Quantum lozenge tiling and entanglement phase transition

2025-04-29 0 0 4.85MB 13 页 10玖币

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Zhao Zhang1,2 and Israel Klich3

1Department of Physics, University of Oslo, P.O. Box 1048 Blindern, N-0316 Oslo, Norway

2SISSA and INFN, Sezione di Trieste, via Bonomea 265, I-34136, Trieste, Italy

3Department of Physics, University of Virginia, Charlottesville, VA, USA

While volume violation of area law has

been exhibited in several quantum spin

chains, the construction of a correspond-

ing ground state in higher dimensions, en-

tangled in more than one direction, has

been an open problem. Here we con-

struct a 2D frustration-free Hamiltonian

with maximal violation of the area law.

We do so by building a quantum model

of random surfaces with color degree of

freedom that can be viewed as a collec-

tion of colored Dyck paths. The Hamil-

tonian may be viewed as a 2D generaliza-

tion of the Fredkin spin chain. It relates

all the colored random surface conﬁgura-

tions subject to a Dirichlet boundary con-

dition and hard wall constraint from be-

low to one another, and the ground state

is therefore a superposition of all such clas-

sical states and non-degenerate. Its en-

tanglement entropy between subsystems

undergoes a quantum phase transition as

the deformation parameter is tuned. The

area- and volume-law phases are similar

to the one-dimensional model, while the

critical point scales with the linear size

of the system Las Llog L. Further it is

conjectured that similar models with en-

tanglement phase transitions can be built

in higher dimensions with even softer area

law violations at the critical point.

1 Introduction

Entanglement entropy (EE) and its scaling has

been a central theme of quantum many-body

physics, not only because entanglement is a

unique feature in the quantum world by itself,

but also for their crucial role in determining the

Zhao Zhang: zhaoz@uio.no

computational complexity of the numerical sim-

ulations of quantum many-body systems, indi-

cation of topological order and understanding

of the holographic principle and black hole en-

tropy. While EE of a generic eigenstate in the

Hilbert space is shown to scale with the systems

size [1], EE of the ground states of gapped lo-

cal Hamiltonians are generally observed to obey

the so-called area law, scaling with the size of the

boundary. A milestone in the study of area-law

has been Hastings’ rigorous proof of the result

in one-dimensional systems [2]. Recently, a sim-

ilar result in two-dimension has been proven for

frustration-free models [3]. While area-law has

been ubiquitous in gapped systems, plenty of ex-

amples of area-law violation has also been found

in various gapless systems. (1+1)-dimensional

critical system described by a conformal ﬁeld the-

ory has EE of logarithmic scaling [4]. Entan-

glement entropy of free fermions with a Fermi

sea in dimension dscales as Ld−1log L[5]. In

one dimension, lattice models with even more se-

vere violation has also been found in a class of

frustration-free (FF) models dubbed Motzkin and

Fredkin spin chains with up to volume-law scaling

of EE [6,7,8,9,10,11,12]. Despite the plethora

of 1D lattice models with such violations, their

2D counterparts are yet to be discovered. In this

manuscript, we give the ﬁrst successful example

of such a model with exotic entanglement scaling

on a two-dimensional lattice.

A crucial ingredient of the Motzkin and Fred-

kin models is that, by employing next-nearest-

neighbor interactions and boundary conditions,

they allow a well-deﬁned height representation

that carries long-range entanglement which local

spins could not. The ﬁrst diﬃculty to general-

ize this to two dimension is to make sure the

height function does not give rise to any ambi-

guity when counting around a closed loop in the

lattice. This problem is intrinsically avoided in

Accepted in Quantum 2024-10-08, click title to verify. Published under CC-BY 4.0. 1

arXiv:2210.01098v3 [quant-ph] 8 Oct 2024

the U(1) Coulomb gas phase, naturally emerged

in constrained Hilbert space of fully packed dimer

or loop covering models [13,14,15]. In Ref. [15],

a ﬁrst attempt was made to build two models

using fully packed loops on a square lattice, but

the entanglement entropy computed there obeys

area law, as the ground state is a projected en-

tangled pair state contracted by the local con-

straints in Hilbert space. In the combinatorics

and classical statistical mechanics literature, the

fully packed dimer conﬁgurations of a honey-

comb lattice are mapped to random tilings of

lozenges, and are extensively studied in the con-

text of limit shape behavior and arctic curves [16],

and are recently used to study two dimensional

Kardar–Parisi–Zhang (KPZ) growth [17]. Moti-

vated by the resemblance between the Fredkin

spin chains and the models of asymmetric simple

exclusion process (ASEP), it is promising that a

quantum version of lozenge tiling could lead to

exotic scalings of the height function that facil-

itate area violation of EE. A 2D generalization

to the Fredkin moves between lozenge tilings was

ﬁrst proposed in Ref. [18] as a classical stochastic

model. In this manuscript, we incorporate those

dynamics into a quantum Hamiltonian with pro-

jection operators that annihilate a ground state

of uniform superposition of lozenge tilings. In

principle, one could compute the bipartite entan-

glement entropy of such a quantum model us-

ing the information of the limit shape across the

interface between the two subsystems and the

ﬂuctuation around it to compute the entangle-

ment entropy scaling in the thermodynamic limit.

Moreover, one can also perform q-deformations

to the Hamiltonian and obtain entanglement en-

tropy of q-weighted superposition from results in

q-enumerations of lozenge tilings [19].

A yet richer model can be deﬁned when the

local Hilbert space is enlarged by introducing a

color degree of freedom to the dimers. Two-

and multi-color dimer models has been studied

both on the square lattice [20] and honeycomb

lattice [21]. Color-code models have also been

heavily studied in the context of exactly solvable

model of topological order [22], as generalizations

of Kitaev’s toric code [23], topological defect [24],

and as a candidate for fault-tolerant quantum

computation [25]. Recently, color degrees of free-

dom have been used to generalize the extensively

entangled rainbow chain [26,27] to a quasi-1D

lattice embedded in 2D space with inhomoge-

neous coupling strength decaying from the center

[28]. For 1D spin chains, the dependency of en-

tanglement entropy scaling on the number of local

degrees of freedom has been studied with an N-

component partially integrable chain [29], which

can be viewed as a low-energy eﬀective Hamilto-

nian of a related quasi-2D spin ladder model [30].

In particular, the entanglement entropy of inte-

grable excited states was shown to decompose

into a part that scales only with the size of the

system and one that scales also with the num-

ber of the dimensionality of local Hilbert space.

Perhaps the power of color degrees of freedom in

drastically changing the scaling behavior of en-

tanglement entropy is only fully unleashed when

they are facilitated by a notion of the height vari-

able, as has been manifested in the Motzkin and

Fredkin spin chains. In both those cases and the

present model, EE also decomposes into a con-

tribution from color degrees of freedom and the

ﬂuctuating height degree of freedom. The leading

contribution comes from the entanglement among

color degrees of freedom, but its scaling behavior

is determined by the average shape of the random

surface described by the height function, which

can be deformed continuously and undergoes a

phase transition between diﬀerent scaling behav-

iors across a critical point.

The rest of the paper is organized as follows.

In Sec. 2, we deﬁne the lattice, Hilbert space and

Hamiltonian of the model of quantum lozenge

tiling, and show that its unique ground state is

a superposition of random surfaces subject to a

hard wall constraint from below. In Sec. 3, the

ground state EE is shown to decompose into con-

tributions from the color and height degrees of

freedom separately, and the former is propor-

tional to the average cross-sectional area under-

neath the random surface. Sec. 4takes the scal-

ing limit of the height function and provides intu-

itive arguments for various scaling behaviours of

EE for diﬀerent deformation parameter. Rigorous

mathematical theorems on the critical scaling be-

havior was referenced to establish the existence of

an entanglement phase transition. Sec. 5brieﬂy

discusses the analogous but quantitatively diﬀer-

ent EE scaling of ground states at critical point

of quantum tiling models in higher dimensions.

Sec. 6gives a conclusion as well as several possi-

ble future directions to pursue.

Accepted in Quantum 2024-10-08, click title to verify. Published under CC-BY 4.0. 2

2 Quantum lozenge tiling

+1 +1

-1 -1

-1

(b)

(a)

(c)

Figure 1: (a) Mapping from s2-colored dimer coverings

on honeycomb lattice to lozenge tilings with an s-colored

line along each pair of parallel sides. (b) Convention of

height change between plaquette centers and vertices of

lozenges: along the positive direction of x,y,z axes height

increases by 1. (c) The s2-coloring of dimers can be

mapped from a two-component s-coloring of lines, one

directed along each edge of the lozenge. The mapping

for dimers oriented along the other two direction can be

obtained by rotations.

Lozenge tiling can be viewed as a covering of

the faces of triangular lattice, where each tile

occupies two adjacent triangular faces and each

triangular face of the lattice is covered by one

lozenge and one lozenge only. We call this tri-

angular lattice Λ, and its dual, hexagonal lattice

Λ∗. Each face of the triangular lattice maps to

a vertex in the dual honeycomb lattice, and a

lozenge tiling therefore maps to a dimer covering

of the hexagonal lattice. Our model further al-

lows the lozenge tiles or equivalently dimers on

the hexagonal lattice to come in s2diﬀerent pos-

sible colors, as is shown in Fig. 1(a). On each

edge of the honeycomb lattice, the local degrees

of freedom are either uncovered, or covered by a

dimer in one of the s2internal states. Further-

more, the global Hilbert space is constrained by

the fully packed dimer covering condition, which

means that each vertex is covered by one dimer

and one dimer only. The constraint can be real-

ized by the Hamiltonian

H0=X

r∈Λ∗

(

i=1

nr,r+ei−1)2,(1)

where nr,r+eiis the number operator of the dimer

Figure 2: A coloring of the minimal volume tiling con-

ﬁguration of a lattice of linear size L= 20. The lattice

resides within the arctic circle of the usual lozenge tiling

of a hexagonal boundary, where the height can ﬂuctu-

ate. In the scaling limit, the boundary of the lattice

approaches the arctic circle, and has constant height 0

along the boundary, where as in the discrete case, the

height ﬂuctuates between ±1

2. The dashed line marks

the cut into the shaded and unshaded subsystems. The

three-dimensional eﬀect is understood with light shining

against the y-axis slightly tilted downward. This partic-

ular tiling is also known as the rhombille tiling in the

mathematical literature.

(b)

(a)

Figure 3: (a) A coloring of the maximal volume tiling

conﬁguration of a lattice of linear size L= 20. In the

thermodynamic limit, the surface approaches a dome

shape. (b) One slice of the x-z plane which intersects

with the maximal surface in (a) giving a colored Dyck

path. The corresponding path in (a) is thickened.

living on the edge along the eidirection at site

r. The constraint of fully packed dimers allows

a well-deﬁned height function on the dual trian-

gular lattice. We adopt the height change con-

Accepted in Quantum 2024-10-08, click title to verify. Published under CC-BY 4.0. 3

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QuantumlozengetilingandentanglementphasetransitionZhaoZhang1,2andIsraelKlich31DepartmentofPhysics,UniversityofOslo,P.O.Box1048Blindern,N-0316Oslo,Norway2SISSAandINFN,SezionediTrieste,viaBonomea265,I-34136,Trieste,Italy3DepartmentofPhysics,UniversityofVirginia,Charlottesville,VA,USAWhilevolumeviolati...

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