OOrthogonal Quantum Many-body Scars
Hongzheng Zhao, Adam Smith, Florian Mintert, and Johannes Knolle
3, 4, 1 Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom School of Physics and Astronomy, University of Nottingham,University Park, Nottingham NG7 2RD, United Kingdom Department of Physics TQM, Technische Universit¨at M¨unchen,James-Franck-Straße 1, D-85748 Garching, Germany Munich Center for Quantum Science and Technology (MCQST), 80799 Munich, Germany
Quantum many-body scars have been put forward as counterexamples to the Eigenstate Thermal-ization Hypothesis. These atypical states are observed in a range of correlated models as long-livedoscillations of local observables in quench experiments starting from selected initial states. Thelong-time memory is a manifestation of quantum non-ergodicity generally linked to a sub-extensivegeneration of entanglement entropy, the latter of which is widely used as a diagnostic for identi-fying quantum many-body scars numerically as low entanglement outliers. Here we show that, byadding kinetic constraints to a fractionalized orthogonal metal, we can construct a minimal modelwith orthogonal quantum many-body scars leading to persistent oscillations with infinite lifetimecoexisting with rapid volume-law entanglement generation. Our example provides new insights intothe link between quantum ergodicity and many-body entanglement while opening new avenues forexotic non-equilibrium dynamics in strongly correlated multi-component quantum systems.
Introduction.–
Rapid experimental progress in statepreparation and controlability, e.g., in cold-atoms [1–7],ion-traps [8, 9] or superconducting circuits [10, 11], haveenabled the exploration of non-equilibrium dynamics ofquantum many-body systems in well-isolated settings.The Eigenstate Thermalization Hypothesis (ETH) pro-vides a basic framework for understanding how these sys-tems approach thermal equilibrium under unitary time-evolution [12–14]. It states that local observables calcu-lated in generic eigenstates look thermal, i.e. only dependon the eigenenergy. Consequently, in a quantum quenchprotocol the memory of initial states is quickly lost. Thisergodicity is also manifest in the extensive (volume-law)scaling of the entanglement entropy of eigenstates as afunction of sub-system size as confirmed by numerous nu-merical studies [15]. Recently, the research focus shiftedto systems which deviate from the ETH paradigm, forinstance integrable systems [16, 17] or disordered many-body localized phases (MBL) [18–20]. These systems arecapable of retaining certain memory of initial states asobserved in quench experiments [4, 7, 11, 21–23]. Thisnon-ergodicity is also manifest in the non-extensive (area-law) entanglement scaling of MBL eigenstates [20].It came as a surprise when a Rydberg atom experimentreported the breaking of ergodicity in a clean and non-integrable system as observed via long-lived oscillationsafter quenching from specific initial states [6]. Subse-quently, it was proposed that it is the kinetic constraintswhich lead to special eigenstates dubbed quantum many-body scars [24–35]. These ETH-violating states may ap-pear throughout an otherwise ETH-obeying spectrumand perfect scarred states populate only a small isolatedsector of the entire many-body Hilbert space [36].In general, quantum many-body scars are expected toshow sub-extensive entanglement scaling, which indeed is routinely used to detected them numerically as entropy-outliers [37]. In the Rydberg experiment, simple productstates were chosen as initial states which have substantialoverlap with the scar states, leading to persistent oscilla-tions and low entanglement generation [26].Here, we address the intriguing question of whetherlong-lived coherent oscillations can coexist with rapidvolume-law entanglement generation in a standardquench-setup of a clean non-integrable quantum many-body system? The intuitive answer should be No becausewithin the ETH paradigm volume-law entanglement nor-mally goes hand-in-hand with ergodic behaviour. Simi-larly, in all known examples of persistent oscillations, theunderlying quantum many-body scars only lead to a sub-extensive entanglement [37]. However, in strongly corre-lated quantum many-body systems, ETH can be violatedin unexpected ways and we answer the above question af-firmatively by constructing a concrete counter-example.Multi-component systems have been suggested toevade thermalization via inter-species interactions, forexample in heavy-light particle mixtures [38–40]. Theoriginal idea is that even in the absence of quenched dis-order, the light particles localize on the disordered back-ground of slow heavy particles which themselves can bein thermal equilibrium. Subsequently, a general class ofnon-ergodic phases known as quantum disentangled liq-uids (QDLs) were proposed [41], in which some degreesof freedom (d.o.f.) exhibit area-law entanglement, whileothers are volume-law entangled. QDL-like behavior hasby now been identified in a range of correlated systems,e.g., in lattice gauge models [42–47], the half-filled Hub-bard model [48] and frustrated quantum magnets [49, 50],but none provide an answer to our main question.We will provide a basic construction of a QDL show-ing infinitely long-lived persistent oscillations and rapid a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b volume-law entanglement generation. Our starting pointis the Orthogonal Metal (OM) model of Ref. [51]. OMswere introduced as a particularly simple example of anon-Fermi liquid in which the original physical d.o.f. splitinto separate components, each carrying fractions of theoriginal quantum numbers and having its own dynam-ics. OMs were named orthogonal because basic spectralfunctions of the physical d.o.f. can be zero despite thepresence of a well defined Fermi sea and non-zero conduc-tivity originating from one of the components. We willshow how to combine the fractionalization mechanism ofthe OM with kinetically constrained tunneling [52–55] forrealizing orthogonal quantum many-body scars. The Model.–
Our starting point is the following one-dimensional Hamiltonian with two species of particles H = − (cid:88) i (cid:16) hσ xi − ,i σ xi,i +1 ( − c † i c i + g z σ zi,i +1 σ zi +1 ,i +2 + Jσ zi,i +1 (cid:17) − t (cid:88) i (cid:16) n i − c † i σ zi,i +1 c i +1 n i +2 + H.c. (cid:17) . (1)The c † i operators create physical spinless fermion d.o.f., atsite i , and the σ i,i +1 operators represent a spins- back-ground field positioned at the links between site i and i + 1 of a one dimensional lattice. A similar model with astandard unconstrained fermionic hopping and vanishinginteraction constant g z had been introduced as a basicsolvable example of the OM [51]. We will show that theaddition of the density dependence of hopping processesintroduces kinetic constraints [52, 53] which is key forinducing non-ergodicity.While the original d.o.f. are strongly coupled, we canshow explicitly how the different sectors emerge. First, aduality transformation [56, 57] maps the link-spins σ toa site-spin τ as τ zi = σ xi − ,i σ xi,i +1 , τ xi τ xi +1 = σ zi,i +1 , suchthat the Hamiltonian reduces to H = − (cid:88) i (cid:16) Jτ xi τ xi +1 + hτ zi ( − c † i c i + g z τ zi τ zi +2 (cid:17) − t (cid:88) i (cid:16) n i − c † i τ xi τ xi +1 c i +1 n n +2 + H.c. (cid:17) . (2)Next, we can define new composite d.o.f. for fermions f r = τ xr c r and dual spins ˜ τ zi = τ zi ( − f † i f i , ˜ τ xi = τ xi .Finally, in the new variables the Hamiltonian separatesinto two components H = H ˜ τ + H f H ˜ τ = − (cid:88) i (cid:0) J ˜ τ xi ˜ τ xi +1 + h ˜ τ zi + g z ˜ τ zi ˜ τ zi +2 (cid:1) ,H f = − t (cid:88) i (cid:16) n i − f † i f i +1 n i +2 + H.c. (cid:17) , (3)a constrained f − fermion system and a non-integrable˜ τ − spin chain. A crucial point is that the dynamics ofthe physical variables is a combination of these separatesectors. We note, that one can introduce different typesof interactions while retaining the separability. For in-stance, since the fermionic particle number n i are the Decouple
T hermal
F rozen
Active
We consider the tensor-product of c − fermions and σ − spins as | (cid:105) = | ψ (cid:105) c ⊗ | S (cid:105) σ , where | S (cid:105) σ = |↑↑ . . . ↑(cid:105) σ and | ψ (cid:105) c represents the fermionic Fock state. For sim-plicity, we focus on a chain of length L with threefermions, and the initial state involves one building block | (cid:105) c and empty sites | . . . (cid:105) c of length L −
6, i.e., | ψ (cid:105) c = | . . . (cid:105) with periodic boundary conditions.The qualitative results are unaltered for a finite fermiondensity as long as the kinetic constraints lead to isolatedacive sites, e.g. here sites 3 and 4, separated by frozensegements.Fig. 2 depicts the dynamics of local observables (or-ange) and nearest neighbor correlations (blue) for physi-cal c − fermions and σ − spins in panel (a) and (b), respec-tively. By construction, the fermionic occupation (cid:104) n i (cid:105) ex-hibits persistent coherent oscillation only for i = 3 , H f to the basis | (cid:105) f , | (cid:105) f in-
Energy
The peculiar features of orthogonal quan-
5. This is in sharp contrast toknown quantum many-body scars, which normally ex-hibit exceptionally low entropy in accordance with theirnon-ergodic dynamical behavior.
Separability breaking.–
It might seem that our resultshinge on the fine-tuned construction of Eq. (1), but they remain valid in the presence of generic perturbationswhich break the separability of the Hamiltonian into sep-arate terms H f and H ˜ τ . This can be demonstrated withthe example of the perturbation g x (cid:80) i σ xi − ,i σ xi,i +1 whichis lacking the factor ( − c † i c i as in the original modelEq. (1). As such, although the mobility of c − fermionsremains restricted to two active sites, their dynamics can-not be separated anymore from the background σ − spins.These spins can then act as a thermalizing bath for thefermions.The overlap between initial states and eigenstates |(cid:104) | E α (cid:105)| is shown in Fig. 4 (a), and the mean energy E = − . c − fermions for perturba-tion strength g x = 0 .
1. The oscillations of the fermionicoccupation is not infinitely long-lived anymore but decayswith a finite lifetime eventually saturating to a constantvalue suggesting thermalization. The same behavior alsoappears for correlation functions, which decay towardszero at long times. Thus, when perturbing away fromthe fine-tuned construction our perfect orthogonal scarsturn into orthogonal scars with long but not persistentoscillations.
Disucssion.–
We have presented a simple model of or-thogonal quantum many-body scars by combining kineticconstraints with the fractionaliztion mechanism of theorthogonal metal, which demonstrates that non-ergodicdynamics can coexist with a rapid volume-law entangle-ment entropy generation in a standard quench protocol.The key elements of our model are potentially realis-able in experiments. For example, the density-dependenttunneling can be obtained in cold atomic gases via modu-lating external magnetic fields periodically in the vicinityof Feshbach resonances [61, 62], or in the presence of dom-inant nearest-neighbor density-density interaction [54].The spin background minimally coupled to the fermionscan be realized by exploiting the tool boxes in develop-ment for lattice gauge theory simulations [1, 8, 63, 64].Our results enrich the phenomenology of quantummany body scars and of quantum disentangled liquids.It will be worthwhile to explore whether the orthogonalscars of our model can appear in other correlated multi-component systems. Similarly to the fractionalizationof the OM, which can emerge as a low energy featurecaptured by slave-particle descriptions of more genericsystems [51], also the kinetic constraints can emerge aseffective interactions. Hence, potential candidates for re-alising orthogonal quantum many-body scars are for ex-ample multi-orbital Hubbard systems and the possibilityto alter their dynamics via Floquet engineering opens an-other promising pathway.The one-dimensional example presented here can bereadily generalized to higher dimensions by combiningdensity assisted tunneling with the two-dimensional OMconstruction which is considerably richer because thebackground spin sector can be topologically ordered [51].An alternative direction for future research is to explorethe interplay between kinetic constraints and local gaugesymmetry. For example, combining the disorder-freelocalization mechanism [42–47] with kinetic constraintscould potentially lead to non-ergodic gauge field dynam-ics.In conclusion, we expect that the interplay of fraction-alization and kinetic constraints in multi-component sys-tems will lead to more surprises and unexpected non-equilibrium physics.
Acknowledgements.–
We acknowledge helpful discus-sion with Andrea Pizzi and Markus Heyl. H.Z. was sup-ported by a Doctoral-Program of the German AcademicExchange Service (DAAD) Fellowship. A.S. was sup-ported by a Research Fellowship from the Royal Commis-sion for the Exhibition of 1851. We acknowledge supportfrom the Imperial-TUM flagship partnership. [1] B. Yang, H. Sun, R. Ott, H.-Y. Wang, T. V. Zache, J. C.Halimeh, Z.-S. Yuan, P. Hauke, and J.-W. Pan, Nature , 392 (2020).[2] A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli,R. Schittko, P. M. Preiss, and M. Greiner, Science ,794 (2016).[3] S. Scherg, T. Kohlert, P. Sala, F. Pollmann, I. Bloch,M. Aidelsburger, et al. , arXiv preprint arXiv:2010.12965(2020).[4] J.-y. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio-Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch,and C. Gross, Science , 1547 (2016).[5] J. Zhang, P. Hess, A. Kyprianidis, P. Becker, A. Lee,J. Smith, G. Pagano, I.-D. Potirniche, A. C. Potter,A. 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Bukov, SciPost Phys , 97 (2019). Numerical details
All of the dynamics and the specturm are obtainedby Exact Diagonalization(ED) by Python package QuS-pin [65, 66]. As analyzed in the main content, thedensity-dependent tunneling for c − fermion significantlyrestricts its dynamics. Therefore, one can employ sucha feature to enlarge the system size simulatable by EDwith reduced fermionic basis, even when the separabilityis broken in the presence of non-vanishing g x as in Fig.4. For instance, the basis for c − fermion with three par-ticles and L sites indeed only involves two Fock states as | ψ (cid:105) c = | . . . (cid:105) and | ψ (cid:105) c = | . . . (cid:105)(cid:105)