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arXiv:1202.1687v1 [cond-mat.str-el] 8 Feb 2012

Antiferromagnetic order in systems with doublet Stot= 1/2 ground states

Sambuddha Sanyal,1Argha Banerjee,1Kedar Damle,1and Anders W. Sandvik2

1Department of Theoretical Physics, Tata Institute of Fundamental Research, Mumbai 400005, India.

2Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachussetts 02215, USA.

We use projector Quantum Monte-Carlo methods to study the Stot = 1/2 doublet ground states

of two dimensional S = 1/2 antiferromagnets on a L×L square lattice with an odd number of sites

Ntot = L2. We compute the ground state spin texture Φz(? r) = ?Sz(? r)?↑ in |G?↑, the Sz

component of this doublet, and investigate the relationship between nz, the thermodynamic limit

of the staggered component of this ground state spin texture, and m, the thermodynamic limit of

the magnitude of the staggered magnetization vector of the same system in the singlet ground state

that obtains for even Ntot. nzand m would have been equal if the non-zero value of Sz

caused the direction of the staggered magnetization vector to be fully pinned in the thermodynamic

limit. By studying several different deformations of the square lattice Heisenberg antiferromagnet,

we establish that this is not the case. For the sizeable range of m accessed in our numerics, we

find a univeral relationship between the two, that is independent of the microscopic details of the

lattice level Hamiltonian and can be well approximated by a polynomial interpolation formula:

nz≈ (1

2−

spin texture Φz(? r) is itself dominated by Fourier modes near the antiferromagnetic wavevector

in a universal way.On the analytical side, we explore this question using spin-wave theory, a

simple mean field model written in terms of the total spin of each sublattice, and a rotor model for

the dynamics of ? n. We find that spin-wave theory reproduces this universality of Φz(? r) and gives

nz= (1−α−β/S)m+(α/S)m2+O(S−2) with α ≈ 0.013 and β ≈ 1.003 for spin-S antiferromagnets,

while the sublattice-spin mean field theory and the rotor model both give nz=

antiferromagnets. We argue that this latter relationship becomes asymptotically exact in the limit

of infinitely long-range unfrustrated exchange interactions.

tot= 1/2

totin |G?↑

3−ab

4)m + am2+ bm3, with a ≈ 0.288 and b ≈ −0.306. We also find that the full

1

3m for S = 1/2

PACS numbers: 75.10.Jm 05.30.Jp 71.27.+a

I. INTRODUCTION

Computational studies of strongly correlated systems

necessarily involve an extrapolation to the thermody-

namic limit from a sequence of finite sizes at which cal-

culations are feasible. Understanding,1and at times

reducing,2these finite-size corrections to the thermody-

namic limit is thus an important aspect of any such cal-

culation. For instance, the best estimates of m, the mag-

nitude of the ground state N´ eel order parameter in the

thermodynamic limit of the two-dimensional S = 1/2

square lattice Heisenberg antiferromagnet rely on a se-

quence of Lx×Lysystems with even length Lx(Ly) in the

x (y) direction and periodic boundary conditions in both

directions.3,4Other studies suggest2that it is some times

advantageous to use “cylindrical” samples with periodic

boundary conditions in one direction and pinned bound-

ary conditions in the other direction, whereby spins are

held fixed by the use of pinning fields on one pair of

edges—this choice also allows for a very accurate deter-

mination of ground-state parameters such as m for spe-

cific values2of the aspect ratio Ly/Lx.

All these approaches focus on systems with an even

number of spin-half variables; this choice allows the

ground-state of the finite system to lie in the sin-

glet sector favoured by unfrustrated antiferromagnetic

interactions.5Although not commonly used, another

choice is certainly possible: Namely, one could in princi-

ple consider antiferromagnets on a L × L square lattice

with an odd number Ntot = L2of spin-half moments.

Such a system is expected to have a doublet ground state

with total spin Stot= 1/2. Focusing on the Sz

member |G?↑ of this doublet, one could examine the

ground state spin texture defined by Φz(? r) ≡ ?Sz

(where ?...?↑ refers to expectation values in |G?↑), and

use the antiferromagnetic component of this spin texture,

defined as

tot= 1/2

? r?↑

nz=

1

Ntot

?

? r

η? r?Sz

? r?↑, (1)

to obtain information about the antiferromagnetic order-

ing in the system (here η? r= +1 on the A sublattice and

−1 on the B sublattice).

Clearly, nzprovides a measure of antiferromagnetic

order that is quite distinct from the conventional order

parameter m, which can be defined, e.g., according to

m2=

1

Ntot

?

? r? r′

η? rη? r′??S? r·?S? r′?0, (2)

where ?...?0denotes averages in the singlet ground state

realized for even Ntot.The relationship between the

thermodynamic-limit values of nzand m is a fundamen-

tal aspect of the spontaneously broken SU(2) symmetry

of the N´ eel state. However, not much is known about

it beyond the fact that nzis significantly smaller than

m for the nearest neighbour Heisenberg antiferromagnet

on ths square lattice.6Here, we provide a more detailed

characterization of this relationship.

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2

Our basic result is that nzis determined in a universal

way by the value of m. In other words, nzplotted against

m for severaldifferent deformations of the S = 1/2 square

lattice Heisenberg antiferromagnet falls on a single curve

which defines a universal function that is insensitive to

the microscopic details of the model Hamiltonian. This

universal function is well-approximated by a polynomial

interpolation formula:

nz≈ (1

3−a

2−b

4)m + am2+ bm3, (3)

with a ≈ 0.288 and b ≈ −0.306. In addition, we also find

that the full spin texture Φz(? r) is dominated by Fourier

modes near the antiferromagnetic wave-vector in a uni-

versal way independent of microscopic details. We show

that this universality is captured by spin-wave theory,

which also predicts

nz= (1 − α − β/S)m + (α/S)m2+ O(S−2),

with α ≈ 0.013 and β ≈ 1.003 for spin-S antiferromag-

nets. In addition, we explore two other ways of thinking

about this universal function. One of them is a mean

field theory formulated in terms of the total spin of each

sublattice, while the other approach is in terms of a quan-

tum rotor Hamiltonian for the N´ eel vector ? n of a system

with an odd number of sites. Both these give

nz=m

(4)

3

(5)

for S = 1/2 antiferromagnets, which is close to the

observed relationship but not exactly right. We argue

that this latter estimate (Eqn. 5) will become asymp-

totically exact in the limit of infinitely long-range un-

frustrated exchange interactions. In this limit, we also

expect m → 1/2, and our polynomial fit to the universal

function nz(m) was therefore constrained to ensure that

nz→ m/3 when m → 1/2.

The outline of the rest of the paper is as follows: In Sec-

tion II we define various deformations of the square lat-

tice S = 1/2 Heisenberg antiferromagnet. In Section III,

we outline the projector quantum Monte Carlo (QMC)

method used in this study, and then discuss in some de-

tail our QMC results for nzas well as the full spin tex-

ture Φz(? r), focusing on the universal properties alluded

to earlier. In Section IV, we outline three analytical ap-

proaches to the relationship between nzand m. The first

is a large-S spinwave expansion, within which we calcu-

late the ground state spin texture Φz(? r) and its antiferro-

magnetic Fourier component nzto leading O(1/S) order,

and demonstrate that such a calculation also yields the

universality properties summarized earlier, but does not

provide a quantitatively accurate account of the QMC

results for Φz(? r) or nz(m). The second is a mean-field

theory formulated in terms of the total spin of each sub-

lattice. And the third approach is in terms of a quantum

rotor Hamiltonian which is expected to correctly describe

the low-energy tower of states for odd Ntot. In Section V,

we conclude with some speculations about a possible ef-

fective field theory approach to the calculation of Φz(? r).

J

J

J

J

J′

J2

J2

FIG. 1: An illustration of the interactions present in the JJ′

(left panel) and JJ2(right panel) model Hamiltonians. In this

illustration, black bonds denote exchange interaction strength

of J, while a red bond represents exchange strength of J′(J2)

in the left (right) panel

ij

ij

kl

ij

kl

mn

JQQ

23

FIG. 2:

tonians. A thick bond denotes a bipartrite projector acting

on that bond. All possible orientations of these bond and

plaquette operators are allowed.

Bond and plaquette operators in JQ model Hamil-

II.MODELS

We consider four deformations of the square lattice

S = 1/2 nearest neighbour Heisenberg antiferromagnet;

all four retain the full SU(2) spin rotation symmetry of

the original model.

The first of these models is the coupled-dimer antifer-

romagnet, in which there are two kinds of nearest neigh-

bour interactions J and J′, as shown in Fig. 1 (left panel),

where the ratio α = J′/J can be tuned from α = 1 to

α = αc≈ 1.90 at which collinear antiferromagnetic order

is lost.7The Hamiltonian for this system reads:

HJJ′ = J

?

?ij?

Si· Sj+ J′?

?ij?′

Si· Sj, (6)

where ?ij? (?ij?

connected by a black (red) bond (see Fig. 1). Another

deformation of the Heisenberg model, the JJ2model, has

additional next nearest neighbour Heisenberg exchange

interactions J2, as shown in Fig. 1 (right panel). The

Hamiltonian reads

′) denotes a pair of nearest neighbour sites

HJJ2= J

?

?ij?

Si· Sj+ J2

?

??ij??

Si· Sj,(7)

where ??ij?? denotes a pair of next nearest neighbour

sites. Both these are amenable to straightforward spin-

wave theory analyses, and the coupled dimer model can

also be studied numerically to obtain numerically exact

results even for very large sizes due to the absence of any

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sign problems in Quantum Monte Carlo studies. How-

ever, exact numerical results on the JJ2 model are re-

stricted to small sizes since Quantum Monte Carlo meth-

ods encounter a sign problem when dealing with next-

nearest neighbour interactions on the square lattice.

In addition, we study two generalizations that involve

additional multispin interactions; the “JQ” models.8,9Of

these, the JQ2model has 4-spin interactions in addition

to the usual Heisenberg exchange terms, and is defined

by the Hamiltonian

HJQ2= −J

?

?ij?

Pij− Q2

?

?ij,kl?

PijPkl, (8)

where the plaquette interaction Q2involves two adjacent

parallel bonds on the square lattice as shown in Fig. 2

(middle panel) and

Pij=1

4− Si· Sj

(9)

is a bipartite singlet projector. The first term in Eqn. 8

is just the standard Heisenberg exchange. Similarly, the

JQ3model has 6-spin interactions and is defined by the

Hamiltonian

HJQ3= −J

?

?ij?

Pij− Q3

?

?ij,kl,nm?

PijPklPnm, (10)

where the plaquette interactions now involve three ad-

jacent parallel bonds on the square lattice, as shown in

Fig. 2 (right panel). The products of singlet projectors

making up the Q2and Q3terms tend to reduce the N´ eel

order of the ground state, and, when sufficiently strong,

lead to a quantum phase transition into a valence-bond-

solid state.8,9Here we stay within the N´ eel state in both

models, and study universal aspects of this state as the

N´ eel order is weakened.

III. PROJECTOR QMC STUDIES

We use the total spin-half sector version10of the

valence-bond basis projector QMC method11,12to study

L×L samples with L odd and free boundary conditions.

We compute Φz(? r) and nzin such samples for the JJ′

model and JQ models in their antiferromagnetic phase.

We also study the same models on L×L lattices with L

even and periodic bondary conditions using the original

singlet sector valence bond projector QMC method. In

both cases we use the most recent formulation with very

efficient loop updates.10,12Our system sizes range from

L = 11 to L = 101, and projection power scales as L3

to ensure convergence to the ground state. We perform

? 105equilibration steps followed by ? 106Monte Carlo

measurements to ensure that statistical and systematic

errors are small.

Data for nzfrom a sequence of L × L systems with L

odd shows that nzextrapolates to a finite value in the

0 0.020.04

0.06

0.08

0.1

1/L

0.03

0.04

0.05

0.06

0.07

0.08

nz

m2

FIG. 3: An illustrative example of finite size corrections of nz

and m2, observed in the antiferromagnetic phase of the JJ′

model(J′= 1.8). Note the non-monotonic behaviour of finite

size corrections for nz, which is fitted to a cubic polynomial.

In contrast, finite size data for m2is well described by a linear

dependence on 1/L.

L → ∞ limit as long as the system is in the antiferro-

magnetic phase. However, we find that the approach of

this observable to the thermodynamic limit has a non-

monotonic behaviour. To obtain accurate extrapolations

to infinite size, it is therefore necessary to fit the finite size

data to a third-order polynomial in 1/L. We find that the

coefficient for the leading 1/L term in this polynomial is

rather small; this is true for all the models studied here,

as long as they remain in the antiferromagnetic phase. In

Fig.. 3 and Fig. 4, we show examples of this behaviour of

the finite size corrections in nz. In these figures, we also

show the approach to the thermodynamic limit for m, as

measured in a sequence of periodic L×L systems with L

even. We find that in complete contrast to the behaviour

of nz, m extrapoloates monotonically to the thermody-

namic limit, with a dominant 1/L dependence—this is

consistent with previous studies of the structure factor

in square lattice antiferromagnets12(however, with spa-

tially anisotropic couplings, one can also observe strong

non-monotonicity in m13).

The non-zero value of nzin the thermodynamic limit

clearly reflects the long-range antiferromagnetic order

present in the system and a partial breaking of the SU(2)

symmetry (due to the fact that we study only one mem-

ber of the doublet ground state). For periodic systems,

the same long range antiferromagnetic order is captured

by the non-zero value of m in the large L limit—and a

calculation of m (through ?m2?) for the odd-L systems

with periodic boundaries would of course lead to the same

value. However, since m ?= nz, the full staggered mag-

netization is not forced to lie along the z spin axis, and

it is interesting to ask: What is the relationship between

these two measures of antiferromagnetic order? Our nu-

merical data are unequivocal as far as this relationship is

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0 0.020.04

0.06

0.08

0.1

1/L

0.02

0.03

0.04

0.05

0.06

nz

m2

FIG. 4: Another illustrative example of finite size corrections

of nzand m2, observed in the antiferromagnetic phase of JQ2

model at Q2 = 1.0. Again, note the non-monotonic behaviour

of finite size corrections for nz, which is fitted to a cubic

polynomial (only L > 20 data used in the fit). In contrast,

finite size data for m2is well described by a linear dependence

on 1/L.

concerned, as is clear from Fig. 5, which shows a plot of

nzversus m in the thermodynamic limit of the JJ′, JQ2

and JQ3 models. Here each point represents the result

of a careful extrapolation similar to the examples shown

in Fig. 3 and Fig. 4, and provides an accurate estimate of

the corresponding thermodynamic limits for nzand m.

From this figure, it is clear that nzis a universal func-

tion of m independent of the microscopic structure of the

Hamiltonian. To model this universal function, we use a

polynomial fit that is constrained to ensure that nz→m

when m →

come clear in Sec. IV. We find (Fig. 5) that the QMC

results for nz(m) are fit well by the following functional

form:

3

1

2; the rationale for this constraint will be-

nz(m) = (1

3−a

2−b

4)m + am2+ bm3,(11)

with a ≈ 0.288 and b ≈ −0.306.

If one views this universal relationship as being a prop-

erty of the low energy effective field theory of the anti-

ferromagnetic phase, one is led to expect that the full

spatial structure of the spin texture Φz(? r) should also be

universal. More precisely, one is led to expect that this

texture is dominated in a universal way by Fourier com-

ponents near the antiferromagnetic wavevector. To test

this, we compare the spin texture in the JJ′model and

the JQ3model, choosing the strengths of the J′interac-

tion and the Q3interaction so that both have the same

value of m, and therefore the same value of nz. This is

shown in Fig. 6, which shows that these very different mi-

croscopic Hamiltonians have spin-textures whose Fourier

transform falls on top of each other at and around the

antiferromagnetic wavevector.

0.1

0.15

0.2

0.25

0.3

0.35

m

0.2

0.25

0.3

0.35

0.4

nz/m

JQ2 model

JJ′ model

JQ3 model

f(m) = 1_

3 - a_

2 - b_

4 + am + bm2

a = 0.288 ± 0.022, b = -0.306 ± 0.031

FIG. 5: Extrapolated thermodynamic values of nzfor three

different models of antiferromagnets on an open lattice, plot-

ted as function of staggered magnetisation m for the same

models on periodic lattices.The former is clearly an uni-

versal function of the later. This universal function can be

well approximated by a polynomial fit constrained to en-

sure that nz(m) → m/3 in the limit of m →

(1/3 − a/2 − b/4)m + am2+ bm3, with a ≈ 0.288 and

b ≈ −0.306.

1

2: nz≈

IV. ANALYTICAL APPROXIMATIONS

We now present three distinct analytical approaches

to understanding these numerical results presented in the

previous section: First, we develop a spin-wave expansion

that becomes asymptotically exact for large S14. Sec-

ond, we explore a mean-field theory written in terms of

the total spin of each sub-lattice. Finally, we describe an

alternative approach in which the low-energy antiferro-

magnetic tower of states of a spin-1/2 antiferromagnet is

described by a phenomenological rotor model17adapted

to the case of a system with odd Ntot.

A.Spin-wave expansion

The leading order spin-wave calculation proceeds as

usual by using an approximate representation of spin

operators in terms of Holstein-Primakoff bosons. The

resulting bosonic Hamiltonian is truncated to leading

(quadratic) order in boson operators to obtain the first

quantum corrections to the classical energy of the system.

As is standard in the spin wave theory of N´ eel ordered

states, we start with the classical N´ eel ordered config-

uration with the N´ eel vector pointing along the ˆ z axis,

which corresponds to Sz

? r= η? rS. We then represent the

spin operators at a site ? r of the square lattice in terms

of canonical bosons to leading order in S as follows: For

sites ? r belonging to the A sublattice we write

S+

? r=

√2Sb? r; Sz

? r= S − b†

? rb? r,(12)

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while on sites ? r belonging to the B sublattice we write

√2Sb? r; Sz

S−

? r=

? r= −S + b†

? rb? r.(13)

The number of bosons at each site thus represents the

effect of quantum fluctuations away from the classical

N´ eel ordered configuration.

To quadratic order in the boson operators, this ex-

pansion yields the following spin wave Hamiltonian in

the general case (with arbitrary two-spin exchange cou-

plings):

Hsw = ǫclS2+S

2b†Mb , with

?A? r? r′ B? r? r′

?b? r

M? r? r′ =

B? r? r′ A? r? r′

?

b? r=

b†

? r

?

. (14)

Here ǫclS2is the classical energy of the N´ eel state, M

in the first line is a 2Ntot dimensional matrix specified

in terms of Ntot dimensional blocks A and B, and b is

a 2Ntot dimensional column vector as indicated above.

Elements of A and B can be written explicitly as

A? r? r′ = (ZU

B? r? r′ = JU

? r− ZF

? r? r′.

? r)δ? r? r′ + JF

? r? r′, (15)

(16)

In the above, JF

tween two sites? r and? r′belonging to the same sub-lattice,

JU

? r? r′ are the Heisenberg exchange couplings between sites

belonging to different sublattices, and

? r? r′ are Heisenberg exchange couplings be-

ZU

? r

=

?

?

? r′

JU

? r? r′, (17)

ZF

? r

=

? r′

JF

? r? r′. (18)

The effects of quantum fluctuations on the classical

N´ eel state can now be calculated by diagonalizing this

Hamiltonian by a canonical Bogoliubov transformation

S which relates the Holstein-Primakoff bosons b to the

bosonic operators γ corresponding to spin-wave eigen-

states

b = SΓ,Γµ=

?γµ

γ†

µ

?

, (19)

where S is a 2Ntot dimensional matrix that transforms

from b which creates and destroys bosons at specific lat-

tice sites ? r to Γ which creates and destroys spin-wave

quanta in specific spin-wave modes µ. Naturally, we must

require that Hswbe diagonal in this new basis. We rep-

resent this diagonal form as

Hsw= ǫclS2+S

2Γ†DΓ , (20)

where

D =

?Λ 0

0 Λ

?

, (21)

with Λ denoting the diagonal matrix with the Ntotposi-

tive spin wave frequencies λµon its diagonal.

To construct a S that diagonalizes Hswin the Γ basis,

we look for 2Ntotdimensional column vectors

yµ=

?uµ

vµ

?

, (22)

which satisfy the equation

Myµ= ǫµIyµ

(23)

with positive values of ǫµequal to the positive spin-wave

frequencies λµfor µ = 1,2,3...Ntot. Here uµand vµare

Ntotdimensional vectors,

I =

?10

0 −1

?

,(24)

and 1 is the Ntot× Ntot identity matrix. With these

yµin hand, one may obtain Ntot additional solutions

to Eqn. 23, this time with negative ǫNtot+µ = −λµ by

interchanging the roles of the Ntot dimensional vectors

uµand vµin this construction. In other words, we have

yNtot+µ=

?vµ

uµ

?

,(25)

with µ = 1,2,3...Ntot.

We now construct S by using these yµ(with µ =

1,2,3...2Ntot) as its 2Ntotcolumns:

S =?y1,y2,y3...y2Ntot?

.(26)

Clearly, this choice of S satisfies the equation

MS = ISID(27)

Furthermore, the requirement that the Bogoliubov trans-

formed operators γ obey the same canonical bosonic com-

mutation relations as the b operators implies that S must

satisfy

S†IS = I ,(28)

This constraint is equivalent to “symplectic” orthonor-

malization conditions:

(uµ)†uν− (vµ)†vν= δµν,

(uµ)†vν− (vµ)†uν= 0 ,

(29)

for µ,ν = 1,2,3...Ntot. It is now easy to see that Eqn 27

and Eqn 28 guarantee that Hswis indeed diagonal in the

new basis, since

b†Mb = Γ†S†MSΓ = Γ†S†ISIDΓ = Γ†DΓ .

For periodic samples, it is possible to exploit the trans-

lational invariance of the problem and work in Fourier

space to obtain these spin-wave modes and their wave-

functions and calculate m = S − ∆

(30)

′correct to leading

Page 6

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2.53 3.5

q

10-6

10-5

10-4

10-3

10-2

10-1

100

|Sz(q,q)|

J′= 1.5

Q3= 0.04

2.533.5

q

10-4

10-3

10-2

10-1

100

|Sz(q,π)|

J′= 1.5

Q3= 0.04

FIG. 6: Fourier transform (with antiperiodic boundary conditions assumed for convenience) of the numerically computed (for

JJ′and JQ3 model with L = 65, S = 1/2 ) Φz(? r) along cuts passing through the antiferromagnetic wavevector (π,π). Note

the universality of the results in the neighbourhood of the antiferromagnetic wavevector, which in any case accounts for most

of the weight in Fourier space.

order in the spin-wave expansion—as these results are

standard and well-known,15we do not provide further

details here. On the other hand, the corresponding re-

sults for L × L samples with free boundary conditions

and NA = NB+ 1 do not seem to be available in the

literature, and our discussion below focuses on this case.

We begin by noting that the non-zero entries in A only

connect two sites belonging to the same sublattice, while

those in B always connect sites belonging to opposite

sublattices. As a result of this, the solutions to the equa-

tion for yµcan also be expressed in terms of a single

function fµ(? r) defined on sites of the lattice. To see this,

we consider an auxillary problem of finding ˜ ǫµsuch that

the operator A − B − ˜ ǫµη? r has a zero mode fµ(? r) (as

before, η? ris +1 for sites belonging to the A sublattice,

and −1 for sites belonging to the B sublattice).

This auxillary problem has Ntotsolutions correspond-

ing to the Ntot roots ˜ ǫµ of the polynomial equation

det(A−B −˜ ǫµη? r) = 0; these ˜ ǫµcan be of either sign. To

make the correspondence with the positive ǫµ solutions

(uµ,vµ) (with µ = 1,2...Ntot) of the original equation

Myµ= ǫµIyµ, we now note that

?fµ|A − B|fµ? = ˜ ǫµNµ

(31)

where

Nµ≡

?

rA

|fµ(rA)|2−

?

rB

|fµ(rB)|2. (32)

Since A−B is a positive (but not positive definite) oper-

ator, this implies that ˜ ǫµhas the same sign as Nµfor all

non-zero ˜ ǫµ. To make the correspondence with the pos-

itive ǫµ≡ λµsolutions (µ = 1,2...Ntot) of the original

problem, we can therefore make the ansatz

uµ

? rA= fµ(rA)/?Nµ,uµ

rB= 0(33)

vµ

rB= −fµ(rB)/?Nµ,vµ

rA= 0

if Nµ> 0, or the alternative ansatz

uµ

rB= −fµ(rB)/?−Nµ,uµ

vµ

rA= 0 (34)

rA= fµ(rA)/?−Nµ,vµ

Here, rA (rB) denotes sites belonging to

the A (B) sublattice of the square lattice. This ansatz

clearly ensures that the yµ(with µ = 1,2..Ntot) obtained

in this manner satisfy the original equation with positive

ǫµ≡ λµand are appropriately normalized.

Atlhough this approach is not the one we use in our ac-

tual computations (see below), it provides a useful frame-

work within which we may discuss possible zero frequency

spin-wave modes, i.e λµ0= 0 for some µ0: A mode µ0

with λµ0= 0 clearly corresponds to a putative zero eigen-

value of the operator A − B. From the specific form of

A − B in our problem, it is clear that such a zero eigen-

value does indeed exist, and fµ0(? r), the corresponding

eigenvector of A − B, can be written down explicitly as

fµ0(? r) = 1

rB= 0

if Nµ < 0.

(35)

Since this corresponds to the root ˜ ǫµ0= 0 of the auxillary

problem, it can in principle be used to obtain a pair of

zero frequency modes ǫµ0and ǫµ0+Ntotfor the original

problemof finding ǫµand yµthat satisfy Myµ= ǫµIyµ.

However, we need to ensure that the symplectic or-

thonormalization conditions (Eqn. 30) are satisfied by

our construction of the corresponding yµ0and yµ0+Ntot.

This is where the restriction to a Ntot = L × L lattice

Page 7

7

with NA= NB+ 1 enters our discussion. For this case,

Nµ0= NA− NB = 1, and we are thus in a position

to write down properly normalized zero-mode wavefunc-

tions:

uµ0

? rA= fµ0(rA),uµ0

rB= −fµ0(rB),vµ0

rB= 0(36)

vµ0

rA= 0 ,

and

uNtot+µ0

r

vNtot+µ0

r

= vµ0

= uµ0

r,

r .

(37)

[Parenthetically, we note that the question of zero fre-

quency spinwave modes for the more familiar case with

NA = NB and periodic boundary conditions has been

discussed earlier in the literature14and will not be con-

sidered here.]

Thus, the equation Myµ= ǫµIyµhas a pair of zero

modes related to each other by interchange of the u and

v components of the mode, and it becomes necessary to

regulate intermediate steps of the calculation with a stag-

gered magnetic field ˆ zǫhη? rwith infinitesimal magnitude

ǫh> 0 in the ˆ z direction. Denoting the corresponding A

by Aǫh, we see that Aǫh− B is now a positive definite

operator and does not have a zero eigenvalue. Indeed,

it is easy to see from the foregoing that the correspond-

ing eigenvalue now becomes non-zero, yielding a positive

spin-wave frequency λǫh

µ0= Ntotǫh. One can also calcu-

late the O(ǫh) term of fǫh

to fµ0(? r) in a non-singular way as ǫh → 0, from which

one can obtain the corresponding yµ0(ǫh) analytically in

this limit. Thus, the contribution of the zero mode to all

physical quantities can be obtained in the presence of a

small ǫh> 0, and the ǫh→ 0 limit of this contribution

can then be taken smoothly and analytically at the end

of the calculation.

In our actual calculations, we use this analytical un-

derstanding of the zero frequency spin wave mode to

analytically obtain the properly regularized zero mode

contribution to various physical quantities, while using a

computationally convenient approach to numerically cal-

culate the contribution of the non-zero spin wave modes.

To do this, we rewrite Eqn. 23 for µ = 1,2,3...Ntotas

µ0(? r) and check that fǫh

µ0tends

(A + B)φµ= λµψµ

(A − B)ψµ= λµφµ

(38)

where

φµ= uµ+ vµ

ψµ= uµ− vµ.

(39)

This implies

(A − B)(A + B)φµ= λµ(A − B)ψµ= λ2

(A + B)(A − B)ψµ= λµ(A + B)φµ= λ2

We now decompose

µφµ(40)

µψµ (41)

A − B = K†K.(42)

where

K =√ωU.(43)

with ω the diagonal matrix with diagonal entries given

by eigenvalues of the real symmetric matrix A−B, and U

the matrix whose rows are made up of the corresponding

eigenvectors.

With this decomposition, we multiply Eqn 41 by K

from the left to obtain

K(A + B)K†χµ= λ2

µχµ.(44)

with χµ= Kψµ. From the solution to this equation, we

may obtain the φ as

φµ= (K†)χµ/λµ. (45)

and thence obtain ψµusing Eqn 39. In order to ensure

the correct normalization of the resulting uµ,vµ, we im-

pose the normalization condition

(χµ)†χµ= λµ. (46)

Thus our computational strategy consists of obtaining

eigenvalues of the symmetric operator K(A+B)K†, and

using this information to calculate the yµand thence

the Bogoliubov transform matrix S. Notwithstanding the

normalization used in Eqn 46, the zero mode with λµ0=

0 causes no difficulties in this approach, since we work in

practice with the projection of K(A+B)K†in the space

orthogonal to the zero mode. This is possible because

we already have an analytic expression correct to O(ǫh)

for yµ0(ǫh) and yNtot+µ0(ǫh) corresponding to this zero

mode, and do not need to determine these two columns

of S by this computational method.

We use this procedure to calculate the zero tempera-

ture boson density as

?b†

? rb? r? = lim

ǫh→0

Ntot

?

µ=1

?vµ

? r(ǫh)?2. (47)

In this expression, one may use the numerical procedure

outlined above to obtain the contribution of all µ ?= µ0

directly at ǫh= 0, while being careful to use our analyti-

cal results for vµ0(ǫh) to obtain the limiting value of the

contribution from µ = µ0. This gives

?b†

? rAb? rA? =

?

µ?=µ0

(vµ

? rA)2

(48)

?b†

iBbiB? = 1 +

?

µ?=µ0

(vµ

? rB)2

(49)

Here, the distinction between sites on the A and

B sublatticesarisesin

limǫh→0vµ0

tice, while limǫh→0vµ0

sublattice.

thisfinalresult because

? r(ǫh) = −1 for ? r belonging to the B sublat-

? r(ǫh) = 0 for ? r belonging to the A

Page 8

8

Knowing the average boson number at each site gives

us the first quantum corrections to the ground state ex-

pectation value ?Sz(? r)?:

?Sz(? r)? = η? r(S − ?b†

This result for the spin-wave corrections to the ground

state spin texture then allows us to write nz

limL→∞(?

nz= S − ∆

where ∆ represents the leading spin-wave correction to

the classical value for nz.

In order to obtain nzreliably in this manner, it is im-

portant to understand the finite size scaling properties

of ∆ for various values of J

model and J2/J in the model with next-nearest neigh-

bour interactions. In Fig. 7, we show a typical exam-

ple of this size dependence. As is clear, we find that ∆

has a non monotonic dependence on L: ∆ initially in-

creases rapidly with size, and, after a certain crossover

size L∗, it starts decreasing slowly to finally saturate to

its asymptotic value. This non-monotonic behaviour is

qualitatively similar to that observed in the finite size

extrapolations of nzfrom our QMC data earlier. To ex-

plore this unusual size dependence further and reliably

extrapolate to the thermodynamic limit, we analyze the

contributions to ∆ from the spin-wave spectrum in the

following way: We note that there is always a mono-

tonically and rapidly convergent O(1) contribution to ∆

from the lowest frequency spin-wave mode, whose spin-

wave frequency scales to zero as 1/Ntot (for any finite

Ntot, this is not an exact zero mode of the system). We

dub this the ‘delta-function contribution’ and its thermo-

dynamic limit is easy to reliably extrapolate to. In ad-

dition, there is a ‘continuum contribution’ coming from

all the other spin-wave modes, each of which contributes

an amount of order O(1/Ntot). This contribution con-

verges less rapidly to the thermodynamic limit, and also

happens to be non-monotonic: it first increases quickly

with increasing size, and then starts decreasing slowly to

finally saturate to the thermodynamic limit.

The delta-function contribution can be fit best to a

functional form

? rb? r?)(50)

=

? rη? r?Sz(? r)?)/Ntotas

(51)

′/J in the striped interaction

Fδ(L) = bδ+cδ

L−aδ

L2+dδ

L3, (52)

with the dominant 1/L2term accounting for the mono-

tonic increase with L, while the continuum contribution

is fit to

Fc(L) = bc+cc

L−ac

L2+dc

L3, (53)

whereby the size dependence is predominantly deter-

mined by the competition between the term proportional

to 1/L which decreases with increasing L, and the term

proportional to 1/L2which increases with increasing L.

20 3040

5060

708090

L

0.4992

0.4994

0.4996

0.4998

0.5

∆

δ

Spin-wave theory

Fδ

aδ= 0.476, bδ= 0.50000, cδ= -0.00038, dδ= 0.85

2030 40

5060

7080 90

L

0.237

0.238

0.239

0.24

0.241

∆

c

Spin-wave theory

Fc

ac= 20.6, bc= 0.2350, cc= 0.63, dc= 147.0

2030 40

50 60

70 80

L

0.736

0.737

0.738

0.739

0.74

0.741

∆

Spin-wave theory

Fδ+Fc

FIG. 7: A typical example of the finite size scaling of the delta-

function and continuum contributions to ∆. Note the mono-

tonically increasing size dependence of the delta-function con-

tribution, and the non-monotonic and more slowly converging

nature of the continuum contribution. Due to this difference

in their behaviour, we find it more accurate to separately fit

each of these contributions to a polynomial in 1/L and use

this to obtain the thermodynamic limit of the total ∆. Here

Fδ/c(L) = bδ/c+ cδ/c/L − aδ/c/L2+ dδ/c/L3.

Page 9

9

This gives rise to non-monotonic behaviour whereby the

continuum contribution first increases rapidly and then

decreases slowly beyond a crossover length L∗to finally

saturate to its infinite volume limit. We also find that the

length L∗gets larger as we deform away from the pure

square lattice antiferromagnet, making it harder to ob-

tain reliable extrapolations to the thermodynamic limit.

Using such careful finite-size extrapolations to obtain

∆ for various values of J2/J and J

result with ∆′calculated analytically. Specifically, we

now ask if the universality seen in our QMC results is re-

flected in these semiclassical spin-wave corrections to nz

and m. The answer is provided by Fig. 8, which shows

that the numerically obtained spin-wave corrections ap-

parently satisfy a universal linear relationship

′/J, we compare the

∆ − ∆′≈ 1.003 + 0.013∆′

(54)

as one deforms away from the pure square lattice anti-

ferromagnet in various ways.

What does this imply for nz(m) to leading order in

1/S? To answer this, we note that

nz

m= 1 −∆ − ∆′

S

+ O(S−2) (55)

Using our numerically established universal result to re-

late ∆ − ∆′to ∆′and thence to m itself, we obtain the

universal relationship

nz= αm + βm2

(56)

with α ≈ 0.987 − 1.003/S and β ≈ 0.013/S. However,

being a large-S expansion, spin-wave theory is unable to

give a quantitatively correct prediction for nz(m) for the

S = 1/2 case.

Finally, we use our spin-wave predictions for the

ground-state spin texture to look at the Fourier trans-

form of the spin-texture for various deformations of the

pure antiferromagnet. The results are shown in Fig. 9,

which demonstrates that spin-wave theory also predicts

that the Fourier transform of the spin-texture near the

antiferromagnetic wave-vector is a universal function of

the wavevector; this provides some rationalization for the

observed universality of the Fourier transformed spin tex-

ture seen in our QMC numerics.

B. Sublattice-spin mean-field theory

We now turn to a simple mean-field picture in terms

of the dynamics of the total spins?SA and?SB of the A

and B sublattices respectively. When NA = NB+ 1,

it is clearly appropriate to assume that the total spin

quantum number of?SAis SB+ 1/2 while the total spin

quantum number of?SBshould be taken to be SB, where

SB= NB/2 tends to infinity in the thermodynamic limit.

In this mean-field treatment, we assume that?SAand

?SB are coupled antiferromagnetically in the effective

1.1

JJ′ model

JJ2 model

∆-∆′ = 1.003 + 0.013 ∆′

0 0.2 0.4

0.6

0.8

∆′

0.9

0.95

1

1.05

∆-∆′

FIG. 8: ∆−∆′, the difference between the leading spin wave

corrections to nzand m, plotted against the leading spin-wave

corrections ∆′to m for the JJ

the text.

′and JJ2 models described in

Hamiltonian that describes the low energy part of the

spectrum:

HMF = JMF?SA·?SB

(57)

with JMF > 0. Within this mean-field treatment, the

Stot = 1/2, Sz

tot= 1/2 ground state that we focus

on in our numerics is thus the Stot = 1/2, Sz= 1/2

state obtained by the quantum mechanical addition of

angular momenta SB and SB+ 1/2. Within this mean-

field theory, nzis modeled as the expectation value of

(Sz

B)/Ntot in this state, which can be readily ob-

tained in closed form using the following standard re-

sult for the minimum angular momentum state |J =

j1− j2,mJ? state obtained by the addition of angular

momenta j1and j2(with j1≥ j2):

?j1,m1;j2,m2|J,mJ? = ρJcJ,mJ

A− Sz

m1,m2

(58)

with

ρJ=

?

(2J + 1)!(2j2)!

(2j1+ 1)!

(59)

and

cJ,mJ

m1,m2= (−1)j2+m2[(j1+ m1)!((j1− m1)!]1/2[(j2+ m2)!(j2− m2)!(J + mJ)!(J − mJ)!]−1/2

(60)

Page 10

10

for m1+ m2= mJand cJ,mJ

In our case, j1 = SB + 1/2, j2 = SB, J = 1/2,

mJ = 1/2, and nz= ?m1− m2?J,mJ/Ntotcan therefore

be readily calculated to obtain

m1,m2= 0 otherwise.

nz=

?2

3SB+1

2

?

/Ntot

(61)

within this phenomenological approach.

On the other hand, when NA = NB, we may also

calculate m2= ?(?SA−?SB)2?J=0/N2

sublattice-spin approach

totwithin the same

m2= (4S2

B+ 4SB)/N2

tot. (62)

This allows us to compute the ratio nz/m in the thermo-

dynamic limit:

nz=1

3m + O

?

1

Ntot

?

(63)

Is there a limit in which this sublattice-spin mean-field

theory is expected to give exact results? To answer this,

we note that the sublattice-spin model represents the

Hamiltonian of an infinite-range model in which every A

sublattice-spin interacts with every B sublattice-spin via

a constant (independent of distance) antiferromagnetic

exchange coupling JMF. Thus, our mean-field theory is

expected to become asymptotically exact in the limit of

infinitely long-range unfrustrated couplings. In this limit,

we also expect m → 1/2, and thus, our mean field theory

predicts that nz→ m/3 when m → 1/2. This is the

constraint that we built into our choice of polynomial fit

for nz(m) in Sec. III.

C. Quantum rotor Hamiltonian

When any continuous symmetry is broken, the corre-

sponding order parameter variable becomes very “heavy”

in a well-defined sense.14The long-time, slow dynamics

of this heavy nearly classical variable is controlled by

an effective “mass” that diverges in the thermodynamic

limit.

For a N´ eel ordered magnet, the order parameter is the

N´ eel vector ? n. In the usual case of an antiferromagnet

with an even number of S = 1/2 moments, the low-

energy effective Hamiltonian that controls the orienta-

tional dynamics of the N´ eel vector ? n is

Hrotor=

?L ·?L

2χNtot

(64)

where?L is the angular momentum conjugate to the

“quantum rotor” coordinate ˆ n ≡ ? n/|? n|, χ is the uniform

susceptibility per spin, and Ntot is the total number of

spins.

What about our case with NA= NB+ 1 and an odd

number of spins Ntot? Following earlier work on quantum

rotor descriptions of insulating antiferromagnets doped

with a single mobile charge-carrier17, we postulate that

the correct rotor description of our problem is in terms of

a rotor Hamiltonian in which?L is replaced by the angular

momentum operator?L′conjugate to a quantum rotor

coordinate ˆ n that now parametrizes a unit-sphere with a

fundamental magnetic monopole at its origin.18In other

words, we postulate a low-energy effective Hamiltonian

H1/2

rotor=

?L

2χNtot

′·?L

′

(65)

where the superscript reminds us that the lowest allowed

angular momentum quantum number l of the modified

angular momentum operator?L

In the notation of Ref 18, the angular wavefunction of

the l = 1/2, ml= 1/2 ground state of this modified rotor

Hamiltonian is the monopole harmonic Y1/2,1/2,1/2(θ,φ).

To model ?nz?↑, we must compute the expectation value

?cos(θ)?1/2,1/2,1/2and multiply this result by m ≡ |? n|.

To do this we note that

′is l = 1/2.

|Y1/2,1/2,±1/2(θ,φ)|2=

1

4π(1 ± cos(θ)) , (66)

which immediately implies

?nz?↑= m

?

dcos(θ)dφcos(θ)|Y1/2,1/2,1/2(θ,φ)|2=1

3m

(67)

Thus, a more general phenomenological approach that

goes beyond sublattice-spin mean-field theory but ignores

all non-zero wavevector modes also gives

nz=m

3

. (68)

Since our QMC data show clear deviatons from this re-

sult, we conclude that such non-zero wavevector modes

are essential for a correct calculation of the universal

function nz(m).

V. DISCUSSION

A natural question that arises from our results is

whether the universal ground state spin texture we have

found here can be successfully described using an effective

field theory approach of the type used recently by Eggert

and collaborators for studying universal aspects of the

alternating order induced by missing spins in two dimen-

sional S = 1/2 antiferromagnets.19This approach uses

a non-linear sigma-model description of the local anti-

ferromagnetic order parameter, with lattice scale physics

only entering via the values of the stiffness constant ρs

and the transverse susceptibility χ⊥, and the presence of

the vacancy captured by a local term in the action. An

analogous treatment for our situation would need two

things—one is a way of restricting attention to averages

Page 11

11

2.53 3.5

q

10-6

10-5

10-4

10-3

10-2

10-1

100

|Sz(q,q)|

J2= 0.3

J′= 6.0

2.53 3.5

q

10-4

10-3

10-2

10-1

100

|Sz(q,π)|

J2= 0.3

J′= 6.0

FIG. 9: Fourier transform (with antiperiodic boundary conditions assumed for convenience) of the spin-wave result for Φz(? r)

(assuming S = 3/2 and calculated using L = 75 for JJ2 and JJ′model) along cuts passing through the antiferromagnetic

wavevector (π,π). Note the nearly universal nature of the results in the neighbourhood of the antiferromagnetic wavevector,

which in any case accounts for most of the weight of the transformed signal.

in the Stot = 1/2 component |G?↑ of the ground state

doublet, and the other is an understanding of the right

boundary conditions or boundary terms in the action, so

as to correctly reflect that fact that our finite sample has

open boundaries. We leave this as an interesting direc-

tion for future work, which may shed some light on the

role of non-zero wavevector modes that were left out of

the rotor description of the earlier section.

VI.ACKNOWLEDGEMENTS

We thank L. Balents, A. Chernyshev, M. Metlit-

ski, S. Sachdev, R. Shankar and R. Loganayagam

for useful discussions.

ported by Grants DST-SR/S2/RJN-25/2006 and IFC-

PAR/CEFIPRA Project 4504-1, and that of AWS by

NSF Grant No. DMR-1104708. The numerical calcula-

tions were carried out using computational resources of

TIFR. AWS gratefully acknowledges travel support from

the Indian Lattice Gauge Theory Initiative at TIFR.

The work of KD was sup-

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