Dynamical localization transition of string breaking in quantum spin chains

The fission of a string connecting two charges is an astounding phenomenon in confining gauge theories. The dynamics of this process have been studied intensively in recent years, with plenty of numerical results yielding a dichotomy: the confining string can decay relatively fast or persist up to extremely long times. Here, we put forward a dynamical localization transition as the mechanism underlying this dichotomy. To this end, we derive an effective string breaking description in the light-meson sector of a confined spin chain and show that the problem can be regarded as a dynamical localization transition in Fock space. Fast and suppressed string breaking dynamics are identified with delocalized and localized behavior, respectively. We then provide a further reduction of the dynamical string breaking problem onto a quantum impurity model, where the string is represented as an"impurity"immersed in a meson bath. It is shown that this model features a localization-delocalization transition, giving a general and simple physical basis to understand the qualitatively distinct string breaking regimes. These findings are directly relevant for a wider class of confining lattice models in any dimension and could be realized on present-day Rydberg quantum simulators.

Introduction.-Theefficient implementation of gauge theories is a central target in quantum simulation [1][2][3][4][5][6], with some remarkable experimental realizations achieved in recent years [7][8][9][10][11][12][13][14][15][16][17][18].However, the intrinsic structure of gauge theory still poses formidable technical challenges.Simultaneously, quantum spin chains, which are more amenable to quantum simulation, have been shown to be a versatile platform to emulate lattice gauge theory phenomenology.This has led to recent intensive efforts to investigate the structure of the gauge vacuum and outof-equilibrium transport properties under the influence of confinement in this setting .Yet, various aspects of such phenomena remain to be elucidated.In particular, numerical studies of dynamical string breakingwhere a string connecting two charges decays due to pair production [41,42]-suggest a dichotomy for the fate of the confining string: its fission can occur relatively fast or be substantially delayed.
In this Letter, we discuss how these observations can be interpreted in terms of an underlying dynamical localization transition.In this picture, the localized phase corresponds to a regime with a long-lived (prethermal) string, while the delocalized phase to fast string breaking.First, we show via exact diagonalization in quantum Ising chains that two qualitatively different string dynamics are separated by a sharp threshold in the longtime behavior of dynamical quantities.In particular, we study the survival probability and the half-chain entanglement entropy, with the former quantity serving as a direct diagnostics of string breaking.We then derive an effective model for the breaking of a short string by projecting onto a reduced subspace that captures resonant decay channels in the limit of vanishing transverse field.Within this effective description, string breaking can be understood as a dynamical localization problem in Fock space.Next, this description is heuristically generalized to a quantum impurity model, where the string is effectively represented by a few-level system coupled to a meson bath.We show that this model features a dynamical localization-delocalization transition, with both sides of the transition explaining the observed string breaking regimes.This description, independent of microscopic details, provides a general and simple physical basis to understand dynamical string breaking.Finally, we discuss how our results can be applied to a wider class of confining lattice models in any dimension, and potential implementations with Rydberg quantum simulators.
String dynamics in quantum Ising chains.-Weconsider the quantum Ising model in both transverse (h x ) and longitudinal (h z ) fields, whose Hamiltonian for L spins on the ring reads where σx/z i are the Pauli matrices at site i, and J > 0 is the strength of a ferromagnetic coupling.The model ( 1) is of paramount importance in various fields-from statistical mechanics and condensed matter [43,44] to highenergy physics [45][46][47].Further, it can be naturally realized in present-day Rydberg quantum simulators [48][49][50][51][52][53], and solid-state materials [54].Both integrability and Z 2 symmetry are broken by a finite h z , which induces a confining potential between pairs of domain wall (DW) excitations (provided that h x < J).In this scenario, pairs of DWs form bound, mesonlike states.String breaking dynamics can then be probed by studying the stability of one such object under the unitary evolution generated by the Hamiltonian (1).Below, we review the main aspects of this process (see also Ref. [24]), in the confining regime with controlled quantum fluctuations h x ≪ J.
A dichotomy between distinct string breaking dynamics is revealed in a simple experimentally feasible quantum quench protocol.The system is initially prepared in a state with an Ising electric-field string of ℓ ↓-spins (in the σ z basis) that connects two DWs, on top of the vacuum, i.e., |ψ string (ℓ Next, the real-time evolution of the system in Eq. ( 1) is studied at finite h x /J and h z /J.Two qualitatively different dynamical string breaking scenarios are illustrated in Figs.1(a) and 1(b), for an initial string of length ℓ = 4.The dynamics are shown in terms of the local DW density, ν i (t) = 1 2 ⟨ Î − σz i (t)σ z i+1 (t)⟩, defined on the bonds between consecutive lattice sites.In Fig. 1(a), a rapid production of new DW pairs occurs inside the string, eventually leading to its decay and emission of lighter mesons.A subsequent proliferation of DW pairs throughout the whole chain eventually restores translation invariance, in agreement with the fact that the system (1) is ergodic and thermalizing at late times [55].This fast string breaking dynamics can be understood as a consequence of underlying resonances that arise for commensurable (J, h z ) [24,56].In sharp contrast, the rapid string breaking dynamics is surprisingly absent in Fig. 1(b), up to the accessed long time O(10 2 J −1 ), which is also beyond the timescale for light meson kinetics t ≫ J/h 2 x .Based on general thermalization arguments (as mentioned above), the latter regime must be understood only as a prethermal phenomenon [34].
The scenarios above have been observed in both quantum spin models [21,24,25] and low-dimensional lattice gauge theories [57][58][59][60][61][62].However, a general picture of how these systems cross from one regime over to the other remains to be provided.As a first step in this quest, we study the long-time behavior of the string survival probability and the half-chain entanglement entropy where |ψ(t)⟩ is the time-evolved many-body wave function and ρA (t) = Tr B [|ψ(t)⟩⟨ψ(t)|] is the reduced density matrix computed on one half of the chain (cutting through the middle of the string and the opposite point on the periodic chain).We compute long-time averages as O = 1 In our calculations we take Jt f = 10 4 , and Jt i = 3Jt sb , where Jt sb ≡ π 2(hx/J) 2 is a typical timescale for string breaking [24].The longtime averages of the quantities in Eqs. ( 2) and ( 3) are shown in Figs.1(c) and 1(d), for various values of h z /J and h x /J.We observe a sharp threshold-defined by the point where P = 0.5-, which roughly scales linearly with (h x /J) 2 [inset in Fig. 1(c)], and separates a regime where the string breaks (P ∼ 0) from one in which it persists up to the accessed timescales (P ∼ 1).The behavior of S shows that string breaking is characterized by a significant amount of entanglement, while in suppressed string breaking dynamics entanglement production is strongly diminished.
String breaking as a localization problem in Fock space.-Wenow derive an effective description of the above phenomenology.Let us fix ℓ = 4, as before.For the considered parameter regime h x ≪ h z ∼ J, one can systematically project out sectors of the Hilbert space that do not participate in resonant decay channels, by applying a Schrieffer-Wolff transformation [63,64] to (1), see Ref. [65] for details.Here, the relevant physical subspace is formed by the direct sum of the "string" sector and the "1-meson pair" sector, see Fig. 2(a).The former sector is spanned by the kets with a string of size ℓ = 4, labelled by the site index j of the first ↓-spin.The second sector comprises configurations with exactly two 1-meson particles: even), is the relative distance between the two ↓-spins.
The resulting effective model reads: and repulsive contact interaction [U = 9v/2] of the 1-meson particles; and Ĥλ couples the two relevant sectors with amplitude λ = −3v; see Fig. 2(a) and Ref. [65] for details.The latter term is responsible for the processes of pair creation and recombination, and therefore, crucial for string breaking.
The energy spectrum of this model is shown in Figs.2(b) and 2(c), for different choices of parameters.In Fig. 2(b) all bands are close in energy, while in Fig. 2(c) a large gap separates an isolated band (associated to string modes) from the rest.In the latter case, string modes are strongly localized.This is quantified by the colorbar in Figs.2(b) and 2(c), which shows the value of the inverse participation ratio (IPR) of individual energy eigenstates: where {|n⟩} are eigenstates of Ĥeff and {|a⟩} preferential basis states.Localized behavior of |n⟩ occurs when IPR(n) ≃ 1, while IPR(n) vanishes as 1/D in the maximally delocalized case, where D is the Hilbert space dimension [66].
The evolution of the survival probability [Eq.( 2)], corresponding to the two cases above, is shown in Figs.2(d) and 2(e).While in the former case, the string eventually breaks (P ∼ 0), in the latter it survives (P ∼ 1) up to long times.The spectra in Figs.2(b) and 2(c) are hence identified with fast and suppressed string breaking dynamics, respectively.String breaking can thus be seen as a dynamical localization problem in the Fock-space graph in Fig. 2(a), where the string localizes if it is not resonantly coupled to the continuum of 1-meson pairs.Quantitative agreement with the dynamics in the full Ising model is also observed in Figs.2(d) and 2(e), which can be systematically improved by decreasing h x /J [65].
Quantum impurity model picture.-Theabove picture resembles localization phenomena in quantum impurity models (QIMs) [67][68][69].This is the basis for a further reduction of the string breaking problem.Let us consider an elementary string breaking/fusion process: where a string (i) gets cut near its edges via pair creation, yielding a metastable configuration (ii), and eventually, a shorter string plus a 1-meson (iii) [58,62].We encode the different configurations of this basic process in the internal states of a spin-1 system ("impurity").Concretely, we map the symmetric and antisymmetric string states onto the impurity states |S z = ±1⟩, respectively, and the state where the string has been cut and a lighter meson radiated onto |S z = 0⟩.The impurity is also locally coupled to a meson bath in analogy to the picture in Fig. 2(a).This motivates a QIM with Hamiltonian where Ĥimp = (M − µ)( Ŝz ) 2 + Λ Ŝz , contains the string mass term M , a chemical potential µ accounting for 1 +h.c., couples the impurity with the bath such that if |S z = 0⟩ a meson at site 1 is created, and whenever |S z = ±1⟩ a meson at that site is annihilated [70].
A schematic of this mapping is shown in Fig. 3(a) for a short string that can decay into two shorter strings.The latter can be emitted into a meson bath, if the impurity-bath coupling is resonant, leaving the impurity in its "vacuum" state.Otherwise, the shorter strings can recombine back into a longer string, avoiding its decay.This QIM picture thus offers a distilled abstraction of the effective graph model in Fig. 2(a).We note, however, that the mapping between these two models is not exact.Yet, as shown below, both models have significant similarities both in the behavior of their eigenstates as well as in the dynamics of the impurity and the string.
The QIM in Eq. ( 6) features a localizationdelocalization transition, explicitly shown in the single-meson limit.In this limit the spin can be replaced by two hard-core bosons, and due to particle number conservation, exact diagonalization is possible for large system sizes [65].Focusing on the IPR of individual eigenstates, we see that the impurity mode can abruptly localize when varying Λ/T above a critical Λ c /T , see Fig. 3(b), while the IPR of bulk eigenstates always vanishes (not shown).A standard finite-size scaling analysis [72], see Fig. 3(c), yields Λ c /T = 1.41 (1) and critical exponents ζ = −1.02(5), ν = 1.00 (5), for the considered parameters.
Such localization-delocalization transition underlies and governs two qualitatively different spin dynamics, see Fig. 3(d).
Here we plot the longtime averaged spin autocorrelation function M(Λ) = lim t→∞ where ⟨•⟩ Λ denotes the expectation value at a given Λ/T .This quantity plays an equivalent role to the survival probability for the spin chain [Eq.( 2)], and likewise, it vanishes on the delocalized side of the transition, while it approaches unity as we ramp up Λ/T , above the localization transition point.
Our conclusions are restricted to the lattice as we have only considered a bounded spectrum of excitations.We expect our observations to hold beyond the limit h x ≪ J, as long as there exist values of h z /J for which certain decay channels lead to faster dynamics than in other regimes.We note that our effective descriptions are valid only within the prethermal timescale of the localized regime.Also, further localization transitions may occur around other resonance points of the spin chain, which could involve longer decay paths [24,32], and hence, would require to consider an impurity with more internal levels.Regarding the Fock-space graph model, we note that adding higher-order corrections could reshape the transition path and change the criticality.Nevertheless, as the effect of such higher-order terms is just a renormalization of hopping amplitudes [25], we expect the physics to remain qualitatively unaltered far from the localization transition point and deep in the two phases.
Discussion and outlook.-Weexpect our main results to be relevant for a wider class of confining theories in one and higher dimensions.In effect, what seems to be crucial in the applicability of the QIM picture is that the system retains rotational symmetry, with the radial coordinate effectively defining a one-dimensional problem, when integrating out the rotation degree of freedom [68].Fermionic bound states (e.g., baryons) could also be accounted for by changing the statistics of the bath [68,69].Finally, our observations can be experimentally realized with current quantum technologies.In particular, Rydberg atoms offer a well suited platform, in which both the initial string states and the target unitary dynamics can be implemented in a highly controllable way [48][49][50][51][52][53].
Data availability.The data shown in the figures is available on Zenodo [73].
Acknowledgments.We thank T.

PERTURBATION THEORY FOR THE EFFECTIVE MODEL IN THE FOCK-SPACE GRAPH
In this section, we give a detailed derivation of the effective model in Eq. ( 4).We restrict ourselves to the limit of weak transverse fields, h x ≪ J.In this limit, it is convenient to write the Hamiltonian (1) as where and regard Ĥ1 as a small perturbation.Within this setting, we can perform a Schrieffer-Wolff (SW) transformation [63], i.e., a unitary transformation e Ŝ to eliminate off-diagonal terms that do not preserve Ĥ0 , order by order in the strength of the perturbation.By choosing the generator of the transformation Ŝ = Ŝ1 + Ŝ2 + • • • , such that [ Ŝ1 , Ĥ0 ] = − Ĥ1 , we get the following second-order Hamiltonian [32,64]: where P ↑(↓) i = ( 1 ± σz i )/2 is the projector onto ↑ (↓) at site i, and σ± i = (σ x i ± iσ y i )/2, are spin-1 2 raising and lowering operators.We have also introduced a succinct notation for the relevant energy scales: ∆ 0 ≡ h 2 x /(2h z ), ∆ ± ≡ h 2 x /(2h z ± 4J).The ℓ-meson energies (with respect to the Ising vacuum energy), are obtained, to leading order, from the unperturbed Hamiltonian Ĥ0 and the diagonal terms on the r.h.s. of Eq. (S3).We get x /J 3 ), (S4) We now focus on the situation of interest considered in the main text.Namely, h x ≪ h z = J and an initial string of size ℓ = 4.The breaking of the string will necessarily yield the formation of two lighter mesons.Within the description above, the resonant decay channel (neglecting for a moment perturbative (h x /J) 2 corrections) is into states with exactly two 1-meson particles, as it can be easily verified from Eqs. (S4) and (S5), for the relevant parameters.Hence, to the leading order, we can project the parent Hamiltonian onto the subsector of states with a single 4-meson particle ("string") and the subsector of two 1-meson states.By doing this, we get the effective model illustrated in Fig. 2 of the main text.Let us now explicitly write down the different terms of the this effective model.In the first place, we have the string and the 1-meson energies, which are given by The latter term gives the first contribution in Eq. ( 4).Namely, with the same notation for the ket |S j ⟩ as in the main text.On the other hand, the 1-meson energy gives the diagonal part of Ĥmesons in Eq. ( 4), namely, Ĥdiag mesons |j, d⟩ = 2m 1 |j, d⟩.We note however that this term has to be modified so as to include a hardcore repulsive interaction between 1-meson particles, which arises from the second and third terms on the r.h.s. of Eq. (S3), namely, U ≡ Next, we have to consider the off-diagonal terms on the r.h.s. of Eq. ( S3).The first one of them, proportional to ∆ + − ∆ 0 , acts as a hopping term for the 1-meson particles.This energy scale, therefore, defines the kinetic energy of such particles, v ≡ −(∆ + − ∆ 0 ) = h 2 x 3J .Hence, Ĥmesons in Eq. ( 4), acts on a 1-meson pair ket |j, d⟩, as follows: The last term on the r.h.s. of Eq. (S3), plays a crucial role in our description, for it describes the decay of a string of length ℓ = 4 into a 1-meson pair, with a relative distance d = 3 and the position of the first ↓-spin in the latter configuration coinciding with that of the first ↓-spin in the former string configuration.This term also describes the reverse process of particle recombination into a single 4-meson particle.These two process are schematically illustrated by the green bond in Fig. 2(a).The numerical coefficient in front of this term thus sets the energy scale for the coupling Ĥλ , namely, λ ≡ We note that the fifth term on the r.h.s. of Eq. (S3), yields no contribution when projecting onto the physically relevant subsectors for the case at hand.Furthermore, even though the derivation above holds strictly for h z = J, it is possible to readily adjust the expressions above for h z ∼ J, by adding a straightforward h z -dependent correction.
Since the system is translationally invariant, it is convenient to work in Fourier space.Using |k, d⟩ = where k can take L possible values in the Brillouin zone −π < k ≤ π, and the Kronecker delta in the last term is needed to appropriately couple a string configuration with the relevant state with two 1-mesons, as pointed out above.While the effective model derived in this section is asymptotically exact as we let h x /J → 0 for the point h z = J, it also yields accurate results for h z ̸ = J.This is not only illustrated in Fig. 2(e) in the main text, but also in Fig. S1, which displays the absolute error (w.r.t. the full Ising model) of the time-integrated string survival probability for values of h z /J that lead to localized string dynamics.We can see that our reduced model shows systematic convergence towards the exact dynamics in the Ising model, as we decrease h x /J and let h z /J → 1 (while keeping the ratio hz−1 h 2 x fixed).
Let us also note that when we apply the SW transformation to rotate the Hamiltonian in order to eliminate off-diagonal terms, we should also rotate the initial state: As a final remark, we comment on how the localization-delocalization transition depends on the value of the coupling term in Eq. (S11).This is illustrated in Fig. S2, where we introduce a new parameter B modulating the strength of the said term.As the coupling between the impurity and the bath is decreased, we observe that the critical value of Λ/T approaches 2 (for the specific parameters used here, in particular, µ/T = 2).Further, the transition becomes sharper and sharper.This can be easily understood from the cartoon picture drawn in Fig. 3(a), which precisely illustrates the limiting case of B/T → ∞.At any rate, the overall qualitative result remains unaltered.

1 FIG. 2 .
FIG. 2. Effective graph model for string breaking dynamics.(a) Schematic of the effective model for a short string (ℓ = 4).Spin configurations in yellow represent string states (rest mass m4 = 12J), whereas configurations in blue represent 1-meson pairs with energy 2m1 and hopping amplitude v = h 2x /3J (red bonds).These are the configurations involved, to leading order, in the resonant decay of the string (green bonds).For illustration, confined DW pairs are depicted in some configurations as red dots joined by a wiggly line.String breaking can thus be thought of as a diffusion problem in the Fock-space graph.(b),(c) Energy spectrum of the effective model for hz = J and hz = 1.02J, respectively.In (b) all bands have a similar energy, while in (c) there is a gap between the "string" band and the continuum of 1-meson pairs.Colorbar shows the IPR of individual eigenstates [Eq.(5)], exhibiting a strong localization of the string modes in the latter case.(d),(e) Time evolution of the survival probability [Eq.(2)], for the respective parameters, both in the effective and full Ising model.Note the log scale in the horizontal axis and the time in unit of v. Parameters: L = 16, hx = 0.1J.

FIG. 3 .
FIG. 3. Localization-delocalization transition in the QIM.(a) Minimal string breaking/fusion as a three-level system (impurity), coupled to a bath.The impurity-bath coupling may or may not be resonant, yielding hybridization (as depicted here)-corresponding to string decay-or localization, respectively.(b) IPR of the impurity mode as a function of Λ/T , for various system sizes, displaying a localization-delocalization transition.(c) Data collapse of the data in (b) using a standard finite-size scaling ansatz within the package pyfssa [71], yielding Λc/T = 1.41(1) and critical exponents ζ = 1.02(5), ν = 1.00(5).(d) Long-time average of the spin autocorrelation function M(Λ), as a function of Λ/T with N = 800.The dashed line indicates the transition point Λc/T ≈ 1.41.Parameters: M/T = 10, µ/T = 2.

10 FIG 1 h 2 x=
FIG. S1.Convergence of the effective model in the localized regime.Absolute error of the time-integrated string survival probability.The effective model becomes more accurate as hx/J is reduced and hz/J approaches 1, with hz −1 h 2 x = 10 kept fixed.The values of the confining field hz/J are such that they lead to localized string dynamics.Both hx and hz are in units of J.In all plots, we use L = 14 and ℓ = 4.

)
FIG. S2.Localization-delocalization transition in the generalized quantum impurity model for varying strength of the coupling.IPR of the lowest-lying eigenstate of the model in Eq. (S11) as a function of Λ/T .Here we explore the effect of adding a further free parameter B modulating the strength of the coupling term in our phenomenological description [last term in Eq. (S11)].As the T /B is reduced the critical value of Λ/T converges to 2, as it can be readily understood from the cartoon picture in Fig.3(a).
Chanda, M. Dal-monte, P. Karpov, and M. Tsitsishvili for discussions and feedback on this work.This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 853443).