Topological Quantum Many-Body Scars in Quantum Dimer Models on the Kagome Lattice

Julia Wildeboer,1 Alexander Seidel,2, ∗ N. S. Srivatsa,3 Anne E. B. Nielsen,3, † and Onur Erten1

1Department of Physics, Arizona State University, Tempe, Arizona 85287-1504, USA2Department of Physics, Washington University, St. Louis, Missouri 63130, USA

3Max-Planck-Institute for the Physics of Complex Systems, D-01187 Dresden, Germany

We utilize a general strategy to turn classes of frustration free lattice models into similar classes containingquantum many-body scars within the bulk of their spectrum while preserving much or all of the original sym-metry. We apply this strategy to a well-known class of quantum dimer models on the kagome lattice with alarge parameter space. We discuss that the properties of the resulting scar state(s), including entanglement en-tropy, are analytically accessible. Settling on a particular representative within this class of models retaining fulltranslational symmetry, we present numerical exact diagonalization studies on lattices of up to 60 sites, givingevidence that non-scar states conform to the eigenstate thermalization hypothesis. We demonstrate that bulkenergies surrounding the scar are distributed according to the Gaussian ensemble expected of their respectivesymmetry sector. We further contrast entanglement properties of the scar state with that of all other eigenstates.

Properties of strongly interacting quantum systems awayfrom equilibrium are attracting a lot of attention in contempo-rary condensed matter theory. Progress in experiments [1–5]now allows for the preparation and study of quantum many-body systems that are well isolated from the environmentthereby giving access to non-equilibrium phenomena. Onesuch phenomenon is given by the so-called quantum-manybody scar states that were recently identified to be responsi-ble for the unusual dynamics unexpectedly observed in one-dimensional Rydberg atom systems [1, 6–8].

Progress concerning theoretical studies poses an interestingand challenging task since the widely employed statistical me-chanics tools fail to capture and describe relevant properties inout-of-equilibrium systems, e.g., the concept of the eigenstatethermalization hypothesis (ETH) breaks down. Moreover, theETH [9–13] postulates that generic closed quantum many-body systems exhibit ergodicity. Nowadays it is widely knownthat there are several important exceptions to this paradigmincluding but not limited to strong ergodicity breaking many-body localized states [14–16] and so-called weak ergodicitybreaking quantum many-body states [6, 7, 17–25], where onlya finite number of eigenstates, the so-called scar states, breakergodicity while the majority of states respects the ETH.

In this Letter, we focus on the latter case. Multiple possi-ble scenarios are being investigated in the current literature.Progress has predominantly been made in one-dimensionalsystems such as the celebrated PXP-model [6, 7, 19–22, 25–28] realized in the Rydberg atoms experiment [1]. Furtheradvances were made by analytically constructing scar eigen-states [17, 18] in Affleck-Kennedy-Lieb-Tasaki (AKLT) spinchains [29] and in the fractional quantum Hall thin-torus limit[28]. Recently a few 2D systems have come under investiga-tion [23, 24, 30]. The literature currently offers several pos-sible scenarios with respect to the mechanism giving rise tothe quantum scars phenomenon, ranging from proximity tointegrability [20], “embedded” SU(2) dynamics [19, 31] andmagnon condensation [25]. At present, there seems to be ascarcity of models on two-dimensional lattices with transla-tional invariance and isolated quantum many-body scars thatare numerically well-documented in terms of level statistics

2

(b)(a)

FIG. 1. (a) The three-site unit-cell (shaded region) of the kagomelattice. (b) A possible dimer covering and the analogous arrow rep-resentation first introduced by Zeng and Elser [32]: The number ofincoming arrows at each triangle must be even (0 or 2), and dimersare associated to links between two incoming arrows.

and entanglement entropy of the scar state and surroundingeigenstates. Indeed, numerical studies are often limited bythe size of the configuration space involved, particularly so inhigher dimensions. In the present work, we examine a simplestrategy to introduce an analytically known scar state givenany class of frustration free Hamiltonians, of which there aremany examples in the literature. Given this, we focus on quan-tum dimer models for their relatively moderate (though stillexponential) scaling between system size and Hilbert spacedimension. This is particularly true on the corner-sharingkagome lattice we will work on. This turns out to have the ad-ditional advantage of giving analytic access to entanglementproperties of the scar state itself that are typically beyond an-alytic reach. Our main results are as follows: (i) Following ageneral strategy, we construct a class of quantum dimer mod-els on the kagome lattice containing quantum many-body scarstates in their spectrum that provably violate the ETH, havingsub-volume entanglement. (ii) Making use of the favorableHilbert-space scaling of kagome dimer models, we numeri-cally demonstrate that the remaining states in the spectrumthermalize by analysing their level statistics and entanglemententropy.

Quantum dimer models. — Rokhsar and Kivelson intro-duced quantum dimer models (QDMs) [33] for the sake ofcapturing the essential topological features of the short-rangedvariety of Anderson’s resonating valence bond states in amodel that is tractable. Originally designed to advance the

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∣∣∣∣∣ pppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppprr r r rr rr r r rrr r

r rrr ∣∣∣∣∣ pppppppppppppppppppppppppppppppppp

pppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppprr r r rr rr r r rrr

rrr ∣∣∣∣∣ pppppppppppppppppppppppppppppppppp

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rr ∣∣∣∣∣ pppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppprr r r rr rr r r rrrr ∣∣∣∣∣ pppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppprr r r rr rr r r rrrr r

r ∣∣∣∣∣ pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppp pppppppppppppppppppprr r r rr rr r r rr

rr ∣∣∣∣∣ pppppppppppppppppppppppppppppppppp

pppppppppppppppppppppppppp

ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppprr r r rr rr r r rrr r ∣∣∣∣∣ pppppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppprr r r rr rr r r rr

.

.

nloop type Ain A1 A2 A3 A4 A5 A6 A7 A8

# of links 6 8 8 10 8 10 10 12rrot 1 3 6 3 6 6 6 1

TABLE I. A list of all eight possible loops surrounding the central hexagon within a star-shaped cell, up to rotational symmetry. Thetotal number of rotationally related loops rrot is shown for completeness (e.g., rrot = 1 for the hexagon and the star and rrot > 1 for theother shapes). Each dimerization realizes exactly one of the resulting 32 loops, where the links of the loop alternate between occupied andunoccupied, yielding two possible realizations via dimers for each loop. The hexagons shown in Fig. 1(b) are surrounded by loops of type A4

and A8, respectively.

understanding of high-temperature superconductors, quantumdimer models have played an increasing role in describingnew and unusual emergent phenomena in many-body systems[34–38]. This includes, in particular, studies on many-bodylocalization in constrained systems [16]. We now proceed bysummarizing some key features of the quantum dimer modelon the kagome lattice introduced by Misguich et al. [36], be-fore introducing a variant of this model that displays quantummany-body scars in its spectrum.

The QDM is defined on a Hilbert space of distinct orthonor-mal states that represent the allowed hard-core dimer cover-

ings of the lattice such that each site participates in exactlyone dimer between nearest neighbors. The Hamiltonian isthen defined by local matrix elements between dimer states,where we distinguish “potential” terms, V , that are diagonalin the dimer basis and associate an interaction energy with var-ious local arrangements of dimers, and “kinetic” terms, t, thatfacilitate a local rearrangement of a small number of dimers.This Letter solely focuses on the kagome lattice where all lo-cal interactions take place within twelve-site star-shaped cells,Table I.

Graphically, the Hamiltonian is presented as:

H =∑

A

[−t1

(∣∣∣∣ pppppppppppppppppppp ppppppppppqq q q qq qq q q qqq qq qqq ⟩⟨ pppppppppppppppppppp ppppppppppqq q q qq qq q q qqq qq qq

q ∣∣∣∣+

∣∣∣∣ pppppppppppppppppppp ppppppppppqq q q qq qq q q qqq qq qqq ⟩⟨ pppppppppppppppppppp ppppppppppqq q q qq qq q q qqq qq qq

q ∣∣∣∣)+ V1

(∣∣∣∣ pppppppppppppppppppp ppppppppppqq q q qq qq q q qqq qq qqq ⟩⟨ pppppppppppppppppppp ppppppppppqq q q qq qq q q qqq qq qq

q ∣∣∣∣+

∣∣∣∣ pppppppppppppppppppp ppppppppppqq q q qq qq q q qqq qq qqq ⟩⟨ pppppppppppppppppppp ppppppppppqq q q qq qq q q qqq qq qq

q ∣∣∣∣)+∑rrot

[− t2

(∣∣∣∣ pppppppppp pppppppppppppppppppp ppppppppppqq q q qq qq q q qqqqq q⟩⟨ pppppppppppppppppppppppppppppp ppppppppppqq q q qq qq q q qq

qqq q∣∣∣∣+

∣∣∣∣ pppppppppppppppppppppppppppppp ppppppppppqq q q qq qq q q qqqqq q⟩⟨ pppppppppp pppppppppppppppppppp ppppppppppqq q q qq qq q q qq

qqq q∣∣∣∣)+ V2

(∣∣∣∣ pppppppppp pppppppppppppppppppp ppppppppppqq q q qq qq q q qqqqq q⟩⟨ pppppppppp pppppppppppppppppppp ppppppppppqq q q qq qq q q qq

qqq q∣∣∣∣+

∣∣∣∣ pppppppppppppppppppppppppppppp ppppppppppqq q q qq qq q q qqqqq q⟩⟨ pppppppppppppppppppppppppppppp ppppppppppqq q q qq qq q q qq

qqq q∣∣∣∣)

]

+ . . .

(. . .

)∑rrot

−t8(∣∣∣∣ pppppppppp pppppppppppppppppppppppppppppp pppppppppp ppppppppppqq q q qq qq q q qq ⟩⟨ pppppppppppppppppppp pppppppppppppppppppp pppppppppp pppppppp

ppqq q q qq qq q q qq ∣∣∣∣+

∣∣∣∣ pppppppppppppppppppp pppppppppppppppppppp pppppppppp ppppppppppqq q q qq qq q q qq ⟩⟨ pppppppppp pppppppppppppppppppppppppppppp pppppppppp ppppppppppqq q q qq qq q q qq ∣∣∣∣)+ V8

(∣∣∣∣ pppppppppp pppppppppppppppppppppppppppppp pppppppppp ppppppppppqq q q qq qq q q qq ⟩⟨ pppppppppp pppppppppppppppppppppppppppppp pppppppppp ppppppppppqq q q qq qq q q qq ∣∣∣∣+

∣∣∣∣ pppppppppppppppppppp pppppppppppppppppppp pppppppppp ppppppppppqq q q qq qq q q qq ⟩⟨ pppppppppppppppppppp pppppppppppppppppppp pppppppppp pppppppp

ppqq q q qq qq q q qq ∣∣∣∣)

]. (1)

Magenta bonds indicate occupancy by dimers. In the above,we sum over all 12-site star plaquettes of the lattice. All ki-netic terms execute “resonance moves” along one of 32 loopscontained within the star, such that occupied links alternatealong the loop, and the move changes the occupancy alongthe loop (cf. Table I). It is easy to see that each dimer cover-ing results in precisely one such move being possible per star[36]. The potential terms associate an energy with the associ-ated loop.

For toroidal topology, i.e., periodic boundary conditions(PBCs), dimer configurations can be classified according to

winding numbersWx andWy . Dimer configurations with dif-ferent winding numbers are thought of as belonging to differ-ent topological sectors and cannot be connected by local res-onance moves of dimers. To determine the winding numberWx (Wy) one considers a horizontal (vertical) line around thetorus which intersects the links. Wx (Wy) is then the parity ofthe number of dimers intersected.

The special choice t1 = . . . = t8 = V1 = . . . = V8 > 0is an instance of a Rokhsar-Kivelson (RK) point. Here, theground state is the equal amplitude superposition of all ad-

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Thick2

FIG. 2. Labeling convention for the links of the 12-site star-shapedcells of the kagome lattice. The 12-site kagome star consists of atotal of 18 links. Internal links that make up the central hexagon arelabeled by link indices from 1 to 6, while links 7, . . . , 18 refer toexternal links. Internal (external) links are associated with internal(external) angles. We show the internal angles φ1 = π/6 and φ2 =π/2 associated with the links 1 and 2, respectively. External anglesare also illustrated and differ from adjacent internal ones by ± π

12,

e.g. φ7 = φ1 − π12

and φ8 = φ1 +π12

.

missable dimer coverings

|Ψ〉 =∑D

|D〉 , (2)

where, for PBCs the sum may be restricted to one topologicalsector, thus leading to a four-fold ground state degeneracy. Onthe kagome lattice, this RK-point lies in the interior of a Z2

topological phase [36, 39, 40] and is fully integrable [36], ow-ing to the fact that the sums of the operators in (1) associatedto any given star will commute for different stars. Further-more, for the kagome lattice, the entanglement entropy of thestates (2) can be analytically calculated and shown to displayarea law entanglement entropy [41].

The scar kagome dimer model. — The goal of this Letter isto design a system made of dimer degrees of freedom on thekagome lattice that admits quantum many-body scar states inits spectrum. We begin by observing that the states (2) areannihilated by the Hamiltonian (1) not only at the special in-tegrable point ti = Vi = 1, but whenever ti = Vi. This isso because each local term associated with ti = Vi annihi-lates Eq. (2). Moving away from the integrable point whilepreserving ti = Vi destroys the integrability (all eigenstatesexcept Eq. (2) will not be known analytically), but preservesthe fact that Eq. (2) is an exact zero energy mode. For posi-tive ti = Vi, all associated local operators thus have a commonground state in Eq. (2). This is then also the ground state ofH , the latter being the sum of these local operators. It is thencommon to call H a frustration free Hamiltonian.

The following strategy is expected to work generally forfrustration free Hamiltonians (though not always while pre-serving all symmetries): We introduce ti = Vi ≡ αi, and

choose the αi different and not all of the same sign. Eq. (2)is still a zero mode of the resulting Hamiltonian, but it is nota ground state, but rather a state somewhere in the middle ofthe spectrum. We establish that this state is a true quantummany-body scar by observing the following properties. First,the state itself satisfies area-law entanglement, despite beinghighly excited. This is usually inferred from the fact that itis the ground state of some local Hamiltonian, and it is ana-lytically provable for the kagome lattice state (2) consideredhere. We further show numerically that the surrounding statesin the energy spectrum behave “generically”, i.e., have muchlarger entanglement entropy (expected to be volume law in thethermodynamic limit), and satisfy the expected level statisticsappropriate to the respective symmetry sector they lie in. Thisin particular means that the Hamiltonian is not “special” in thesense of integrability.

Explicitly, we introduce a scar dimer model Hamiltonian asfollows

Hscar =∑

A

32∑l=1

αl(|Dl〉 −

∣∣Dl

⟩) (〈Dl| −

⟨Dl

∣∣) . (3)

The sums in (3) go over all 12-site kagome stars and overall 32 loop coverings. Dl and Dl represent the dimeriza-tions associated with loop l. We could easily follow the strat-egy described above while preserving all lattice symmetries.However, the level statistics we are interested in make senseonly within symmetry sectors, as there is no level repulsionrule between different sectors. To avoid an over-abundance ofsymmetry sectors, we preserve only translational symmetryby choosing

αl = C +∑

l′∈loop

sin (5 · φl′ + δ) . (4)

which simulates the influence of a substrate with 5-fold rota-tional symmetry. The φ′is are angles associated with each linkthat the respective loop covers. They are defined to be

φj =

(2j − 1)π6 for j = 1, . . . , 6

φ j−52− π

12 for j = 7, 9 . . . , 17

φ j−62

+ π12 for j = 8, 10 . . . , 18 ,

(5)

see Figure 2. We choose C = −0.05 to make dimer-loopswith inversion symmetry contribute, and δ = 0.1 to renderthe mirror axes of the “substrate” different from those of thelattice.

Level statistics. — While the scar state of the model andits properties are analytically under control, we proceed bynumerically investigating the genericity of its other levels.We focus on (translational) symmetry sectors with time re-versal symmetry, which contain the scar state. Fig 3 showsthe distribution of energy eigenvalues for a 60 site kagomelattice with PBCs, within the zero-momentum sector that hasthe scar state located roughly in the middle of the spectrum(see Fig 4). Here, we work within the (Wx,Wy) = (0, 0)topological sector and use an unfolding technique to bin the

4Srivatsa’

s plot

FIG. 3. Distribution of energy levels for time reversal invariant zero-momentum, (Wx,Wy) = (0, 0) topological sector, which contains ascar state (2). The inset shows the 60-site kagome lattice used in thecalculation. An unfolding technique using 4378 groups containing12 energies each has been used for binning the data (cf., e.g., Ref.[42]). The resulting data closely resemble a GOE distribution (solidcurve), indicating that almost all states thermalize.

data (cf., e.g., [42]). One observes that the distribution iswell described by the Gaussian orthogonal ensemble (GOE),as expected for generic real matrices. This can be quanti-fied further as follows [43, 44]. Introducing level spacingssn = En − En−1, one defines quantities rn = sn/sn−1and rn = min(rn, 1/rn). With this, we find the average ofthe quantities rn over the energy slice described, and overall symmetry-inequivalent time-reversal invariant momentumsectors within the (Wx,Wy) = (0, 0) topological sector tobe 〈r〉 = 0.5333. This is quite close to the exact value ofrGOE = 0.5359 [12], and markedly different from the cor-responding value rPoisson = 0.3863 [43] for the Poisson dis-tribution. The average rn value tends to require larger sam-ples owing to the possibility of small denominators, but at〈r〉 = 1.7626 is likewise very close to the exact value ofrGOE = 1.7781. Had we at least retained inversion symme-try, all symmetry sectors would be described by real matri-ces, and one would expect to find similar values in all sec-tors. However, inversion being absent, there are time-reversalnon-invariant momentum sectors in this model, not contain-ing the scar state (2), which, for sufficiently generic models,can be expected to be described by the Gaussian unitary en-semble. To test this, we carried out the analogous analysis forthese sectors, finding r = 0.5996 and r = 1.3709, again veryclose to the exact values rGUE = 0.60266 and rGUE = 1.3607.These findings lend strong support to the hypothesis that themajority of the high energy states in the spectrum of Hamilto-nian (3) are ergodic, i.e., they thermalize.

Entanglement entropy. — To complement the above find-ings, we calculate bipartite entanglement entropy for all statesof the scar-containing symmetry sector (fixing also the topo-

FIG. 4. The von Neumann entanglement entropy for all states withinthe zero momentum sector of topological winding numbers (0, 0)for a 48 site kagome lattice, bi-partitioned into two 24-site “ribbons”(inset). The scar state has SvN = 7 and is marked by a blue star.Thermalizing eigenstates of similar energy are well separated andhave SvN ≈ 10.

logical sector) for a 48 site kagome lattice with PBCs. By theirdefinition, quantum many-body scar states belong to the bulkof the spectrum while simultaneously violating the ETH, i.e.,they fail to thermalize and display low (sub-volume) entan-glement behavior. In contrast, generic high-energy states dothermalize and exhibit a volume-dependent entanglement be-havior. We find that this contrast is starkly displayed alreadyon the 48 site lattice, which we cut into two 24 site ribbonswrapping around the torus (Fig. 4, inset). For simplicity, indoing so we regard the arrows of the Zeng-Elser representa-tion of permissible dimerizations of the kagome lattice as thephysical local degrees of freedom (Fig. 1(b)). For the rib-bon described, whose boundary passes eight unit cells on eachside, and in the presence of the topological sector constraint,one may show that the (base 2) von Neumann entanglemententropy, SvN = −

∑w w log2 w, equals 7. Here, the sum goes

over the eigenvalues of the local density matrix of the ribbon.Fig. 4 clearly shows that the scar state (blue star) is isolatedfrom the rest of the spectrum (purple dots) in terms of its muchlower entanglement as compared to surrounding bulk energyeigenstates. This establishes the state (2) as a bona fide quan-tum many-body scar.

Conclusion. — We investigated a general approach to turn-ing classes of frustration free lattice Hamiltonians into onescontaining isolated quantum many-body scars in their spec-trum while retaining most or all symmetries. In addition,the introduction of disorder is straightforward. We appliedthis strategy to a two-dimensional quantum dimer model onthe kagome lattice, retaining full translational symmetry. Wedemonstrated that this model contains an exactly known quan-tum many-body scar with analytically accessible entangle-ment properties. We established that the remainder of the

5

eigenstates and energy spectrum exhibit no “fine-tuned” be-havior. Specifically, for a 60-site kagome lattice, we showedthat bulk energies conform to the Gaussian ensembles ex-pected for their respective symmetry sectors, and we cal-culated von Neumann entanglement entropies for all stateswithin the scar-sector of a 48 site kagome lattice, exposingthe scar’s isolated character. Interesting questions for the fu-ture include the scar’s fate under generic perturbations. Due totheir quality of being numerically manageable on fairly large-size lattices, quantum dimer models of the type introducedhere should become a fertile playground for investigations ofthis kind. Moreover, the original, frustration-free quantumdimer model stabilizes a Z2 topological phase. This lends atopological character to our scar states, whereas the nature ofthe ground states of our “scarred” quantum dimer models iscurrently unknown. This is true for the particular instance ofthe model that we studied closely, and even more so withinthe associated large parameter-space of αi-couplings. We arehopeful that these interesting questions will stipulate furtherinvestigations.

AS is indebted to Z. Nussinov for insightful discussions.OE acknowledges support from NSF-DMR-1904716.

∗ Corresponding author: seidel@physics.wustl.edu† On leave from Department of Physics and Astronomy, Aarhus

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