Scaling in the Fan of an Unconventional Quantum Critical Point

0.25 MB·English

by  Roger G. Melko

Scaling in the Fan of an Unconventional Quantum Critical Point
Roger G. Melko1,2 and Ribhu K. Kaul3
1Department of Physics and Astronomy, University of Waterloo, Ontario, N2L 3G1, Canada
2Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge TN, 37831
3Department of Physics, Harvard University, Cambridge MA, 02138
(Dated: February 1, 2008)
Wepresentresultsofextensivefinite-temperatureQuantumMonteCarlo simulationson aSU(2)
symmetric S = 1/2 quantum antiferromagnet with a four-spin interaction [Sandvik, Phys. Rev.
Lett. 98, 227202 (2007)]. Our simulations, which are free of the sign-problem and carried out
on lattices containing in excess of 1.6×104 spins, indicate that the four-spin interaction destroys
8
the N´eel order at an unconventional z = 1 quantum critical point, producing a valence-bond solid
0
paramagnet. Ourresults are consistent with the‘deconfined quantumcriticality’ scenario.
0
2

n ResearchintothepossiblegroundstatesofSU(2)sym- l k
a
J metric quantum antiferromagnets has thrived over the T m
9 last two decades, motivated to a large extent by the un- oble i j
dopedparentcompoundsofthecupratesuperconductors. Pr Quantum Critical Fan
] In these materials, the Cu sites can be well described as gn
el S = 1/2 spins on a two-dimensional (2D) square lattice Si VBS Neel
- that interact with an anti-ferromagnetic exchange, the
r -0.02 0 0.02 0.04 0.06 0.08 0.1
t archetypalmodel for which is the Heisenberg model. By J/Q
s
. now,itiswellestablished[1]thatthegroundstateofthis
t
a model with nearest-neighbor interaction has N´eel order FIG. 1: (color online) Schematic of the proposed T-J/Q
m
that spontaneously breaks the SU(2) symmetry. Two phase diagram of the JQ model. Large-scale finite-T sim-
- logicalquestions immediately arise: Whatpossible para- ulations presented here substantiate the following: (i) The
d magneticgroundstatescanbereachedbytuningcompet- T =0N´eelorderpresentfor J/Q≫1isdestroyedat aQCP
n (J/Q≈0.038);(ii)Inthe‘quantumcriticalfan’,thereisscal-
ing interactions that destroy the N´eel state? Are there
o
ing behavior characteristic of a z =1 QCP; (iii) An accurate
c universal quantum-critical points (QCP) that separate
estimateofthescaling dimensionoftheN´eelfieldestablishes
[ these paramagnets from the N´eel phase?
that thistransition isnot in theO(3) universality class; and,

2 (iv)TheparamagneticgroundstateforsufficientlysmallJ/Q
v An answer to the first question is to disorder the N´eel isaVBS.IntheQMCbasisusedhere,theregionwithQ<0
1 state by the proliferation of topological defects in the is sign problematic. The inset shows how the frustrating Q
6 N´eel order parameter [2]. It was shown by Read and term is written in terms of bondson a plaquette.
9
Sachdev [3] that the condensation of these defects in
2
the presence of quantum Berry phases results in a four-
. with a frustrating four-spin interaction,
7
folddegenerateparamagneticgroundstate,whichbreaks
0
square-latticesymmetryduetotheformationofacrystal 1 1
7 H =J S ·S −Q (S ·S − )(S ·S − ), (1)
JQ i j i j k l
0 ofvalencebonds–avalence-bondsolid(VBS)phase. An X X 4 4
hiji hijkli
: answer to the second question was posed in recent work
v
by Senthil et al.[4], where the possibility of a direct con- whereindicesarearrangedasintheinsetofFig.1. Using
i
X tinuousN´eel-to-VBStransitionwasproposed. Thenatu- aT =0projectorQuantumMonteCarlo(QMC)method
r ralfieldtheoreticdescriptionofthis‘deconfinedquantum onlatticessizesupto32×32[6],Sandvikshowedthatthe
a
criticalpoint’iswrittenintermsofcertainfractionalized four-spin interaction destroys N´eel order and produces a
fields that are confined on either side of the QCP and VBS phase at J/Q∼0.04. Close to this critical value of
become ‘deconfined’ precisely at the critical point. As J/Q, scaling in the spin and dimer correlation functions
is familiar from the general study of QCPs, these frac- suggests a continuous transition with anomalous dimen-
tionalexcitationsareexpectedtoinfluencethephysicsin sions of the N´eel and VBS order parameters equal, with
a large fan-shaped regionthat extends above the critical a common value η = 0.26(3). In this Letter, we explore
point at finite-T [5] (see Fig. 1). the candidate N´eel-VBS QCP in the full T −J/Q phase
diagramonlargelatticesusingacomplementaryfinite-T
Itisclearlyofgreatinteresttofindmodelsthatharbor QMC technique, the Stochastic Series Expansion (SSE)
a direct N´eel-VBS QCP and that can be studied with- method with directed loops [7]. The SSE QMC allows
out approximation on large lattices. Currently, the best access to the physically important quantum critical fan
candidate is the ‘JQ’ model, introduced by Sandvik [6], (see Fig. 1), and admits high-accuracy estimates for the
which is an S = 1/2, SU(2) invariant antiferromagnet spinstiffness,ρ ,andtheuniformsusceptibility,χ . The
S u
2
0
10 LηN−1SN
0.0152mV0.0004 10-1
L=12
2mN0.01 J/Q=0.01 L=16 LηN−2χN
0 J/Q=0.034 L=24
0 0.02 L=32
J/Q=0.036 -2
10 L=48
J/Q=0.038
0.005 L=64
J/Q=0.040
L=80
J/Q=0.042 L=128
J/Q=0.1 -3
10
00 0.02 0.04 0.06 0.08 10-1 100 101 102
1/L LzT
FIG. 2: (color online) T → 0 converged N´eel (main) and
FIG.3: (coloronline)CriticalityoftheN´eelfieldatJ =0.038:
VBS (inset) order parameters as a function of 1/L. Dashed
collapse of the N´eel structure factor (S ) and susceptibility
linesarequadraticfitsthatillustrate thefinitecondensatein N
(χ ) with z = 1 and η = 0.35, determining the universal
tyh=e ocr1dxecr2ed(ilpluhsatsreast.edThfoersJo/liQd (=red0.)04li0n)e, iwshaerfietct2o=thze+forηmN founnNtchtieoxnsaXndS(yx)axaensd).XTχh(Nex)on(ulypfittopnaorna-muentievrerfsoarlbsocathleSfactaonrds
is expected at the critical coupling. In fitting to the nine L N
χ is η , theanomalous dimension of theN´eel field.
values for each J/Q, we find a minimum in the chi-squared N N
value (per degree of freedom) of 3.1 for J/Q = 0.040, with
c2 ≈ 1.35(1). For J/Q = 0.038, the chi-squared value is 3.9,
with c2 ≈ 1.37(1). All other J/Q produce much larger chi- S(r) · S(r+xˆ) as well as the traceless symmetric ten-
square (greater than 10). sor constructed from Si(r)Sj(r+xˆ). Structure factors
for the N´eel and VBS phases are constructed from these
correlation functions by Fourier transformation at equal
scaling of these observables provides strong evidence for time, S [q] = [exp(−iq · r)Cz (r,τ =0)]/N ,
N,V r N,V spin
a continuous z =1 transition in the JQ model. from which the orPder parameters are defined at the ob-
Basis and Sign of Matrix Elements: A priori, it is un- served ordering wavevectors: m2N,V = SN,V[qN,V]/Nspin.
clear that SSE simulations of HJQ are free of the notori- Zero-frequency susceptibilities (χN and χV) are con-
ous sign-problem: a fluctuating sign in the weights used structedby integratingoverallτ andFourier transform-
in the QMC sampling. In the SSE, finding an orthog- ing in space to the ordering vectors, qN,V.
onal basis in which all off-diagonal matrix elements of Examination of the full q-dependent structure factors
theHamiltonianarenon-positivesolvesthesign-problem. indicate the presence of sharp ordering wavevectors in
A simple unitary transformation on the Sz basis (a π- S [q = (π,π)] for large J/Q and S [q = (π,0) or
N N V V
rotation about the z-axis on one sub-lattice) results in a (0,π)](thelatterinthecasewherethecorrelatorismea-
new basis in which, for J,Q > 0, all off-diagonal matrix sured with yˆ) for large Q/J [8], confirming the N´eeland
elements of HJQ are non-positive, allowing sign-problem VBSphasesobservedinRef.[6]. AsshowninFig.2,T →
freesimulations(Fig.1). Wenotethatthisnon-positivity 0 convergeddata scales convincingly to a non-zero value
condition is also the main ingredient in the proof of for m2 at J/Q = 0.1 and for m2 at J/Q = 0.01. The
N V
the Marshall sign theorem, allowing us to infer that the critical coupling appears to occur between J ≈ 0.038
c
ground state of HJQ for J,Q>0 must be a spin-singlet. and 0.040 (we set Q = 1 fixed throughout), such that
As shown below, this singlet state changes from N´eel at as J is approached from above (below) the extrapo-
c
Q≪J to VBS at J ≪Q. lated N´eel (VBS) order parameter is suppressed. Very
Numerical Results: Using the SSE QMC, we studied near J , both order parameters vanish within our er-
c
various physical observables in the JQ model on finite- ror bars, while a power law with no y-intercept fits the
size lattices of linear dimension L (with number of spins N´eel data with high accuracy. More specifically, at J ,
c
Nspin = L2). Particular attention was paid to the scal- scaling arguments require SN ∝ L1−ηNXS(LzT/c) and
ing ofthe spinstiffness ρs =∂2E0/∂φ2 (E0 is the energy χN ∝ L2−ηNXχ(LzT/c), with ηN the anomalous dimen-
andφisatwistintheboundaryconditions)andtheuni- sion of the N´eel field. In Fig. 3, we verify this scaling
form spin susceptibility χ =h( Sz)2i/TN . In the behavior and determine the universal functions X and
u i i spin S
Sz basis used here, it is easy toPmeasure the correlation Xχ. BothanalysesillustratedinFigs.(2,3)giveaconsis-
functions Cz(r,τ) = hSz(r,τ)Sz(0,0)i and Cz(r,τ) = tent estimate of η ≈ 0.35(3). This value is larger than
N V N
h[Sz(r,τ)Sz(r+xˆ,τ)][Sz(0,0)Sz(xˆ,0)]i. WhileCz isthe the resultofη ≈0.26(3)fromRef. [6]. While the exact
N N
correlationfunctionoftheN´eelorderparameter,theVBS source of this discrepancy is unclear due to the entirely
orderisindicatedbyCz,whichisthecorrelationfunction differentmethodsusedtoextracttheexponents,wenote
V
ofthe compositeoperatorSz(r)Sz(r+xˆ),receivingcon- that (a) our analysis does not involve extra fit param-
tribution from both the standard VBS order parameter eters from the inclusion of sub-leading corrections, and
3
J/Q=0.042 J =0.039 0.02
J/Q=0.040 0.5 c 0.015
J/Q=0.038 0.50.4 0.01
0.006
J/Q=0.036 0.3 ν=0.68 0.005 0.015
J/Q=0.034 -100-50 0 50 -100-50 0 50
J/Q=0.032 J−JcL1/ν J−JcL1/ν
χu 0.004 1e-04 Jc Jc
a 0 0.4
ρsL χuL 0.01
-0.0001
0.002 -0.0002 L=24
L=32
0.032 0.036 0.04 L=40
J/Q 0.3 L=48
0 L=64 0.005
0 0.05 0.1 0.15 0.2
T
0.02 0.03 0.04 0.02 0.03 0.04
FIG. 4: (color online) Finite-T uniform susceptibility, for a J/Q J/Q
L = 128 system near J . Error bars are much smaller than
c
the symbol size. For the region 0.08 ≤T ≤0.18, the data is
FIG. 5: (color online) Zoom-in of the spin stiffness and sus-
highly linear, and a straight-line fit for J/Q=0.038 (shown)
ceptibility close to the expected critical point, taken for sim-
interceptstheoriginwithinerrorbars. Interceptsofstraight-
ulations cells of size L = 1/T. Data in the inset is scaled to
line fits for all data sets are in the inset. From the slope of
getthebestcollapseforthelargestsystemsizes,whichoccurs
thelinear-T behavior we obtain A /c2 =0.0412(2).
χ for J ∼0.039(1) and ν ∼0.68(4).
c
(b) the collapse of both S and χ takes place overtwo
N N
χ . The inset shows how the extracted value of the y-
andahalfordersofmagnitudeofLT withonlyonecom- u
intercept,a(fromafittotheforma+bT),changessignas
mon fit parameter, η ; both facts give us confidence in
N
our estimate. The critical scaling of Cz is more compli- thecouplingistuned,consistentwith0.036≤Jc ≤0.040
V
and demonstrating to high precision the z =1 scaling.
cated; due to the aforementioned mixing-in of two order
parameters, Cz is expected to receive two independent Turning to study Eqs. (2,3) further, one may hold the
V
power-law contributions. Indeed, it is difficult to disen- first argument of the universalfunctions fixed by setting
tangletheseindividualcontributionsonthelimitedrange L = 1/T (assuming z = 1 as indicated above). In order
oflatticessizesavailable,precludingusfromverifyingthe to achieve this, we performed extensive simulations on
proposal [6] that η =η . lattices sizes up to L = 1/T = 64, illustrated in Fig. 5.
N V
We now turn to an analysis of the scaling properties According to Eqs. (2,3), data curves for Lρs and Lχu
of χ and ρ in the hypothesized quantum critical fan plotted versus J should show a crossing point with dif-
u s
region of Fig 1. χu and ρs, being susceptibilities of con- ferent L precisely at Jc. We find that for relatively large
served quantities have no anomalous scaling dimension, sizes (32≤L≤64) the crossing point converges quickly
and hence at finite-T and L in the proximity of a scale- in the interval 0.038 ≤ J ≤ 0.040. The insets show the
invariant critical point, assuming hyper-scaling: data collapse when the x-axis is re-scaledto gL1/ν (with
ν = 0.68). We note that with the inclusion of small
ρ (T,L,J) = T Y LzT,gL1/ν , (2) sub-leading corrections (of the form aω/Lω), the cross-
s Ld−2 (cid:18) c (cid:19) ing point and data collapse of ρ and χ can be made
s u
1 LzT consistent, at the expense of two more fit parameters,
χ (T,L,J) = Z ,gL1/ν , (3)
u TLd (cid:18) c (cid:19) even for much smaller system sizes than illustrated [9].
In contrast to the U(1) symmetric JK model [11], where
where g ∝ (J −J )/J . At criticality (g = 0), it is easy theabsenceofaT-linearχ andacrossinginthedatafor
c c u
to see that Y(x → 0,0) = A /x and Z(x → ∞,0) = ρ L castdoubtonits interpretationas a z =1QCP,the
ρ s
A xd/z, where Y(x,y) and Z(x,y) are universal scal- presentdataforthisSU(2)symmetricmodelgivesstrong
χ
ing functions andA ,A areuniversalamplitudes ofthe support for a z =1 QCP between 0.038≤J ≤0.040.
χ ρ
quantum critical point; c is a non-universalvelocity. Finally, we hold the second argument of the scaling
At criticality and L → ∞, one can show from Eq. (3) functions [Eqs. (2,3)] constant by tuning the system to
that χ = Aχ Td/z−1; i.e. for a z = 1 transition, χ g = 0. One then expects a data collapse for ρ /T and
u cd/z u s
should be T-linear and have a zero intercept on the y- Lχ whentheyareplottedasafunctionofLzT (withz =
u
axis at T = 0 [10]. In Fig. 4, χ data for an L = 128 1). Fig. 6 shows this collapse for simulations carried out
u
system is presented. Within our error bars, this data is with extremely anisotropic arguments LT, varying over
L → ∞ converged for the region of T shown; at smaller almost three orders of magnitude. There is an excellent
T the finite-size gap causes an exponential reduction in data collapse over 8 orders of magnitude of the range of
4
103 which the N´eel-VBS transition is described by the non-
102 Ld−2ρ TLdχ compactCP1 field theory. All of the qualitative observa-
s u
T
101 tions above, including an unusually large ηN [13] agree
with the predictions of this theory. Indeed, our esti-
100
mate of η ≈ 0.35 [Fig. 3] is in remarkable numerical
10-1 N
agreement with a recent field-theoretic computation [14]
10-2 LL==1126 of this quantity, which finds ηN = 0.3381. With regard
10-3 L=24 to other detailed quantitative comparisons,we havepro-
L=32
10-4 L=48 videdthefirststepbycomputingmanyuniversalquanti-
10-5 L=64 ties, Xχ(x), XS(x), Y(x,0), Z(x,0) andAρ Aχ ≈0.075
L=80
10-6 L=128 [Fig.6]intheJQmodel. Analogouscompuptationsinthe
CP1model,althoughcurrentlyunavailable[15]arehighly
10-7
0.1 1 10 100 desirable to further demonstrate that the JQ model re-
LzT
alizes this new and exotic class of quantum criticality.
FIG. 6: Scaling of χ and ρ at J =0.038≈J , with z =1
u s c WeacknowledgescintillatingdiscussionswithS.Chan-
andd=2. TheseplotsaretheuniversalfunctionsY(x,0)and
Z(x,0) up to the non-universal scale factor c on the x-axis. drasekharan, A. del Maestro, T. Senthil, and especially
The expected asymptotes (see text) are plotted as dashed S. Sachdev and A. Sandvik. This research (RGM)
lines Y(x→0,0)=A /x and Z(x→∞,0)=A xd/z. From wassponsoredbyD.O.E.contractDE-AC05-00OR22725.
ρ χ
fitstothedata,wefindA /c2 =0.041(4)andA c=0.37(3), RKK acknowledges financial support from NSF DMR-
χ ρ
allowing us to estimate a universal model-independent num- 0132874, DMR-0541988 and DMR-0537077. Computing
ber associated with theQCP, AρpAχ ≈0.075(4). resources were contributed by NERSC (D.O.E. contract
DE-AC02-05CH11231), NCCS, the HYDRA cluster at
Waterloo,andthe DEASandNNIN clustersatHarvard.
theuniversalfunctions,withnofitparameters. Thisdata
together with that in Fig. 4 provide our most striking
evidence for the existence of a QCP with z = 1 in the
proximity of J/Q≈0.038.
Discussion: In this paper we have presented extensive [1] E. Manousakis, Rev.Mod. Phys. 63, 1 (1991).
data for the SU(2) symmetric JQ model which indicates [2] F. D.M. Haldane, Phys.Rev.Lett. 61, 1029 (1988)
that the N´eel order (present when J ≫ Q) is destroyed [3] N. Read and S.Sachdev,Phys. Rev.B 42, 4568 (1990).
[4] T. Senthilet al. Science 303, 1490 (2004); Phys.Rev. B
ata continuousquantumtransitionasQ is increased[6].
70, 144407 (2004).
Inthefinite-T quantumcriticalfanabovethisQCP,scal-
[5] S. Sachdev, Quantum Phase Transitions (Cambridge
ing behavior is found that confirms the dynamic scaling University Press, New York,1999).
exponentz =1tohighaccuracy. Theanomalousdimen- [6] A. W. Sandvik,Phys.Rev.Lett. 98, 227202 (2007).
sionoftheN´eelfieldatthistransitionisdeterminedtobe [7] O. F. Sylju˚asen and A. W. Sandvik, Phys. Rev. E 66,
η ≈ 0.35(3), almost an order of magnitude more than 046701 (2002); R. G. Melko and A. W. Sandvik, Phys.
N
its value of 0.038 [12] for a conventionalO(3) transition. Rev. E72, 026702 (2005).
[8] The VBS order is also visible in measurements of corre-
For sufficiently largevalues of Q we find that the system
lation functions between off-diagonal terms [9].
enters a spin-gapped phase with VBS order. To the ac-
[9] R. K. Kauland R.G. Melko, unpublished.
curacyofoursimulations,ourresultsarefully consistent [10] A. V.Chubukovet al., Phys.Rev.B 49, 11919 (1994).
withadirectcontinuousQCPbetweentheN´eelandVBS [11] A. W. Sandvik and R. G. Melko, cond-mat/0604451
phases, with a critical coupling between J/Q ≈ 0.038 (2006); Ann.Phys.(NY),321, 1651 (2006).
and J/Q ≈ 0.040. Although our finite size study can- [12] M. Campostrini et al.,Phys. Rev.B 65, 144520 (2002).
not categorically rule out a weak first-order transition, [13] O. I. Motrunich and A. Vishwanath, Phys. Rev. B 70,
075104 (2004).
we have found no evidence for double-peaked distribu-
[14] Z. Nazario and D. I. Santiago, Nucl. Phy. B 761, 109
tions, indicating an absence of this sort of first-orderbe-
(2007)
havior on the relatively large length scales studied here. [15] A. Kuklov et al., Annals of Physics 321, 1602 (2006);
It is interesting to compare our results to the only the- found a discontinuous transition in a U(1) deformation
ory currently available for a continuous N´eel-VBS tran- oftheCP1model.ResultsrelevanttotheSU(2)invariant
sition: the deconfinedquantumcriticalityscenario[4],in HJQ are so far unavailable (see however[13]).