Diffusive-to-ballistic crossover of symmetry violation in open many-body systems
DDiffusive-to-ballistic crossover of symmetry violation in open many-body systems
Jad C. Halimeh and Philipp Hauke INO-CNR BEC Center and Department of Physics,University of Trento, Via Sommarive 14, 38123 Povo (TN), Italy (Dated: October 2, 2020)Conservation laws in a quantum many-body system play a direct role in its dynamic behavior.Understanding the effect of weakly breaking a conservation law due to coherent and incoherent errorsis thus crucial, e.g., in the realization of reliable quantum simulators. In this work, we perform exactnumerics and time-dependent perturbation theory to study the dynamics of symmetry violation inquantum many-body systems with slight coherent (at strength λ ) or incoherent (at strength γ )breaking of their local and global symmetries. We rigorously prove the symmetry violation to be adivergence measure in Hilbert space. Based on this, we show that symmetry breaking genericallyleads to a crossover in the divergence growth from diffusive behavior at onset times to ballistic orhyperballistic scaling at intermediate times, before diffusion dominates at long times. More precisely,we show that for local errors the leading coherent contribution to the symmetry violation cannot beof order lower than ∝ λt while its leading-order incoherent counterpart is typically of order ∝ γt .This remarkable interplay between unitary and incoherent gauge-breaking scalings is also observedat higher orders in projectors onto symmetry (super)sectors. Due to its occurrence at short times,the diffusive-to-ballistic crossover is expected to be readily accessible in modern ultracold-atom andNISQ-device experiments. CONTENTS
I. Introduction 1II. Preamble 3A. Definition of symmetry violation 3B. Glossary 3C. Summary of main findings 4III. Meaning of the symmetry violation 4A. Symmetry violation as a divergence measurein Hilbert space 5B. Symmetry violation as overlap betweengauge-transformed states 5IV. Extended Bose–Hubbard model 5A. Model and quench protocol 5B. Symmetric initial state 7C. Multiple quantum coherences 8D. Unsymmetric initial state 9V. Z lattice gauge theory 9A. Model and quench protocol 10B. Quench dynamics 11C. Variations of jump operator 12D. Other initial states 13E. Multiple quantum coherences 13F. Variations of relative strength of incoherenterrors 14G. Dynamics under gauge protection 15H. Dynamics under particle loss 16I. Decoherence starting from equilibrium 16VI. U(1) quantum link model 18VII. Conclusions and outlook 19 Acknowledgments 20A. Time-dependent perturbation theory 201. Coherent terms 202. Leading incoherent terms 223. Mixed terms 224. Higher-order incoherent terms 23B. Numerics specifics 231. Implementational details 232. Running average versus raw signal 24References 24 I. INTRODUCTION
Symmetries play a quintessential role in nature, frombeing the progenitors of different phases of matter tomanifesting conservation laws in and out of equilibrium. The group-theoretic notion of symmetries, i.e., definingsymmetry as invariance under a group transformation,has become a cornerstone in the development of modernphysics. Symmetries are also used to simplify a physicalproblem by reducing its effective Hilbert space based onwhat symmetry sectors the relevant physics takes placein (see glossary in Sec. II B for our nomenclature). Thesesectors involve global or local, as well as both continuousor discrete symmetries, and are manifestations of conser-vation laws. In the case of continuous symmetries, theassociated conservation laws are formalized in Noether’stheorem, which asserts an equivalence between them andthe continuous symmetries in a physical system. As anexample, the Fermi– and Bose–Hubbard models host aglobal U(1) symmetry, which translates into the conser-vation of particle number. Nevertheless, this equivalence a r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p is not restricted to continuous symmetries, as their dis-crete counterparts can also endow the system with con-served quantities. For example, coordinate inversion (P),charge conjugation (C), and time reversal (T) are dis-crete symmetries equivalent to the conservation of spa-tial, charge, and time parities, respectively. Whereas(continuous and discrete) global symmetries give rise tothe conservation of global charge , local symmetries leadto local charges being conserved. A prime example isgauge invariance, which is known as Gauss’s law inquantum electrodynamics, and forms the principal prop-erty of gauge theories. By default, conservation of localcharges leads to conservation of the global charge, butthe converse is not necessarily true.Given the richness afforded to physics by symmetries,a fundamental question is the effect of weakly breakingan underlying symmetry of a model on its subsequentdynamics. Even though traditionally studied in systemswith weak integrability breaking, prethermalization hasrecently been generalized to nonintegrable systems withperturbative coherent breaking of global and localsymmetries.
Interestingly, in the case of weak break-ing of the local gauge symmetry in a lattice gauge theoryof N matter sites, a prethermalization staircase arises inthe dynamics of the gauge violation (see Sec. II A for defi-nition), composed of prethermal plateaus at timescales ∝ λ − s , with s = 0 , , , . . . , N/ λ the strength of gaugeinvariance-breaking unitary errors. This rich prethermalbehavior has the property of delaying the timescale offull gauge violation exponentially in system size, and itcan also be observed in other local observables. Alsoin case of slight breaking of global symmetries, the equi-librating dynamics is strongly affected. This can occurwhen conservation laws due to, e.g., integrability are per-turbatively broken.
In this case, the initial equili-bration to a generalized Gibbs ensemble (GGE)steady state is replaced by thermal equilibrium at a latertimescale ∝ λ − , with λ the strength of the perturbativeintegrability-breaking term, as can be shown by kineticBoltzmann-like equations that can be derived throughemploying time-dependent perturbation theory startingfrom the GGE steady state. This behavior is not re-stricted to quenched systems, but can also be seen inweakly interacting driven models.
As these examplesillustrate, coherent errors can drastically change the dy-namical properties of a quantum many-body system.Going beyond unitary closed-system dynamics, deco-herence has been a central topic of research in quan-tum many-body physics.
Its mitigation is a nec-essary capability to achieve reliable quantum comput-ers, because these devices rely on the principles of su-perposition and entanglement, and are therefore par-ticularly sensitive to interactions with the environment.Examples abound such as 1 /f -noise in superconduct-ing quantum interference devices (SQUIDs) that con-stitutes a dominant adverse effect on superconduct-ing qubits, CMB-photon noise in superconductingcryogenic detectors, and thermomechanical motion in microwave cavity interferometers. In open quantummany-body systems, the effects of decoherence havebeen studied on light-cone dynamics and the spread ofcorrelations, and in a recent experiment a ballistic-to-diffusive crossover in quantum transport has been ob-served due to environmental noise in a 10-qubit networkof interacting spins. Moreover, the effect of decoherenceon weakly driven quantum many-body systems have alsobeen studied in the context of long-time steady statesin the presence of approximately conserved quantities,where it is shown that a GGE state can also arise. Therefore, like their coherent counterparts, incoherentsymmetry-breaking errors due to decoherence can funda-mentally change the properties and behavior of a physicalsystem.From a technological point of view, in modernultracold-atom experiments that aim to quantum-simulate a given target model, it is of critical im-portance to reliably implement certain conservation lawsdue to both local and global symmetries. As global sym-metries define the fixed points of renormalization-groupflow, they decisively influence quantum phase diagrams.For example, particle-number conservation in the formof a global U(1) symmetry is crucial in the transitionbetween the superfluid and Mott-insulator phases in theBose–Hubbard model, a paradigm phase transitionof cold-atom experiments. Similarly, local gauge sym-metries have fundamental consequences such as mass-less photons and a long-ranged Coulomb law, buttheir realization in quantum simulators requires carefulengineering—in contrast to fundamental theories of na-ture such as quantum electrodynamics or quantum chro-modynamics, in quantum devices they are not given byfundamental laws. This has recently generated a surge ofresearch investigating unitary errors in lattice gauge the-ories that compromise gauge invariance, including waysof protecting against them.
As these examples high-light, the existence of coherent and incoherent symmetry-breaking errors in realistic setups necessitates a rigorousunderstanding of their influence on the dynamics of quan-tum many-body systems. Particularly relevant is to see ifsuch errors are controlled insomuch that one may extractthe ideal theory dynamics despite their presence.In this paper, we investigate the effect of experimen-tally motivated incoherent global and local symmety-breaking errors on the dynamics of quantum many-bodysystems. We demonstrate that the symmetry violation—the expectation value of the symmetry generator or itssquare, which has been often used in the past to esti-mate the effects of errors—is a rigorous divergence mea-sure in Hilbert space that quantifies the deviation ofthe state from the target symmetry sector. We exploitthis insight to show, using exact numerics and rigorousproofs in time-dependent perturbation theory, the exis-tence of a diffusive-to-ballistic crossover in the dynamicsof the symmetry violation as a result of competition ofthese errors with their coherent counterparts. These re-sults extend and generalize upon the findings presentedin Ref. 84, which considered quenches from a separablegauge-invariant initial state in a Z gauge theory. By pre-senting a thorough analysis of various sources of errorsand various different model scenarios, our results providea guideline for quantum-simulation experiments on noisyintermediate-scale quantum (NISQ) devices that aim torealize target models with a given symmetry.The rest of the paper is organized as follows. In Sec. II,we define the symmetry violation, clarify our nomencla-ture in a short glossary, and provide a summary of ourmain results. In Sec. III, we rigorously explain the phys-ical meaning of the symmetry violation by equating itwith a divergence measure in Hilbert space and by relat-ing it to the decrease of the overlap between states thatdiffer only by a symmetry transformation. Our findingsand conclusions are then illustrated on three main mod-els: the extended Bose–Hubbard model in Sec. IV, theZ lattice gauge theory in Sec. V, and the U(1) quantumlink model in Sec. VI. We conclude and discuss possiblefuture directions in Sec. VII. Appendix A contains ourdetailed derivations in time-dependent perturbation the-ory that explain the various scalings seen in our exactdiagonalization results. Appendix B provides details onour numerical implementation. II. PREAMBLE
Before entering in the details of our work, we definein this section our main figure of merit, the symmetryviolation, we provide a short glossary for nomenclatureclarity, and we present a concise summary of our mainfindings.
A. Definition of symmetry violation
The motivation behind our work is the assessment ofquantum many-body systems in modern experimentalsettings where realistically coherent and incoherent errorswill always be present at least to a perturbative degree.These errors may break target local and global symme-tries, which may or may not be desired in the experiment,but where a thorough understanding of their effects onthe dynamics is nevertheless advantageous. These effectscan be qualitatively and quantitatively studied by cal-culating dynamics of the symmetry violation and otherrelevant observables.The symmetry violation ε is defined as the expecta-tion value of a (local or global) symmetry generator with respect to a target symmetry sector (see glossary inSec. II B), or the square of this expectation value, andis often used to estimate the effect of slightly breakinga symmetry. To formalize its definition, let us con-sider a many-body model described by the Hamiltonian H with a local symmetry generated by the operators G j , with j a lattice-site index (for the local gauge sym-metries considered below, matter particles reside on the lattice sites and gauge fields on the links in between). Theeigenvalues of G j are the local charges g j , and a givencombination of them on the lattice classifies a gauge-invariant sector g = { g , g , . . . , g N } . We select a tar-get gauge-invariant sector g tar = { g tar1 , g tar2 , . . . , g tar N } ,and call a given state gauge-invariant or symmetric iff G j ρ = g j ρ , ∀ j . We prepare the system in an initialstate ρ , which may be gauge-invariant or not. Gaugeinvariance restricts dynamics within a gauge sector, andso in case ρ lies in a given gauge-invariant sector g , thenthe system will remain in this sector for all times if thedynamics is solely due to H , because due to the gaugeinvariance of the latter, [ H , G j ] = 0 , ∀ j . In the presenceof unitary or incoherent gauge-breaking errors, the gaugeviolation generically will spread across various gauge sec-tors g , and can in general be quantified as ε ( t ) = 1 N (cid:88) j Tr (cid:110) ρ ( t ) (cid:2) G j − g tar j (cid:3) (cid:111) , (1)where ρ ( t ) is the density matrix of the time-evolved sys-tem at time t . The motivation behind this measure liesin the assumption that the system is desired to residewithin the target gauge-invariant sector g tar . Thus, anycoherent or incoherent errors during the preparation of ρ or the subsequent dynamics that take the system awayfrom g tar will make ε as defined by Eq. (1) nonzero.Much the same way, this definition can be extended tothe case of global-symmetry models, with the only caveatbeing that there the deviation is across global-symmetrysectors, each of which consists of all states with a givenfixed value of the global charge (see glossary in Sec. II B).We select a target global-symmetry sector defined by thetotal global charge g tar . The system is prepared in an ini-tial state ρ which may be in the target sector g tar or not.The global symmetry is generated by the operator G , andwe denote the Hamiltonian of the global-symmetry modelas H , i.e., [ H , G ] = 0. The initial state ρ is said to besymmetric iff Gρ = gρ . Consequently, the symmetryviolation becomes ε ( t ) = 1 N Tr (cid:110) ρ ( t ) (cid:2) G − g tar (cid:3) (cid:111) . (2)The normalization with N is chosen since G typically isan extensive quantity such as the total particle number,see, e.g., Eq. (9) below. B. Glossary
Symmetry sectors and symmetric states.
In caseof a local symmetry, a state ρ is said to be gauge-invariantor symmetric iff G j ρ = ρG j = g j ρ, ∀ j where G j arethe local-symmetry generators of the gauge group atmatter sites j with eigenvalues g j that depend on thegauge symmetry of the model. A given set of values g = { g , g , . . . , g N } constitute a gauge-invariant sector.In the case of a global symmetry, a state ρ is symmetriciff Gρ = ρG = gρ , where G is the generator of the globalsymmetry, and its eigenvalue g denotes the global chargeof the corresponding symmetry sector. As a concrete ex-ample, in the Bose–Hubbard model G can be chosen asthe total particle number. In this case, a given symmetrysector g would consist of all Fock states | n , n , . . . , n N (cid:105) where the individual on-site particle numbers n j sum to (cid:80) Nj =1 n j = g . Target symmetry sector.
In an experiment, it is of-ten desired to prepare the system and restrict its dynam-ics within a given symmetry sector in a local- or global-symmetry model. This is called the target symmetrysector. The symmetry violation measures how far off astate ρ is from this target symmetry sector. Gauge-invariant supersector.
Whereas both localand global symmetries have sectors, in our nomencla-ture only local symmetries have supersectors. A givensupersector α is the set of all gauge-invariant sectors g = { g , g , . . . , g N } that have a total violation α withrespect to the target gauge-invariant sector g tar . Thisdefinition can be formalized as (cid:80) Nj =1 ( G j − g tar j ) p ρ = αρ ,with p = 1 for the Z LGT and p = 2 for the U(1) QLM.In the case of the Z LGT discussed in Sec. V and a targetgauge-invariant sector g tar = , a gauge-invariant super-sector can be defined by the set of gauge-invariant sectors g each carrying M violations with respect to g tar = .For notational brevity, these supersectors are then de-noted by M and the projectors onto them by P M , witheach then being a sum of all projectors onto the con-stituent sectors; cf. Eq. (18). C. Summary of main findings
The main result of our work is the crossover in theshort-time dynamics of the symmetry violation from adiffusive spread through symmetry sectors caused by in-coherent errors to a ballistic spread caused by coherenterrors. Formally, we consider a system ideally describedby a Hamiltonian H , which has either a local or globalsymmetry generated by the local or global operators G j or G , respectively, with j indicating a site in the as-sociated lattice. In practice, these symmetries will becompromised in an experiment without unrealistic fine-tuning and isolation from the environment. We repre-sent the coherent errors by the Hamiltonian λH , with λ their strength, while we model decoherence (dissipationand dephasing) using a Lindblad master equation withthe coupling strength to the environment denoted by γ (see further below for the specific terms used).We prepare the system in an initial state ρ and quenchat t = 0 with H + λH in the presence of decoherence.We demonstrate that the symmetry violation yields adivergence measure for the quantum state across sym-metry sectors, enabling us to identify an increase as t a with diffusive ( a = 1), ballistic ( a = 2), and hyperballis-tic scaling ( a > ∝ γt . The leading order of the coherentcontribution depends on the structure of H and ρ . If ρ is symmetric or is a generic eigenstate of H , then thecoherent contribution cannot be of an order lower than ∝ λ t . If ρ is neither symmetric nor a generic eigen-state of H , then theoretically the leading order of thecoherent contribution can be ∝ λt , but we find this tohappen only in two rather pathological cases and one en-gineered (artificial) case. The first pathological scenariois when ρ is the ground state of H + λ i H (cid:48) with λ i (cid:54) = 0and H (cid:48) a highly nonlocal Hamiltonian, and the second iswhen H (cid:48) is local but H is highly nonlocal. If H = H (cid:48) ,the contribution ∝ λt vanishes identically to zero, andthen the order of the coherent contribution cannot belower than ∝ λt . We find this also to be the case if H (cid:48) and H are both local, or mildly nonlocal, but notequal. One may also get the scaling ∝ λt by artificiallyengineering the initial state in a common eigenbasis of H and the symmetry generator to be an arbitrary su-perposition of eigenstates degenerate with respect to H but not with respect to the symmetry generator. Con-versely, one can also construct specific combinations ofinitial state and coherent gauge breaking that result inhyperballistic expansion across symmetry sectors.Therefore, it is shown that in quantum many-bodysystems with small coherent and incoherent symmetrybreaking, a crossover from diffusive to ballistic spreadtakes place in the short-time dynamics of the symmetryviolation and certain other observables. The crossovertime is t ∝ γ/λ (or t ∝ γ /λ in case of hyperballis-tic spread) for initial states that reside in a symmetrysector or are generic (possibly unsymmetric) eigenstatesof H , and t ∝ γ/λ for unsymmetric initial states thatare not eigenstates of H . Our work presents an exten-sion to and generalization of Ref. 84, which has studiedthis crossover in a Z lattice gauge theory starting in agauge-invariant initial product state. Qualitatively, wefind that our main findings are similar for systems withbreakings of local or of global symmetries. III. MEANING OF THE SYMMETRYVIOLATION
In this section, we imbue the symmetry violation withmathematical and physical meaning, as a divergencemeasure across symmetry sectors in case of local or globalsymmetries, as well as the overlap between states thatdiffer only by a symmetry transformation. This sectiongeneralizes the discussion of Ref. 84, which consideredlocal gauge symmetries, and provides further details.
A. Symmetry violation as a divergence measure inHilbert space
We follow a similar reasoning as in Ref. 86, where ameasure for divergence in Hilbert space has been definedin the context of many-body localization. We first con-sider gauge-symmetry models, and start by defining themean-square displacement across the gauge sectors, D (2) g tar ( t ) = (cid:88) g d ( g , g tar ) Tr (cid:8) ρ ( t ) P g (cid:9) , (3)where P g is the projector onto gauge-invariant sector g , and g tar is the target gauge-invariant sector (seeSec. II A). Using the definition d ( g , g tar ) = [ g − g tar ] ,we can rewrite this expression as D (2) g tar ( t ) = Tr (cid:110) ρ ( t ) (cid:88) g P g (cid:2) g − g tar (cid:3) (cid:111) = (cid:88) j Tr (cid:110) ρ ( t ) (cid:2) G j − g tar j (cid:3) (cid:111) (4)= (cid:88) j (cid:104) (cid:2) G j − g tar j (cid:3) (cid:105) . By comparison with Eq. (1), we obtain ε ( t ) = 1 N D (2) g tar ( t ) . (5)Thus, commonly used measures of gauge violation, inU(1) as well as Z gauge theories, rigorously de-fine a mean-square displacement across gauge sectors asencapsulated in Eq. (4). One can immediately apply ex-actly the same reasoning to global symmetries, whereonly g needs to be replaced by the scalar g representingthe global-charge sector (and analogously for g tar and g tar ).In this sense, ε ( t ) ∝ γt is associated with a diffu-sive spreading of the wavefunction across gauge (global-charge) sectors in local- (global-) symmetry models, withdiffusion constant ∝ γ , while ε ( t ) ∝ λ t is indicative of aballistic spreading, a characteristic property of coherentquantum dynamics. B. Symmetry violation as overlap betweengauge-transformed states
We can further understand the gauge violation asquantifying how fast the overlap diminishes betweenstates that differ only by a gauge transformation. Letus again begin by considering the case of local symme-try. By definition, a gauge-invariant state should beoblivious to a local gauge transformation generated bya unitary U ( φ ) = (cid:81) Nj =1 e iφ j G j , with φ = { φ , . . . , φ N } and φ j arbitrary and independent angles. To quantifyby how much gauge-breaking errors compromise gaugeinvariance, one can use the overlap between the state of interest and the transformed state, which for a pure statereads C = | (cid:104) ψ ( t ) | U ( φ ) | ψ ( t ) (cid:105) | . The rate with which theoverlap reduces under a gauge transformation is η j = − ∂ C ∂φ j = (cid:104) G j (cid:105) − (cid:104) G j (cid:105) , (6)which is nothing else but the variance of the localGauss’s-law generator G j .For a homogeneous system, and assuming without re-striction of generality g tar = , definition (1) yields η j = ε − (cid:104) G j (cid:105) . For the Z gauge theory consideredbelow, where G j = 2 G j , we further have η j = ε (2 − ε )[upon normalizing ε by adding an inconsequential factorof 1 / ε . For smallgauge violations in a homogeneous Z gauge theory, botheven coincide (apart from an inconsequential factor of2), as they do in a homogeneous U(1) gauge theory with (cid:104) G j (cid:105) = 0. This insight lends further physical motivationfor ε as a good measure of gauge violation. As a sidenote, for mixed states, η j is given by the quantum Fisherinformation of the Gauss’s-law generator, which for ther-mal states, e.g., can be measured through linear-responsesusceptibilities or engineered quench dynamics. Again, these considerations can be immediatelyadapted to global symmetries. In this case, the unitarytransformation is defined as U ( φ ) = e iφG and the rate ofchange is η = − ∂ C ∂φ = (cid:104) G (cid:105) − (cid:104) G (cid:105) . For translationally-invariant systems with g tar = 0 and (cid:104) G (cid:105) = 0, using defi-nition (2) this relation becomes η = N ε . IV. EXTENDED BOSE–HUBBARD MODEL
In this section, we numerically evaluate the interplay ofcoherent and incoherent breakings of a global symmetry.To this end, we consider the paradigmatic example of theextended Bose–Hubbard model (eBHM).
A. Model and quench protocol
The eBHM considered here is defined on a one-dimensional spatial lattice with N sites and assumingperiodic boundary conditions, and is described by theHamiltonian H = − N (cid:88) j =1 (cid:0) J a † j a j +1 + J a † j a j +2 + H.c. (cid:1) + U N (cid:88) j =1 n j ( n j −
1) + W N (cid:88) j =1 n j n j +1 , (7)where a j is the bosonic annihilation operator at site j sat-isfying the canonical commutation relations [ a j , a l ] = 0
Domain wall
Markovian bath
Markovian bath
Figure 1. (Color online). Initial product states prepared athalf-filling ( g tar = N/ N = 6 sites with periodic bound-ary conditions for this model. (a) A staggered product statewhere odd sites are empty and each even site contains onehard-cose boson. (b) A domain-wall product state where theleft half of the lattice has a hard-core boson at each site andthe right half of the chain is empty. and [ a j , a † l ] = δ j,l . The eBHM is nonintegrable at finitenonzero values of J , J , U , and W . Generically, it hastwo integrable points: the atomic limit of J = J = 0and the free-boson limit of U = W = 0. In our case, weadditionally impose a hard-core constraint on the bosons,through which the term ∝ U does not play any role inthe dynamics, so we can remove it from the Hamiltonianin Eq. (7) (formally, the hard-core constraint amountsto setting U = ∞ ). This leads to another integrablepoint at J = 0, where the eBHM becomes equivalent tothe XXZ model. To avoid any effects due to integra-bility breaking, we therefore set J = 1, J = 0 .
83, and W = 0 .
11 in our numerics, although we have checkedthat other generic values of these parameters yield thesame conclusions. In all our results for the eBHM we usea periodic chain with N = 6 sites.The eBHM conserves the total particle number because (cid:80) j [ H , a † j a j ] = 0. This translates to the eBHM hostinga global U(1) symmetry. In a realistic experiment, theimplementation of this model will suffer from coherentand incoherent errors that in the best case slightly breakthis symmetry. The coherent errors are described by aHamiltonian λH , where λ denotes the strength of theseerrors, and dynamics in the presence of decoherence ismodeled by the Lindblad master equation ˙ ρ = − i [ H + λH , ρ ]+ γ (cid:88) j (cid:16) L j ρL † j − (cid:8) L † j L j , ρ (cid:9)(cid:17) , (8)where L j = a j are the jump operators describing the dis-sipation of our system with the environment, and γ isthe environment-coupling strength. Below, we test vari- (a) (b) (c) Figure 2. (Color online). Quench dynamics of the symmetryviolation in the closed ( γ = 0) extended Bose–Hubbard modelin the presence of unitary symmetry-breaking errors λH andstarting in a symmetric initial state. The staggered initialproduct state shown in Fig. 1(a) is used for (a) and (b), whilethe domain-wall product state in Fig. 1(b) is used as the initialstate in (c). System size is N = 6 sites, and periodic boundaryconditions are assumed. H is composed of single-body termsin (a), while it is a two-body error in (b) and (c). Generically,the initial increase of symmetry violation is ∝ ( λt ) (a,c),while specific combinations of initial state and error terms canincrease the order of the short-time behavior, to λ t in thecase of panel (b). Within our simulation times, the maximalvalue of the violation in all cases is ∝ λ , which is the sameorder in λ at which the violation grows at short times. ous examples of initial state and terms for H .In the following, we prepare our system in an initialstate ρ and solve Eq. (8) for the dynamics of the sym-metry violation ε ( t ) = Tr (cid:8) G ρ ( t ) (cid:9) , G = 1 N (cid:20) N (cid:88) j =1 a † j a j − g tar (cid:21) , (9)where g tar = N/ B. Symmetric initial state
Let us first prepare our system in the staggered initialproduct state shown in Fig. 1(a). This state is symmetricbecause it lies in the half-filling sector: G ρ = 0. Thesubsequent time evolution of the symmetry violation inEq. (9) without decoherence under the unitary errors λH = λ N (cid:88) j =1 (cid:0) a j + a † j (cid:1) , (10)is shown in Fig. 2(a). The error term in Eq. (10) woulddescribe, e.g., the coupling of the system to another in-ternal state (call its associated annihilation operator b ),which is occupied by a condensate that we assume tobe evenly spread across the entire lattice. Then, a Rabiflop between the two states gives H = (cid:80) j (cid:0) b † a j + a † j b (cid:1) ≈ (cid:80) j (cid:0) (cid:104) b (cid:105) ∗ a j + (cid:104) b (cid:105) a † j (cid:1) . Without restriction of generality,we set the phase of the condensate to 0. Thus, we get H = (cid:104) b (cid:105) (cid:80) j (cid:0) a j + a † j (cid:1) , thereby achieving Eq. (10) by ab-sorbing (cid:104) b (cid:105) in the definition of the unitary-error strength λ . The short-time scaling in Fig. 2(a) is ∼ ( λt ) . Aswe show in Sec. A 1 through time-dependent perturba-tion theory (TDPT), this is the lowest order of coherentcontributions to the symmetry violation in the case of asymmetric initial state.However, one can engineer the initial state or coherenterror to make higher orders the leading contributions. Asan example, we consider the same initial state shown inFig. 1(a), but consider the two-body unitary errors λH = λ N (cid:88) j =1 (cid:0) a j a j +1 + a † j a † j +1 (cid:1) . (11)Similarly to the motivation behind the error term ofEq. (10), one can imagine immersing an optical latticeinto a condensate of biatomic molecules, where bonding ∝ a j a j +1 gives a molecular annihilation operator m . Then, a term that drives the coupling to the molecu-lar condensate is H = (cid:80) j (cid:0) m † a j a j +1 + a † j a † j +1 m (cid:1) ≈(cid:104) m (cid:105) (cid:80) j (cid:0) a j a j +1 + a † j a † j +1 (cid:1) , which by absorbing (cid:104) m (cid:105) intothe definition of λ , we achieve the error term in Eq. (11).When ρ is the staggered initial state of Fig. 1(a), thecoherent contribution λ t Tr {G H ρ H } to the symme-try violation is finite in case of the unitary errors inEq. (10), but vanishes in case of the unitary errors ofEq. (11). The reason is that the staggered initial occupa-tion yields H ρ = 0. As shown in Sec. A 1, the next lead-ing contribution ∝ λ t Tr {G H H ρ H H } now domi-nates, where it does not vanish because H induces tun-neling processes that can bring bosons on adjacent sites,allowing H to act on ρ without destroying it. This isexactly what we see in Fig. 2(b). Such an increase ofthe mean-square displacement with a power of t largerthan 2 is a hallmark of hyperballistic expansion, and setsthe crossover timescale to t ∝ ( γ/λ ) in the presence ofdecoherence (see below). (a) (b) Figure 3. (Color online). Quench dynamics of the symmetryviolation in the N = 6-site open eBHM starting in a staggeredinitial product state and in the presence of the single-bodyunitary errors of Eq. (10) at strength λ . The coupling to theenvironment is at strength γ with the jump operator L j = a j at each site. (a) Symmetry violation over time with fixed λ at various values of γ . (b) Symmetry violation over timewith fixed γ at various values of λ . Note how decoherence,regardless the value of λ , takes the symmetry violation toits maximal value g /N = 1 / t ∝ γ/λ where thesymmetry violation goes from a diffusive spread ∼ γt for t (cid:46) γ/λ to ballistic spread ∼ λ t for t (cid:38) γ/λ . Nevertheless, the short-time scaling ∼ λ t is not ageneric feature of the unitary errors of Eq. (11), but israther a combination of such errors and the fact that westart in the staggered initial state. Indeed, if we considerthese same pairing errors but instead start in, say, thedomain-wall initial state shown in Fig. 1(b), then theleading coherent contribution to the symmetry violationis again ∝ λ t , as shown in Fig. 2(c). The reason isthat a domain-wall initial state already has bosons onadjacent sites, and thus H can act on it nontrivially.In what follows, we take ρ as the staggered initial stateand include decoherence through jump operators L † j . Weconsider first in Fig. 3 the unitary errors of Eq. (10). Ascan be shown in TDPT (see Appendix A 2), the leadingincoherent contribution to the symmetry violation in caseof a symmetric initial state is ∝ γt (cid:80) j Tr {G L j ρ L † j } . Assuch, at times t (cid:46) γ/λ , the symmetry violation shownin Fig. 3 scales diffusively ∼ γt , before being overtakenby the ballistic spread ∼ λ t due to the leading coher-ent contribution. Note that the maximal value ≈ / (a) (b) Figure 4. (Color online). Same as Fig. 3 but for the two-body error of Eq. (11). The initial state can play a nontrivialrole in the short-time dynamics of the symmetry violation.In this case, the crossover is from diffusive to hyperballistic spread and occurs at t ∝ ( γ/λ ) . This behavior is qualita-tively different from the case of the single-body unitary errorsin Eq. (10), and is due to the fact that H in Eq. (11) anni-hilates ρ when it acts on it: H ρ = 0, which leads to thecontribution ∝ λ t vanishing, and thus the next leading or-der ∝ λ t sets in for t (cid:38) ( γ/λ ) . Decoherence allows thesymmetry violation to reach its maximal value of 1 / tive difference here is that the diffusive scaling ∼ γt dueto incoherent errors is overtaken by hyperballistic spread ∼ λ t , due to the leading coherent contribution, at acrossover time t ∝ ( γ/λ ) . Again here in the presenceof decoherence the symmetry violation attains its maxi-mal value of 1 /
4, even though in the purely unitary caseit does not.
C. Multiple quantum coherences
In order to further quantify the effects of deco-herence especially in the late-time dynamics of ourquenches, we analyse multiple quantum coherences(MQC). These are experimentally accessible quantitiesthat have been used to study quantum coherences innuclear magnetic resonance imaging, as well as de-coherence effects on correlated spins and many-bodylocalization. Moreover, they have also been con-nected to multipartite entanglement and out-of-time-ordered correlators, and they have been measuredin trapped-ion experiments.
Let us call ∆ g the difference in global charge betweentwo global-symmetry sectors. The associated MQC is (a) (b) Figure 5. (Color online). Multiple quantum coherences I ∆ g (solid blue curves) at fixed coherent- and incoherent-errorstrengths λ and γ (see gray boxes for exact values), respec-tively, in the quench dynamics of the extended Bose-Hubbardmodel starting in the staggered initial state of Fig. 1(a). Forreference, we also show the symmetry violation (solid redcurve) as well as the MQCs in the purely coherent case ( γ = 0;same color but dotted lines). In (a) the coherent errors aresingle-body terms given by Eq. (10). Decoherence compro-mises all I ∆ g> , causing them to settle into steady-state val-ues lower than those in the purely coherent case. However, I does not deviate much from its coherent steady-state value,suggesting that decoherence does not affect quantum coher-ences within each sector much. In (b) the coherent errorsare the two-body terms given by Eq. (11). Even though I behaves the same as in the case of single-body errors, I ∆ g> here are nonzero only in the case of even ∆ g and settle intosteady-state values larger than those in the purely coherentcase. then defined as I ∆ g = Tr (cid:8) ρ † ∆ g ρ ∆ g (cid:9) , (12a) ρ ∆ g = (cid:88) g P g +∆ g ρP g , (12b)where P g is the projector onto the sector of global charge g . In Fig. 5, we show the MQC for various ∆ g for bothunitary errors in Eqs. (10) and (11) at fixed λ and γ . Forthe single-body coherent errors of Eq. (10), we see thatodd values of ∆ g give rise to a finite MQC as shown inFig. 5(a). This behavior is plausible, as in this case afirst-order process in H removes or adds a single bosonon a given site j . We see that the MQC is dominatedby processes within the same sector (corresponding to∆ g = 0). Whereas the symmetry violation (red curve)starts at t = 0 to grow ∼ γt , the MQCs, being measuresof quantum coherences between global-symmetry sectors,do not show any scaling related to incoherent processes (a) (b) Figure 6. (Color online). Starting in the unsymmetric groundstate ρ of H + λ i H with H given in Eq. (10). (a) Quenchdynamics of the symmetry-violation change in the case of aclosed eBHM chain with N = 6 matter sites. Since ρ isunsymmetric, the leading coherent contribution to the sym-metry violation is ∝ λt . Note here that the steady-statevalue is ∝ λ rather than ∝ λ as in Fig. 2 when the initialstate is symmetric. (b) Decoherence brings about a diffusive-to-ballistic crossover at an earlier timescale γ/λ < γ/λ thanthe case of a symmetric initial state. at early times. Rather, their growth is ∼ ( λ t ) ∆ g . Ascan be expected in case of decoherence, the I ∆ g with∆ g > t ≈ /γ as compared to thesteady-state value of the purely coherent case. In con-trast, I exhibits an increase in its values after an initialdecrease, finally settling at roughly the coherent steady-state value. This behavior suggests that even though de-coherence compromises coherences between different sec-tors, those within the same sector are not affected muchby the decoherence studied here.By changing H to Eq. (11), the above picture changes,as shown in Fig. 5(b). First, MQCs due to odd ∆ g areidentically zero, as there can be no coherent processes be-tween sectors differing by an odd number of bosons— H as per Eq. (11) removes or adds two neighboring bosonssimultaneously. Furthermore, due to the staggered oc-cupation of ρ , the MQCs scale ∼ ( λt ) ∆ g . Whereas I behaves the same as in the case of the single-body er-ror, I ∆ g> behave fundamentally differently in case ofthe two-body error in Eq. (11). Interestingly, at t ≈ /γ they begin to decrease in value, but at later times theyincrease again and settle at a value larger than that oftheir purely coherent dynamics, represented by dottedlines in Fig. 5. This suggests that decoherence populatessectors that cannot be populated by H alone. Indeed,in the case of purely unitary dynamics with only coher-ent errors as in Eq. (11), we have checked that (cid:104) P g (cid:105) = 0identically for odd g . In the case of decoherence, how- ever, (cid:104) P g (cid:105) acquire nonzero values, and this allows then,through H , coherence within and between sectors withodd global charges, but separated by an even ∆ g . Thisis quite counterintuitive in that decoherence here seemsto help in building up quantum coherences by allowingaccess to previously inaccessible sectors. D. Unsymmetric initial state
We have so far considered only initial states lying inthe half-filling sector. Let us now consider an initial state ρ that is unsymmetric , i.e., (cid:2) G , ρ (cid:3) (cid:54) = 0. In particular,we consider ρ to be the ground state of the Hamiltonian H + λ i H , where λ i = 0 . H is given by Eq. (10). Such ascenario may arise when aiming at adiabatically prepar-ing the ground state of H in the presence of errors. Ashas been done until now, we quench ρ with H + λH in the presence of decoherence under the jump operators L j = a j .The ensuing dynamics of the change in symmetry vi-olation | ∆ ε | is shown in Fig. 6. In the case of no deco-herence ( γ = 0), shown in Fig. 6(a), | ∆ ε | ∼ λt at shorttimes rather than λ t n (even n ≥
2) as in the case ofa symmetric initial state; cf. Figs. 2–4. As explained inSec. A 1 through TDPT, the coherent contribution to thesymmetry violation ∝ λt , given by Eq. (A5f) always van-ishes when the initial state is symmetric or an eigenstateof H , but not when ρ is unsymmetric yet not an eigen-state of H as in the case of Fig. 6. Moreover, the contri-bution ∝ λt in Eq. (A5d) completely vanishes in this case,since the errors H in preparing ρ are the same as thosein the subsequent dynamics (see Appendix A 1). Notehow, consequently, the steady-state value at which | ∆ ε | settles is ∝ λ . Upon introducing decoherence ( γ > t ∝ γ/λ occurs taking the spread of the symmetric violation fromdiffusive ∼ γt to ballistic ∼ λt . Note that the diffusive-to-ballistic crossover here occurs at an earlier timescalethan that in the case of a symmetric initial state. Finally,as in the case of a symmetric initial state, the symmetryviolation reaches its maximal value of g /N = 1 / ρ is unsymmetric. V. Z LATTICE GAUGE THEORY
In this Section, we present results that supplementthose on a Z gauge theory presented in Ref. 84 in var-ious ways: by studying the effect of decoherence underdifferent Lindblad operators including those for particleloss; by starting in various initial states including gauge-noninvariant ones; by investigating the effect of dissipa-tion and dephasing set at different environment-couplingstrengths; by analyzing the addition of energy-penaltyterms on the dynamics; and by adding further results onMQCs under decoherence.0 L g1 ,
Matter-site
Domain wall
Figure 7. (Color online). Symmetric initial product statesused in the dynamics of the Z LGT in the presence of co-herent errors. (a) A staggered product state where odd mat-ter sites are empty and each even matter site contains onehard-cose boson, with the electric fields along links pointingfrom odd to even matter sites, thereby satisfying Gauss’s lawat each local constraint. (b) A “domain-wall” product statewhere the left half of the lattice has a hard-cose boson at eachsite and the right half of the chain is empty. The electric fieldsalong the links are oriented such that Gauss’s law is satisfiedat each local constraint.
A. Model and quench protocol
A recent experiment has employed a Floquet setupto successfully implement a building block of the Z LGTdescribed by the Hamiltonian H = N (cid:88) j =1 (cid:2) J a (cid:0) a † j τ zj,j +1 a j +1 + H.c. (cid:1) − J f τ xj,j +1 (cid:3) , (13)where N is the number of matter sites, a j is the anni-hilation operator of a hard-core boson at site j obey-ing the canonical commutation relations [ a j , a l ] = 0 and[ a j , a † l ] = δ j,l (1 − a † j a j ), and the Pauli matrix τ x ( z ) j,j +1 represents the electric (gauge) field at the link betweenmatter sites j and j + 1. The first term of Eq. (13) rep-resents assisted matter tunneling and gauge flipping atstrength J a , which, e.g., forms the essence of Gauss’s lawin quantum electrodynamics. The electric field’s energyis given by J f . In this work, we adopt periodic boundaryconditions, which means our effective system size is 2 N ,and we set J a = 1 and J f = 0 .
54 throughout our paper,even though we have checked that our conclusions arenot restricted to these values.Gauge invariance is embodied in local conservationlaws, the generators of which are G j = 1 − ( − j τ xj − ,j (1 − a † j a j ) τ xj,j +1 , (14)where [ H , G j ] = 0 , ∀ j . As discussed above, the eigenval-ues g j of G j are known as local charges , and a set of their values g = { g , g , . . . , g N } defines a gauge-invariant sec-tor (see Sec. II A). A gauge-invariant supersector M isdefined as the set of gauge-invariant sector that satisfy (cid:80) j g j = 2 M (see Sec. II B).In the implementation of the Z LGT without un-realistic fine-tuning, coherent error terms emerge with[ H , G j ] (cid:54) = 0. Here, inspired by the effective coherenterrors of the building block of Ref. 76, we assume the er-rors to have the form of unassisted matter tunneling andgauge flipping, which can be formalized as λH = λ N (cid:88) j =1 (cid:104)(cid:0) c a † j τ − j,j +1 a j +1 + c a † j τ + j,j +1 a j +1 + H.c. (cid:1) + a † j a j (cid:0) c τ zj,j +1 − c τ zj − ,j (cid:1)(cid:105) . (15)The strength of these errors is given by λ , and the coef-ficients c ... depend on a dimensionless driving parame-ter χ that is tunable in the experiment of Ref. 76. Thespecific expressions for these coefficients can be found inAppendix B. Just as in the joint submission, we showhere results for χ = 1 .
84, but we have also checked thatour results hold for various values of χ within the rangefound in the Floquet setup of Ref. 76.As for the case of a global symmetry discussed above,the system is prepared in an initial state ρ , which at t = 0 is quenched by H + λH and decoherence is turnedon. The subsequent dynamics is computed using theLindblad master equation˙ ρ = − i [ H + λH , ρ ]+ γ N (cid:88) j =1 (cid:16) L m j ρL m † j + L g j,j +1 ρL g † j,j +1 − (cid:8) L m † j L m j + L g † j,j +1 L g j,j +1 , ρ (cid:9)(cid:17) , (16)where L m j and L g j,j +1 are the jump operators couplingthe matter and gauge fields, respectively, to the environ-ment at strength γ . We are interested in dynamics of thegauge violation, supersector projector, electric field, andstaggered boson number, given respectively by, ε ( t ) = Tr (cid:8) G ρ ( t ) (cid:9) , G = 1 N (cid:88) j (cid:2) G j − g tar j (cid:3) , (17) (cid:104)P M ( t ) (cid:105) = Tr (cid:8) ρ ( t ) P M (cid:9) , P M = (cid:88) g ; (cid:80) j g j =2 M P g , (18) m x ( t ) = 1 N (cid:12)(cid:12)(cid:12) Tr (cid:110) ρ ( t ) (cid:88) j τ xj,j +1 (cid:111)(cid:12)(cid:12)(cid:12) , (19) n stag ( t ) = 1 N (cid:12)(cid:12)(cid:12) Tr (cid:110) ρ ( t ) (cid:88) j ( − j a † j a j (cid:111)(cid:12)(cid:12)(cid:12) . (20)Note that the form of the gauge violation in Eq. (17) isspecific for the Z LGT. It is a special case of Eq. (1)using g j = 2 g j and dropping an irrelevant factor of 2.1 (a) (b) (c) (d) (e) (f) Figure 8. (Color online). (a) Full unitary dynamics of the quench scenario illustrated in Fig. 7(a). Two prethermal plateausat timescales λ − and J a λ − (see insets) appear as explained numerically and analytically through a Magnus expansion inRefs. 13 and 14. (b) Complementary results to those of the joint submission Ref. 84, where here we fix γ and scan λ . Theconclusion remains the same, with the gauge violation exhibiting diffusive scaling ε ∼ γt at short times for sufficiently small λ , and ballistic scaling ∼ ( λt ) at sufficiently large λ , with the timescale of the crossover from the former to the latter being t ∝ γ/λ . Upper inset shows the second (and final in the case of N = 4 matter sites) prethermal timescale getting compromisedat smaller values of λ at which γ = 10 − dominates. Lower inset shows that the first prethermal timescale is more resilientthan the second, as its timescale persists for smaller values of λ . (c,d) Same as (b) but for the supersector projectors P and P . As we can see, the crossover from the diffusive to ballistic regime is again at t ∝ γ/λ , but whereas P shows the samescaling orders as ε , P ∼ γ t in the diffusive regime and P ∼ λ t in the ballistic regime. (e,f) Influence of gauge violationon local observables. As exemplified by (e) the staggered particle density and (f) the electric field, the dynamics is practicallyindistinguishable from the decoherence-free model before the timescale ∝ γ − . Note how the electric field shows no diffusivebehavior at the onset as the gauge violation and supersector projectors do. This is because the corresponding correction γt L ρ to the unitary part of the density matrix makes a vanishing contribution to the electric field, as explained in Sec. A 2. B. Quench dynamics
As shown in Ref. 84, the coexistence of unitary andincoherent gauge-breaking processes leads to competingtimescales due to λ > γ > LGT.While incoherent gauge-breaking processes yield a singletimescale 1 /γ , coherent errors can generate a sequence or staircase of prethermal plateaus with timescales J s − a /λ s with s = 0 , , , . . . , N/ These coherent timescalesarise due to unitary dynamics in a gauge theory as a re-sult of resonances between different gauge-invariant sec-tors coupled through H , as can be shown through a Mag-nus expansion. In the case of the initial states shown inFig. 7 with N = 4 matter sites, this means two plateaus at timescales 1 /λ and J a /λ (the one at timescale ∝ λ does not appear in this case ), at which maximal vio-lation is attained; see Fig. 8(a) for the staggered initialproduct state shown in Fig. 7(a).The picture changes significantly when γ >
0; seeFig. 8(b). When λ = 0 in this case, the gauge viola-tion accumulates diffusively as ε ∼ γt until it reaches amaximal value of unity at t ≈ /γ . This gauge violationdue to purely incoherent gauge-breaking processes showsno signatures of prethermalization. The picture startsto change for λ > γ , as then the prethermal plateau attimescale ∝ /λ can still appear unaffected by the deco-herence, which becomes dominant for t (cid:38) /γ ; see lowerinset of Fig. 8(b). The effects of decoherence on the sec-ond plateau, which in the purely unitary case appears at2timescale ∝ J a /λ , are more prominent as can be seen inthe upper panel of Fig. 8(b). This plateau survives onlywhen λ /J a (cid:38) γ as then the final prethermal timescale ∝ J a /λ appears earlier than the decoherence timescaleof /γ .It is interesting to examine again the short-time scalingof the gauge violation for finite λ in Fig. 8(b). The be-havior concurs with the conclusions of Ref. 84, where weobserve the diffusive scaling ε ∼ γt for t (cid:46) γ/λ , whilefor later times t (cid:38) γ/λ the violation is dominated bycoherent errors and ε ∼ ( λt ) . Intriguingly, we thus findin general two regimes where incoherent errors dominateat finite λ : the first at evolution times t (cid:46) γ/λ and thesecond for t (cid:38) /γ . At intermediate times, the coher-ent gauge-breaking processes dominate when λ > γ , andboth prethermal plateaus appear for N = 4 matter siteswhen λ > γJ a .Let us now again look at the dynamics of the super-sector projectors in the presence of decoherence. This isshown for the projectors onto the supersectors M = 2 and M = 4 in Fig. 8(c,d). Congruent to the conclusions ofthe joint submission, we get the same short-time scal-ing for the supersector projectors P and P , with thecrossover from the diffusive regime where (cid:104)P (cid:105) ∼ γt and (cid:104)P (cid:105) ∼ γ t at t (cid:46) γ/λ to the ballistic regime where (cid:104)P (cid:105) ∼ λ t and (cid:104)P (cid:105) ∼ λ t at t (cid:38) γ/λ . The deteriora-tion of the first prethermal timescale can also be observedin these quantities, and that of the second prethermalplateau is observed in (cid:104)P (cid:105) . The larger λ is, the greaterthe integrity of the prethermal plateaus, with the firstplateau exhibiting greater resilience as it survives smallervalues of λ than its second counterpart. Interestingly, atlong times t (cid:38) /γ , both projectors relax to the values (cid:104)P (cid:105) ≈ .
125 and (cid:104)P (cid:105) = (cid:104)P (cid:105) ≈ .
75, meaning that (cid:104)P (cid:105) / (cid:104)P (cid:105) = (cid:104)P (cid:105) / (cid:104)P (cid:105) = 6, which is equal to the ra-tio of number of gauge sectors in each supersector. Thisgenerally does not happen in the case of no decoherence( γ = 0), but in the presence of decoherence at any γ > H and H both have global U(1) symme-try, and there is only dephasing on the matter fields] andthe electric field in Fig. 8(f). One cannot discern any dif-fusive behavior at early times in these observables (evenby looking at the deviation from the fully unitary case).A deeper reason may be that these local observables arenot related to a divergence measure through the gaugesectors, contrary to the gauge violation (see Sec. III A).The leading-order (in γ ) correction to the unitary part ofthe density matrix is γt L ρ makes a vanishing contribu-tion to both of these observables as discussed in Sec. A 2.In the quench dynamics of the joint submission andFig. 8, we have focused on λ >
0. For completeness, weshow in Fig. 9 the effect of γ on the dynamics of gauge vi- (a) (b) Figure 9. (Color online). Dynamics of (a) the gauge violationand (b) the staggered matter field for λ = 0 at various val-ues of the environment-coupling strength γ (see legend). Thebehavior is qualitatively similar to that of Figs. 8(a,e), re-spectively, albeit here there is no signature of the prethermalplateaus in the gauge violation since λ = 0. olation and staggered boson number, considering quenchdynamics for the same initial state as in Fig. 7(a) butfor λ = 0. The gauge violation [Fig. 9(a)] spreads diffu-sively in the gauge sectors scaling as ε ∼ γt at short timesbefore settling into a maximal-violation steady state at t ≈ /γ . The staggered boson number [Fig. 9(b)] be-haves much the same way as in the case of λ > t ≈ /γ , with its temporal average decaying ∼ ( γt ) − at late times. C. Variations of jump operator
So far, we have included dissipation in the gauge fieldsas governed by the jump operator L g j,j +1 = τ zj,j +1 . Tocorroborate the generality of our results, we study theeffect of a different dissipative jump operator L g j,j +1 = τ − j,j +1 at various values of γ in the presence of co-herent gauge-breaking terms at strength λ = 10 − J a .We show the associated gauge-violation and supersector-projector dynamics in Fig. 10. The qualitative pictureis unchanged, and we see that the diffusive-to-ballisticcrossover is also at t ∝ γ/λ , with scaling ∼ γt ( ∼ γ t )in the diffusive regime and scaling ∼ λ t ( ∼ λ t ) in theballistic regime in the short-time dynamics of the gaugeviolation and (cid:104)P (cid:105) ( (cid:104)P (cid:105) ).3 (a) (b) (c) Figure 10. (Color online). Same as Fig. 1 of Ref. 84 but with adifferent jump operator on the gauge links. Quench dynamicsof (a) the gauge violation, (b) supersector projector P , and(c) supersector projector P in the open Z LGT starting inthe gauge-invariant initial state of Fig. 1(a), in the presenceof the unitary gauge-breaking errors of Eq. (15) at strength λ = 10 − J a , and coupling to the environment at strength γ with the jump operators L m j = a † j a j and L g j,j +1 = τ − j,j +1 onmatter sites and gauge links, respectively. The diffusive-to-ballistic crossover is again at t ∝ γ/λ . D. Other initial states
Furthermore, our conclusions remain unaltered forother initial states. Whereas in the joint submissionRef. 84 and hitherto in this paper our initial statehas been the staggered product state in Fig. 7(a), inFig. 11 we show the gauge-violation and supersector-projector dynamics for the “domain-wall” product statein Fig. 7(b). Again, here we include coherent gauge-breaking terms at strength λ = 10 − J a and study theeffects of decoherence at various values of the environ-ment coupling γ , with the jump operators L m j = a † j a j and L g j,j +1 = τ zj,j +1 . The qualitative picture is identicalto that of Fig. 1 in Ref. 84 and Fig. 11. (a) (b) (c) Figure 11. (Color online). Same as Fig. 10 but startingin the gauge-invariant “domain-wall” initial product state ofFig. 7(b) and with L g j,j +1 = τ zj,j +1 . E. Multiple quantum coherences
In the Z LGT, the MQC are a generalization of thosegiven in Eq. (12), and read I ∆ g = Tr (cid:8) ρ † ∆ g ρ ∆ g (cid:9) , (21a) ρ ∆ g = (cid:88) g P g +∆ g ρP g , (21b)where now they measure quantum coherences betweengauge-invariant sectors and not just supersectors as inthe case of the eBHM (see Sec. IV C). Since the deviationsbetween sectors are vectors, the MQC spectra F φ = (cid:88) ∆ g I ∆ g e − i (cid:80) j φ j ∆ g j (22)depend on N angles φ = { φ , φ , . . . , φ N } . In the jointsubmission Ref. 84, we provide results for the MQC andtheir spectra, for a given choice of angles, at λ = 10 − J a and γ = 10 − J a . Let us now look at these results butwith no decoherence, i.e., λ = 10 − J a and γ = 10 − J a .The corresponding results are shown in Fig. 12. Inthe absence of decoherence, the MQC over evolutiontime and settle at a steady-state value at long times4 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) Figure 12. (Color online). Same as Fig. 2 of Ref. 84 but withno decoherence, and additionally showing the MQC spectraat different angles. (a) Dominant MQC. (b-j) MQC spectra,with rows from top to bottom F ( φ , , φ , F ( φ , φ , , F (0 , φ , φ , tJ a = 10 , tJ a = 10 , and tJ a = 10 . Unsurprisingly, in the absenceof decoherence, there is no decay of the intensities, and thebandwidth of the MQC spectra increases with time, in con-trast to the case of decoherence in Fig. 2 of Ref. 84, wheredecoherence diminishes the spectrum. [Fig. 12(a)], in contrast to the case with decoherence ofRef. 84, where their temporal averages decay ∼ ( γt ) − at t > γ − . The spectrum also behaves fundamentally dif-ferently. Whereas in the case with decoherence the spec-trum almost vanishes for t (cid:38) γ − , in the closed-systemcase it is maximal in this temporal regime.Our choice of the MQC-spectrum angles in the maintext is neither special nor unique. Due to symmetry, wehave F ( φ , , φ ,
0) = F (0 , φ , , φ ) , (23a) F ( φ , φ , ,
0) = F (0 , , φ , φ ) , (23b) F (0 , φ , φ ,
0) = F ( φ , , , φ ) . (23c) (a) (b) (c) (d) (e) (f) Figure 13. (Color online). Same as Fig. 2(c-e) of Ref. 84 butfor different angles of the MQC spectrum, where again λ =10 − J a and γ = 10 − J a . We show F ( φ , φ , ,
0) at evolutiontimes (a) t = 10 /J a , (b) t = 10 /J a , and (c) t = 10 /J a , andwe show F (0 , φ , φ ,
0) at evolution times (d) t = 10 /J a ,(e) t = 10 /J a , and (f) t = 10 /J a . The results show thatdecoherence diminishes the spectrum, in agreement with theconclusion from Fig. 2(c-e) of the joint submission. For completeness, we show in Fig. 13 for the MQCspectra F ( φ , φ , ,
0) and F (0 , φ , φ ,
0) in the pres-ence of gauge-breaking coherent and incoherent errorsat strengths λ = 10 − J a and γ = 10 − , where the initialstate is that of Fig. 7(a) and the dynamics is governed byEq. (16). Similarly to their counterpart F ( φ , , φ , F ( φ , φ , ,
0) and F (0 , φ , φ ,
0) diminish in thepresence of decoherence.It is interesting to compare these findings to the dy-namics of the MQC with a different jump operator. Assuch, we again start in the staggered initial state ofFig. 7(a) and set the jump operator L g j,j +1 = τ − j,j +1 .Once again, we set λ = 10 − J a and γ = 10 − J a , with L m j = a † j a j . (The dynamics of the gauge violation andsupersector projectors for this quench protocol are shownin Fig. 10.) The corresponding MQC results are shownin Fig. 14. Unlike the case of L g j,j +1 = τ zj,j +1 (seeFig. 2 of Ref. 84), the MQC do not decay to zero when L g j,j +1 = τ − j,j +1 , but rather saturate at finite steady-state values. This behavior depends on the fixed point ofthe Liouvillian superoperator. Indeed, we have checked(not shown) that when the “domain-wall” product stateshown in Fig. 7(b) is the initial state, the MQC take onthe same steady-state values as those shown in Fig. 14. F. Variations of relative strength of incoherenterrors
Let us now investigate the effect of turning on the de-phasing and dissipation at different strengths γ m and γ g ,respectively. The Lindblad master equation generalizes5 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) Figure 14. (Color online). Same as Fig. 2 of Ref. 84 butwith L g j,j +1 = τ − j,j +1 and including additional angles for theMQC spectra. (a) Dominant MQC. (b-j) MQC spectra, withrows from left to right F ( φ , , φ , F ( φ , φ , , F (0 , φ , φ ,
0) and columns from left to right tJ a = 10 , tJ a = 10 , and tJ a = 10 . In contrast to the case of L g j,j +1 = τ zj,j +1 as in Fig. 2 of Ref. 84, here the MQC donot decay to zero, but rather saturate at finite steady-statevalues. This behavior depends on the fixed points of the Li-ouvillian superoperator. to ˙ ρ = − i [ H + λH , ρ ]+ N (cid:88) j =1 (cid:104) γ m (cid:16) L m j ρL m † j − (cid:8) L m † j L m j , ρ (cid:9)(cid:17) + γ g (cid:16) L g j,j +1 ρL g † j,j +1 − (cid:8) L g † j,j +1 L g j,j +1 , ρ (cid:9)(cid:17)(cid:105) . (24)Interestingly, the dephasing strength γ m has little effecton the short-time dynamics of the gauge violation, whichat short times t (cid:46) γ g /λ scales as ε ∼ γ g t , and at in-termediate times γ g /λ < t (cid:46) /λ scales as ε ∼ ( λt ) due to the dominance of coherent gauge-breaking terms. In fact, it can be shown in TDPT (see Sec. A 2) thatthe contribution to the gauge violation due to dephasingat short times vanishes. However, we find that both γ m and γ g have a significant effect on the later timescale atwhich decoherence dominates at maximal violation, withthis timescale being roughly 1 / max { γ g , γ m } , as can beseen in the lower insets of Fig. 15(a,b). In particular, de-phasing incurs a nonperturbative effect on the prethermalplateaus, as shown in the lower inset of Fig. 15(a). (a) (b) Figure 15. (Color online). Dynamics of the gauge-invarianceviolation at different environment-coupling strengths γ m forthe dephasing on matter fields and γ g for the dissipationon gauge links, with fixed strength λ = 10 − J a of coherentgauge-breaking processes. (a) Gauge violation at various val-ues of γ m for fixed value of γ g = 10 − J a . (b) Gauge violationat various values of γ g for a fixed value of γ m = 10 − J a . Dis-sipation clearly shows an effect on the gauge violation at shorttimes, whereas dephasing does not. However, at late times de-phasing also has a clear effect on the prethermal plateaus, asshown in the lower inset of (a). G. Dynamics under gauge protection
Recently, several theoretical works have proposedto use gauge protection to suppress processes driv-ing the system out of its initial gauge-invariantsector, and the principle has beendemonstrated experimentally for a U(1) gauge theory. The basic idea is to introduce a suitable energy-penaltyterm, which for a Z gauge theory reads V H G = V (cid:88) j G j , (25)where V controls the protection strength. For suffi-ciently large V , the associated gauge violation due to H is suppressed by ( λ/V ) , and the ensuing dynam-ics is perturbatively close to a renormalized version of6the ideal gauge theory. Using as quench Hamilto-nian H = H + λH + V H G , and numerically solving therespective Lindblad master equation˙ ρ = − i [ H + λH + V H G , ρ ]+ γ N (cid:88) j =1 (cid:16) L m j ρL m † j + L g j,j +1 ρL g † j,j +1 − (cid:8) L m † j L m j + L g † j,j +1 L g j,j +1 , ρ (cid:9)(cid:17) (26)at fixed values of γ and λ , we obtain the gauge-violationdynamics shown in Fig. 16. We find that a finite V sup-presses only coherent contributions to the gauge viola-tion, but not incoherent ones. This finding is not sur-prising as the dissipative errors in our work are modelledby a Markovian Lindblad master equation that couplesstates regardless of their energy differences, and which isthus oblivious to energy penalties.Interestingly, for intermediate values of the protectionstrength ( V = J a at λ = 10 − J a ), we see that aftergoing from diffusive ( ε ∼ γt ) at t (cid:46) γ/λ to ballistic( ε ∼ λ t ) dynamics at t (cid:38) γ/λ , the gauge violationagain exhibits diffusive behavior before settling into amaximal-violation steady state at t ≈ /γ . The larger V is, the shorter is the intermediate ballistic regime. Atsufficiently large V , coherent errors are almost completelysuppressed and the ballistic regime vanishes, with thegauge violation scaling as ε ∼ γt for all times t (cid:46) /γ . H. Dynamics under particle loss
Until now, within Sec. V we have considered only de-phasing in the matter fields in order to allow for theconservation of particle number, thereby enabling us toachieve larger system sizes (see Appendix B for furtherdetails). Here, we consider also dissipation in the matterfields. In particular, we will choose L m j = a j while alsousing L g j,j +1 = τ zj,j +1 and fixing λ = 10 − J a . Again, weconsider the initial state in Fig. 7(a) for N = 2 mattersites (here, N = 4 matter sites is numerically intractablefor the evolution times we need to reach in our calcula-tions), and solve Eq. (16) for λ = 10 − J a using severalvalues of γ . The corresponding results for the gauge-violation dynamics are shown in Fig. 17(a). For ease ofcomparison, we repeat these results for the case of de-phasing on matter fields with jump operators L m j = a † j a j in Fig. 17(b), just as in all the results before this point.Aside from the absence of a second prethermal plateaudue to the halved matter-site number, the results are verysimilar to those in Fig. 1(b) of Ref. 84, and the qualita-tive behavior is identical: the gauge violation displays acrossover from diffusive scaling ε ∼ γt to ballistic scaling ε ∼ λ t at t ∝ γ/λ when γ < λ . As such, we can con-clude that our results in Ref. 84 are general, and are notrestricted by considering only dephasing in the matterfields. (a) (b) Figure 16. (Color online). Using the setup of Fig. 7(a) un-der decoherence, we add a protection term V H G = V (cid:80) j G j such that the coherent quench is actuated by H + λH + V H G ,which suppresses only the coherent gauge-breaking errors, buthas no effect on the gauge violation due to Markovian deco-herence. (a) At fixed λ = 10 − J a and γ = 10 − J a (bluecurves), we see that at large enough protection strength V ,the diffusive scaling ε ∼ γt seen at short times emerges againbefore the maximal violation is reached, and, in some cases,after the ballistic scaling ε ∼ ( λt ) has appeared at intermedi-ate times. For reference, we also show the gauge violation for λ = 0 and γ = 10 − J a (red curve), which exhibits only scal-ing ε ∼ γt before saturating at its maximal value for t (cid:38) /γ .At nonzero λ and without energy protection, the gauge viola-tion scales ε ∼ γt only at early times t (cid:46) γ/λ before scaling ε ∼ ( λt ) at intermediate times t (cid:38) γ/λ , and finally reachingthe maximal violation at t ∝ min { λ − N/ , γ − } . (b) The sameas panel (a) but for λ = 10 − J a . I. Decoherence starting from equilibrium
Aside from quench dynamics, we also study the ef-fect of decoherence through jump operators L m j = a † j a j and L g j,j +1 = τ zj,j +1 on the ground state of an LGTwith N = 4 matter sites and periodic boundary condi-tions. For this aim, we prepare our system in the groundstate of H , and switch on decoherence at t = 0 accord-ing to the Lindblad master equation (16) with λ = 0.Such a scenario may occur, e.g., when variational statepreparation is used to achieve a good approximationto a gauge-invariant initial ground state, which is thenstored in the quantum computer and thus subjected todecoherence. As we do not project the ground state intothe target gauge-invariant sector g tar , the gauge viola-tion at t = 0 starts at a finite nonzero value. At t > ε ∼ γt at short times until the system set-tles into a maximal-violation steady state at t ≈ /γ , asshown in Fig. 18(a).It is also instructive to investigate the projectors onto7 (a) (b) Figure 17. (Color online). Dynamics of gauge violation in theZ LGT with N = 2 matter sites in the presence of dissipationin the gauge fields with jump operators L g j,j +1 = τ zj,j +1 and (a)dissipation in the matter fields with jump operators L m j = a j and (b) dephasing in the matter fields with jump operators L m j = a † j a j , just as in all the results for the Z LGT beforenow. Coherent gauge breaking is at strength λ = 10 − J a .The behavior is qualitatively identical whether the matterfields are subjected to dephasing or dissipation. The factthat we have only N = 2 matter sites here brings about onlya single prethermal plateau instead of two as in the case of N = 4 matter sites. the three relevant gauge-invariant supersectors P M =0 , , (the supersector projectors P M with odd M are of zeronorm in the half-filling global-symmetry sector of the Z LGT). Figure 18(b) shows the three projectors that giverise to nonzero expectation values in the case of N = 4matter sites. Interestingly, the steady-state expectationvalues are proprotional to the number of gauge-invariantsectors within the associated supersector. Indeed, for N = 4 matter sites, the supersector P contains six differ-ent gauge sectors, while the two supersectors representedby P and P each contains a single gauge-invariant sec-tor. As shown in the inset, (cid:104)P (cid:105) ≈ . (cid:104)P (cid:105) = (cid:104)P (cid:105) ≈ . (cid:104)P (cid:105) / (cid:104)P (cid:105) = (cid:104)P (cid:105) / (cid:104)P (cid:105) = 6. Thisindicates that with decoherence the system evolves atlate times into a steady state where the gauge violationhas spread into an equal distribution over all gauge sec-tors. This is qualitatively and quantitatively identicalto the behavior of these projectors at late times in thequench dynamics with decoherence shown in Fig. 1(c,d)of Ref. 84, despite here the initial state and dynamic pro-cess being fundamentally different. Hence, in these re-sults decoherence erases the memory of the initial state.For completeness, we additionally present the associatedresults for the MQC intensity I { , , , } in Fig. 19(a) andstaggered particle density in Fig. 19(b). Starting at itsground-state value, each of these observables shows a de- (a) (b) Figure 18. (Color online). (a) Time evolution of gauge-invariance violation after starting in the ground state of the Z LGT and switching on decoherence with strength γ at t = 0.Note that the ground state of H is not gauge-invariant, aconsequence of the fact that different gauge-invariant sectorsof H have energy overlaps. Similarly to the case of quenchdynamics, the gauge violation at short times scales ∼ γt be-fore plateauing at its maximal value. (b) Projectors onto thethree accessible gauge-invariant supersectors. Their steady-state expectation values are proportional to the number ofconstituent gauge-invariant sectors. (a) (b) Figure 19. (Color online). Same scenario as in Fig. 18: westart in the ground state of H and switch on decoherence at t = 0. Running averages of the (a) MQC intesity I ∆ g = { , , , } and (b) staggered boson number are shown. Unlike the gaugeviolation and supersector projectors, these observables showno trace of diffusive behavior at short times, but they decay ∼ ( γt ) − due to decoherence for t (cid:38) /γ . (a) (b) Figure 20. (Color online). Gauge violation over evolutiontime after starting in the (gauge-noninvariant) ground state of H and quenching with H + λH with Markovian decoherencethrough jump operators L m j = a † j a j and L g j,j +1 = τ zj,j +1 . (a)In the case of γ = 0 (no decoherence), the pre-onset andfinal prethermal plateaus appear and the violation at earlytimes scales ∼ λ t , because ρ is the ground state of H .(b) When decoherence is turned on, a diffusive-to-ballisticcrossover appears at timescale ∝ γ/λ taking the violationfrom a diffusive spread ∼ γt to a ballistic scaling ∼ λ t . (a) (b) Figure 21. (Color online). Same as Fig. 20 but with λ i = 0 . ρ is the ground state of H + λ i H . The initialstate ρ is again gauge-noninvariant, but not a ground stateof H . This makes the leading order of coherent contribution ∝ λt , which identically vanishes in the case of λ i = 0 ofFig. 20. This therefore leads to a crossover from a diffusivespread ∼ γt to a ballistic scaling ∼ λt at an earlier timescale t ∝ γ/λ < γ/λ . Note here in the purely coherent case (a)how there are three prethermal plateaus instead of just twoas in the case of Fig. 20. cay ∼ ( γt ) − in its temporal average at a time t ≈ /γ .The electric field (not shown) behaves also qualitativelythe same.A more interesting scenario is starting in ground stateof H + λ i H , which may become relevant in situationswhere a pre-quench state-preparation protocol is alreadysubject to gauge-breaking errors. Not only is the initialstate here gauge-noninvariant, the presence of H allowsfor a competition between coherent errors and their in-coherent counterparts due to decoherence.We first consider the case when λ i = 0, i.e., we havemanaged to prepare the system in the ground state ofthe ideal gauge theory without any coherent errors, butwe shall assume that upon quenching, unitary errors λH will be present. This scenario may appear whenthe preparation follows one protocol, e.g., a variationalprinciple, while the quench dynamics is studied withanother, e.g., an analog quantum-simulation scheme.The ensuing dynamics of the gauge-violation change isshown in Fig. 20(a) for the case without decoherence butwith finite λ >
0. The gauge violation grows ∼ λ t atearly times, with all lower-order coherent contributionsvanishing identically as rigorously explained in Sec. A 1through TDPT. The gauge violation exhibits the pre-onset and final plateaus at timescales ∝ ∝ J a /λ ,respectively, but the onset plateau at timescale ∝ /λ ,prominent in the case of gauge-invariant states, is miss-ing here. Maximal violation occurs at the final prether-mal timescale ∝ J a /λ . Upon introducing decoherencethrough jump operators L m j = a † j a j and L g j,j +1 = τ zj,j +1 in Fig. 20(b) at fixed λ , a crossover emerges at t ∝ γ/λ from a diffusive spread ∼ γt in the gauge violation toa ballistic spread ∼ λ t , similarly to the generic be-havior we find when starting in a gauge-invariant initialstate. Moreover, decoherence compromises the prether-mal plateaus, with its effect more apparent on the laterplateaus.On the other hand, when λ i (cid:54) = 0, i.e., when ρ isgauge-noninvariant but also not the ground state of H ,a lower-order coherent contribution ∝ λt that vanishesin the case of λ i = 0, now becomes finite, and thus thegauge-violation change scales ∼ λt at early times in thecase of no decoherence, as shown in Fig. 21(a). Inter-estingly, here we find all three prethermal plateaus: pre-onset at timescale t ∝
1, onset at t ∝ /λ , and final at t ∝ J a /λ . By switching on decoherence through jumpoperators L m j = a † j a j and L g j,j +1 = τ zj,j +1 at a fixed valuesof λ , we see that a crossover appears where the gauge-violation difference goes from a diffusive scaling ∼ γt toa ballistic spread ∼ λt at the timescale t ∝ γ/λ . Asexpected, decoherence also compromises the prethermalplateaus in this case. VI.
U(1)
QUANTUM LINK MODEL
To further demonstrate the generality of our findings,we now study the gauge-violation dynamics in the U(1)9 (a) (b)
Figure 22. (Color online). A system with N = 2 mattersites and periodic boundary conditions, prepared in a gauge-invariant initial state with zero bosons and a N´eel configura-tion of the electric field is quenched with H + λH in thepresence of decoherence through jump operators L m j = σ zj and L g j,j +1 = τ − j,j +1 both at environment-coupling strength γ . Decoherence compromises the prethermal plateaus, and infact leads the violation to a maximal value not attained bythe coherent errors is alone. This is due to the absence ofresonance between a few of the gauge-invariant sectors in theU(1) QLM, which does not have an analog in the Z LGT.This maximal value depends on the fixed points of the Liouvil-lian superoperator. The short-time dynamics is qualitativelythe same as in the generic case of the Z LGT when the ini-tial state is gauge-invariant: a diffusive-to-ballistic crossoverarises at t ∝ γ/λ where the gauge violation goes from adiffusive spread ∼ γt to a ballistic spread ∝ λ t . quantum link model (QLM) given by H = N (cid:88) j =1 (cid:104) − J (cid:0) σ − j τ + j,j +1 σ − j +1 + H.c. (cid:1) + µ σ zj (cid:105) , (27)where the Pauli matrix σ + j is the creation operator of aparticle on site j , while the Pauli matrix τ + j,j +1 ( τ zj,j +1 )on link j, j + 1 represents the gauge (electric) field. Here,we consider a lattice of N = 2 matter sites and periodicboundary conditions. The Gauss’s-law generator is G j = ( − j (cid:0) τ zj − ,j + σ zj + τ zj,j +1 + 1 (cid:1) . (28)and has eigenvalues g j = − , , ,
2. This model has beenthe subject of recent ultracold-atom experiments.
We calculate the quench dynamics of this model in thepresence of decoherence through jump operators L m j = σ zj and L g j,j +1 = τ − j,j +1 , both at environment-couplingstrength γ , by solving the Lindblad master equation (16) with λH = λ (cid:88) j (cid:0) σ − j σ − j +1 + σ + j σ + j +1 + τ xj,j +1 (cid:1) , (29)which describes unassisted matter tunneling and gaugeflipping, and we consider the jump operators L m j = σ zj and L g j,j +1 = τ + j,j +1 , although we have checked that otherchoices of the jump operators yield the same qualitativepicture. Moreover, the gauge-invariant initial state ρ has zero matter particles and a N´eel configuration of theelectric field. In our numerics, we have set J = 1 and µ = 0 .
05, though we have checked that our conclusionsare independent of this particular choice of parameters.The ensuing time evolution of the gauge violation isshown in Fig. 22. Focusing first on the long-time dy-namics, we again see how decoherence compromises theprethermal plateau, but, more dramatically than in thecase of the Z LGT, it drives the gauge violation to alarger value. This is because the U(1) QLM has a largernumber of gauge-invariant sectors (due to an eigenvalue g j having four possible values) some of which do not haveresonances through H with one another—unlike the Z LGT, here the value of g j restricts the possible valuesthat g j − and g j +1 can take. In the presence of deco-herence, resonances between these gauge-invariant sec-tors are facilitated, and this is what leads to a largerlong-time violation compared to the purely unitary-errorcase. Furthermore, as can be seen in Fig. 22, the short-time dynamics and the diffusive-to-ballistic crossover areidentical to those for the Z LGT and the eBHM, furthervalidating the generality of our results. Indeed, as we rig-orously show in Sec. A through TDPT, our conclusionsare valid for any many-body system with local of globalsymmetries.
VII. CONCLUSIONS AND OUTLOOK
In this work, we have considered the dynamics of quan-tum systems where a symmetry is slightly broken, withspecial focus on the interplay between coherent and dis-sipative errors. To obtain a general unifying picture, wehave performed extensive numerical studies and analyt-ical derivations, considering a broad range of scenariosincluding dynamics starting from initial product statesand from ground states, as well as a variety of mod-els with global symmetries [global U(1) symmetry inan extended Bose-Hubbard model corresponding to to-tal particle-number conservation] and local symmetries[Z and U(1) gauge symmetries corresponding to Gauss’slaw].From these, several generic features emerge. First,the symmetry violation—the expectation value of thesymmetry generator—generically reveals a short-timecrossover from diffusive to ballistic or even hyperballisticmean-square displacement across symmetry sectors. Sec-ond, for purely coherent errors interference effects canprevent the symmetry violation to reach its theoretical0maximum, even in the long-time limit. Decoherence canlift these interference effects. As a consequence, the dy-namics typically is dominated by decoherence at earlyand late times, while it is dominated by coherent errorsin an intermediate time window. Third, the MQC is apowerful tool to reveal this complex interplay by quanti-fying the coherence between symmetry sectors encapsu-lated in the quantum state. Counterintuitively, we findsituations where the addition of decoherence to coherenterrors can increase the MQCs.Our findings will be highly relevant for quantum simu-lation experiments on NISQ devices. They illuminate thegeneral behavior with which the dynamics of a quantummany-body system under slight symmetry breaking de- teriorates from the ideal model with intact conservationlaws. These enable to estimate, e.g., target time scalesthat experimental technology needs to achieve in orderto observe desired phenomena. ACKNOWLEDGMENTS
This work is part of and supported by the Interdis-ciplinary Center Q@TN — Quantum Science and Tech-nologies at Trento, the DFG Collaborative Research Cen-tre SFB 1225 (ISOQUANT), the Provincia Autonoma diTrento, and the ERC Starting Grant StrEnQTh (Project-ID 804305).
Appendix A: Time-dependent perturbation theory
In our results, we have seen that the symmetry violation transitions from diffusive behavior ε ∼ γt at short timesto ballistic behavior ε ∼ λ m t n with integers m ≥ n ≥ Let us first rewrite Eqs. (8) and (16) in the concise form˙ ρ = ( S + γ L ) ρ, (A1a) S ρ = − i [ H, ρ ] , (A1b) L ρ = N (cid:88) j =1 (cid:16) L j ρL † j − (cid:8) L † j L j , ρ (cid:9)(cid:17) . (A1c)where, without loss of generality, we have used the jump operators L j without distinguishing between those actingon matter or gauge fields as that is inconsequential in the following derivations. Accordingly, the exact solution toEq. (A1a) is ρ ( t ) = e ( S + γ L ) t ρ . (A2)The Taylor expansion of this solution is ρ ( t ) = ∞ (cid:88) n =0 ( S + γ L ) n t n n ! ρ = (cid:26) ∞ (cid:88) n =1 (cid:20) S n + γ n − (cid:88) m =0 S m LS n − m − + γ n − (cid:88) m =1 n − m − (cid:88) k =0 S n − m − k − LS k LS m − + O ( γ ) (cid:21) t n n ! (cid:27) ρ . (A3)
1. Coherent terms
It is important to recall here that γ is not the only perturbative parameter in this problem, because there arecoherent errors at perturbative strength λ encapsulated within the unitary processes of S . In the case of γ = 0, onecan show that the leading order in gauge violation scales as ε ∼ ( λt ) at short times t (cid:46) /λ when starting in asymmetric state or a generic eigenstate of H , or as ε ∼ λt when starting in an unsymmetric state that is also notan eigenstate of H . This can be seen by considering the first- and second-order (in S ) terms of Eq. (A3) for γ = 0.If we were to write S = S + λ S , where S contains all the processes due to H while S all those due to H , thecorresponding approximate density matrix would be ρ ( t ) ≈ (cid:20) t S + 12 t S + λt S + 12 λt (cid:16) S S + S S (cid:17) + 12 λ t S (cid:21) ρ . (A4)1Employing for convenience the symmetry-violation operator G [see Eqs. (9) and (17)], it is straightforward to deriveTr (cid:8) G ρ (cid:9) = 0 for all ρ in target symmetry sector , (A5a) t Tr (cid:8) GS ρ (cid:9) = − it Tr (cid:8)(cid:2) G , H (cid:3) ρ (cid:9) = 0 , ∀ ρ , (A5b) t Tr (cid:8) GS ρ (cid:9) = − t Tr (cid:8) G H H ρ − G H ρ H + G ρ H H (cid:9) = 0 , ∀ ρ , (A5c) λt Tr (cid:8) GS ρ (cid:9) = − iλt Tr (cid:8)(cid:2) G , H (cid:3) ρ (cid:9) , (A5d) λt Tr (cid:8) GS S ρ (cid:9) = − λt Tr (cid:8) G H (cid:2) H , ρ (cid:3) + (cid:2) ρ , H (cid:3) H G (cid:9) = 0 , ∀ ρ , (A5e) λt Tr (cid:8) GS S ρ (cid:9) = − λt Tr (cid:8) G H (cid:2) H , ρ (cid:3) + (cid:2) ρ , H (cid:3) H G (cid:9) . (A5f)In the Eqs. (A5) that identically vanish for any initial state ρ , we have used the cyclic property of the trace, and thefact that [ H , G ] = 0. The remaining term in Eq. (A4) makes the following contribution to the symmetry violation:12 λ t Tr (cid:8) GS ρ (cid:9) = − λ t Tr (cid:8) G H H ρ − G H ρ H + G ρ H H (cid:9) (cid:54) = 0 in general . (A6)In case ρ is symmetric, then Eqs. (A5d) and (A5f) vanish since G ρ = ρ G = const. × ρ , which means thatthe leading nonvanishing coherent contribution to the gauge violation is, at lowest order, ∝ λ t due to Eq. (A6).This is shown numerically in Fig. 1(b) of Ref. 84, and also in, e.g., Figs. 8, 10, and 11 of this work for the Z LGT,Fig. 22 for the U(1) QLM, and in Figs. 2(a,c) and 3 for the eBHM. Note how in Figs. 2(b) and 4 the leading coherentcontribution to the gauge violation is ∝ λ t rather than ∝ λ t . This is due to a special interplay between thestaggered symmetric initial state shown in Fig. 1(a) and the two-body error term of Eq. (11). In that case, since ρ is symmetric, Eq. (A6) reduces to λ t Tr {G H ρ H } . Moreover, H ρ H vanishes as H removes or adds bosons ontwo adjacent sites simultaneously, while ρ has a staggered boson occupation. The next leading term in the Taylorexpansion of the density matrix, ∝ λ t H H ρ H H , does not vanish since H actuates tunneling, leading to H ρ H containing filled adjacent sites.In case ρ is unsymmetric, then theoretically Eq. (A5d) does not vanish in general. However, our numericalinvestigations suggest that this happens in one artificial and two pathological cases. The first pathological case iswhen ρ is an eigenstate of a highly nonlocal Hamiltonian that does not commute with G . The second is when H isitself highly nonlocal and does not commute with either ρ or G . Usually in modern experimental setups the coherentgauge-breaking errors in the preparation of ρ and during the dynamics are local. Our numerical checks reveal thatin such realistic situations the coherent contribution ∝ λt usually vanishes. Indeed, when the unitary errors in thepreparation of ρ and during the dynamics are due to the same term H , then Eq. (A5d) vanishes identically. Let ρ be the ground state of H + λ i H with λ i (cid:54) = 0, while the unitary quench dynamics is actuated by H + λH . One canderive λt Tr (cid:8)(cid:2) G , H (cid:3) ρ (cid:9) = λt Tr (cid:8) G (cid:2) H , ρ (cid:3)(cid:9) = λλ i t Tr (cid:8) G (cid:2) H + λ i H , ρ (cid:3)(cid:9) = 0 , (A7)where we have invoked Eq. (A5b). This is why the leading coherent order in such a case is ∝ λt , as can be seenin Fig. 6 for the eBHM and Fig. 21 for the Z LGT. Thus, when ρ is unsymmetric due to generic experimentalpreparation errors H , the nonvanishing leading coherent contribution to the symmetry violation is ∝ λt due toEq. (A6).To illustrate the artificial case, let us assume we are in a common eigenbasis of H and G . Then if in this eigenbasisthere are eigenstates that are degenerate with respect to H but not to G , then an arbitrary superposition of theseeigenstates, itself still an eigenstate of H , is no longer an eigenstate of G , thereby possibly leading to a finitecontribution ∝ λt due to Eq. (A5d). Realistically, such a state seems difficult to prepare in experiment. Moreover,when ρ is an eigenstate of H , no matter how artificially engineered, Eq. (A6) completely vanishes by noting thecyclic property of the trace and that [ H , ρ ] = 0. Therefore, when the initial state is a generic eigenstate of H , theleading coherent contribution to the gauge violation is ∝ λ t , i.e., the same as that for a symmetric initial state.This is indeed what we see in Fig. 20.In order to explain the short-time scalings of the supersector projectors in the ballistic regime in Figs. 8, 10, and 11of this work and Fig. 1(c,d) of Ref. 84 in the case of the Z LGT, let us focus on the case of a symmetric initial state,as is used in these aforementioned results. If we replace G with P in Eqs. (A5b)–(A6), we will arrive at the sameconclusions since [ G , P ] = [ H , P ] = 0, and because P includes violations, with respect to the target gauge sector g tar = , due to first-order processes in H , meaning that λ t Tr (cid:8) P H ρ H (cid:9) (cid:54) = 0 in general, which explains itsballistic behavior ∼ λ t at early times.2For the same reasons, it is also found that Eqs. (A5b)–(A5f) will all hold if we replace G with P , but differently,we would get λ t Tr (cid:8) P H ρ H (cid:9) = 0, because H ρ H does not involve any second-order processes in H , and the(super)sector M = 4 includes only such processes (this order could become nonzero for a less localized H that breaksfour local symmetry generators simultaneously). For similar reasons, coherent terms ∝ λ cannot involve second-orderprocesses in H on both sides of ρ at the same time (the associated terms would be H H H ρ , H H ρ H , andtheir Hermitian conjugates), and their contribution to (cid:104)P (cid:105) vanishes. Thus, for the error terms considered in thiswork the nonvanishing leading-order coherent contribution for P at early times is λ t Tr (cid:8) P H H ρ H H (cid:9) / (cid:54) =0 in general, which explains its scaling ∼ λ t in the ballistic regime.
2. Leading incoherent terms
In the presence of decoherence, the dominant correction to the unitary part of the density matrix at leading order of γ is γt L ρ , as can be seen for n = 1 (and, consequently, m = 0) in Eq. (A3). The contribution to the gauge violation ε [see Eq. (17)] at short times due to the term γt L ρ is γt Tr (cid:8) GL ρ (cid:9) = γt (cid:88) j Tr (cid:26) G L j ρ L † j − G L † j L j ρ − G ρ L † j L j (cid:27) (cid:54) = 0 in general , (A8)regardless of whether ρ is symmetric or not. Indeed, the term γt L ρ involves only incoherent gauge-breakingprocesses, and its contribution will lead to diffusive scaling ∼ γt in the gauge violation. This diffusive behavior willdominate over the leading-order coherent gauge breaking ∝ λ t for evolution times t (cid:46) γ/λ in case of a symmetricinitial state or a generic eigenstate of H , as shown in Fig. 1(b) of Ref. 84 and Figs. 8, 10, 11, 15, 17, and 20 in caseof the Z LGT, Fig. 22 for the U(1) QLM, and Fig. 3 for the eBHM. In the case of a generic (i.e., not pathologicalor artificial; see discussion in Sec. A 1) unsymmetric initial state, the diffusive scaling ∼ γt will dominate over theleading-order coherent contribution ∝ λt for t (cid:46) γ/λ , as seen in Fig. 6 for the eBHM and Fig. 21 for the Z LGT.However, this crossover time can be made even earlier, such as in the case of hyperballistic scaling ∝ λ t shown inFig. 4, where it becomes t ∝ ( γ/λ ) . By replacing G with P in Eq. (A8), it is straightforward to see that the samedominant incoherent contribution to P is also ∝ γt , and it will therefore show diffusive scaling ∼ γt at early times.Replacing G with the supersector projector P in Eq. (A8), we see that when ρ is symmetric or a generic eigenstateof H , P cannot scale ∼ γt in the diffusive regime. We will come back to this later. However, if ρ is unsymmetric(but not a generic eigenstate of H or another observable commuting with G ) with finite support in the supersector M = 2, then P can show diffusive behavior ∼ γt at early times.We can also explain from Eq. (A8) why when dissipation and dephasing have different environment-couplingstrengths γ g and γ m , respectively, the gauge violation at short times scales diffusively as ε ∼ γ g t , with dephas-ing having no effect as shown in Fig. 15. In the case of dephasing, L m j = a † j a j does not create a violation in thesystem because (cid:2) G , a † j a j (cid:3) = (cid:2) G l , a † j a j (cid:3) = 0 , ∀ j, l , and so the associated contribution γ m t (cid:80) j Tr (cid:8) G a † j a j ρ a j a † j (cid:9) = 0,where we recall that ρ lies in the target symmetry sector g tar = . As such, the only remaining contribution fromEq. (A8) is γ g t (cid:80) j Tr (cid:8) G L g j,j +1 ρ L g † j,j +1 (cid:9) , which does not vanish in general, because ∃ l : (cid:2) G l , L g j,j +1 (cid:3) (cid:54) = 0 in the caseof dissipation. The supersector projector (cid:104)P (cid:105) exhibits the same behavior as the violation.In contrast, the term γt L ρ in the Taylor expansion of Eq. (16) leads to a vanishing contribution to (cid:104)P (cid:105) becauseTr (cid:8) P L m(g) j ρ L m(g) † j (cid:9) = 0 as the jump operators drive the system into the supersector M = 2. As discussed inSec. A 1, this is similar to the reason why Tr (cid:8) P H ρ H (cid:9) = 0, as it involves first-order processes in H , which drivethe system into the supersector M = 2, and thus (cid:104)P (cid:105) cannot show scaling ∼ λ t either.It is important to note here that leading-order (in γ ) corrections to the density matrix in Eq. (A3) also includeterms that are quadratic in time, and involve the term γt ( LS + SL ) ρ , which can be rewritten as γt ( LS + SL ) ρ = (cid:2) γt ( LS + S L ) + γλt ( LS + S L ) (cid:3) ρ . (A9)The purely incoherent gauge-breaking term γt ( LS + S L ) ρ in Eq. (A9) on the gauge violation ε will always bedominated by that ∝ γt , which so far we have seen is a generic feature of the symmetry violation in presence ofdecoherence. As such, generically we will not see scaling ∼ γt in the gauge violation.
3. Mixed terms
We now shift our attention to the component of Eq. (A9) where unitary and incoherent gauge-breaking processesmix: γλt ( LS + S L ) ρ . For this contribution to dominate over that ∝ γt , we must have t > /λ , which is anyway3beyond the perturbative regime as then prethermalization kicks in. Automatically this means that in generic situationsthe contribution ∝ γλt will not dominate over any of the (hyper)ballistic scalings that dominate over γt after thecrossover time in the gauge violation.
4. Higher-order incoherent terms
We have thus far understood why the dominant scaling in the gauge violation is ε ∼ γt at times t (cid:46) γ/λ , beyondwhich we see in the ED results that the gauge violation scales as ε ∼ λ t up until evolution times t ≈ /λ for asymmetric initial state. We also understand why the latter scale is not compromised by terms ∝ γt or ∝ γλt .One remaining term that merits investigation is γ t L ρ as it pertains to the supersector projector P , to which itscontribution is12 γ t Tr (cid:8) P L ρ (cid:9) = 12 γ t (cid:88) j Tr (cid:26) P L (cid:18) L j ρ L † j − L † j L j ρ − ρ L † j L j (cid:19)(cid:27) = 14 γ t (cid:88) j,l Tr (cid:110) P (cid:16) L l L j ρ L † j L † l − L l L † j L j ρ L † l − L l ρ L † j L j L † l − L † l L l L j ρ L † j − L j ρ L † j L † l L l + L † j L j ρ L † l L l (cid:17)(cid:111) , which is nonzero in general. This becomes the dominant incoherent contribution to P , because it involves terms withsecond-order violating processes (quadratic in jump operators) on each side of ρ . This explains exactly why (cid:104)P (cid:105) exhibits the scaling (cid:104)P (cid:105) ∼ γ t at times t (cid:46) γ/λ before scaling as (cid:104)P (cid:105) ∼ λ t for t (cid:38) γ/λ for γ (cid:46) λ , as shown inFig. 1(d) of Ref. 84 and Figs. 8, 10, and 11 of this work for the Z LGT.Finally, we note that even though the contribution ∝ γ t to the gauge violation does not necessarily vanish, it willalways be dominated by that ∝ γt in generic situations. Appendix B: Numerics specifics
In this Appendix we provide details pertaining to our numerical implementation. First we provide the exactexpressions we used for the coefficients c n in H of Eq. (15), which read c = (cid:88) k> N ( χ ) k (cid:2) J − k − ( χ ) J − k − ( χ ) + J k ( χ ) J k +1 ( χ ) − J k − ( χ ) J k − ( χ ) − J − k ( χ ) J − k +1 ( χ ) (cid:3) , (B1a) c = (cid:88) k> N ( χ ) k (cid:2) J − k +1 ( χ ) J k − ( χ ) + J − k ( χ ) J k − ( χ ) − J k +1 ( χ ) J − k − ( χ ) − J k ( χ ) J − k − ( χ ) (cid:3) , (B1b) c = (cid:88) k> N ( χ ) k (cid:2) J k − ( χ ) + J k − ( χ ) − J − k − ( χ ) − J − k − ( χ ) (cid:3) , (B1c) c = (cid:88) k> N ( χ ) k (cid:2) J − k +1 ( χ ) + J − k ( χ ) − J k +1 ( χ ) − J k ( χ ) (cid:3) , (B1d)where J q ( χ ) is the q th -order Bessel function of the firstkind and we use a normalization factor N ( χ ) to ensurethat (cid:80) n =1 c n = 1, in order to make the strength of theunitary gauge-breaking term independent of χ , and solelydependent on λ .
1. Implementational details
All results presented in this work have been calcu-lated using our in-house exact diagonalization toolkitLaGaDyn, where we have also performed benchmarkswith QuTiP.
We solve Eq. (16) by rewriting it as˙˜ ρ = M ˜ ρ, (B2)where we have matricized the equation of motion such4 Figure 23. (Color online). Same as Fig. 1 of Ref. 84 butshowing the raw signal rather than its temporal average. Thequalitative picture is unaltered. The fluctuations in the rawsignal are completely suppressed at times t (cid:38) /γ . that ˜ ρ is a flattened version of the density matrix ρ wherethe latter’s columns in left-to-right order are stacked ontop of each other, and M is the corresponding Lind-bladian superoperator encapsulating all relevant unitaryand incoherent processes from Eq. (16) in the resulting
H ⊗ H space, where H is the Hilbert space of our modelof interest. For the time evolution, we solve˜ ρ ( t ) = e M t ˜ ρ , (B3)using our exact exponentiation routine. We opt to usethe latter instead of common methods based on itera-tive solutions of ordinary differential equations in orderto be able to reliably achieve the large evolution times displayed in our results.As mentioned in the main text, for numerical feasibil-ity we have chosen in the main results to turn on onlydephasing rather than dissipation on the matter fields.This leads to retaining a global U(1) symmetry in theform of particle-number conservation, because both H and H also conserve it. Taking into account the hard-core boson constraint and staying in the half-filling sectorallows for reducing the number of states in the system’sHilbert space H from 2 N to (cid:96) = 2 N (cid:0) NN/ (cid:1) . However, ourdynamics of Eq. (B3) is not solved in H (whose size is (cid:96) × (cid:96) ), but rather in H ⊗ H (whose size is (cid:96) × (cid:96) ).
2. Running average versus raw signal
In this work and Ref. 84, all results have shown the run-ning temporal average A ( t ) = (cid:82) t ds A ( s ) /t of all quanti-ties A ( t ). This is done in order to suppress fluctuationsin the raw signal A ( t ), which are prominent especiallyin the case of purely unitary dynamics. For comparison,we provide in Fig. 23 the raw data of the gauge-violationdynamics from Fig. 1(b) of Ref. 84 in the Z LGT withcoherent and incoherent gauge-invariance breaking. Ascan be seen, the qualitative picture remains exactly thesame. We see that decoherence itself behaves similar tothe running average in that it fully suppresses fluctua-tions for times t (cid:38) /γ . That may be seen as anotherinstance of the effect of diffusion, in contrast to oscilla-tions that are typical of coherent wave-like dynamics. S. Sachdev, Quantum Phase Transitions (Cambridge Uni-versity Press, 2001), ISBN 9780521004541, URL https://books.google.de/books?id=Ih_E05N5TZQC . L. Balents, Nature , 199 (2010), URL https://doi.org/10.1038/nature08917 . L. Savary and L. Balents, Reports on Progress in Physics , 016502 (2016), URL https://doi.org/10.1088%2F0034-4885%2F80%2F1%2F016502 . K. Brading and E. Castellani,Symmetries in Physics: Philosophical Reflections(Cambridge University Press, 2003), ISBN9781139442022, URL https://books.google.de/books?id=fB38Q9D-WdwC . A. Zee, Quantum Field Theory in a Nutshell (PrincetonUniversity Press, 2003), ISBN 9780691010199, URL https://books.google.de/books?id=85G9QgAACAAJ . H. Tasaki, Journal of Physics: Condensed Mat-ter , 4353 (1998), URL https://doi.org/10.1088%2F0953-8984%2F10%2F20%2F004 . H. A. Gersch and G. C. Knollman, Phys. Rev. ,959 (1963), URL https://link.aps.org/doi/10.1103/PhysRev.129.959 . M. E. Peskin and D. V. Schroeder,An Introduction To Quantum Field Theory (CRCPress, 2018), ISBN 9780429983184, URL https://books.google.de/books?id=9EpnDwAAQBAJ . J. D. Jackson and L. B. Okun, Rev. Mod. Phys. ,663 (2001), URL https://link.aps.org/doi/10.1103/RevModPhys.73.663 . T. Cheng and L. Li, Gauge Theory of Elementary Particle Physics,Oxford science publications (Clarendon Press, 1984),ISBN 9780198519614, URL https://books.google.it/books?id=lk8GEzVNb10C . K. Mallayya, M. Rigol, and W. De Roeck, Phys. Rev. X , 021027 (2019), URL https://link.aps.org/doi/10.1103/PhysRevX.9.021027 . S. Ray, J. R. Anglin, and A. Vardi (2020), 2009.01491,URL https://arxiv.org/abs/2009.01491 . J. C. Halimeh and P. Hauke (2020), 2004.07248, URL https://arxiv.org/abs/2004.07248 . J. C. Halimeh and P. Hauke (2020), 2004.07254, URL https://arxiv.org/abs/2004.07254 . M. Moeckel and S. Kehrein, Phys. Rev. Lett. ,175702 (2008), URL https://link.aps.org/doi/10.1103/PhysRevLett.100.175702 . M. Eckstein, M. Kollar, and P. Werner, Phys. Rev. Lett. , 056403 (2009), URL https://link.aps.org/doi/10.1103/PhysRevLett.103.056403 . M. Kollar, F. A. Wolf, and M. Eckstein, Phys. Rev. B , 054304 (2011), URL https://link.aps.org/doi/10.1103/PhysRevB.84.054304 . M. Tavora and A. Mitra, Phys. Rev. B , 115144 (2013),URL https://link.aps.org/doi/10.1103/PhysRevB.88.115144 . N. Nessi, A. Iucci, and M. A. Cazalilla, Phys. Rev. Lett. , 210402 (2014), URL https://link.aps.org/doi/10.1103/PhysRevLett.113.210402 . F. H. L. Essler, S. Kehrein, S. R. Manmana, and N. J.Robinson, Phys. Rev. B , 165104 (2014), URL https://link.aps.org/doi/10.1103/PhysRevB.89.165104 . B. Bertini, F. H. L. Essler, S. Groha, and N. J. Robinson,Phys. Rev. B , 245117 (2016), URL https://link.aps.org/doi/10.1103/PhysRevB.94.245117 . M. Fagotti and M. Collura (2015), 1507.02678. P. Reimann and L. Dabelow, Phys. Rev. Lett. ,080603 (2019), URL https://link.aps.org/doi/10.1103/PhysRevLett.122.080603 . M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Phys.Rev. Lett. , 050405 (2007), URL https://link.aps.org/doi/10.1103/PhysRevLett.98.050405 . M. Rigol, Phys. Rev. Lett. , 100403 (2009), URL https://link.aps.org/doi/10.1103/PhysRevLett.103.100403 . M. Rigol, Phys. Rev. A , 053607 (2009), URL https://link.aps.org/doi/10.1103/PhysRevA.80.053607 . L. Vidmar and M. Rigol, Journal of Statistical Me-chanics: Theory and Experiment , 064007 (2016),URL https://doi.org/10.1088%2F1742-5468%2F2016%2F06%2F064007 . F. H. L. Essler and M. Fagotti, Journal of Statisti-cal Mechanics: Theory and Experiment , 064002(2016), URL https://doi.org/10.1088%2F1742-5468%2F2016%2F06%2F064002 . M. A. Cazalilla and M.-C. Chung, Journal of Statisti-cal Mechanics: Theory and Experiment , 064004(2016), URL https://doi.org/10.1088%2F1742-5468%2F2016%2F06%2F064004 . J.-S. Caux, Journal of Statistical Mechanics: Theory andExperiment , 064006 (2016), URL https://doi.org/10.1088%2F1742-5468%2F2016%2F06%2F064006 . K. Mallayya and M. Rigol, Phys. Rev. Lett. ,070603 (2018), URL https://link.aps.org/doi/10.1103/PhysRevLett.120.070603 . M. Stark and M. Kollar (2013), 1308.1610. L. D’Alessio, Y. Kafri, A. Polkovnikov, andM. Rigol, Advances in Physics , 239 (2016),https://doi.org/10.1080/00018732.2016.1198134, URL https://doi.org/10.1080/00018732.2016.1198134 . A. Lazarides, A. Das, and R. Moessner, Phys. Rev. Lett. , 150401 (2014), URL https://link.aps.org/doi/10.1103/PhysRevLett.112.150401 . E. Canovi, M. Kollar, and M. Eckstein, Phys. Rev. E , 012130 (2016), URL https://link.aps.org/doi/10.1103/PhysRevE.93.012130 . H. D. Zeh, Foundations of Physics , 69 (1970), URL https://doi.org/10.1007/BF00708656 . M. Schlosshauer, Rev. Mod. Phys. , 1267 (2005),URL https://link.aps.org/doi/10.1103/RevModPhys.76.1267 . F. Yoshihara, K. Harrabi, A. O. Niskanen, Y. Nakamura,and J. S. Tsai, Phys. Rev. Lett. , 167001 (2006), URL https://link.aps.org/doi/10.1103/PhysRevLett.97.167001 . K. Kakuyanagi, T. Meno, S. Saito, H. Nakano, K. Semba,H. Takayanagi, F. Deppe, and A. Shnirman, Phys. Rev. Lett. , 047004 (2007), URL https://link.aps.org/doi/10.1103/PhysRevLett.98.047004 . R. C. Bialczak, R. McDermott, M. Ansmann,M. Hofheinz, N. Katz, E. Lucero, M. Neeley,A. D. O’Connell, H. Wang, A. N. Cleland, et al.,Phys. Rev. Lett. , 187006 (2007), URL https://link.aps.org/doi/10.1103/PhysRevLett.99.187006 . J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara,K. Harrabi, G. Fitch, D. G. Cory, Y. Nakamura, J.-S.Tsai, and W. D. Oliver, Nature Physics , 565 (2011),URL https://doi.org/10.1038/nphys1994 . H. Wang, C. Shi, J. Hu, S. Han, C. C. Yu, and R. Q.Wu, Phys. Rev. Lett. , 077002 (2015), URL https://link.aps.org/doi/10.1103/PhysRevLett.115.077002 . P. Kumar, S. Sendelbach, M. A. Beck, J. W. Free-land, Z. Wang, H. Wang, C. C. Yu, R. Q. Wu,D. P. Pappas, and R. McDermott, Phys. Rev. Applied , 041001 (2016), URL https://link.aps.org/doi/10.1103/PhysRevApplied.6.041001 . P. K. Day, H. G. LeDuc, B. A. Mazin, A. Vayonakis, andJ. Zmuidzinas, Nature , 817 (2003), URL https://doi.org/10.1038/nature02037 . C. A. Regal, J. D. Teufel, and K. W. Lehnert, NaturePhysics , 555 (2008), URL https://doi.org/10.1038/nphys974 . D. Poulin, Phys. Rev. Lett. , 190401 (2010), URL https://link.aps.org/doi/10.1103/PhysRevLett.104.190401 . J. Marino and A. Silva, Phys. Rev. B , 060408 (2012),URL https://link.aps.org/doi/10.1103/PhysRevB.86.060408 . B. Descamps, Journal of Mathematical Physics ,092202 (2013), https://doi.org/10.1063/1.4820785, URL https://doi.org/10.1063/1.4820785 . J.-S. Bernier, P. Barmettler, D. Poletti, and C. Kollath,Phys. Rev. A , 063608 (2013), URL https://link.aps.org/doi/10.1103/PhysRevA.87.063608 . J. C. Halimeh, M. Punk, and F. Piazza, Phys. Rev. B , 045111 (2018), URL https://link.aps.org/doi/10.1103/PhysRevB.98.045111 . J.-S. Bernier, R. Tan, L. Bonnes, C. Guo, D. Poletti, andC. Kollath, Phys. Rev. Lett. , 020401 (2018), URL https://link.aps.org/doi/10.1103/PhysRevLett.120.020401 . C. Maier, T. Brydges, P. Jurcevic, N. Trautmann,C. Hempel, B. P. Lanyon, P. Hauke, R. Blatt, andC. F. Roos, Phys. Rev. Lett. , 050501 (2019), URL https://link.aps.org/doi/10.1103/PhysRevLett.122.050501 . Z. Lenarˇciˇc, F. Lange, and A. Rosch, Phys. Rev. B , 024302 (2018), URL https://link.aps.org/doi/10.1103/PhysRevB.97.024302 . F. Lange, Z. Lenarˇciˇc, and A. Rosch, Phys. Rev. B , 165138 (2018), URL https://link.aps.org/doi/10.1103/PhysRevB.97.165138 . C. Gross and I. Bloch, Science , 995 (2017), ISSN 0036-8075,https://science.sciencemag.org/content/357/6355/995.full.pdf,URL https://science.sciencemag.org/content/357/6355/995 . M. Lewenstein, A. Sanpera, and V. Ahufinger,Ultracold Atoms in Optical Lattices: Simulating quantum many-body systems(OUP Oxford, 2012), ISBN 9780191627439, URL https://books.google.de/books?id=Wpl91RDxV5IC . P. Hauke, F. M. Cucchietti, L. Tagliacozzo, I. Deutsch,and M. Lewenstein, Reports on Progress in Physics ,082401 (2012). F. Zhou and C. Wu, New Journal of Physics , 166 (2006),URL https://doi.org/10.1088%2F1367-2630%2F8%2F8%2F166 . I. Maruyama, T. Koide, and Y. Hatsugai, Phys. Rev. B , 235105 (2007), URL https://link.aps.org/doi/10.1103/PhysRevB.76.235105 . A. Roy and K. Saha, New Journal of Physics , 103050(2019), URL https://doi.org/10.1088%2F1367-2630%2Fab4da0 . K. Donatella, A. Biella, A. L. Boit, and C. Ciuti (2020),2004.12883. M. Greiner, O. Mandel, T. Esslinger, T. W. H¨ansch, andI. Bloch, Nature , 39 (2002), URL https://doi.org/10.1038/415039a . K. Melnikov and M. Weinstein, Phys. Rev. D ,094504 (2000), URL https://link.aps.org/doi/10.1103/PhysRevD.62.094504 . L.-C. Tu, J. Luo, and G. T. Gillies, Reports on Progress inPhysics , 77 (2004), URL https://doi.org/10.1088%2F0034-4885%2F68%2F1%2Fr02 . E. Zohar and B. Reznik, Phys. Rev. Lett. ,275301 (2011), URL https://link.aps.org/doi/10.1103/PhysRevLett.107.275301 . E. Zohar, J. I. Cirac, and B. Reznik, Phys. Rev. Lett. , 125302 (2012), URL https://link.aps.org/doi/10.1103/PhysRevLett.109.125302 . D. Banerjee, M. Dalmonte, M. M¨uller, E. Rico, P. Ste-bler, U.-J. Wiese, and P. Zoller, Phys. Rev. Lett. , 175302 (2012), URL https://link.aps.org/doi/10.1103/PhysRevLett.109.175302 . E. Zohar, J. I. Cirac, and B. Reznik, Phys. Rev. Lett. , 055302 (2013), URL https://link.aps.org/doi/10.1103/PhysRevLett.110.055302 . P. Hauke, D. Marcos, M. Dalmonte, and P. Zoller, Phys.Rev. X , 041018 (2013), URL https://link.aps.org/doi/10.1103/PhysRevX.3.041018 . K. Stannigel, P. Hauke, D. Marcos, M. Hafezi, S. Diehl,M. Dalmonte, and P. Zoller, Phys. Rev. Lett. ,120406 (2014), URL https://link.aps.org/doi/10.1103/PhysRevLett.112.120406 . S. K¨uhn, J. I. Cirac, and M.-C. Ba˜nuls, Phys. Rev. A , 042305 (2014), URL https://link.aps.org/doi/10.1103/PhysRevA.90.042305 . Y. Kuno, K. Kasamatsu, Y. Takahashi, I. Ichinose,and T. Matsui, New Journal of Physics , 063005(2015), URL https://doi.org/10.1088%2F1367-2630%2F17%2F6%2F063005 . Y. Kuno, S. Sakane, K. Kasamatsu, I. Ichinose, andT. Matsui, Phys. Rev. D , 094507 (2017), URL https://link.aps.org/doi/10.1103/PhysRevD.95.094507 . A. S. Dehkharghani, E. Rico, N. T. Zinner, and A. Ne-gretti, Phys. Rev. A , 043611 (2017), URL https://link.aps.org/doi/10.1103/PhysRevA.96.043611 . J. C. P. Barros, M. Burrello, and A. Trombettoni, ArXive-prints (2019), 1911.06022, URL https://arxiv.org/abs/1911.06022 . C. Schweizer, F. Grusdt, M. Berngruber, L. Barbiero,E. Demler, N. Goldman, I. Bloch, and M. Aidelsburger,Nature Physics , 1168 (2019), URL https://doi.org/10.1038/s41567-019-0649-7 . J. C. Halimeh and P. Hauke, Phys. Rev. Lett. ,030503 (2020), URL https://link.aps.org/doi/10.1103/PhysRevLett.125.030503 . B. Yang, H. Sun, R. Ott, H.-Y. Wang, T. V. Zache, J. C.Halimeh, Z.-S. Yuan, P. Hauke, and J.-W. Pan, ArXive-prints (2020), 2003.08945, URL https://arxiv.org/abs/2003.08945 . J. C. Halimeh, R. Ott, I. P. McCulloch, B. Yang, andP. Hauke (2020), 2005.10249, URL https://arxiv.org/abs/2005.10249 . J. C. Halimeh, H. Lang, J. Mildenberger, Z. Jiang, andP. Hauke (2020), 2007.00668, URL https://arxiv.org/abs/2007.00668 . S. V. Mathis, G. Mazzola, and I. Tavernelli (2020),2005.10271, URL https://arxiv.org/abs/2005.10271 . H. Lamm, S. Lawrence, and Y. Yamauchi (2020),2005.12688, URL https://arxiv.org/abs/2005.12688 . M. C. Tran, Y. Su, D. Carney, and J. M. Taylor (2020),2006.16248, URL https://arxiv.org/abs/2006.16248 . J. C. Halimeh, V. Kasper, and P. Hauke (2020),2009.07848, URL https://arxiv.org/abs/2009.07848 . Note that continuous symmetries have Lie-algebra genera-tors, but group theory does define generators for discretesymmetries as well, and these can be connected to Liealgebras in certain cases.
In this work, the term “gen-erator” is used to denote the operator of the quantityconserved under the associated symmetry. P. Hauke and M. Heyl, Phys. Rev. B , 134204 (2015),URL https://link.aps.org/doi/10.1103/PhysRevB.92.134204 . D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Rev.Mod. Phys. , 021001 (2019), URL https://link.aps.org/doi/10.1103/RevModPhys.91.021001 . P. Hauke, M. Heyl, L. Tagliacozzo, and P. Zoller, NaturePhysics , 778 (2016), URL https://doi.org/10.1038/nphys3700 . R. C. de Almeida and P. Hauke (2020), 2005.03049, URL https://arxiv.org/abs/2005.03049 . T. D. K¨uhner and H. Monien, Phys. Rev. B ,R14741 (1998), URL https://link.aps.org/doi/10.1103/PhysRevB.58.R14741 . T. D. K¨uhner, S. R. White, and H. Monien, Phys. Rev. B , 12474 (2000), URL https://link.aps.org/doi/10.1103/PhysRevB.61.12474 . O. Dutta, M. Gajda, P. Hauke, M. Lewenstein, D.-S. Lhmann, B. A. Malomed, T. Sowi´nski, and J. Za-krzewski, Reports on Progress in Physics , 066001(2015), URL https://doi.org/10.1088%2F0034-4885%2F78%2F6%2F066001 . C. Kollath, G. Roux, G. Biroli, and A. M. Luchli, Jour-nal of Statistical Mechanics: Theory and Experiment , P08011 (2010), URL https://doi.org/10.1088%2F1742-5468%2F2010%2F08%2Fp08011 . H. P. Breuer and F. Petruccione,The Theory of Open Quantum Systems (Oxford Uni-versity Press, 2002), ISBN 9780198520634, URL https://books.google.de/books?id=0Yx5VzaMYm8C . D. Manzano, AIP Advances , 025106 (2020),https://doi.org/10.1063/1.5115323, URL https://doi.org/10.1063/1.5115323 . J. Baum, M. Munowitz, A. N. Garroway, and A. Pines,The Journal of Chemical Physics , 2015 (1985),https://doi.org/10.1063/1.449344, URL https://doi.org/10.1063/1.449344 . J. Baum and A. Pines, Journal of the American ChemicalSociety , 7447 (1986), URL https://pubs.acs.org/doi/abs/10.1021/ja00284a001 . C. M. S´anchez, R. H. Acosta, P. R. Levstein, H. M.Pastawski, and A. K. Chattah, Phys. Rev. A ,042122 (2014), URL https://link.aps.org/doi/10.1103/PhysRevA.90.042122 . G. A. ´Alvarez and D. Suter, Phys. Rev. Lett. ,230403 (2010), URL https://link.aps.org/doi/10.1103/PhysRevLett.104.230403 . G. A. ´Alvarez, D. Suter, and R. Kaiser, Science ,846 (2015), URL https://science.sciencemag.org/content/349/6250/846 . M. G¨arttner, P. Hauke, and A. M. Rey, Phys. Rev. Lett. , 040402 (2018), URL https://link.aps.org/doi/10.1103/PhysRevLett.120.040402 . R. J. Lewis-Swan, S. R. Muleady, and A. M. Rey (2020),2006.01313.
M. G¨arttner, J. G. Bohnet, A. Safavi-Naini, M. L. Wall,J. J. Bollinger, and A. M. Rey, Nature Physics , 781(2017), URL https://doi.org/10.1038/nphys4119 . E. Zohar, A. Farace, B. Reznik, and J. I. Cirac, Phys.Rev. Lett. , 070501 (2017), URL https://link.aps.org/doi/10.1103/PhysRevLett.118.070501 . L. Barbiero, C. Schweizer, M. Aidelsburger, E. Dem-ler, N. Goldman, and F. Grusdt, Science Advances (2019), URL https://advances.sciencemag.org/content/5/10/eaav7444 . U. Borla, R. Verresen, F. Grusdt, and S. Moroz, Phys.Rev. Lett. , 120503 (2020), URL https://link.aps.org/doi/10.1103/PhysRevLett.124.120503 . C. Kokail, C. Maier, R. van Bijnen, T. Brydges, M. K.Joshi, P. Jurcevic, C. A. Muschik, P. Silvi, R. Blatt, C. F.Roos, et al., Nature , 355 (2019), URL https://doi. org/10.1038/s41586-019-1177-4 . U.-J. Wiese, Annalen der Physik , 777 (2013),https://onlinelibrary.wiley.com/doi/pdf/10.1002/andp.201300104,URL https://onlinelibrary.wiley.com/doi/abs/10.1002/andp.201300104 . D. Yang, G. S. Giri, M. Johanning, C. Wunder-lich, P. Zoller, and P. Hauke, Phys. Rev. A ,052321 (2016), URL https://link.aps.org/doi/10.1103/PhysRevA.94.052321 . A. Mil, T. V. Zache, A. Hegde, A. Xia, R. P. Bhatt, M. K.Oberthaler, P. Hauke, J. Berges, and F. Jendrzejewski,Science , 1128 LP (2020), URL http://science.sciencemag.org/content/367/6482/1128.abstract . A. Altland and B. Simons,Condensed Matter Field Theory, Cambridge booksonline (Cambridge University Press, 2010), ISBN9780521769754, URL https://books.google.de/books?id=GpF0Pgo8CqAC . J. C. Halimeh et al. (in preparation).
J. Johansson, P. Nation, and F. Nori, ComputerPhysics Communications , 1760 (2012), URL . J. Johansson, P. Nation, and F. Nori, ComputerPhysics Communications , 1234 (2013), URL . T. F. Havel, Journal of Mathe-matical Physics , 534 (2003),https://aip.scitation.org/doi/pdf/10.1063/1.1518555,URL https://aip.scitation.org/doi/abs/10.1063/1.1518555 . K. H. Mariwalla, Journal of Mathematical Physics , 114(1966), https://doi.org/10.1063/1.1704797, URL https://doi.org/10.1063/1.1704797https://doi.org/10.1063/1.1704797