.

VII. DIMERIZATION AND VORTICES
Having discussed the stable phase, it remains to analyze the transitions out of this phase (for J > 0), and also the separate universal behavior in the measurement only model (J = 0).We discuss the first of these issues here and the second in the next section.
To capture the transition into the area-law phase, which is a disordered phase for the Q field, we need to take into account vortices (which proliferate in the disordered phase). 18Vortices introduce an additional coupling in the (crudest approximation to the) RG flows, which is the vortex fugacity.We begin by discussing the symmetry properties of this coupling.In this section and 18 Recall that a vortex is a pointlike singularity in spacetime (instanton).Encircling the vortex, Q describes a nontrivial path in SO(N ) associated with the nontrivial element of π 1 (SO(N )) = Z 2 .We may worry about how to make sense of a vortex in the replica limit: one way to make the idea more precise may be via an equivalent supersymmetric formulation of the NLσM [36].Various kinds of topological defects arise in replica NLσMs including domain walls [41,48] and vortices [66,67,73].the next, we draw heavily on the analogy with instantons in the XXZ spin chain and related systems [87][88][89].Following a standard heuristic picture (see e.g.Ref. [83]), we may imagine an expansion of the partition function in terms of the number n of vortices.Schematically, (73) The kth vortex is located at position (x k , t k ): we take x to be a lattice coordinate taking values x ∈ Z. 19 Each vortex costs a fugacity y x .The magnitude of this fugacity will be sensitive to the way the theory is cut off near the vortices (this magnitude will be independent of x, by translation invariance, if ∆ = 0).However, the sign structure of y x carries universal information.We argue that The same kind of alternation holds for vortices (instantons) in the easy-plane XXZ chain, with the sign factor being (−1) 2Sx .The minus sign arises from the Berry phase terms in the lattice action (for the coherent states path integral) when we consider configurations with vortex singularities.The argument is of a standard kind and is given in Appendix H. Importantly, the argument applies independently of the microscopic structure near the core of the vortex.Next (in a standard argument [90]) we may imagine coarse-graining pairs of sites to define a coarse-grained vortex fugacity y.Heuristically, y ∝ y x + y x+1 .In the case where N F is odd, this leads to a cancellation.The cancellation is perfect if ∆ = 0, but if ∆ ̸ = 0 (so that translational symmetry by one lattice spacing is broken, and y x ̸ = y x+1 ) it is not perfect.At small ∆ we therefore expect where b is a constant.Conversely, for even N F , the fugacity of a vortex is independent of its lattice position j.Therefore, in this case y ̸ = 0 even when ∆ = 0, and by symmetry (where the discussion in Sec.VIII suggests that c is negative and d is positive for N F = 2).The expansions above are for small ∆ and do not rule out a nonmonotonic dependence at larger ∆.We see that in the N F -odd models with ∆ ̸ = 0 and in the N F -even models with generic ∆, it is possible for the bare vortex fugacity to be nonzero.In order to address the transition out of the stable nontrivial phase, let us now consider an approximate RG involving y and g.
For N = 1 the critical fixed point is not at small g or small y, preventing a controlled perturbative calculation.However, the topology of the flows can be determined in a limit where 2 − N is treated as a small parameter, in the spirit of the epsilon expansion.This was discussed for the S N −1 sigma model (the classical ferromagnet with O(N ) symmetry) near N = 2 and d = 2 in [91] and for replica NLσMs in [66,67] (where sign effects for y like those discussed above are relevant).The key point is that in all these hierarchies of models, the point N = 2 is an XY model, with the well-known Kosterlitz-Thouless RG equations in which y = 0 is a fixed line.Assuming analyticity in N , the beta functions can then be expanded Here F (g R ) is the derivative of the beta function for g with respect to ε at ε = 0 and y = 0.At small g R it is given by the perturbative result quoted in Eq. 49.The values of g R of interest now are not necessarily small, but for a qualitative picture of the flows what matters is the sign structure of F (g R ) which we take to be as suggested by the perturbative beta function [91].Fig. 7 shows the schematic RG flow obtained from these equations.The structure of these flows matches the "upper" part of the phase diagram in Fig. 2 (the limit J → 0 will be discussed in the next section).The axes are in heuristic correspondence between the two figures: recall from Eq. 46 that g −1 B increases with J 2 /Γ, and from Eq. 75 that the sign of y is equal to that of ∆ (at least for small enough ∆).
For large g −1 R , a small vortex fugacity y is irrelevant and the couplings flow towards the stable fixed point at (g −1 R , y R ) = (+∞, 0) that we have already discussed.There are phases where g −1 R flows to small values and y flows to large positive or negative values, which we identify with disordered phases.The analysis of Sec.VI makes clear that a disordered phase corresponds to an area-law entangled phase.
Two green points denote the fixed points (unstable in one direction) which control the transitions from the nontrivial phase (the pink region, which flows to y = 0, g −1 R = ∞) to the disordered phases (note that the lower boundary of this figure corresponds to an arbitrary value of g −1 R , and does not correspond to the J = 0 boundary of the fermion model phase diagram).
The transitions between the nontrivial and trivial phases are controlled by a pair of unstable fixed points (at positive and negative y), shown in red in Fig. 7.When N → 2, these fixed points approach the y = 0 line, giving the conventional BKT flow diagram.For N < 2 the flow structure is different, and for example the fixed points at finite y have a finite value for the correlation length exponent ν. 20 By contrast, an "annealed" aproximation to the Majorana dynamics gives a Kosterlitz Thouless transition with ν = ∞ [25].
Finally, given that at this fixed point there are no marginal flows, we expect a model that is on the phase transition line to exhibit the entanglement scaling dictated by conformal invariance [32,55], i.e. S n,A ∝ ln L, rather than (ln L) 2 , cf.Sec.VI.

VIII. MEASUREMENT-ONLY MODEL (J = 0)
The so(N ) × so(N ) replica symmetry of the Majorana chain is generic, in the sense that it is the minimal continuous symmetry shared by any model of free fermions. 21owever, a larger so(2N ) replica symmetry is present both in the measurement-only limit Γ = 0 and in the unitary limit J = 0 [25].In this section we explore the consequences of this symmetry for the measurement-only line. 22he feature responsible for the additional symmetry on the J = 0 line is not the absence of unitaries, but rather a bipartite structure: by examining the replicated Hamiltonian, one may check that so(2N ) is present whenever it is possible to group the physical Majoranas into two subsets, A and B, such that the unitary hopping only involves pairs iγγ from the same set, and measurements are only of pairs iγγ from opposite sets (see endnote for examples 23 ).
In the present model, the symmetry is made manifest by a sign change γ (σa) j,µ → (σ) j γ (σa) j,µ , so that, in the block decomposition of the local 2N × 2N matrix degree of freedom S j (Eq.27), the off-diagonal blocks are redefined as Q j → (−1) j Q j while L j and R j are unaffected.The Hamiltonian is then SO(2N ) transformations act on S by conjugation.This is an antiferromagnet.For N = 2 one can check that it reduces to the conventional SU(2) Heisenberg chain with spin S = 2N F .We have allowed for dimerization in the measurement rates, Γ j = Γ(1 + (−1) j ∆).
In the semiclassical limit, or in a coherent-states representation of the path integral, S becomes an antisymmetric orthogonal matrix that lives in the manifold SO(2N )/U(N ) (see Sec. IV A).In the model with J > 0 discussed in Sec.IV, the Hamiltonian favored ordered states in which only Q was nonzero, and we integrated out L and R to obtain the theory for Q only.Here the larger symmetry means we must retain the full target space for the low-energy theory, with is therefore an NLσM on SO(2N )/U(N ) [39][40][41].
By symmetry, the Lagrangian for such a theory is expected to take the standard form with two couplings, namely the inverse stiffness g and a Θ angle.The bulk physics is invariant under Θ → Θ + 2π. 24The values of these couplings should in principle be obtained at large N F by taking a continuum limit of the coherent states path integral [64,87].We do not perform this calculation here, instead conjecturing the form of Θ on the basis of the N = 2 special case.The latter reduces to the SU(2) chain with spin N F /2, for which the NLσM mapping at large spin is well known and gives the sphere sigma model with a Θ term.
The simplest assumption that matches with this case is that Θ = N F π in the undimerized model, with Θ = N F (π + O(∆)) for weak dimerization ∆.Finally if ∆ is varied all the way from −1 to 1 the topological angle varies monotonically from 0 to 2πN F [76].
The SO(2N )/U(N ) model with N → 0 arises as a description of Anderson localization problems in symmetry class D [39-41, 44-46, 48-51].Localization models in this class have a rich phase diagram with various kinds of transition [36]: in addition to a nontrivial dependence on the Θ angle there is a stable metallic phase when N → 0 (also, two slightly different models arise 25 ).Here we expect a simpler phase diagram, in particular because the perturbative beta function [36] shows that unlike the N → 0 case the N → 1 case does not have a metallic phase.
Instead, the simplest conjecture, given the instability of the g = 0 fixed point, is that the flows in the (Θ, g) plane at N → 1 resemble those of various other replica NLσMs, including the Pruisken sigma model for the Integer Quantum Hall transition (a celebrated example of the effect of a Θ term [36,92]).If so, then the theory is gapped for all g, except on the lines with Θ = π mod 2π (which are preserved under RG, by parity symmetry).On these lines the model flows, for any initial g, to a critical fixed point that is unstable in the Θ direction.
At N F = 1 this leads to a single critical point on the J = 0 axis, at ∆ = 0.This is consistent with the numerical results of Ref. [12].In general it leads to N F critical points on this axis, as varying ∆ causes the Θ angle to cycle N F times through the value π mod 2π (in analogy to the spin-N F /2 chain [76][77][78][79]).
We also conjecture that the Θ = π fixed point is unstable to the symmetry-breaking perturbation induced by turning on J.The considerations above then yield the phase diagram structure for small J that was proposed in Sec.V. 26 However, we emphasize that these are conjectures: in particular it is not ruled out that the stability properties of the Θ = π fixed point could be different (we will return to this in the future).
Finally let us briefly note the analogy between the RG flows described above and those in the Majorana loop model [6,22,27,93] (see Sec. V).The Majorana loop model has a measurement-only line described by the CP M −1 NLσM with a Θ term in the limit M → 1 [94,95].If the measurements are un-dimerized, then Θ is equal to π, and this is a critical point between two disentangled phases.Adding SWAP unitaries gives a symmetry-breaking perturbation in the NLσM, which induces a flow to the RP M −1 NLσM, which has a metallic phase [67,96].The RP M −1 NLσM allows Z 2 vortex defects, and the sign of the vortex fugacity is inherited from the sign of Θ − π in the parent CP M −1 model.These vortices can drive a phase transition to a trivial phase.This structure of flows is similar to that proposed above, with the SO(2N )/U(N ) NLσM at Θ = π playing the role of the CP M −1 NLσM, and the SO(N ) NLσM playing the role of the RP M −1 NLσM.However, there are also basic differences between the two kinds of measurement model, cf.Sec.V E.

IX. CONCLUSIONS
In the context of a simple Majorana Hamiltonian, we have argued that monitored free fermions give rise to problems in critical phenomena that can be viewed formally as sitting (at replica number "N " equal to 1) in between two well-studied classes of critical points: zerotemperature phase transitions in spin chains, described by NLσMs with values of "N " that are greater than one, and Anderson transitions in eigenstates of disordered Hamiltonians, described by the limit N → 0. Our aim in this paper has been to give an analytically controlled derivation of an effective field theory for a generic model and to use it to analyze the entanglement properties of the phases and transitions.
Concretely, our approach was to map the generator of the dynamics for moments of the density matrix (prior to normalization) to an so(2N ) spin chain, which at large N F is in a semiclassical limit, allowing a controlled reduction to a long-wavelength theory.
The nonlinear sigma model was particularly useful in the stable nontrivial phase, where the calculation of the entanglement is asymptotically exact, and agrees with with numerical simulations made with the quantumstate-diffusion method.The calculation illustrates that the nonlinear sigma model stiffness 1/g (analogous to a conductivity in a localization problem) functions as a scale-dependent strength of entanglement.The NLσM also gives a picture for the structure of critical points, via an ϵ = 2 − N expansion in the generic case, or via transitions between different Θ-vacua in Majorana models with a certain bipartite structure for interactions and measurements, in particular in a model in which the dynamics involves only noncommuting measurements [6,97].This structure of RG flows for these sigma models explains the previously observed [22] similarity between the phase diagram of the Majorana loop model and more generic models, despite these problems being in distinct universality classes.
We emphasize that various kinds of connection between either free monitored [10] or free non-Hermitian [8,98] dynamics and replica Lagrangians have been developed previously.Ref. [10] mapped a regime of monitored Dirac fermions in the spatial continuum onto a non-Hermitian Sine-Gordon theory.A Majorana model closer to ours was considered in Ref. [98] in the context of the SYK 2 model, with a non-Hermitian Hamiltonian rather than measurements.There a different regime was considered, where fluctuations were suppressed, so that the effective description was different.Most relevant to our work is Ref. [8], which provided a symmetry classification of Gaussian random tensor networks (observing the connection with nonunitary time evolution operators) and proposed the DIII sigma model with N → 0 as a description of generic Gaussian tensor networks on grounds of symmetry.
The relevant limit of the target space for NLσMs for monitored systems is N → 1.It will be interesting to explore the RG fixed points [42] relevant to these N → 1 sigma models (with various symmetries), using simulations with Born's rule or with the state diffusion formalism to compute exponents and flow diagrams, and to map out the relation to different monitored models [3,[99][100][101][102][103].
For the specific models studied here, it also remains to perform a fuller numerical analysis of the phase diagrams in order to check the conjectured structure for general N F .
In this work we have focussed on a simple case where the Hamiltonian couplings fluctuate as white noise, since this led to a particularly simple effective model.In the future it will be interesting to study models where the hopping amplitudes either have a deterministic part (nonzero mean), or are nonrandom (so that the randomness is only from measurements, as for example in [10]).In these cases the model does not reduce microscopically to a purely bosonic model.However, one could employ tools from Anderson localization to derive NLσMs in the long-wavelength limit [64,104].
Note added : While finalizing this manuscript, a closely related work appeared on the arXiv [105], studying monitored free fermions dynamics with the same symmetry, albeit using a different microscopic model.
writing dµ G (J, M ) to denote the product of the various Gaussian measures at different positions and time-steps.
Here the last factor of Tr ρJ,M ensures that quantum trajectory are chosen according to Born's rule.The key of the replica trick lies in the observation that the average over J and M can be most easily performed by computing for N ≥ n and finally performing an analytic continuation to N → 1.In this approach the fundamental object is ρ (N ) = E G ρ⊗N J,M (t) (Eq.13), which through appropriate contractions can yield all terms of the form A12.
Averages are most simply computed in the "folded" respresentation (via the Choi-Jamio lkowski isomorphism).Fixing a basis {|k⟩} of the Hilbert space H, we can identify any operator Ô with a state |O⟩ in H ⊗ H via In this formalism At this stage the average can simply be taken using the cumulant expansion at each time step.It is straightforward to see that this yields result consistent with the averaging rules given in the main text Eqs. 5 and 10.However, in order to write an explicit expression for K (N ) , we need to define the tensor product by mapping to a bosonic Fock space.

Appendix B: Replica formalism for Majorana fermions
The convention that we used to define replicated Majorana operators is an extension of the one used in Ref. [61], which for the reader's convenience we summarize here.For simplicity, in our treatment, we will assume that LN F is a multiple of 8.
While the tensor product of fermionic Fock spaces is not well-defined, due to the anticommutation relations, we can construct a bosonic Hilbert space by means of a Jordan-Wigner transformation.For this purpose, we impose some ordering on the indices (j, µ) of the Majoranas operators and label them with a single index k, e.g. by defining j k = ⌊(k − 1)/N F ⌋ + 1 and µ k − 1 = (k − 1) (mod N F ).We can then identify the LN F Majorana fermions with spin operators of a spin-1/2 chain with LN F /2 sites: In terms of the bosonic (spin) Hilbert space H B , tensor products are well defined and we can consider replicas H 2N = (H B ) ⊗2N and Pauli operators acting on them with V = X, Y, Z and I denoting the identity on the space H B .We then proceed to define replicated Majoranas operators in the extended space.As an intermediate step, we introduce the operators χ σa k , where a (1 < a < N ) and σ = ± label the different replicas: Here γ k denote the Majorana operators, which act on the bosonic Hilbert space according to Eq. B1.Finally γ * k denotes the complex conjugate of γ k w.r.t.some basis -the actual basis choice is inconsequential.Note that the operators χ (σa) k cannot be interpreted as Majorana operators in the enlarged space, as χ-operators acting on different replicas commute.This can however be fixed, by adding a Klein factor among different replicas.We define the Klein factors as and the replicated Majoranas One can then verify that the operators γ (α) k with α = σa indeed form a set of Majorana operators, i.e. (γ The specific choice of Majorana operators above turns out to be very convenient both to compute averages of (K ⊗ K * ) N , as we now discuss, and to characterize the boundary states needed to write the entanglement entropies (see Appendix C).
We are now set to compute the average evolution for N replicas E G (K J,M ⊗ K * J,M ) ⊗N .We can start by observing that in the replicated space the time-evolution along a quantum trajectory can be expressed in terms of a new continuum Hamiltonian playing a role analogous to (7) In terms of the χ-operators this takes the form j+1,ν . (B10) Note that σ multiplies M µν j (t) and not J µν j (t), viz. the part of the Hamiltonian produced by weak measurements and not the one produced by unitary evolution.This is unlike what happens in replicas of real bosonic Hamiltonians, and it happens because the complex conjugation of the Majorana operators is absorbed in the definition of the operators χ. (However, it is possible to move the σ from one term to the other in the replicated Hamiltonian Eq. 22 by a sign redefinition of γ operators on a sublattice.)By working separately on the case σ = ±, one can then show that χ j+1,ν , therefore obtaining Eq. 22. Finally, integrating over J and M , as explained in the previous Appendix, and using the cumulant expansion we arrive at Eqs. 21 and 23.

Appendix C: Boundary states and their stabilizers
In this section we introduce the boundary states relevant for the study of entanglement entropies reported in the main text.Finally, we characterize them in terms of quadratic Majorana stabilizers.More of their properties, related to the SO(2N ) irrep they belong to, are reported in Appendix D.
We begin with the simplest one, corresponding to the initial density matrix, which we take to be ρ(t = 0) = 2 −LN F /2 I. (C1) After taking N replicas ρ ⊗N (t = 0) and mapping operators to states of a doubled Hilbert space (see Eq. A13), this gives us as initial state |I⟩, which can be defined in terms of its bosonic stabilizers.The identity operator satisfies V k IV k = I where V = X, Y, Z and k labels sites j and flavors µ.This then translates into This implies that as can be verified by rewriting the γ-operators in terms of Pauli matrices, and using that for Pauli matrices V T = V * .The relevant boundary states at final time can be characterized in a similar fashion.For concreteness, we discuss the boundary state for the computation of where A includes the sites from j = ℓ + 1 to j = L, and ρ A (t) = Tr Ā ρ(t) is obtained by tracing the density matrix over Ā -the complement of A. [Note that the computation of the entropy of the overall state is obtained by setting ℓ = 0.] The computation above can be written as where C A,n is defined in terms of its action on bosonic operators.For any operator V Ā supported in Ā, C A,n behaves like I. Instead for operators V A supported in A, C A,n acts as a cyclic permutation.Equivalently, after mapping operator to states through Eq.A13, we have Here |C A,n ⟩ can be described by its stabilizers of Pauli matrices Finally, by re-expressing the Majoranas in terms of Pauli operators, we have the following relations for a < n and k ∈ A.
We can equivalently characterize the boundary states by the expectation value of S on each site.For all boundary states |B⟩⟩ discussed above we will have that ⟨L⟩ = ⟨R⟩ = 0. Finally, the expectation value of Q depends on the boundary state.For the initial state |I⟩, we have the expectation value of Q is the N × N identity matrix.Finally, for |C A,n ⟩, and a site in A, the structure of Q = q n , reported in Eq. 59.

Appendix D: Properties of the boundary states
We now proceed to show that the set of stabilizers above fix the so(2N ) irrep on every site, and within this it holds that S T S = 1 + O(1/N F ).
We begin by observing that it is enough to characterize the state |I⟩, since the other boundary states can be obtained through local rotations from |I⟩, e.g. by rotating the γ + as specified by the orthogonal matrix ⟨Q⟩.
To characterize the irrep of |I⟩, we begin by considering the N F = 1 case, where we will show that the state lies within the spin representation.For this purpose, we recall that the Dynkin diagram of so(2N ) (viz.D N ) is given by with each dot associated to a simple root α n .Given an orthonormal basis (w.r.t. the Killing form) {e n } the simple roots can be chosen to be α j = e j −e j+1 for j < N and α N = e N −1 + e N .For a single site and flavor, such an orthonormal basis is naturally represented by the set e n → X n = iγ +n γ −n /2.We can then note that the state |I⟩ has weight 0 in all α n with n < N and weight 1 in α N .Finally, we show that |I⟩ is an highest-weight state, i.e. it is annihilated by all raising operators E k defined by For the purpose of verifying this, we write the E k explicitly in terms of Majorana bilinears Rewriting Eq.C3 as it is immediate to see that E k |I⟩ = 0 for all k.This therefore proves that |I⟩ lies in the spin representation.Note that this irrep is 2 N −1 dimensional and corresponds to the states with R = +1.[The remaining 2 N −1 states, which are not involved in the physics of the problem have R = −1 lie in the conjugate representation.]The case of larger N F follows immediately from the case N F = 1.Given the tensor-product structure of |I⟩ across different flavors, we have that in this case |I⟩ is a highest-weight state with weights 0 w.r.t.all α n (n < N ) and weight N F w.r.t.α N .
Finally, we show that for states |ψ⟩ within this irrep we have To this end, we will show that Eq.D5 holds when acting on the state |ψ⟩ = |I⟩.From this state, the property can be lifted to the whole irrep, by noting that an arbitrary state |ψ⟩ within the irrep can be written as W |I⟩⟩ for some W representing a rotation O ∈ SO(2N ), then To show that the property above holds for |ψ⟩ = |I⟩, we express S T S as Then, using Eq.D4 we can transform any polynomials in γ + and γ − into ones involving only γ + .By doing this, we obtain where the equivalence above means that the LHS and RHS are equal when applied on the state |I⟩.This therefore concludes the proof of Eq.D5.The quadratic Casimir of the representation can be obtained as Tr S T S, which yields Eq. 31.
In addition, we can further analyze higher-order Casimir.Most of these are already specified by the condition (D5), however, the space of orthogonal antisymmetric matrices is split in two disconnected components distinguished by the Pfaffian of Pf(S): a polynomial of degree 2N of its entries that is invariant under S → O T SO for O ∈ SO(2N ).Due to its invariance properties, Pf(S) commutes with all the generators of so(2N ) and therefore with H.At large N F we can see that as it is true for any polynomial in the entries of S of degree ≤ 2N , so that in the irrep of interest.
Appendix E: Path integral from coherent states In Sec.IV we used a semiclassical (large N F ) analysis to argue that there exists a path integral representation in terms of a field S(x, t) that lives in the space of antisymmetric SO(2N ) matrices.Here we confirm, using generalized coherent states (see e.g.Ref. [63] and references therein) that the representation of the path integral in terms of the field S(x, t) makes sense for any value of N F , not necessarily large (with x initially taking values on the spatial lattice).This is consistent with our expectation that the same infra-red field theories and universality classes apply at both small N F and large N F , even though for small N F we do not have quantitative control over nonuniversal constants when we take the continuum limit.
We consider a single site j.Here our starting point is the state |I⟩.By definition, acting on |I⟩ with O ∈ SO(2N ) rotations generates the whole irreducible representation the state belongs to.More precisely we act on |I⟩ with a unitary transformation ŴO that are the image of O ∈ SO(2N ) matrices under the representation map.The states ŴO |I⟩, commonly called coherent states, form an overcomplete basis of the irrep and can therefore be used to construct a path integral representation of the problem at hand.It is a standard results that the identity (on the irrep) can be represented as an integral of the form where dµ H (O) is the Haar measure on the group SO(N ), with some proportionality constant that is irrelevant for the subsequent treatment.To see this, we can note that the integral commutes with any unitary ÛO in the representation; therefore, by Schur's lemma, the integral is proportional to the identity on the irrep.We further note that the states ŴO |I⟩ can be identified, up to a phase, by viz. the expectation value of the spin operator Ŝ on them.
In this appendix only, we distinguish between operators and c-numbers by putting a hat on the former.Here J 0 is the matrix The value of S uniquely defines the state up to a phase, given that the states in the same irrep of I are Gaussian by construction and are therefore uniquely specified by the expectation values of quadratic Majorana operators, which are contained in Ŝ.Since the coherent states are defined by S, up to a phase, we can write the resolution of the identity as an integral over S, where |S⟩ is defined by fixing some choice of phase convention, and µ(S) is the SO(2N )-symmetric measure for S induced by µ H (O). The path-integral representation for H can then be obtained through conventional means by trotterizing the time-evolution operator and inserting resolutions of the identity (E4).The degrees of freedom in this path integral are orthogonal antisymmetric matrices, i.e. whose blocks satisfy Eq. 33, continuously connected to J 0 , i.e. with Pf(S) = (−1) N (N −1)/2 .This path integral has the schematic form Here N F Ω[S j ] is the Berry phase term for the j-th site, see e.g.Refs.[63,65] (it is proportional to N F because the states |S⟩ are product states over the flavors, and therefore the Berry phase is additive for the flavors).The quantity H[{S j (t ′ )}] appearing above is obtained by replacing the spin operators Ŝj with the c-number fields S j (t).We will not need the explicit form of the Berry phase, but it is important that the lattice action above is proportional to N F .Ultimately this is what allows a controlled derivation of a continuum theory at large N F (Sec.IV B).
The second important point is the precise target space for S. We show that it is isomorphic to the quotient space SO(2N )/U(N ), where the isomorphism is given by the Here we think of U (N ) as a subgroup of SO(2N ).For this purpose we use a standard embedding of U (N ) in SO(2N ) (see e.g.Ref. [106]) where X and Y are the real and imaginary parts of the U(N ) matrix U , respectively.One can explicitly verify that the image of a unitary matrix is orthogonal.By continuity of the determinant and connectedness of U(N ), then the image lies within SO(2N ).From these observations, we can see that the image of U(N ) is the set of Finally, from this one can see that the map O → S = O T J O O is bijective.In fact, any matrix S, which is antisymmetric admits a spectral decomposition of the form with Λ diagonal and O ∈ SO(N ).S being orthogonal and with Pfaffian 1, we can further choose Λ = 1.Finally, it is immediate to verify that the matrices O are unique up to multiplication by a matrix in the image of U(N ), therefore showing that the manifold os S is isomorphic to SO(2N )/U(N ).One route to the continuum theory would be via a derivative expansion of the lattice action Eq.E5 -see e.g.Ref. [65] for similar derivations for spin chains.In the main text, we instead obtained the continuum theory by analyzing the equations of motion.The ultimate result should be the same.quantum-state-diffusion formalism [58][59][60] where H(t) = L−1 j=1 J j (t)iγ j γ j+1 (for our numerical simulation we chose open boundary conditions).Here dξ j are real random variables satisfying where we employed Itô notation [107], while ⟨γ j γ j+1 ⟩ t = ⟨ψ t |γ j γ j+1 |ψ t ⟩ .(F3) In the following, we will also set for the staggered value of Γ j at odd/even position.Using a standard derivation [108], one can show that Eq.F1 describes the continuous-time limit of the discrete dynamics discussed in Appendix A. This formulation is convenient from the numerical point of view, because one does not need to sample measurement outcomes according to the Born rule.Note that at the first relevant order in Itô calculus, we may split the infinitesimal time evolution into three parts: a unitary step where j ∈ e (j ∈ o) means that the sum is restricted to j even (j odd).
For an initial Gaussian state [26], the state of the system remains Gaussian at all times, and the dynamics can be simulated efficiently.Indeed, Gaussian states satisfy Wick theorem, and can be described entirely in terms of the covariance matrix In our numerics, we initialize the system in the vacuum state |0⟩ associated with the fermions c j = (γ 2j−1 − iγ 2j )/2.Its covariance matrix reads Here and throughout this Appendix, we order the Majoranas as γ = (γ 1 , γ 3 , . . ., γ 2M −1 , γ 2 , γ 4 , . . .γ 2M ).Given a Gaussian state |ϕ⟩, its covariance matrix can be computed if we know a Gaussian unitary operator O s.t.For what follows, it is important to note that the state (F9) is annihilated by the operators where the M × M matrices U , V are defined by Note that the matrices U and V in the LHS are welldefined since O is a real matrix.In addition, because of unitarity, it must be The algorithm works as follows.We first discretize time and, for all time steps, we compute the orthogonal matrix O(t) corresponding to an operator O(t) s.t.|ψ t ⟩ = O(t) |0⟩ (clearly, O(0) = 1 1).The infinitesimal time interval dt is replaced by a finite Trotter time ∆t, which is taken to be sufficiently small (we have always checked robustness of our data with respect to decreasing ∆t).
Let us first consider the unitary step (F5).Denoting by |ψ t ⟩ the state of the system at time t and by O(t) the corresponding orthogonal matrix, the state of the system is updated as where W = exp(−iH∆t), with J j,j+1 (t) drawn from the random Gaussian distribution with average 0 and variance J 2 /(∆t).Therefore, the matrix O(t) is updated as where W = e 2 j,j+1 Jj,j+1(t)(Ej,j+1−Ej+1,j )∆t , (F16) and we introduced the matrices E j,k defined by (E j,k ) α,β = δ j,α δ β,k .Note that in Eq.F15 we have right matrix multiplication.Next, we consider the non-unitary step (F6) corresponding to weak measurements of the odd Majorana pairs.Again, we denote by |ψ t ⟩ the state of the system right before this step, and by O(t) the corresponding orthogonal matrix.After this step, the state is updated as In Eq.F20, U and V are defined as in (F12), where O is the orthogonal matrix O(t).
Since D † = D * = D, we have However, because D is not a unitary operator, Ũ and Ṽ in Eq.F20 do not satisfy the second relation in Eq.F13, and therefore do not define an orthogonal matrix.However, since |ψ ′ t ⟩ is annihilated by b µ , it is also annihilated by different operators that can be written as bµ = T µν b ν as long as T is a non-singular matrix.This condition can be most conveniently formulated as Next, we show that Q 11 , and Q 21 satisfy both the relations (F13).First, by the property of the QR decomposition we immediately have Next, plugging Eq.F26 into Eq.F23 we have Because the matrix (F26) is full-ranked, R 11 is invertible and so necessarily It follows that we can update the orthogonal matrix where By construction, O ′ (t) is orthogonal and |ψ ′ t ⟩ = O ′ (t) |0⟩.Finally, let us consider the non-unitary step (F6) corresponding to weak measurements of the even Majorana pairs.This can be implemented by simply shifting the covariance matrix by one, and repeating the step described above for the non-unitary step (F6) corresponding to weak measurements of the odd Majorana pairs.The shift corresponds to the matrix multiplication After the QR-decomposition step described above is applied, yielding an orthogonal matrix, we finally shift back performing a right multiplication by the matrix S.
For completeness, we also briefly recall how the Rényi entropies can be obtained from the covariance matrix [109].Let us denote by Γ ℓ the matrix obtained by selecting the rows and columns j, k = 1, . . .2ℓ from Eq. F7.Γ ℓ is real and antisymmetric and its 2ℓ eigenvalues form ℓ pairs of imaginary numbers ±iν j , with ν j ∈ R. The Rényi and von Neumann entanglement entropies of the subsystem associated with the first 2ℓ Majoranas read Computing the Berry phase for a given winding reduces to the spin-1/2 problem.First, we note that the relevant coherent states factorize between different flavors, so N F just appears as an exponent in the form e iN F Ω [w] .[From now on we consider a single site, so suppress the site index.]Next we note that, for the specified Q(t), the coherent state spans a 2-dimensional subspace.
We can then map the space of the 2N Majorana modes into a collection of N − 1 spin-1/2, e.g.we can identify iγ +(a+1) γ −(a+1) with X a , the X Pauli matrix on the ath site.The fact that we can encode the 2N Majorana modes into N −1 spins rather than N is due to the conservation of R = a (iγ +a γ −a ), which halves the dimension of the Hilbert space.Through this mapping W O |I⟩ corresponds to a 2-dimensional space where X a = 1 for all a > 1, whereas the state of the first spin depends on O.
From the mapping above we now recognize that the coherent states W O |I⟩ correspond to states of a spin-1/2.Computing the expectation values of X 1 , Y 1 and Z 1 = −iX 1 Y 1 as a function of θ, we have that the Berry Using the explicit form of the Berry phase for spins [64] we see that the Berry phase only depends on the winding number m of the angle modulo 4π, and that e iΩ(1) = −1, thus completing our argument.

Figure 3 .
Figure 3. Schematic phase diagram for NF = 2 with the same color coding as Fig. 2, i.e. areas, lines, and points with the same colors are represented by the same infra-red field theory in the bulk.

4 Figure 5 .
Figure5.Simulations of Eq. 68 for Γ = 1, ∆ = 0, J = 2.For a given quantum trajectory we average Sn,A over the time-interval t ∈[40,80], and we also average over more than 400 independent quantum trajectories.The errorbar is the standard error over the set of distinct quantum trajectories.Dashed faded lines and solid lines are data obtained with timestep dt = 5 • 10 −3 and dt = 2.5 • 10 −3 respectively.Left: Steady-state entanglement Sn,A(L) vs. ln L. Right: [n/(n + 1)]∆Sn,A vs. ln L. The dashed gray line shows the predicted universal slope, Eq. 72.

2 Figure 6 .
Figure 6.The of the Right panel of Fig. 5, but for the smaller measurement rate Γ = 0.1 (with ∆ = 0, J = 2).Averages are performed in the time interval t ∈ [100, 150].See caption of Fig. 5 for further details.

11 ][Q 11 ]
where X is any M × 2M matrix, whose rows form a basis for the row-space of the M × 2M matrixŨ T , Ṽ T .(F25)By definition, we can find such a matrix by performing a QR decompositionŨ Ṽ = Q 11 R 11 Q 21 R 11. for the row-space of the M × 2M matrix (F25).It follows that the state |ψ ′ t ⟩ is also anniµ,j c j + [Q T 21 ] µ,j c † j , j,µ c j + [Q 21 ] j,µ c † j .(F28)

[
We can see this by explicitly by choosing the orthogonal matrix O and its unitary representation W O appearing in the definition of the coherent states |S⟩ = ŴO |I⟩.We choose We have denoted the dimension of the identity matrix as a subscript.]This can be represented by the unitary transformation phase associated with Q(t) is equal to that of a spin-1 The most technical parts of our work are consigned to several appendices.