Entanglement in interacting Majorana chains and transitions of von Neumann algebras

We consider Majorana lattices with two-site interactions consisting of a general function of the fermion bilinear. The models are exactly solvable in the limit of a large number of on-site fermions. The four-site chain exhibits a quantum phase transition controlled by the hopping parameters and manifests itself in a discontinuous entanglement entropy, obtained by constraining the one-sided modular Hamiltonian. Inspired by recent work within the AdS/CFT correspondence, we identify transitions between types of von Neumann operator algebras throughout the phase diagram. We find transitions of the form II$_1\leftrightarrow\,$III$\,\,\leftrightarrow\,\,$I$_\infty$ that reduce to II$_1\leftrightarrow\,\,$I$_\infty$ in the strongly interacting limit, where they connect non-factorized and factorized ground states. Our results provide novel realizations of such transitions in a controlled many-body model.

Introduction.-Entanglement in many-body systems and quantum field theories (QFTs) has recently been explored via a novel take on local operator algebras and axiomatic QFT [1][2][3][4][5][6] (see [7,8] for a review).A fruitful platform for these analyses is the Anti-de Sitter/Conformal Field Theory correspondence [9][10][11] (also known as holography), which relates strongly coupled QFTs in d dimensions with gravity theories on negatively curved spacetimes in d + 1-dimensions.In this context, operator algebras, and especially von Neumann algebras, have recently been leveraged to rigorously describe the entanglement structure of holographic systems [12][13][14][15][16][17][18][19].A direct consequence of these investigations is the algebra typification of the two phases in the Hawking-Page transition [12,13], characterized by factorized and nonfactorized Hilbert spaces, respectively.This state of the art motivates us to study transitions of operator algebras arising in interacting many-body quantum systems whose Hilbert space structure allows for a controlled analysis of entanglement.A useful object which helps in classifying types of algebras is the onesided modular Hamiltonian associated to a given subregion [6].For free fermionic systems this object is uniquely determined by the two-point correlation functions restricted to the subregion [20].We extend these results to interacting fermionic systems by constraining the form of the one-sided modular Hamiltonian in the limit of a large number of on-site fermions.For a wide class of interacting Majorana lattices, we exploit this extension to identify the operator algebras underlying our models, together with their transitions between different regimes of the phase diagram.This paves the way for addressing the classification of algebras, and possible transitions thereof, in previously suggested models for discrete holography, such as O(N )-invariant aperiodic spin chains [21].More precisely, in this Letter, we introduce a lattice model with N Majorana fermions on each site, interacting via a general potential involving multi-body hoppings.In the large N limit, all higher-point functions factorize.Thus, we solve the system exactly by obtaining the two-point correlation function for generic interaction potentials.We showcase the wide applicability of our techniques by considering instances of the model including both finite and infinite chains with nearest-neighbor hopping.We report three main results: First, we derive the entropy of a two-site chain with generic interaction potential.This entropy is fully determined by the correlations between the two sites, which are dictated by the interaction potential via self-consistency.This result can be interpreted both as the thermal entropy of the chain at a finite temperature or as the entanglement entropy of a two-site subregion in a larger chain.Remarkably, we find that the entropy itself does not depend explicitly on the chosen potential.Second, for a four-site chain, we identify two phases of the system characterized by strong and weak correlations within any two-site subsystem of the chain relative to all other correlations in the system.In particular, we identify a regime where the correlation structure indicates the factorization of the ground state.For a vast class of potentials, we show that these phases are connected by a quantum phase transition.Moreover, we determine the entanglement entropy of the halfchain by imposing a constraint on the one-sided modular Hamiltonian.In correspondence with the phase transition, the entanglement entropy exhibits a discontinuity above a critical value of the interaction strength.Third, exploiting the exact solvability of our model, we identify transitions between the local operator algebras underlying the four-site chain.When the correlations within a given subregion are the most relevant in the system, we find a von Neumann algebra of type I ∞ , which in general encodes finite entanglement.In our model, entanglement vanishes, thus signaling factorization.In the opposite regime, the algebra is of type II 1 , associated with infinite entanglement entropy, while still allowing for the definition of a trace functional.The intermediate domain is described by type III algebras, where the entanglement is infinite and a trace functional cannot be defined.Strikingly, in the strongly interacting limit and for an exponential potential in the fermion bilinear, we find that the transition between types II 1 and I ∞ algebras occurs in correspondence with the identified quantum phase transition.This transition then connects a non-factorized ground state to a ground state factorized into a product state.
Hamiltonian and Schwinger-Dyson equations.-Weconsider a lattice of L sites with N Majorana fermions ψ j x at each site, with anti-commutation relations ψ j x , ψ k y = δ jk δ xy .The microscopic Hamiltonian is given by with a general interaction potential h xy , which, without loss of generality, obeys h xy (ξ) = h yx (−ξ) and h xx (ξ) = 0.The theory is invariant under a global O(N ) rotation where O is an orthogonal matrix.It is a lattice counterpart of the Gross-Neveu model [22] with a general interaction potential, and only the bubble diagrams contribute to two-point functions at leading order in 1/N , see the Supplemental Material (SM) [23].When h xy (ξ) ∝ ξ q , the model is equivalent to the replicated Brownian Sachdev-Ye-Kitaev (SYK q ) model in disorder averaging [24].In this work, h xy (ξ) always includes a linear term in ξ.We solve the model (1) by introducing the effective action of two auxiliary bi-local fields, the Green's function G xy (τ 1 , τ 2 ) = 1/N j ψ j x (τ 1 )ψ j y (τ 2 ) and the selfenergy Σ xy (τ 1 , τ 2 ), introduced as a Lagrange multiplier, in the spirit of [25][26][27].Here, τ denotes Euclidean time.We consider the canonical ensemble at temperature 1/β and write the thermal partition function Z = D GD Σe −S E [ G, Σ] with the effective action where G xy (τ ) = −G yx (−τ ) and we have assumed timetranslational invariance.From Z, we may derive the properties of the ground state (when β → ∞), or the thermodynamics of the system at finite temperature.In the large N limit, the saddle point approximation of the path integral leads to the Schwinger-Dyson (SD) equations with τ ij ≡ τ i − τ j , and Σ xy = −Σ yx due to the conditions on h xy mentioned above.This model is exactly solvable in the large N limit, in the sense that all higher-point functions factorize into products of two-point functions, i.e. we have large N factorization.We solve Eqs. ( 3) and ( 4) for general τ by leveraging the fact that the self-energy is solely determined by −2iG xy (0).We can thus solve for G(0) by means of selfconsistency (SC) conditions.Using the form of (3) in Fourier space, SC imposes with ω n = 2π(n+1/2)/β and Σ given by (4).The Green's function G(τ ) is then obtained by inserting G(0) back into (4) and (3).
Two-site chain.-Toprovide an explicit application of our general techniques, we now focus on a system governed by a general Hamiltonian H of the form (1) with L = 2 sites and at a finite temperature 1/β.Without loss of generality, we can absorb β into the general form of the Hamiltonian (1), i.e. set β = 1.Thus, the density matrix reads where Z = Tr(e −H ) and h(ξ) ≡ h 12 (ξ) = h 21 (−ξ).We can exactly solve the SD equation (3) to obtain the Green's function G(τ ) [23].At τ = 0, the solution reads with X ≡ −2iG 12 (0).We can read off the SC equation which can alternatively be derived from (5).In general, we have −1 < X < 1 regardless of the form of the potential.Two relevant regimes of ( 8) are when |X| → 0 and |X| → 1, corresponding to weak and strong correlations between the two sites, respectively.From the path integral in the large N limit, we can study the thermodynamic properties of the system [23].In particular, we find the entropy density S/N ≡ s to be where X satisfies the SC equation (8).The fact that the entropy density function (9) does not explicitly depend on the interaction potential signifies the first main result of this work.The free energy is given by . Remarkably, we see that S → N log 2 + O(1) in the strong correlation limit, and S → 0 in the weak correlation regime [23].Free energy of the four-site chain with potential hx,x+1(ξ) = µx 1 − e Jξ /(2J) for different interaction strengths J.We observe a non-analyticity at r ≡ µa/µ b = 1/2 for J above the critical value Jc = √ 2, signaling a phase transition.The two phases of the system are characterized by the correlation structure given by (10), whose limiting cases for r → 0 and r → ∞ are shown in the two embedded diagrams.
Four-site chain:-We can think of the two-site system as being part of larger chains and take advantage of the results derived above to study the entanglement structure.As an example, we consider an open chain of length L = 4 at zero temperature.Although our techniques are valid for any potential, we consider here a specific instance h x,x+1 (ξ) = µ x 1 − e Jξ /(2J) = h x+1,x (−ξ), where J > 0 is the interaction strength and µ 1 = µ 3 ≡ µ a , µ 2 ≡ µ b denote hopping parameters.All remaining entries of h xy are zero.For convenience, we introduce the hopping ratio r ≡ µ a /µ b .To access the ground state properties in the large N limit, the hierarchy of parameters N ≫ µ b β, µ a β ≫ 1 needs to be taken into account.We solve the SD equation ( 3) and obtain G(τ ) [23], which at τ = 0 reads where the parameter θ is determined by the SC constraint derived from (5), = re J(sin θ−cos θ) .
(11) This transcendental equation may be solved numerically and has a unique solution for J < J c and three solutions for J > J c , where the critical value can be proven analytically to be J c = √ 2 [23].This multi-valuedness of the SC equation indicates that the system exhibits a discontinuous behavior as a function of r, now to be seen as a control parameter.We identify the thermodynamically dominant solutions by minimizing the free energy F obtained from the effective action (2) [23].This free energy is shown in Fig. 1 for different values of the interaction strength below, at, and above the critical point.We see that the free energy exhibits non-analyticity at r = 1/2 for interaction strengths J > J c .Thus, the system is characterized two phases for J > J c and it undergoes a first order quantum phase transition [28] across r = 1/2, the existence of which constitutes the second main result of this work.At the critical point J = J c , this transition is of second order.Let us stress that this phase transition is present for a large class of potentials other than the exponential [23].The two phases differ by the order parameter tan θ (11), which characterizes the correlation structure via (10).Two limiting regimes of this structure when r → 0 and r → ∞ are shown in the insets of Fig. 1.We now turn our attention to the study of entanglement in the four-site model and consider a connected two-site subregion A, which we take to be, e.g., the sites x = 1, 2. The reduced density matrix ρ A of this system can be written as a thermal density matrix of a two-site chain of the form (6), with H now to be thought of as the one-sided modular Hamiltonian.Its explicit form is not known in our case, so we take as an ansatz the general form given in (1).For this ansatz to describe a proper reduced density matrix, ρ A should reproduce the expectation values of local operators in the subregion.In particular, it must reproduce the correlations given by the Green's function (10) restricted to the subregion A. We must therefore impose a constraint for the one-sided modular Hamiltonian at large N , where G xy (0) is given in (10).Eq. ( 12) uniquely determines the modular Hamiltonian only when J = 0 [20].Nevertheless, we can still use it to compute the entanglement entropy for J ≥ 0. Indeed, we have shown that for a density matrix of the form given in (6), the entropy density of the two-site system can be computed for any form of the one-sided modular Hamiltonian ansatz and is given by (9).Because of the constraint (12), s(X) needs to be evaluated on the SC solution X = −2iG 12 (0) = sin θ obtained from (11), with X ∈ [0, 1) since θ ∈ [0, π/2).In this way, even without knowing the explicit form of ρ A , we find that s(X) = S A /N is the entanglement entropy of subregion A. The resulting entanglement entropy density as a function of r (recall X depends on r via (11)) is shown in Fig. 2 for different values of J.The phase transition reflects itself in the entanglement entropy evaluated on SC solutions as it becomes discontinuous for J > J c .
von Neumann algebras.-Motivatedby recent results in holography [12,13,[15][16][17][18], we study the classification of operator algebras associated to subsystems of our four-site model.Operator algebras can generally be classified into three types, denoted as type I, II and III [1,5,8].Based on the standard trace Tr, a type I al- < l a t e x i t s h a 1 _ b a s e 6 4 = " X i W 3 V n + 6 W 9 b g V 5 w r / x p n t 6 H W k S e y Y t j n V f n z X n / G S 0 5 x c 4 x + Q P n 4 x s Z P 5 F C < / l a t e x i t > 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " gebra encapsulates a finite entanglement entropy.Using Tr, entanglement entropy is infinite in both type II and type III algebras.To further distinguish the algebras, a key ingredient is the trace functional, denoted by lower case "tr" (to differentiate it from upper case Tr), which is defined to be a positive, linear and cyclic functional on the algebra [7,8].In particular, type II algebras allow for the definition of such a trace functional, while type III algebras do not.For the technical construction of operators in the algebra, we closely parallel [7].In our system of consideration and in the N → ∞ limit, operators in A A consist of products of finitely many Majoranas located in subregion A. In particular, these will act trivially on countably infinitely many indices j of the Majorana color space.
Based on the results for the ground state entanglement of our model, together with the operators in A A defined above, we identify the operator algebras associated to subregion A in different regimes of the correlation measure X = −2iG 12 (0).This classification is shown in the phase diagram inset in Fig. 2. When X → 1, the entropy S A → 0 [23].This is consistent with our physical intuition since we expect the subsystems to completely factorize in this limit.Thus, we find that A A is a type I ∞ algebra when X → 1 (which implies r → ∞ by (11)).
The index in I ∞ alludes to the infinite dimensionality of the local Hilbert space.When X < 1, the ground state is no longer factorized, and the entropy ( 9) is infinite, therefore ruling out A A being of type I.To specify the type, we resort to the definition of a trace functional tr on A A .When the maximally entangled state |Ψ⟩ is in the Hilbert space gener-ated by the algebra A A , a well-defined trace functional is given by tr(a) ≡ ⟨Ψ|a|Ψ⟩, with a ∈ A A [7]. Importantly, when X = 0, we find that the entanglement entropy in our ground state is infinite and maximal up to subleading corrections in 1/N [23].This implies that our ground state can be mapped to |Ψ⟩ by applying finitely many Majorana operators.Therefore, we conclude that the functional tr defines a proper trace when X = 0 (r = 0 by ( 11)), thus unveiling that A A is of type II 1 only at this point, cf.Fig. 2.
As for the regime 0 < X < 1, corresponding to 0 < r < ∞ by virtue of (11), we find that the entanglement entropy ( 9) is infinite but not maximal at leading order as N → ∞.Therefore, our ground state cannot be mapped to the maximally entangled state |Ψ⟩ by finitely many local operators, and therefore the algebra A A does not admit the definition of a trace [7].This implies that A A is of type III.Recall that the first and second equalities in (11) imposes SC for generic potentials, and therefore the analysis above is valid in the general interacting case.
In the free case J = 0, where the entanglement Hamiltonian ought to be quadratic in the fermions, the classification of the algebras can be attained by studying the spectrum of the modular operator [29][30][31].Given this setup, we are able to compute the large-N spectrum of ∆ [23], finding Spec(∆) = {λ n } n∈Z .
Here, the parameter λ is related to the correlations within the subsystem as λ = 1−X 1+X .When the modular operator has precisely this form, the associated operator algebras are said to be of I ∞ when λ = 0, type II 1 for λ = 1, and type III λ for λ ∈ (0, 1).Such type III λ algebras are known to arise for free fermions on a lattice [32].At finite J, these considerations lead to transitions between operator algebras of type II 1 ↔ III ↔ I ∞ in the phase diagram, cf.Fig. 2. Our setup's analytical tractability enables us to study the limit J → ∞, where the solution to the SC equation (11) for the class of interaction potentials with exponential behavior is X = Θ(r − 1/2), with Θ the Heaviside step function.In this limit, the entanglement entropy is [23].This results in a direct transition of algebras II 1 ↔ I ∞ at r = 1/2, which coincides with the phase transition undergone by the system.The transitions between different types of local operator algebras across the phase diagram provide the third result of our work.
Closed periodic chains.-Toshowcase the generality of our methods, we now consider closed periodic chains.
In particular, we focus on a closed chain consisting of L sites with a Hamiltonian of the form (1) with staggered interaction where h a and h b are generic functions and we have the periodic identification L + x ∼ x.Notice that by defining cells consisting of adjacent sites interacting by h b , we can leverage translational invariance with respect to these cells to solve the model in momentum space [23].We find that the Green's function is determined by an implicit dependence on its own entries.In particular, G is an implicit function of only the correlations within a given cell G 2x,2x+1 (0), and those connecting adjacent cells G 2x−1,2x (0).This dependence manifests itself via the parametrization where v ∈ [−1 , 1].By the aforementioned translational invariance, the correlations entering ( 14) are independent of x.Explicit expressions for G 2x−1,2x (0) and G 2x,2x+1 (0) at zero temperature and in the limit L → ∞ can be found in terms of v itself and read with K(ξ), E(ξ) the complete elliptic integrals of the first and second kind, respectively.In the spirit of our methods, we can impose self-consistency inserting ( 15) into ( 14) and solving for v.The model is thus completely solved once this value of v has been found.Notice that v → −v exchanges G 2x−1,2x (0) and G 2x,2x+1 (0) by virtue of (15), and this amounts to exchanging h a and h b .Since g(1) = 1, the limit v → 1 and v → −1 correspond to maximal correlations between nearest neighbor sites within and across cells, respectively.For periodic chains with L = 4, we can parallel the previous discussions on the entanglement structure and, consequently, on the typification of the operator algebras across the phase diagram.
Conclusions and future work.-Wedetermine the phase structure and entanglement for a large class of Majorana models with O(N ) symmetry in the large N limit.The von Neumann algebras underlying this entanglement structure are summarized in Fig. 2. We provide a class of exactly solvable models that gives rise to non-trivial operator algebra transitions in a highly controllable way.This allows us to track the parameter regimes in which the correlations signal the factorization of the ground state into a product state, as shown in Fig. 1.While here we use correlations for characterizing factorization, a state-based approach consists of using entanglement orbits [18].In spite of the different approaches used, we see a similar relation between factorization and the value of entanglement entropy.Analyzing these similarities is a promising line of future investigations.Intriguingly, the algebra transition we find in the strong coupling limit J → ∞ coincides with a phase transition that connects a factorized and a non-factorized state, similarly to the holographic Hawking-Page phase transition [12,13].Differently from our case (i.e.II 1 ↔ I ∞ ), this is a transition between algebras of type I ∞ and III 1 as function of the temperature.Remarkably, we observe an analogous algebra transition although our model differs from previous works in the context of holography [25][26][27][33][34][35][36], which consider on-site random interactions leading to a non-zero entropy at zero temperature.Further relations to holography can be obtained by including random disorder into our model by attaching an SYK model to each lattice site.This enlarges the phase diagram and we expect the competition between spatially inhomogeneous hoppings and the locally random disorder to change the renormalization group properties of critical points.Additionally, it is promising to investigate the mentioned connections with Brownian SYK [24,[37][38][39].Since our setup allows for general spatially disordered interactions, it is also relevant for further infinite disordered chains [40,41] where the hopping parameters are distributed according to a binary aperiodic sequence.These so-called aperiodic spin chains have recently been considered [21,[42][43][44] as a step towards establishing a holographic duality on discrete spaces.
We are grateful to Souvik Banerjee, Moritz Dorband, Elliott Gesteau, Shao-Kai Jian, Changan Li, René Meyer, Alexey Milekhin, and Sara Murciano for fruitful discussions.This work was supported by Germany's Excellence Strategy through the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter -ct.qmat (EXC 2147, project-id 390858490), and by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Center "ToCoTronics", Project-ID 258499086-SFB 1170, as well as a German-Israeli Project Cooperation (DIP) grant "Holography and the Swampland".ZYX also acknowledges support from the National Natural Science Foundation of China under Grant No. 12075298.We are grateful to the long term workshop YITP-T-23-01 held at the Yukawa Institute for Theoretical Physics, Kyoto University, where a part of this work was done.

Feynman diagrams
The Feynman diagrams are made of the bare propagator G 0 (τ 1 , τ 2 ) = 1 2 sgn(τ 1 − τ 2 ) and the simultaneous but non-local interactions given by the series expansion of the potential where h xy are expansion coefficients.The p = 1, 2, 3 order terms are represented by the interaction vertices shown in Fig. 3.

Feynman diagrams
The Feynman diagrams are made of the bare propagator G 0 (⌧ 1 , ⌧ 2 ) = 1 2 sgn(⌧ 1 ⌧ 2 ) and the simultaneous but non-local interactions given by the series expansion of the potential where the terms of (S.4) and (S.5) are constructed from those in (S.2) through combination and amputation of external legs.From the summation over the color indexes j, k, l, . . ., we can determine the order of the bubble diagrams in the 1/N expansion.The bubble diagrams on the first line (S.4) are order N 0 and give rise to the Schwinger-Dyson (SD) equations ( 3) and (4) in the main text.Thus, all Green's functions reported in the main text receive contributions from these diagrams.The diagrams on the second line (S.5) are order N 1 and therefore are suppressed in the large N limit.These provide sub-leading corrections to the results reported in the main text.3) and (4) in the main text.Thus, all Green's functions reported in the main text receive contributions from these diagrams.The diagrams on the second line of Fig. 4 are order N −1 and therefore are suppressed in the large N limit.These provide sub-leading corrections to the results reported in the main text.

Feynman diagrams
The Feynman diagrams are made of the bare propagator G 0 (⌧ 1 , ⌧ 2 ) = 1 2 sgn(⌧ 1 ⌧ 2 ) and the simultaneous but non-local interactions given by the series expansion of the potential where the terms of (S.4) and (S.5) are constructed from those in (S.2) through combination and amputation of external legs.From the summation over the color indexes j, k, l, . . ., we can determine the order of the bubble diagrams in the 1/N expansion.The bubble diagrams on the first line (S.4) are order N 0 and give rise to the Schwinger-Dyson (SD) equations ( 3) and ( 4) in the main text.Thus, all Green's functions reported in the main text receive contributions from these diagrams.The diagrams on the second line (S.5) are order N 1 and therefore are suppressed in the large N limit.These provide sub-leading corrections to the results reported in the main text.

General case
To solve the SD equations ( 3) and ( 4) in the main text, we work in Fourier space.For a system at finite temperature β −1 , we have where the Fourier transforms of G(τ 1 , τ 2 ) and Σ(τ 1 , τ 2 ) are defined by and ω n = 2π(n + 1/2)/β to account for the antiperiodicity of the fermionic variables.The dependence on the time difference τ 12 ≡ τ 1 − τ 2 is due to the time-translational invariance.The entries of G(τ 12 ) are obtained by taking the inverse Fourier transform of (S.3).In the limit of zero temperature, i.e. β → ∞, the frequencies ω n become a continuous variable ω and we replace n∈Z → β 0 dω in (S.4).In this work, we consider cases where the self-energy in Eq. ( 4) of the main text depends on time through a delta function.This implies that Σ(ω) is independent of the frequency.

Two-site chain
For the two-site chain discussed in the main text, the self-energy reads where we have chosen h(ξ) ≡ h 12 (ξ) = h 21 (−ξ) in the Hamiltonian in Eq. ( 1) of the main text and X ≡ −2iG 12 (0).Inserting the Fourier transform of (S.5) into (S.3),we obtain Exploiting (S.4), we find the matrix of the two-point functions at finite temperature β −1 = 1 for any value of τ > 0 where the sums over n in (S.4) for the various entries are evaluated by exploiting the standard trick of rewriting them as a contour integral of a function with simple poles along the imaginary axis.When τ = 0, (S.7) becomes Eq. ( 7) of the main text.Notice that G(τ ) is not yet fully determined since it depends on the entry G 12 (0) itself.To conclude the derivation, we have to find the value of G 12 (0) as a function of the potential h in the microscopic Hamiltonian.By comparing the off-diagonal components of both sides of (S.7) at τ = 0, we obtain the self-consistency (SC) Eq. ( 8) of the main text.The solution G 12 (0) obtained from the latter is to be plugged into (S.7) to determine the Green's function for any time τ > 0.

Four-site chain
Consider the four-site chain described by the Hamiltonian in Eq. ( 1) of the main text, with a nearest-neighbor interaction potential given by h x,x+1 (ξ) = h x+1,x (−ξ) and all the other components of h xy (ξ) equal to zero.Notice that, although in the main text we have chosen a specific (exponential) form of the potential, our derivation holds for generic functions.In the following analysis, to preserve the symmetries of the example discussed in the main text, we further impose reflection symmetry, i.e. h 12 (ξ) = h 34 (ξ).The expression of the free energy is Inserting its Fourier transform in (S.3), we obtain where we define the shorthand notation b ≡ 2h ′ 23 (−2iG 23 (0))), we notice that the dependence on the frequencies ω n occurs through the new parameter w n ≡ ω n /b and we introduce the parametrization r = 2 tan θ considered also in the main text.We are interested in the ground state correlations of the system, i.e. the limit β → ∞.In this regime, the sums over the frequencies in (S.4) to determine the two-point functions in the time domain become integrals over w ∈ [−∞, ∞].We can perform these integrals, noticing that each entry of (S.9) has four poles given by If τ > 0, we choose the integration contour including the two poles with negative imaginary part.Using the residue theorem, the two-point functions are found to be (S.11) where we introduce s such that 1  2 tan θ = 1 2 √ s 2 − 1, providing a more convenient parametrization for this result.As already seen for the two-site chain above, also in this case the Green's functions depend on their values at τ = 0 via the parameters s, b and, in turn, θ.By evaluating (S.11) at τ = 0 and recalling the relation between s and θ, we obtain Eq. ( 10) of the main text, which leads to the SC condition in Eq. (11).Inserting the solution of the SC condition back into (S.11),we finally determine all the two-point functions in terms of the microscopic parameters of the Hamiltonian and for any value of τ > 0.

Free energy from path integral
We now derive the free energy F of the two-site chain for generic interaction potentials.It is instructive to reinstate the physical temperature parameter 1/β for this purpose.We also write the interaction potential in the form h ′ (ξ) ≡ µ m(ξ) to extract the explicit dependence on the energy scale µ, where m(ξ) is dimensionless and encapsulates the functional dependence of the potential on the fermion bilinear.Using this notation, the Hamiltonian can be formally obtained by integration h(ξ) = µ dξ ′ m(ξ ′ ).We further define the dimensionless quantity μ ≡ µβ for convenience.The free energy is obtained by evaluating the on-shell effective action on solutions to the SC equation given by Eq. ( 8) of the main text.This is because from the path integral, we have (S.12) where the tildes denote integration variables and S E is the Euclidean effective action given in Eq. ( 2) of the main text, which we recall here for convenience Also, recall that h(ξ) ≡ h 12 (ξ) = h 21 (−ξ), and the other components are zero, without loss of generality.Remembering that X = −2iG 12 (0), we can write the third term of the effective action as Therefore, the effective action can be written as (S.15) In the large N limit, we can approximate the path integral in (S.12) by its saddle-point value, which then allows us to write where now the action is to be evaluated on-shell, i.e. on the solutions to the SD equations ( 3) and ( 4) of the main text.Subsequently, S E is to be computed on solutions to the SC condition given by Eq. ( 8) of the main text, which can be written as As discussed in [27], the direct evaluation of the effective action is non-trivial due to the presence of the Pfaffian determinant.To avoid this, we consider the derivative of the action with respect to μ, On-shell, the first two terms on the right hand side vanish and the only remaining contribution is where the on-shell condition imposes X = −2iG 12 (0).In order to obtain back the free energy, we need to integrate the expression above with respect to μ, keeping in mind that, on-shell, the argument of the potential function h(X) ≡ h(X(μ)) is now implicitly dependent on the integration constant, with dependence given through (S.17).However, we can circumvent this by changing the integration variable using dμ µ = dβ = ∂β(X) ∂X dX.From the SC equation (S.17), and recalling that h ′ (X) = µ m(X), we see that ∂β(X) . Therefore, we find the free energy to be given by to be evaluated on those values of X which are solutions to (S.17).Notice that the expression of F is correct up to a constant which is independent of X and μ, which originate from those terms of (S.18) which give zero contribution on-shell.Additionally, the internal energy of the system is given by (cf.(S. 19)) Computing the entropy S at leading order in 1/N , we find The leading 1/N correction can be extracted from the (χ−X) 2 order of the series expansions of the averaged quantities in (S.31).For the entropy, this correction is given by and is therefore of O((1/N ) 0 ) for |X| < 1.Notice that, when X → 1, the subleading corrections to S diverge and the computation above is not valid.The breakdown of the calculation can be physically interpreted as follows.The result (S.32) is obtained by assuming the system is characterized by the density matrix (S.24) and the Hamiltonian h.The form (S.32) is general as long as the system is in a mixed state and stops being valid only if the state becomes pure.Thus, the divergence of the corrections to S in (S.32) suggests that when X → 1, the system is in a pure state.Crucially, if the system is in a pure state, S = 0 to all orders in the 1/N expansion.This is consistent with the fact that lim X→1 s(X) = 0, as discussed in the main text and noticed from (S.27).
As explained in the main text, the corrections (S.32) and the fact that S = 0 when X → 1 can be exploited both as results for the thermal entropy of a two-site chain or for the entanglement entropy of a two-site block within a larger chain.The case of a two-site subsystem in a four-site chain was considered in the main text (see also the following sections of this SM).In that specific instance, we can provide further arguments supporting the fact that S A → 0 when X → 1 (the change of notation for S is due to the fact that in this example we always refer to an entanglement entropy).Indeed, we first recall from Eqs. ( 10) and ( 11) of the main text that X → 1 corresponds to r = µ a /µ b → ∞.By setting µ b = 0 in the Hamiltonian Eq. ( 1) of the main text, and subsequently taking the large N limit, we still obtain the SD equations ( 3) and ( 4) of the main text.In other words, the limits µ a /µ b → ∞ and N → ∞ commute.If µ b = 0 in Eq. ( 1) of the main text, it follows straightforwardly that S A = 0, given that the two parties of the bipartition factorize.Given the commutativity of the limits, we expect that this property also survives if we first take N → ∞ and then µ a /µ b → ∞, i.e.X → 1.The above arguments also match the analysis from the Feynman diagrams.When considering µ b → 0, the self-energy Σ xy with x = 1, 2 and y = 3, 4, including the whole 1/N series expansion in Fig. 4, vanish.It means that the correlations G xy with x = 1, 2 and y = 3, 4, including their 1/N corrections, also vanish in the limit µ b → 0. Therefore, we find that S A → 0 when X → 1 at all orders in 1/N .Summarizing, in this section we showed that the subleading corrections to the entropy of a two-site density matrix in the large N expansion are of order (1/N ) 0 when |X| < 1, while S = 0 at all order in 1/N if X → 1.When representing the entanglement entropy of half of a four-site chain with Hamiltonian in Eq. ( 1) of the main text, these results are crucial ingredients in the classification of the operator algebras in the various regimes of the phase diagram of the model.

Self-consistency equation and critical point
This section provides a detailed analysis of the SC equation for the four-site chain given in Eq. ( 11) of the main text with generic interaction potential obeying h x,x+1 (ξ) = h x+1,x (−ξ).We also define the short-hand notation h x,x+1 (ξ) ≡ h x (ξ).As in the previous section, it is instructive to extract the hopping parameters from the potential and define h ′ x,x+1 (ξ) = µ x m(ξ), where µ 1 = µ 3 = µ a , and µ 2 = µ b denote the hopping parameters on the four-site chain.We define the hopping ratio µa µ b ≡ r.Keep in mind that the function m(ξ) implicitly contains any dependence on the interaction strength J.For concreteness, the reader can follow the analysis of this section with the exponential potential m(ξ) = −e Jξ /2 in mind.The self-consistency equation for this potential is given by r e J(sin θ−cos θ) = 1 2 tan θ .(S.33) We plot solutions to this equation in Fig. 5, where we observe that all curves pass through the point r = 1/2 and tan θ = 1.This is a consequence of θ = π/4 being a fixed point of the equation (S.33), in the sense that sin(π/4) = cos(π/4) and therefore the self consistency equation is automatically solved, independently of the value of J.We also identify a value J ≡ J c = √ 2 below which there are unique solutions and above which the SC has three possible solutions.The fixed point of the SC equation mentioned previously allows us to find the critical value J c where the multi-valuedness of the solutions to the SC equation sets in for general potentials, as we show below.In the general case, we can consider a generic potential h ′ x (ξ) with x = a, b and ξ either sin θ or cos θ, as per Eq. ( 11) . Solutions to the self-consistency equation (S.33) for an exponential potential with different interaction strengths, as function of the hopping ratio r ≡ µa/µ b .Above the critical value J = Jc = √ 2, the equation starts having three possible solutions.Solid lines denote all SC solutions, while the dots represent a subset of all the thermodynamically dominant solutions, which are the ones minimizing the free energy (S.43) (derived in the next section) and shown in Fig. 1 of the main text.For J > Jc the SC solutions which are thermodynamically dominant exhibit a discontinuous jump at r = 1/2, indicating that the systems undergoes a phase transition.The two phases are characterized by their correlation structure, as shown schematically in the insets of Fig. 1 of the main text.
of the main text.The self consistency equation (S.33) specifies the dependence of the hopping ratio r as a function of θ as Notice that now only the function m appears, since the hopping ratio has been extracted explicitly into r.In order to identify the critical point, we need to investigate the extrema and inflection points of the function r(θ) defined in (S.34).We do this by setting its first and second derivative equal to zero.For general functions m, this would yield two equations with three unknowns, namely m(ξ(θ * )), m ′ (ξ(θ * )) and m ′′ (ξ(θ * )), where θ * is the point under consideration which is a solution of (S.34) and ξ can be either the function sin(•) or cos(•).Since all curves for all J pass through the point (1/2, π/4), we can analyze these equations at θ * = π/4 in order to specify J c .Indeed, J c is the value where the function r(θ) has a saddle point, meaning that both its first and second derivative evaluated at this point vanish.The first derivative takes the form Importantly, note that the arguments of 1/ √ 2 arising in (S.35) come from evaluating sin(π/4) = cos(π/4) = 1/ √ 2. When imposing (S. 35) for an explicit potential m(ξ), one should first compute its derivative for a generic argument ξ and subsequently evaluate it at ξ = 1/ √ 2. Indeed, it is only after the insertion of an explicit potential m(ξ) that the parameter J enters, and it is precisely the condition (S.35) which singles out the value of J c for that specific potential.Since we are evaluating functions at a specific value, the constraint (S. 35) is no longer a partial differential equation but rather a simple algebraic equation with solution . The second derivative is given by Indeed, we see that the equation (S.36) is solved automatically by the solution of (S.35) for a generic function m(ξ) when inserting the values r = 1/2 and θ = π/4.Therefore, we conclude that when the potential obeys as in (S.35), this condition singles out a specific value of J, namely J c , for which the solution r(θ) has a saddle point at θ = π/4.Let us take a look at explicit potential examples.For the exponential potential we have m(ξ) = −e Jξ /2, the condition (S.35) is solved straightforwardly and yields J c = √ 2. We can also consider truncations of the series expansion of the exponential, i.e potentials of the form These truncations describe the cases where we restrict the amount of interaction terms in the original Hamiltonian.We can evaluate the condition (S.35) to obtain We thus see that determining the value of J c boils down to finding the roots of an order n polynomial with constant coefficients.Therefore, the critical point J c can be found analytically for all potential of the form given in (S.37).Importantly, the multi-valuedness of the solutions to the SC equation is an indicator that the system undergoes a phase transition, as we identify in more detail in the next section.More precisely, this phase transition can be detected by computing the free energy on the SC solutions, as we present in Fig. 6 for different choices of the potential.

Free energy and phase transition
We now give a derivation of the free energy F for the four-site chain with generic potentials.This derivation follows the same steps as that of Section "Free energy from path integral" for the two-site chain, with the main differences being the effective action and the physical parameters entering the problem.In the case of the four-site chain, the last term of the effective action (S.13) reads (S.40) Evaluating this expression on-shell, the first two terms vanish because the fields G, Σ obey the equations of motion.The only non-vanishing contribution comes from the explicit dependence on the hopping ratio, and thus we have where once again the SC condition imposes X 12 = −2iG 12 (0), similarly to the two-site case.The difference is than now G 12 (0) is the two-point function of the four-site chain, cf.(S.11) at τ = 0 or Eq. ( 10) of the main text.
In order to obtain the free energy, one would need to integrate back over r, albeit in a suitable parametrization.A useful one is in terms of the parameter θ appearing in the SC equation, which gives −2iG 12 (0) = sin θ and for which r is itself dependent on θ and the interaction potential via (S.34).Implementing this change of variables, the general expression for the free energy reads .
(S.42)This expression is somewhat reminiscent of the one obtained in (S.20) for the two-site case.In the latter, due to the simple nature of the two-site system, the integral could be evaluated without explicit knowledge of the functions h or m.For the four-site chain this is no longer the case, and the integral (S.42) can only be evaluated explicitly after choosing a specific form of the interaction potential.For the case of the exponential potential discussed in the main text, we have m(ξ) = −e Jξ /2, and we fix its integral to give the interaction potential mentioned in the main text, i.e. h x (ξ) = µ x dξ ′ m(ξ ′ ) = µx 2J (1 − e Jξ ).Inserting these expressions into (S.42),we obtain the free energy of the four-site chain tan(θ)e J(cos(θ)−sin(θ)) − (tan(θ) + 1)e J cos(θ) + e J , (S. 43) where the last term e J results from evaluating the integration constant at θ = 0, tan(θ 0 )e J(cos(θ0)−sin(θ0)) − (1 + tan(θ 0 )e J cos(θ0) ) θ0=0 = −e J .(S.44)This is the free energy plotted in Fig. 1 of the main text for different values of the interaction strength J. Without the loss of generality, we have set µ b = 1 for all the plots.Recall that the free energy is to be evaluated on solutions θ to the SC equation (S.34), which can be multi-valued for J > J c as discussed in the previous sections of this SM.This multi-valuedness is then also present in the plots for the free energy.The SC solutions which minimize the free energy (S.43) are the thermodynamically dominant ones, and it can be seen from Fig. 5 that these exhibit a discontinuity at r = 1/2 for J > J c .This leads to a non-analyticity of the free energy (S.43), signaling the phase transition.We can identify two phases, one for r > 1/2 and the other for r < 1/2, and the function (tan(θ))(r) reveals the character of each of the phases.In the phase of r < 1/2, tan θ, and thus θ, is small, and therefore the correlations that dominate the system according to the Green's function given in Eq. ( 10) of the main text are G 23 and G 14 .In the phase of r > 1/2, tan θ is large, implying that θ approaches π/2.Again, comparing with the Green's function given in Eq. ( 10) of the main text, this indicates that in this phase the correlations G 12 = G 34 are the dominant ones.This correlation structure is shown in the insets of Fig. 1 of the main text for the limits r → 0 and r → ∞.
For completeness, we evaluate the free energy (S.42) for some examples of truncated polynomial potentials m of the form given in (S.37) and for different values of the interaction strength J.In particular, we consider linear, quadratic, cubic and quartic potential functions m.The resulting free energy computed via (S.42) is shown in the panels of Fig. 6.The critical values J c , if existent, have been obtained through (S.38), and the free energy at these critical values is always displayed, together with values of J below and above the critical one.The linear case is the only one not exhibiting a phase transition, given that its critical interaction strength is J c = ∞.The critical values for the truncated polynomial potentials approach the critical value of the exponential potential when the order is increased.This is expected since the series expansion (S.37) approaches that of the exponential when the order n → ∞.For the other three potentials shown as an example here, the system undergoes a phase transition at r = 1/2 for J > J c , where the two phases are characterized by their correlation structure, as explained above for the case of the exponential potential, cf.In what follows, we consider the four-site chain with Hamiltonian (1) of the main text and potential given by h This choice corresponds to having a free model, for which explicit results for the reduced density matrix are known.This case therefore provides a rich playground to gain insights into the entanglement structure in general, and the properties of the underlying operator algebras in particular.In this free case, the ground state factorizes into contributions from the different species (labeled by j) of Majorana fermions and, due to this fact, the model can be solved for any value of N .Because of the quadratic nature of the model, the reduced density matrix of a two-site subsystem A has the form where the two sites in the subsystem are labeled by 1 and 2, the constant Z ensures the normalization Trρ A = 1 and the 2 × 2 matrix K xy is the only unknown to determine ρ A .Let us notice that (ψ j x ) 2 = 1/2 and the constant resulting We provide details on our analysis of the limit J → ∞, focusing in particular on the case of an exponential interaction potential.Based on Fig. 2 of the main text, it appears that the behavior of the entanglement entropy density S A /N is that of a step function when J → ∞, abruptly going from entanglement S A /N = log 2 to zero entanglement S A /N = 0 at r ≡ µ a /µ b = 1/2.We now make these considerations more precise by analyzing the SC constraint given in Eq. ( 11) of the main text in this limit.We start by recalling that in order to evaluate the entanglement entropy, we need to find the solutions to the SC equation (cf.(S.33) and Eq. ( 11) of the main text) re J(sin θ−cos θ) = 1 2 tan θ , (S.54) For J > J c = √ 2, which is clearly the case if J → ∞, there are three possible solutions, the thermodynamically dominant one being that which minimizes the free energy functional (S.43).For the four-site chain at zero temperature, we know that the SC equation (S.54) has a fixed point which is independent of J, namely θ * = π/4 (cf.discussion below (S.33)).This fixed point also imposes that r = 1/2.Consider now the two cases of slight deviations from this fixed point, namely θ ± = θ * ± ϵ, with ϵ ≪ 1.Note that these deviations automatically imply that we restrict ourselves to values of r slightly above or below 1/2, respectively.For θ + , we have that the exponent in the SC equation is positive, i.e. (sin θ − cos θ) > 0. This implies that e J(sin θ−cos θ) → ∞ as J → ∞.For self-consistency, the r.h.s. of the SC equation (S.54) must therefore also diverge, even though it has no explicit J-dependence.From tan θ → ∞ we extract that the solution to the SC must tend to θ → π/2.Using the relation (S.49) (recall that the latter holds both for free and interacting cases), this implies that κ → −∞, which in turn yields X → 1 after recalling that X = − tanh κ by virtue of (S.48).Thus, the corresponding entanglement entropy vanishes S A → 0 according to (S.23) and Eq. ( 9) of the main text.Analogously, for the case of the deviation θ − , we have the opposite behavior in which e J(sin θ−cos θ) → 0 as J → ∞, implying that the self-consistent solution tends to θ → 0. Again using (S.49) and (S.23) (or Eq. ( 9) of the main text), we see that this implies κ → 0 (X → 0) and therefore entanglement S A /N → log 2. Thus, we have shown that in the limit of J → ∞, the entanglement entropy is maximal (at leading order for N → ∞) in the whole regime 0 ≤ r < 1/2, while it is vanishing for r > 1/2, based on the behavior of the solutions to the SC equation.Remarkably, this phenomenon is unique to the class of potentials with exponential behavior.This is because for any potentials given by a general truncated series expansion of the form (S.37), the behavior of the l.h.s. of (S.54) when J → ∞ does not impose a constraint on the r.h.s..More precisely, for exponential potentials we were left with the function e J(sin θ−cos θ) , which has a clear-cut limiting behavior as J → ∞ based on the value of θ.In contrast, for any truncated potential, the limit J → ∞ would make the highest-power term dominate, such that ) . (S.55) The SC equation (S.54) in this case takes on the form r(tan θ) n = 1 2 tan θ , (S.56) which indeed has uniquely defined solutions given by the smooth function θ = tanh −1 ((2r) 1 1−n ) throughout the whole range of θ.Therefore, we see that for truncated polynomial interaction potentials, the SC solutions exhibit a smooth behavior.Thus, the entanglement entropy does no longer have a discontinuity.We further report evidence as to how the limit J → ∞ leads to a Heaviside step function Θ centered at r = 1/2 for both the correlations X = Θ(r − 1/2) and correspondingly the entanglement entropy S A /N = log 2Θ(1/2 − r) for the case of an exponential interaction potential.Consider the deviation θ + .We know from the above considerations that the self-consistent solution should be given in this regime by θ = π/2.Let us therefore expand the SC equation with respect to a small deviation ε ≪ 1 from this point.At first order, we have Note that this expression is not necessarily an exact solution of the SC, but rather an approximation which is very good in the regimes of r far away from 1/2.Around 1/2, the validity of the approximation increases as J gets larger.
The expression (S.60) formally allows us to set r = 1/2 and then take the limit J → ∞.However, since we are working with an approximation as mentioned above, it is more proper to leave the value of the hopping ratio r free and analyze the overall behavior of the expression (S.59) when J → ∞.It is straightforward to see that lim We thus see that the value of the entanglement entropy for a value of r infinitesimally larger than 1/2 tends to zero in the large J limit.Therefore, the entropy as a whole tends to zero for any r > 1/2 when J → ∞.
In fact, the exact same analysis can be done for the other relevant value of the SC equation, namely the point towards which θ is pushed when slightly deviating from r = 1/2 by −ϵ.In this case, from the considerations of the previous paragraph, the solution to the SC equation is θ = 0. We can again expand the SC equation around this point and insert the resulting dependence θ(r, J) into the formula for the entanglement entropy, obtaining an expression very similar to (S.59) Once again, we see that the value of the entropy density for a value of r infinitesimally smaller than 1/2 tends to log 2. The entropy density as a whole thus tends to this value for all r < 1/2 when J → ∞.

Periodic chain
We consider the closed periodic chains discussed in the dedicated section of the main text.We assume to have a chain with L sites described by the Hamiltonian (1) in the main text with a staggered interaction of the form (13).In our conventions, x = 0, . . ., L − 1 labels the site of the chain and the periodic boundary conditions impose the identification x ∼ L + x for this variable.This model exhibits translational invariance with respect to cells of adjacent sites interacting by h b .More precisely, we can introduce r = 0, . . ., L 2 − 1, such that x = 2r + s, with s ∈ {0, 1}.The chain is then invariant with respect to the translation of the index r.Given x = 2r 1 + s 1 and y = 2r 2 + s 2 , h xy ≡ h s1s2 r1r2 and we can rewrite the derivative of Eq. ( 13) of the main text as h ′ s1s2 r1r2 = M b δ r1r2 ϵ s1s2 + M a (δ r1−r2,−1 δ 1,s1 δ 0,s2 − δ r1−r2,1 δ 0,s1 δ 1,s2 ) , (S.66) where ϵ s1s2 is the totally anti-symmetric symbol such that ϵ 01 = 1 and we have further defined M a ≡ h ′ a −2iG 1,0 r,r+1 (0) , M b ≡ h ′ b −2iG 0,1 r,r (0) , (S.67) exploiting the fact that the components of h ′ s1s2 r1r2 depend on the indices r 1 and r 2 only through their differences due to the aforementioned translational invariance.For convenience, we also denote the spatial dependence of G xy in terms of r 1 , r 2 , s 1 and s 2 .Given that h ′ s1s2 r1r2 depends only on r 1 − r 2 ≡ δr and on the two indices s 1 and s 2 , it is particularly convenient to write the entries of h ′ s1s2 r1r2 uniquely as function of δr and collect them into 2 × 2 matrices in the indices s 1 and s 2 .We thus have where ω is to be thought as being multiplied by the 4 × 4 identity matrix, and in the last step we have used (S.68) and the parametrization in (S.70).Assuming τ > 0 and inserting (S.72) into the second equation of (S.69), we obtain Exploiting also the first equation of (S.69), the Green's functions of the system in its ground state for τ > 0 read where the last expression is obtained in the limit L → ∞ of infinitely many sites along the chain integrating over p = 2k/L, and we have introduced ε(p) ≡ lim L→∞ ε k=pL/2 and φ(p) ≡ lim L→∞ φ k=pL/2 .The functions ε(p) and φ(p) in (S.74) are not explicitly known given that they depend via M a and M b on some of the entries of the Green's function evaluated at τ = 0. Thus, in order to determine the two-point functions and solve the model, we still need to find expressions of ε(p) and φ(p) in terms of the parameters of the microscopic Hamiltonian.Following the spirit ′ xy (τ 12 ) − z dτ 3 Σ xz (τ 13 )G zy (τ 32 ) = δ xy δ(τ 12 ), (3) Σ xy (τ 12 ) = 2i h ′ xy (−2iG xy (0))δ(τ 12 ) ,

2 J = 2 J = 2 <
l a t e x i t s h a 1 _ b a s e 6 4 = " Y i 9 5 o B S D H 6 g 8 J r y 2 H a C s B 4 q D

9 FFigure 1 .
Figure1.Free energy of the four-site chain with potential hx,x+1(ξ) = µx 1 − e Jξ /(2J) for different interaction strengths J.We observe a non-analyticity at r ≡ µa/µ b = 1/2 for J above the critical value Jc = √ 2, signaling a phase transition.The two phases of the system are characterized by the correlation structure given by(10), whose limiting cases for r → 0 and r → ∞ are shown in the two embedded diagrams.

Figure 2 .
Figure2.Entanglement entropy(9) of subregion A on solutions to(11) as function of r ≡ µa/µ b for different coupling J. Solid lines denote all SC solutions, while a sample of physical solutions minimizing F in Fig.1is marked with dots.The phase transition is signaled by the discontinuity for J > Jc.Inset: von Neumann type of the algebra AA in different regimes of the phase diagram.Each type is denoted by a different color, and the black dot (line) represents a phase transition of second (first) order.
xy are expansion coefficients.In the spirit of Feynman diagrams, the p = 1, 2, 3 order terms are represented by the interaction vertices (S.2), each dot indicates the non-local bilinear operator j x j y and the dotted lines indicate the synchronicity of the interacting operators.The Green's function G xy (⌧ 1 , ⌧ 2 ) = D lines indicate the bare propagators and the blob indicates the one-particle irreducible (1PI) diagram.The self energy ⌃(⌧ 1 , ⌧ 2 ) is given by the 1PI diagram

Figure 3 . 1 N j ψ j x ψ j y in the presence of interactions is given by the
Figure 3. Interaction vertices for terms in (S.1) for orders p = 1, 2, 3.Each dot indicates the non-local bilinear operator ψ j x ψ j y xy are expansion coefficients.In the spirit of Feynman diagrams, the p = 1, 2, 3 order terms are represented by the interaction vertices (S.2), each dot indicates the non-local bilinear operator j x j y and the dotted lines indicate the synchronicity of the interacting operators.The Green's function G xy (⌧ 1 , ⌧ 2 ) = D lines indicate the bare propagators and the blob indicates the one-particle irreducible (1PI) diagram.The self energy ⌃(⌧ 1 , ⌧ 2 ) is given by the 1PI diagram

Figure 4 .
Figure 4. 1PI diagram constructed from the diagrams in Fig. 3 through their combination and subsequent amputation of external legs.

− 1 2
x,y dτ h xy (−2iG xy (0)) = −Vol(τ ) 2h a (−2iG 12 (0)) + h b (−2iG 23 (0)) = −Vol(τ )µ b 2r X12 dξm(ξ)+ X23 dξm(ξ) ≡ S h [G](r) , (S.39) where we have exploited the anti-symmetry of G, the property h x,x+1 (ξ) = h x+1,x (−ξ), and the fact that G 12 (0) = G 34 (0).We have also written the potential in the form used in previous sections,h a,b (ξ) = µ a,b ξ dξ ′ m(ξ ′ ),where the two integrals are over the same function m but evaluated at different values ξ = X 12 ≡ −2iG 12 (0) and ξ = X 23 ≡ −2iG 23 (0), in a slight but necessary abuse of notation.Further, note that the integrals over Euclidean time appearing in the effective action (S.13) are trivial and give only a constant contribution, since the self-energy (S.8) contains a Dirac delta in τ , therefore leaving only G(0) present in the action.More precisely, there is a constant (divergent) contribution which we denote as dτ ≡ Vol(τ ) coming from the integrals in the action.Finally, we have extracted a factor of µ b such as to identify the hopping ratio r in the first term of the second line in (S.39).We can now split the total action into three terms−S E /N ≡ S Pf [Σ] + S G [G, Σ] + S h [G](r).In a similar fashion as for the two-site chain, we avoid the evaluation of the Pfaffian by performing a derivative with respect to the dimensionless hopping ratio r,