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arXiv:0905.4286v2 [cond-mat.str-el] 9 Jul 2009

Valence Bond and von Neumann Entanglement Entropy in Heisenberg Ladders

Ann B. Kallin,1Iv´ an Gonz´ alez,2Matthew B. Hastings,3and Roger G. Melko1

1Department of Physics and Astronomy, University of Waterloo, Ontario, N2L 3G1, Canada

2Centro de Supercomputaci´ on de Galicia, Avda. de Vigo s/n, E-15705 Santiago de Compostela, Spain

3Microsoft Research, Station Q, CNSI Building, University of California, Santa Barbara, CA, 93106

(Dated: July 9, 2009)

We present a direct comparison of the recently-proposed valence bond entanglement entropy and

the von Neumann entanglement entropy on spin 1/2 Heisenberg systems using quantum Monte

Carlo and density-matrix renormalization group simulations. For one-dimensional chains we show

that the valence bond entropy can be either less or greater than the von Neumann entropy, hence

it cannot provide a bound on the latter. On ladder geometries, simulations with up to seven legs

are sufficient to indicate that the von Neumann entropy in two dimensions obeys an area law, even

though the valence bond entanglement entropy has a multiplicative logarithmic correction.

Introduction.– Entanglement has arisen as a new

paradigm for the study of correlations in condensed mat-

ter systems.Measurements of entanglement between

subregions, chiefly using entropic quantities, have an ad-

vantage over traditional two-point correlation functions

in that they encode the total amount of information

shared between the subregions without the possibility of

missing “hidden” correlations [1], such as may occur in

some exotic quantum groundstates. For example spin liq-

uid states, where two-point correlation functions decay

at large lengthscales, can exhibit topological order that

is quantified by a “topological entanglement entropy” [2].

This and other entropic measures are typically discussed

in the context of the von Neumann entanglement entropy

(SvN), which for a system partitioned into two regions A

and B, quantifies the entanglement between A and B as

SvN

A = −Tr[ρAlnρA].(1)

Here, the reduced density matrix ρA= TrB|ψ??ψ| is ob-

tained by tracing out the degrees of freedom of B.

The properties of SvNare well-studied in quantum in-

formation theory. In interacting one-dimensional (1D)

quantum systems, exact analytical results are known

from conformal field theories (CFT); they show that,

away from special critical points, SvN

of the boundary between A and B. This so-called area

law [3] is also believed to hold in many groundstates

of two dimensional (2D) interacting quantum systems,

although exact results are scarce [4]. This has conse-

quences in the rapidly-advancing field of computational

quantum many-body theory, where it is known for exam-

ple that groundstates of 1D Hamiltonians satisfying an

area law can be accurately represented by matrix prod-

uct states [5]. Tensor-network states and MERA give

two promising new classes of numerical algorithms [6]

that may allow simulations of 2D quantum systems not

amenable to quantum Monte Carlo (QMC) due to the no-

torious sign problem. However, these simulation frame-

works are constructed to obey an area law; in order

to be represented faithfully by them, a given 2D quan-

tum groundstate must have entanglement properties also

A

scales as the size

obeying the area law [4].

Unfortunately, entanglement is difficult to measure in

2D, due to the fact that QMC does not have direct access

to the groundstate wavefunction |ψ? required in Eq. (1).

In response to this, several authors [7, 8] have introduced

the concept of valence bond entanglement entropy (SVB),

defined for a spin system as

SVB

A

= ln(2) · NA,(2)

where NA is the number of singlets (| ↑↓? − | ↓↑?)/√2

crossing the boundary between regions A and B. Unlike

SvN, SVBcan be accessed easily in the valence-bond ba-

sis projector QMC method recently proposed by Sand-

vik [9]. As demonstrated in Refs. [7, 8], SVBhas many

properties in common with SvN, in particular the rela-

tionship SVB

A

= SVB

regions “un-entangled” by valence bonds. A comparison

of the scaling of SVBfor (critical) 1D spin 1/2 Heisen-

berg chains shows good agreement with analytical results

known from CFT, however in the 2D isotropic Heisenberg

model it displays a multiplicative logarithmic correction

to the area law [7, 8]. If true also for SvN, this would

have negative consequences for the simulation of the N´ eel

groundstate using tensor-network simulations.

In this paper, we compare SVBcalculated by valence-

bond QMC to SvNaccessible through density-matrix

renormalization group (DMRG) simulations of the

Heisenberg model on N-leg ladders. For N = 1, the CFT

central charge calculated by scaling SvNshows excellent

agreement to c = 1, whereas SVBconverges to c < 1.

For N > 1, SVBis systematically greater than SvN, a

trend which grows rapidly with N. An examination of

entanglement defined by bipartitioning multi-leg ladders

shows a logarithmic correction for the valence-bond en-

tanglement entropy, SVB∝ N lnN (in agreement with

Refs. [7, 8]), however for data up to N = 7, the von Neu-

mann entanglement entropy convincingly shows a scaling

of SvN∝ N, thus obeying the area law.

Model and Methods.– We consider the spin 1/2 Heisen-

berg Hamiltonian, given by H =?

sum is over nearest-neighbor lattice sites. Geometries

B, and the fact that SVB

A

= 0 for

?ij?Si·Sj, where the

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2

studied are 1D chains, and multi-leg ladders with length

L and number of legs N. We employ two complemen-

tary numerical techniques, the valence-bond basis QMC

and DMRG, both of which give unbiased approximations

to the ground state of the Hamiltonian at zero temper-

ature. SvN

A

is naturally accessible through the DMRG

algorithm [10]. At each step of the algorithm, the wave-

function of the system is approximated by keeping only

the states with largest coefficients in the Schmidt de-

composition for a given bipartition into regions A and

B ≡ ∁A. To find the basis entering the Schmidt decom-

position for region A, the reduced density matrix ρA is

calculated and diagonalized, thus allowing easy calcula-

tion of Eq. (1). The truncation of the basis implies that

only a lower bound for SvN

Ais calculated, so care must be

taken to ensure that enough of the eigenvalue spectrum is

included to converge SvN

Ato sufficient accuracy; typically

the number of states required is larger than that needed

to converge the energy alone. We have kept up to 1800

states per block for the largest ladders, with truncation

errors under 10−7in all cases.

SVB

A

can be calculated using the valence-bond basis

QMC proposed by Sandvik [9]. The algorithm we use is

the simple single-projector method, with lattice geome-

tries constructed to match those used by DMRG. The

ground state of the system is projected out by repeated

application of a list of nearest-neighbor bond operators,

a number of which are changed each step. The change is

accepted with a probability of 2nb/2nawhere na(nb) is

the number of off-diagonal operators in the current (last

accepted) step. Measured quantities such as energy or

SVB

A

are then calculated by a weighted average over all

the valence bond states obtained by this procedure.

One-dimensional chain.– We consider first the case of

Heisenberg chains (N = 1) of length L, simulating both

open (OBC) and periodic (PBC) boundary conditions.

The DMRG algorithm requires the regions A and B to

be topologically connected, so in 1D the bipartition is

defined by a site index x with sites within the interval

[1,x] ([x + 1,L]) belonging to region A (B) [thus we can

label SAby its site index, S(x)]. We stress that the QMC

and DMRG results are on the same geometry and Hamil-

tonian, and reproduce the same ground state energies;

Figure 1 (and subsequent figures) should be considered

as exact comparisons between SVBand SvN.

The 1D Heisenberg model is known to be critical and

thus can be mapped to a 2D classical Hamiltonian at

its critical point, which can be described by CFT in the

limit L → ∞. To address finite-length chains one can

use the conformal mappings x → x′= (L/π)sin(πx/L)

for PBC, and x → 2x′for OBC. Calculations within

CFT [11] obey SvN(x) = (c/3)ln(x′) + S1for PBC, and

SvN(x) = (c/6)ln(2x′)+ln(g)+S1/2 for OBC, where c is

the central charge of the CFT, S1is a model-dependent

constant, and g is Ludwig and Affleck’s universal bound-

ary term [12].

1

10

0.4

1.06

0.5

0.6

0.7

0.8

0.9

1

1

10

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 102030 40

0.98

1

1.02

1.04

0

0.005

0.01

0.015

0.94

0.96

0.98

1

S(x)

L = 128

OBCPBC

SVB

SvN

L = 100

L = 200

L = 64

1/L

z

c

c

x′

x′

S(x)

FIG. 1:

Heisenberg chain with PBC and OBC. Upper panels show

the entropies as a function of the conformal distance x′=

(L/π)sin(πx/L) for 100-site chains. Lower plots show the

central charge c, obtained by fitting the numerical data to

the CFT result, for several L. For PBC, c is calculated with

the two smallest x′points removed. For OBC, the fits depend

on the number of sites included, z, which we systematically

decrease by removing x′data points from the outside ends of

the chain. c is shown for SvN(closed symbols) and SVB(open

symbols) for system sizes L = 64 (circles), L = 100 (squares),

L = 128 (diamonds), and L = 200 (triangles)

(color online) Entanglement entropies for a 1D

Figure 1 illustrates simulation results in both cases,

the left (right) panels corresponding to PBC (OBC). For

PBC both SVBand SvNappear to fit well to the CFT

result, although SvN> SVB. The regression fit for SvN

shows very good convergence with the central charge pre-

dicted by CFT, c = 1, while the fit for SVByields a lower

value of c than predicted. For OBC both SvNand SVB

split into two branches, the upper (lower) correspond-

ing to an odd (even) number of lattice sites in A. This

reflects a well-known “dimerization” effect induced by

OBC [13]. Notice that contrary to the PBC case, now

SvN< SVB. A regression fit of the lower branch to the

form (c/6)ln(2x′) (inset) shows excellent convergence of

SvNto the central charge predicted by CFT, c = 1, once

finite-size effects and the proximity of the data to the

open boundaries are taken into account. In contrast, SVB

deviates significantly from the CFT result, giving c > 1

when all or most data is included in the fit, changing to

c < 1 as data closest to the open boundary is systemati-

cally excluded [14], e.g. for z = L/2, cL→∞≈ 0.85.

Multi-leg ladders.– Moving away from the 1D chain,

one can add “legs” to the lattice in a systematic way. In

this case the sum over nearest neighbors is extended to

neighbors along rungs as well as along legs. As noted

before, DMRG imposes constraints to the subregion ge-

ometry. In multi-leg ladders we choose to sweep in a 1D

path that visits first bonds sitting in rungs rather than

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3

0

50 100 150 200 250

x

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 100200

x

300400

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1

2

3

15

5

6

7

8

9

10

11

12

13

14

19

21

20

18

4

16

17

23

22

24

1 10 100

1

1.5

2

SVB

SvN

S(x)

S(x)

x′

S(x)

FIG. 2: (color online) Entanglement entropies for 3-leg (left)

and 4-leg (right) ladder systems with OBC and 100 sites per

leg. For odd-leg ladders, S(x) ∝ ln(x′). The left inset shows

S(x) as a function of the conformal distance, x′, on a log

scale. For even-leg ladders, S(x ? ξ) = const. The right inset

shows the site indexing used for multi-leg ladders where the

bipartition A is shaded and labeled by x = 9.

bonds sitting in legs (see Fig. 2). DMRG computational

demands increase dramatically with the number of legs,

so in this paper we restrict ourselves to ladders with OBC

up to N = 7 legs. QMC lacks this limitation and one can

go up to N = 20 with minimal CPU effort.

Figure 2 shows SvNand SVBcalculated for the 3-leg

and 4-leg ladder. As for the OBC 1D chain, SVB> SvN.

Entropy shows different behavior depending on N being

even or odd. Even-leg ladders have a spin-gap [15], and

thus only sites within distances from the boundary be-

tween A and B smaller than the correlation length ξ con-

tribute to the entanglement, yielding S(x ? ξ) = const.

In contrast, odd-leg ladders are gapless, and thus all sites

contribute to the entanglement, yielding S(x) ∝ ln(x′),

which follows the CFT result in analogy to the 1D case.

As can be seen, S(x) splits into branches, with a (quasi-)

periodic structure superimposed over the main depen-

dence on x, the period being N (2N) for even- (odd-) leg

ladders. This reflects the periodicity of the underlying 1D

path through which the algorithm sweeps, and the fact

that valence bonds within the same rung are energetically

favored [15]. The doubled period for odd-leg ladders is

due to the same dimerization effect as in Ref. [13].

Area law in multi-leg ladders.– We can use these results

to address the question of the adherence of the 2D N´ eel

state to an area law. To do so, we define the lattice geom-

etry such that region A is rectangular, cutting a multi-leg

ladder cleanly across all legs, so that the “area” separat-

ing region A and B is equivalent to the number of legs

in the ladder N. We choose the region A to contain 2N2

sites, to have an aspect ratio of order unity. In contrast,

the entanglement entropy of a long narrow region would

be dominated by the behavior of the gapless mode for

odd-leg ladders. The 2:1 aspect ratio makes the region

110

0.3

0.4

0.5

0.6

SvN(L = 100)

S(x = 2N2)/N

N

SVB(L = 4N)

SvN(L = 4N)

SvN

SVB(L = 100)

ff−0.3 (L = 100)

FIG. 3: (color online) Entanglement entropies divided by N,

for N-leg Heisenberg (filled symbols) and free-fermion (open

diamonds) ladders, taken such that the region A includes 2N2

sites. For the Heisenberg model and large N, SVB∝ N lnN,

whereas SvN∝ N. Data for free fermions, SvN

shown for comparison. We show data for ladders with length

(sites per leg) L = 100 and L = 4N.

ff

∝ N lnN, are

length even for all N, reducing even-odd oscillations.

Figure 3 illustrates the simulation results for N-leg lad-

ders. Plotting S(x)/N versus N on a log scale, we obtain

a multiplicative logarithmic correction to the SVBarea

law, in agreement with results from Refs. [7, 8]. How-

ever, the linear slope is not present in the plot of the SvN

data from the DMRG, which convincingly approaches a

constant for large N [16]. Clearly, for SvNthe area law

is indeed obeyed in the N´ eel groundstate, leading one

to conclude that the multiplicative logarithmic correc-

tion occurs in SVBonly. One can compare SVBto data

obtained for free fermions, which has a well-known [17]

logarithmic correction to the area law for SvN(Fig. 3).

In the next section, we explain SVBin the context of the

bond length distribution in the QMC. We note also that,

contrary to the suggestion in Ref. [7], a gapless Goldstone

mode will not give a logarithmic divergence to SvNsince

a gapless bosonic mode in 2D obeys an area law [18]. We

have also done a spin-wave calculation of SvNfor this sys-

tem and found an area law, albeit with S(x)/N ≈ 0.2,

slightly lower than suggested for spin 1/2 in Fig. 3.

Bond Length Distribution.– Sandvik defined the bond

length distribution P(x,y) as the probability of a bond

going from site x to site y, and found that P(x,y) ∼

|x−y|−pwith p ≈ 3 in the N´ eel state [9]. This value of p

gives the logarithmic divergence in SVB, as can be found

by directly calculating

SVB

A

=

?

x∈A,y∈B

P(x,y)ln(2). (3)

We can understand the value of p from a scaling ar-

gument similar to that in Ref. [19]. Consider the num-

ber of bonds of length l exiting a region of linear size l.

For p > 4, this scales to zero and such a state should

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4

have no long-range order. For p = 4, this number is

l-independent, corresponding to a critical state (p = 2

is the critical power in 1D, matching the observed loga-

rithmic behavior of SVB). For p < 2, the bond length

distribution is unnormalized, and all bonds are long, so

that there is no short-range order. p = 3 represents a

state with both long- and short-range order. There is

a heuristic argument that p = 3 corresponds to a state

with Goldstone modes: the correlation function of two

spins is the probability that they lie on the same loop.

This can be estimated by the Green’s function of a L´ evy

flight with power law distributed steps with power 3, re-

producing the power law decay of correlations.

Discussion.– In this paper, we have compared scal-

ing properties of the valence bond entanglement entropy

(SVB) [7, 8] to the von Neumann entanglement entropy

(SvN) in the spin 1/2 Heisenberg model on multi-leg lad-

der geometries, using QMC and DMRG simulations. In

1D, we find that SVBmimics the behavior of SvNclosely,

although it is less than SvNfor periodic chains, and

greater than SvNfor open chains. In addition, fits to

1D conformal field theory, which are excellent for SvN

calculated via DMRG, appear to deviate significantly for

SVBin the large chain-size limit, approaching c < 1 for

both boundary conditions [14].

The fact that SVBcan be either greater or less than

SvNcan be understood through simple examples. Let

|(ij)(kl)...? denote a state in which sites i,j are in a

singlet, sites k,l are in a singlet, and so on. Consider

an 8-site chain, with sites 1 to 4 in region A. Then,

the state |(12)(34)(56)(78)?+|(14)(32)(58)(76)? has van-

ishing SVB

A

since no bonds connect A to B, but non-

vanishing SvN

A

≈ 0.325. On the other hand, consider

a 4-site chain, with sites 1 and 3 in region A. Then, the

state |(12)(34)? + |(14)(32)? has a maximal SVB

to 2ln(2) ≈ 1.386, while SvN

This second state has the maximum possible N´ eel order

parameter: it is the equal amplitude superposition of all

configurations of bonds connecting the two sublattices.

Thus, it is unsurprising that states with N´ eel order show

SVB> SvN, a fact which we have demonstrated numeri-

cally on multi-leg ladder systems with open boundaries.

Defining the boundary between the two entangled re-

gions as being bipartitioned by a cut across all legs on

a ladder, we have shown using DMRG that SvNobeys

the area law in the many-leg limit. Since DMRG can

also accurately measure logarithmic size-dependencies of

SvN(in 1D critical systems), this suggests that simula-

tion procedures similar to those here might enable the

measurement of area-law corrections in SvNas indica-

tors of exotic phases in other models, such as those with

a spinon Fermi surface [20].

The valence bond entanglement entropy harbors a mul-

tiplicative logarithmic correction for the N´ eel ground-

state, which we have shown is caused by the valence-

bond length distribution, and is not present in SvN. It

A, equal

A= ln(3) ≈ 1.099 is smaller.

is clear that SVBis a reasonable measurement of entan-

glement, readily accessible to numerical simulations in

2D and higher, and capable of reproducing the area law

in some gapped groundstates [7, 8]. However, the in-

ability of SVBto provide a bound on SvN(unlike other

measures such as Renyi entropies), along with its discrep-

ancies from SvNin 1D critical systems and the 2D N´ eel

state, must be taken into account in proposals to use SVB

for future tasks such as characterizing topological phases

or studying universality at quantum phase transitions.

Acknowledgments.– The authors thank I. Affleck,

A. J. Berlinsky, N. Bonesteel, A. Del Maestro, A. Feiguin,

M. Fisher, L. Hormozi, and E. Sørensen for useful discus-

sions. This work was made possible by the computing

facilities of SHARCNET and CESGA. Support was pro-

vided by NSERC of Canada (A.B.K. and R.G.M.) and

the NSF under Grant No. NSF PHY05-51164 (I.G.).

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