Exact solution for the lling-induced thermalization transition in a 1D fracton system Calvin Pozderac1Steven Speck1Xiaozhou Feng1David A. Huse2and Brian Skinner1 1Department of Physics Ohio State University Columbus Ohio 43210 USA

2025-04-27 0 0 705.99KB 17 页 10玖币

侵权投诉

Exact solution for the ﬁlling-induced thermalization transition in a 1D fracton system

Calvin Pozderac,1Steven Speck,1Xiaozhou Feng,1David A. Huse,2and Brian Skinner1

1Department of Physics, Ohio State University, Columbus, Ohio 43210, USA

2Department of Physics, Princeton University, Princeton, NJ 08544, USA

(Dated: October 7, 2022)

We study a random circuit model of constrained fracton dynamics, in which particles on a one-

dimensional lattice undergo random local motion subject to both charge and dipole moment con-

servation. The conﬁguration space of this system exhibits a continuous phase transition between

a weakly fragmented (“thermalizing”) phase and a strongly fragmented (“nonthermalizing”) phase

as a function of the number density of particles. Here, by mapping to two diﬀerent problems in

combinatorics, we identify an exact solution for the critical density nc. Speciﬁcally, when evolution

proceeds by operators that act on `contiguous sites, the critical density is given by nc= 1/(`−2).

We identify the critical scaling near the transition, and we show that there is a universal value of

the correlation length exponent ν= 2. We conﬁrm our theoretical results with numeric simulations.

In the thermalizing phase the dynamical exponent is subdiﬀusive: z= 4, while at the critical point

it increases to zc&6.

I. INTRODUCTION

An isolated system with many degrees of freedom is

thermalizing if it is able to dynamically act as a bath

for all of its small subsystems and thus bring them all

to thermal equilibrium with each other. The eigenstate

thermalization hypothesis (ETH) extends these consider-

ations to speciﬁc quantum states, by asserting that when

a large thermalizing system is in an energy eigenstate,

the reduced density operator of each of its small subsys-

tems is the same as in the corresponding standard ther-

mal ensemble (see, e.g., Refs. [1,2] for reviews of ETH).

The last few decades have seen intense interest in sys-

tems and states that fail to thermalize or to obey the

ETH, and which therefore cannot be described by con-

ventional equilibrium thermodynamics even at arbitrar-

ily long times. Some prominent examples include many-

body localized states [3–9] and quantum scar states [10–

25].

The recently-identiﬁed fracton systems [26–35] pro-

vide yet another pathway by which a system can fail to

thermalize. In fracton systems, thermalization can be

avoided because of kinetic constraints on the system’s

dynamics, which prevent the system from exploring the

full set of states consistent with the conserved quanti-

ties. A now-paradigmatic example of fracton dynamics

is that of a one-dimensional system of charges for which

both the charge and dipole moment are conserved. The

dipole moment conservation ensures that a single, iso-

lated charge cannot move freely through the system, un-

less an opposite-facing dipole is simultaneously created

from the vacuum [31,36]. Recent work has shown that

when such a system evolves under local dynamics, the

conﬁguration space associated with a given symmetry

sector can become “fragmented” [37–41]. That is, the

set of all microstates that are consistent with a given

value of charge and dipole moment may separate into

many dynamically disconnected sectors which are mutu-

ally inaccessible by the dynamics. Here we refer to these

dynamically disconnected sectors as “Krylov sectors.”

Fragmentation of the symmetry sector, where it occurs,

may happen in either a weak or a strong way [37,39].

Under weak fragmentation, there is a dominant Krylov

sector that contains the vast majority of states in the

symmetry sector, such that in the limit of inﬁnite sys-

tem size the probability that a randomly-chosen state

is contained within the largest Krylov sector approaches

unity. When there is strong fragmentation, on the other

hand, even the largest Krylov sector contains a vanish-

ingly small fraction of the symmetry sector. If we as-

sume that the dynamics is ergodic within each Krylov

sector, then in the latter case no initial state is able to

thermalize, while in the case of weak fragmentation a

randomly-chosen initial state will, with a probability that

approaches unity in the thermodynamic limit, thermal-

ize. Thus, as a shorthand, throughout this paper we refer

to the transition between strong and weak fragmentation

as the “thermalization transition”.

Initial work on the thermalization transition in fracton

systems focused on the eﬀect of varying the spatial range

`of the operators governing dynamical evolution, or on

varying the size qof the local Hilbert space at each site

[42–46]. When `or qis large enough, the system ther-

malizes under either random dynamics or certain types

of Hamiltonian dynamics, while small `and qprevents

thermalization. In a recent paper, however, Morningstar

et. al. [47] showed that the thermalization transition may

also be eﬀected by changing the ﬁlling of the system for

ﬁxed `and q. Here, as an example, we focus on the

case of a one-dimensional lattice of sites for which the

charge at each site can be any non-negative integer. If

the average ﬁlling nof the lattice satisﬁes n1, then a

typical state consists of rare charges that are well sepa-

rated from each other. Since isolated charges are unable

to move while satisfying the dipole moment constraint

(and since negative values of the charge are forbidden),

this system is unable to evolve under the action of lo-

cal operators, and it fails to thermalize. On the other

hand, when n1 local operators can easily rearrange

arXiv:2210.02469v1 [cond-mat.stat-mech] 5 Oct 2022

charges locally while satisfying the dipole moment con-

straint, and the system thermalizes. Thus, varying the

ﬁlling nallows one to study the thermalization transition

in terms of a continuous variable (unlike `and q, which

are discrete), and thus to identify the critical exponents

and critical scaling associated with the transition.

In this paper, we focus on the model introduced in

the previous paragraph (which we deﬁne more precisely

below), which diﬀers slightly from that of Ref. [47], and

we study the ﬁlling-induced thermalization transition. In

addition to numeric simulations, we provide exact solu-

tions for the size of the symmetry sector and also the

size of what appears to be the largest Krylov sector in

the large-system limit. These solutions, which we obtain

by a mapping to two separate problems in combinatorics,

provide us with exact solutions for the critical ﬁlling nc

as a function of gate size `. Speciﬁcally,

nc=1

`−2.(1)

We are also able to exactly identify the correlation length

exponent ν= 2, which is universal to all models of this

type. Numerical simulations suggest a large dynamical

critical exponent zc&6, consistent with the results in

Ref. [47].

II. MODEL

We consider a “bosonic” system of Nindistinguish-

able particles moving on a 1-D lattice of size Lwith

closed boundary conditions. Each site x, with x=

0,1, ..., (L−1), has an occupation number given by a non-

negative integer, nx= 0,1,2, . . . . The system evolves by

a random sequence of `-site local gates, as illustrated in

Fig. 1. The gates are each chosen randomly from the set

of operators that conserve both the charge Nand the

dipole moment P, with N=Pxnxand P=Pxnxx.

Since the fragmentation of the Hilbert space arises from

the classical charge and dipole moment constraints, we

are able to restrict our attention to an eﬀectively classical

Markov dynamics for which each operator takes a given

charge state (a string of deﬁnite values of nx) to another

given charge state. This approach is equivalent to the the

recently-described “automaton dynamics” method [48–

51].

This restriction of the dynamics to classical charge

states implies that each operator is chosen from a small,

ﬁnite set. For example, in the case `= 3 any allowed op-

erator is a multiple of only two nontrivial actions, which

we denote U3,±. Speciﬁcally, U3,±makes the transforma-

tion {nx−1, nx, nx+1}→{nx−1±1, nx∓2, nx+1 ±1}for

some location x, as illustrated in Fig. 1. In our dynam-

ics, each operator is chosen randomly from one of these

possibilities and then applied to a random set of `con-

tiguous sites. If the operator does not produce a valid

basis state – i.e., if one of the occupation numbers would

become negative – then no operation is applied.

x=0 1 2 3 4 5 6 7 8 9 10 ...

...

L−1L−2L−3

t−→

U3,±

U3,+

U3,−

FIG. 1. An illustration of the dynamics with charge- and

dipole-conserving 3-site gates U3,±. The circuit (above) shows

the sequence of random operations, while the balls (below) il-

lustrate the occupation numbers of the state. Yellow balls

represent the starting positions of particles involved in the

ﬁrst two applied operators, blue balls represent the ﬁnal po-

sitions of these particles, and grey balls show particles that

remain in place. These two operations are the only 3-site

gates for our model.

For a given charge Nand system size L, there is some

ﬁnite number of basis states which all have the same

given dipole moment P. We refer to this set of states as

the symmetry sector. Within the symmetry sector, there

may be states which cannot be evolved into one another

through the application of only local dipole-conserving

gates of size `. For example, in the case L= 5, N= 3,

and P= 6, the two states (1,0,1,0,1) and (0,0,3,0,0)

are dynamically disconnected when `= 3 despite belong-

ing to the same symmetry sector. We refer to each subset

of the symmetry sector for which any pair of states within

the subset can be reached one from another through local

gates as a Krylov sector. The Krylov sectors are depen-

dent on the gate size `, while the symmetry sectors are

not. For example, when `= 2 all Krylov sectors contain

only a single state, since there are no nontrivial 2-site

operators that conserve both charge and dipole moment;

in the limit `=Leach symmetry sector consists of only

a single large Krylov sector, since all possible N- and

P-conserving transformations are possible.

With these deﬁnitions, we can concretely deﬁne a ther-

malized system in terms of the proportion of states within

a symmetry sector that belong to its largest Krylov

sector (LKS). Speciﬁcally, we deﬁne the quantity D=

DLKS/Dsym, where DLKS is the number of basis states

within the largest Krylov sector and Dsym is the num-

ber of basis states in the corresponding symmetry sector.

The thermalized phase is characterized by D→1 in the

limit L→ ∞ with n=N/L held ﬁxed, while the local-

ized phase exhibits instead D→0.

III. SOLUTION FOR THE CRITICAL FILLING

In this section we present results for the size Dsym

of the symmetry sector and the size DKS of a speciﬁc

Krylov sector (which, as we discuss below, is apparently

the largest Krylov sector). By considering the scaling of

Dsym and DKS with the system size L, we are able to

precisely identify the critical density ncassociated with

the thermalization transition. We restrict our attention

primarily to the symmetry sector with dipole moment

P=N(L−1)/2, whose average local charge density is

symmetric about the center of the system. Throughout

this section we focus on the smallest nontrivial gate size,

`= 3; the generalization to larger `is provided in Sec. IV.

A. Scaling conjecture for localized and thermalized

regimes

We begin by conjecturing that in the localized phase,

n < nc, the relative size Dof the LKS is exponentially

small in the system size L, while 1 −Dis exponentially

small in the thermalizing phase, n>nc. This conjec-

ture is supported by numerical observations in Refs. 37

and 39, as well as our own numeric simulations. Under

this conjecture, all Krylov sectors must occupy an ex-

ponentially small fraction of the symmetry sector in the

localized phase. Likewise, in the thermalized phase, all

Krylov sectors other than the LKS occupy an exponen-

tially small fraction of the symmetry sector.

In the remainder of this section we demonstrate the

existence of a particular Krylov sector that occupies a

power-law fraction of the symmetry sector at the ﬁlling

n= 1. Given our scaling conjecture about D, such a

Krylov sector can only exist precisely at the critical ﬁll-

ing. Consequently the value of ncmust be equal to nc= 1

(for gate size `= 3). As we argue below, the Krylov sec-

tor we identify is very likely to be the LKS, which allows

us to study the critical scaling of Dnear the transition.

B. Size of the symmetry sector

In order to identify the critical ﬁlling, we ﬁrst study

how the size of the symmetry sector scales with Land

n. For this question we can exploit an exact analogy

between the number of states in the symmetry sector and

the number of non-decreasing lattice paths in a square

grid that enclose a ﬁxed area. The key idea is that one

can deﬁne a “height ﬁeld” y(x) deﬁned for discrete values

xby y(x) = Pw≤xnw[46,52]. This height ﬁeld has an

endpoint y(L−1) = Nthat is ﬁxed by the total charge,

and an area under the curve Pxy(x) = N(L−1) −P

that is ﬁxed by the dipole moment. Thus the number

of states in the symmetry sector is equal to the number

of such curves with ﬁxed endpoint and ﬁxed area. An

example is shown in Fig. 2.

y(x)

(0,0)

(0, N)

(L−1,0)

(L−1, N)

Area = P= 28

FIG. 2. Analogy between non-decreasing integer lattice paths

and symmetry sector states. The x-axis corresponds to po-

sition and each particle corresponds to a one unit move in

the y-direction. This construction guarantees that the area

bounded between the curve and the y-axis is equal to the

dipole moment of the state.

Fortunately, this latter problem has been studied in

the mathematical literature [53–55]. In the limit of large

Nand L, the number of non-decreasing integer lattice

paths has been shown to follow [53]

Dsym(N, L, P )

'N+L−1

NNP;N(L−1)

2,N(L−1)(N+L)

12 

'√3

πn(n+ 1)L2(n+ 1)(n+1)

nnL

×exp "−6˜

n(n+ 1)L2(L−1)#.(2)

Here N(v;µ, σ2) denotes a normal distribution for the

variable vwith mean µand variance σ2, and ˜

P−N(L−1)/2 is the dipole moment relative to a coor-

dinate system with its origin at the center of the system.

This expression can be roughly understood as follows. If

the dipole constraint (or, in analogy, the area constraint)

is removed, then the number of lattice paths can be found

by straightforward combinatorics to be N+L−1

N. Intu-

itively, symmetric states (lattice paths with area half of

the rectangle) are the most likely, and as L→ ∞ the

likelihood of a given value of ˜

Pfollows a Gaussian distri-

bution.

Notice, in particular, that at n= 1 the value of Dsym

at ˜

P= 0 scales with system size as 4L/L2= 4N/N2.

C. Size of the Krylov sector containing the

uniform state

Now that the asymptotic scaling of the size Dsym of

the symmetry sector is understood, we consider the frac-

C D

3 3

x=0 1 2 3 4

A C

C D

2 4

x=0 1 2 3 4

U3,−U3,+

FIG. 3. Analogy between tournament scoring sequences and

states in the apparent LKS. On the left are tournament graphs

for N= 5 teams, with the score of each team (the number

of outgoing edges) labeled. A given scoring sequence corre-

sponds to a particle distribution, with the number of wins for

each team corresponding to the position xof a particle. In

this analogy, it is clear that U3,±corresponds to ﬂipping the

result of a game between two teams that that have either the

same number of wins (U3,+) or a number of wins that dif-

fer by 2 (U3,−). Note, particles are indistinguishable in our

dynamics and in this ﬁgure are only labeled for clarity.

tion of the symmetry sector that is occupied by a spe-

ciﬁc Krylov sector. In particular, we consider the Krylov

sector containing the uniform state (nx= 1 for all x).

This Krylov sector belongs to the symmetry sector with

N=Land ˜

P= 0. As we now show, for this speciﬁc

Krylov sector we can make use of another exact analogy

to a problem in combinatorics.

In order to ﬁnd the size of the Krylov sector contain-

ing the uniform state, we draw an analogy to a classic

problem in combinatorics: the number of unique scoring

sequences of an N-team round robin tournament [56,57].

A round robin tournament is a directed graph in which

each of the Nnodes (“teams”) is connected to all N−1

other nodes, as shown in Fig. 3. An outgoing (incom-

ing) edge at a particular node corresponds to that team

winning (losing) its match-up with the team at the other

end of the edge. An ordered list of numbers of outgoing

edges from each vertex makes up a “scoring sequence” for

a given tournament graph – that is, a “scoring sequence”

is the rank-ordered record of how many games were won

by each team. We make an analogy to this problem by

relating the number of wins by each of the Nteams to

the positions of the Nparticles in our system.

For example, a tournament in which one team loses

all of their games, one teams wins one game, one teams

wins two games and so on would have a scoring sequence

{0,1,2, ..., N −1}. In our analogy, this sequence corre-

sponds to a single particle at each lattice position. Fur-

thermore, the action of a local gate, U3,±, is analogous to

ﬂipping the outcomes of certain games in the tournament.

If two teams have the same number xof wins (by analogy,

two particles have the same position x) and then the re-

sult of the game between them is ﬂipped, then one team

decreases its win total by 1 and the other team increases

its win total by 1 (one particle hops left to position x−1

and one hops right to position x+ 1). Thus, by ﬂipping

the outcome of this game we have performed a U3,+gate

centered at the position x. Similarly, by ﬂipping the out-

come of a game in which a team with x+ 1 wins defeated

a team with x−1 wins, we can eﬀectively apply a U3,−

gate. While ﬂipping the outcome of some games would

eﬀectively implement longer range gates (if the two teams

involved have a number of wins that is diﬀerent by more

than 2), we show in Appendix Athat there is a one-to-one

mapping between the set of states within this Krylov sec-

tor and the set of scoring sequences, so that the eﬀect of

any such long-ranged gate can be equivalently produced

by a sequence of local gates. Thus, we have shown that

the number of states in the Krylov sector that contains

the uniform state is equivalent to the number of unique

scoring sequences in an N-team round robin tournament.

Having made this analogy, we can understand the size

of this Krylov sector by looking up the result for the

number of unique scoring sequences in the mathematical

literature. Speciﬁcally, Refs. 56 and 57 show that

DKS ∼4N

N5/2(3)

at large N1.

One can now compare Eq. (3) with the size Dsym of

the corresponding symmetry sector, given by Eq. (2).

For the corresponding density n= 1 and dipole moment

P= 0, Eq. (2) gives Dsym ∼4N/N 2, which means that

the Krylov sector containing the uniform state occupies

a fraction D∼1/N1/2of the symmetry sector. From the

exponential scaling conjecture of Sec. III A, a Krylov sec-

tor occuping a power-law fraction of the symmetry sector

can only exist precisely at n=nc. Hence we conclude

that nc= 1.

D. Size of the LKS at n≤nc

We now conjecture that the Krylov sector containing

the uniform state, considered in the previous subsection,

is precisely the LKS at n= 1 and ˜

P= 0. This conjecture

can be checked by explicit numerical enumeration of all

states in the symmetry sector when Lis not too large;

this procedure conﬁrms our conjecture for L≤15.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Exactsolutionforthelling-inducedthermalizationtransitionina1DfractonsystemCalvinPozderac,1StevenSpeck,1XiaozhouFeng,1DavidA.Huse,2andBrianSkinner11DepartmentofPhysics,OhioStateUniversity,Columbus,Ohio43210,USA2DepartmentofPhysics,PrincetonUniversity,Princeton,NJ08544,USA(Dated:October7,2022)Westudy...

展开>> 收起<<

Exact solution for the lling-induced thermalization transition in a 1D fracton system Calvin Pozderac1Steven Speck1Xiaozhou Feng1David A. Huse2and Brian Skinner1 1Department of Physics Ohio State University Columbus Ohio 43210 USA.pdf

共17页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Exact solution for the lling-induced thermalization transition in a 1D fracton system Calvin Pozderac1Steven Speck1Xiaozhou Feng1David A. Huse2and Brian Skinner1 1Department of Physics Ohio State University Columbus Ohio 43210 USA

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: