Susceptibility of the transverse field Ising model on the square lattice

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by  A. Kashuba

Susceptibility of the transverse field Ising model on the square lattice
A. Kashuba
Bogolyubov Institute for Theoretical Physics, 14-b Metrolohichna, Kiev 03680 Ukraine
Susceptibility of the transverse field Ising model on the square lattice is calculated numerically
in the paramagnetic phase in a wide range of temperatures and transverse fields. An expression
with one constant π, that determines both the critical exponent γ and the critical transverse field,
compellinglyrepresentsthedataasymptoticallynearthequantumcriticalpoint,exceptforanarrow
1 classical region close to thephase transition line, and shows two crossovers as temperature varies.
1
0 PACSnumbers: 05.30.Rt,64.60.-i,75.40.Mg
2

n
Quantumfieldtheoryhavingbeenderivedfromanex- wherespin-half~σ residesonsitesxofthesquarelattice.
a x
J plicitly Lorentz invariant model encounters divergences We set the exchange coupling to unity, J = 1, leaving
whenimposingquantummechanics. Alternatively,quan- a single parameter, the strength of the transverse field
6
2 tum models defined on lattices explicitly obey the quan- H, in the Hamiltonian eq.(1). Without an external field
tummechanicsbuttheLorentzinvariance,theconformal theHamiltoniandoescommutewiththeparityoperator:

] symmetry as well as other symmetries emerge, if at all, Pˆ = σz. Accordingly, all eigenstates of this model
l x x
al only in the long wavelength limit [1]. This continuum are diQvided into even and odd states of equal number.
h limitcanbefoundeitherinspecialmasslessphases,such Availableexperimentalrealizationsofthetransversefield
- asquantumantiferromagnets,orinthevicinityofaphase Ising model either include long range interactions [6] or
s
e transitionknownasaquantumcriticalpointatzerotem- are one-dimensional magnetic chains [7].
m perature. Novelemergentphenomenanearquantumcrit- Phase diagram (H,T) of the transverse field Ising
. icalpointsisofgreatinterestincondensedmatterphysics model on the square lattice is shown schematically on
t
a [2]. Phase diagrams near quantum critical points show Fig.1. There are two phases, a paramagnetic phase at
m abundant characteristic crossovers [3] like a pseudogap high temperatures and high transversefields and a mag-
- phenomenon in the high-temperature superconductors. netically ordered along the longitudinal direction phase
d
Symmetry brings in simplicity. First demonstrated by at low temperatures and low transverse fields. There is
n
Maldacena [4], quantitative description of quantum lat- always a non-vanishing magnetization in the transverse
o
c tice models governed by a symmetry may correspond to direction that grows with H. A line of phase transi-
[ a solution of the classical Einstein equation for gravi- tions(H,T (H)),notfoundinthispaper,separatesthese
c
1 tational fields in special settings representing the same two phases. On the right, it ends in a quantum criti-
v symmetry. Itis plausiblethatsuchasolutioninthe case cal point, (Hc,0), and, on the left, in a classical critical
7 of the quantum critical point would be given in terms point,(0,T (0)),thephasetransitionoftheclassicaltwo-
c
7 of a simple, ’school-curriculum’ function. Provided this dimensional Ising model at T0 = 2/asinh(1) =2.269185.
0
simplicity, underpinned by unknown symmetry, precise At the approachesto the phase transition line the longi-
5
numerical simulations may be sufficient to guess the an- tudinal susceptibility diverges according to a power law:
.
1
swer for some physical questions. This paper, based on
0 numericaldata,revealsthatinthe caseofthe transverse T −7/4
1 χ(H,T)=C+(H) 1 , (2)
1 field Ising model on the square lattice the magnetic sus- cl (cid:18)T (H) − (cid:19)
c
: ceptibility canbe representedasasimple expressionina
v
wide range of temperatures and transverse fields around corresponding to the universality class of the classical
i
X the quantum critical point. In particular, both the posi- two-dimensional Ising model [8]. On the line T = 0, the
r tion of the quantum critical point and the critical expo- quantum two-dimensional transverse field Ising model is
a
nentofthemagneticsusceptibilityisspecified. Thereare equivalent to the classicalthree-dimensional Ising model
twocrossoversatT∗ =1/πandT∗ =π intheexpression [9, 10]. At the approaches to the quantum critical point
1 2
for magnetic susceptibility. along the line T = 0 the longitudinal susceptibility di-
The spin-half transverse field Ising model [5] on the verges as
square lattice is anisotropic in the spin space with two
distinctaxis,longitudinalxandtransversez. TheHamil- H −γ
χ(H)=C+ 1 , (3)
tonianincludestheexchangeinteractionbetweennearest Q(cid:18)H − (cid:19)
c
neighbors xy ,theZeemanenergyinthetransversefield
h i
H and the external longitudinal magnetic field h: where γ, found to lie in the interval γ = 1.23...1.24
in many studies both quantum and classical [11–14], is
Hˆ = J σx σx H σz h σx , (1) a critical exponent of the susceptibility in the classical
− x y− x− x
hXxyi Xx Xx three-dimensional Ising model.
2
the lattice Hamiltonianeq.(4) andthe longitudinalmag-
netization eq.(5) onto one embedding of this graph into
the lattice. Both Hˆ and Mˆ do not depend on the par-
g g
ticular embedding. The operator Mˆ connects even and
g
oddstates. TheevenandoddblocksofHˆ aswellasMˆ
g g
have all equal dimension.
Wearriveatthefollowingalgorithm. Weenumerateall
graphs embeddable into the square lattice. There are in
total 1,1,2,4,6,14,28,68,156,399,1012 graphs embed-
dable into the square lattice with the number of edges
1,2,3,4,5,6,7,8,9,10,11correspondingly [16]. For each
graph g the graph Hamiltonian Hˆ = i E (i) i is
g i| i g h |
diagonalized numerically whereas the grPaph magnetiza-
tion iMˆ j is rotated into the new basis. The Gibbs
g
h | | i
FIG. 1: Phase diagram (H,T) of the transverse field Ising thermodynamicaverageofthegraphsusceptibilityreads:
model. Insidethegreenboxthesusceptibilitydatahavebeen
ecnoldleinctgeidn.tRheedq,ulaownteurm,licnreitiiscatlhpeoIisnintgHpch.aBseluter,amnsiidtdiolne,Tlicn(eHis) χg = hiE|Mˆ(gj|)jihjE|Mˆ(gi|)iie−Eg(i)/T/ e−Eg(i)/T
a spurious quantumtransition TQ(H) [singularity in eq.(11)] Xi,j g − g Xi
ending before reaching H = 0 axis. Black, upper, line is a (6)
concept of thequantum-to-classical crossover T∗(H). The Gibbs thermodynamic average of the magnetic sus-
ceptibility on the lattice per one site in the thermody-
namic limit is a sum of the graph susceptibilities:
We calculate numerically the longitudinal susceptibil-
ity using Nickel’s linked cluster expansion method [15]
calledasagraphexpansionbelow. Itappliestoallquan-
χ= Z(g)χg Zg(f)χf , (7)
tumlatticemodelswiththeHamiltonianbeingauniform Xg −Xf∈g
sum over lattice edges:  
where Z(g) is the number of different embeddings of the
H
Hˆ = Hˆe = (cid:20)− 4 σxz +σyz −σxx σyx(cid:21) , (4) graph g into the lattice. f is the sub-graph of the graph
Xe hXxyi (cid:0) (cid:1) g, hence, also embeddable into the lattice. Zg(f) is the
number of different embeddings of the sub-graph f into
andtothetransversefieldIsingmodelinparticular. The
the graph g considered on the lattice.
transverse field Zeeman term as well as the longitudinal
The graph expansion alters in the structure of the
magnetization operator:
Feynman diagrams method where one sums up ’geom-
Mˆ = 1 σx+σx , (5) etry’ in terms of propagatorsand interaction points first
4 x y and gives the result in powers of the coupling. In the
hXxyi (cid:0) (cid:1)
graphexpansion one sums up the interaction in all pow-
for each site can be split into four equal parts assigned ersofthecouplingfirstandgivestheresultingeometrical
to the four incident edges. Our algorithm would not re- patterns, footprints. Also, the graph expansion counts
quire it but in derivation we assume the validity of the largelythe sameprocessesastheLanczosmethodonthe
perturbation theory, for the transverse field Ising model regular6 6 cluster albeit on many thousands of graphs
×
in the limit H . The transverse field Zeeman term bendedandturnedinmanythousandswayswithobserv-
→ ∞
is the unperturbed Hamiltonian whereas the Ising term, ables being given in the thermodynamic limit.
residing on edges, is the perturbation. Usually, pertur- Graphsareclassifiedaccordingtothenumberofedges,
bation terms are arranged by the power. Instead, we referredto hereasaweightw. The sumeq.(7)restricted
assign a footprint i.e. a set of connected edges involved all graphs g with the weight w defines a partial suscep-
in the given perturbation process. Different footprints tibility χ . For each point (H,T) on the phase diagram
w
specify classes of perturbation terms. In general, some thesusceptibilityisgivenbyasumofpartialsusceptibil-
perturbationterms areequalwhile havingdifferent foot- ities, χ(H,T) = χ (H,T). A sequence of numbers
w w
prints. Such footprints are different embedding of the χw(H,T) for w =P1...11 is calculated numerically. In the
same graph into the lattice. Truly distinct classes are paramagnetic phase a typical sequence χ seems to be
w
| |
represented by graphs embeddable into the lattice. The convergent. In the magnetically ordered phase a typical
result of summing up the perturbative series on a graph sequence χ seems to be divergent. At H < H the
w c
| |
isalternativelyfoundusingthematrixquantummechan- sign of χ is positive whereas at H >H the sign of χ
w c w
ics. The Hamiltonian Hˆg and a longitudinal magnetiza- shows typically an irregular pattern. To extrapolate to
tionMˆ ofagraphgareuniquelydefinedasrestrictionof w = whileallowingforaonechangeinthepattern,the
g
∞
3
one approaches the Ising phase transition line and espe-
cially the line T =0 the worse is the quality of the data.
The graph expansion, relying on discreet energy spectra
ofsmallgraphs,isexpectedtobecomeproblematicinthe
limit T 0.
→
In the paramagnetic phase the longitudinal suscepti-
bility is given by the Kubo equation:
2
∞ ∞ dχ
χ(H,T)= [σx(0,0),σx(t,r)] dt= dξ
1 2 Xr Z0 h −i Z0 dξ (9)
where in the spirit of the renormalization group [1] at
3
0 H large distance r 1 the sum proceeds in a scale-wise
1 2 3 4 manner, with t|h|e≫scale ξ = log r. In classical statisti-
T 4 | |
calphysicsthesusceptibilitydensitydχ/dξ isdetermined
byarunningrenormalizationgroupenergy,Hamiltonian,
H (ξ). Asξ growsitapproachesthefixed-pointHamil-
RG
tonianH and, neara criticalpoint,staysin the fixed-
FIG.2: Longitudinalsusceptibilitysortedoutintofivequality FP
point, H (ξ) = H , for a long interval of ξ. Here
grade baskets, darkblue 281 points, blue73 points, pink100 RG FP
points, green 102 points and red 46 points vs (H,T). The a critical part of the susceptibility χC develops. At the
eqs.(10,11,14) is shown as theblack surface. initialtransientscalesaswellasattheexitfromthefixed-
point a regular part of the susceptibility χ develops.
reg
Analogously, we write for the quantum model:
available sequence is split in the middle at some weight
w∗. Then, χw for w <w∗ is summed up whereas χw for χ(H,T)=χQC(H,T)+χreg(H,T) (10)
w w∗ is augmented by a variable z into a polynomial:
≥ Forthequantumcriticalpartofthesusceptibilityweuse
w∗−1 wmax an expression without adjustable parameters:
χ(H,T)= χ (H,T)+ χ (H,T) zw−w∗ (8)
w w
wX=1 wX=w∗ π 2 γ
χ (H,T)= −
QC
(cid:18)γ 1(cid:19) ×
where w = 11. This polynomial is extrapolated us- −
ingthePmaadxeapproximation. Selectingallmiddleweights T γ T γ T2γ +H2 H2 −γ
1 + − c (11)
in the interval 2 w∗ 9 and all degrees of the poly- (cid:18)(cid:18)π(cid:19) (cid:18) −(cid:18)π(cid:19) (cid:19) 1+(T/π)γ (cid:19)
≤ ≤
nomial in the nominator of the Pade approximant while
where the critical transverse field H is related to the
sending z to one gives us forty or so different extrapola- c
critical exponent by two conditions:
tions χ for one point χ(H,T). Their distribution ρ(χ)
i
has a maximum corresponding, probably, to the correct
π 2γ γ2+1
extrapolation χ(H,T), and a tail of those χi that are Hc = γ− 1 , γ2 1 =8−Hc (12)
gone astray when z = 1 comes close to a pole. We dec- − −
imate the extrapolations most distant from the average
or explicitly:
χ , one by one, and stop when four extrapolations re-
i
h i
main. Their average gives us χ(H,T). For a measure 1
γ = π 2+ 328+32π+π2
of quality of thus calculated datum the above procedure 18(cid:16) − p (cid:17)
is repeated twice for wmax = 10 and wmax = 11. The 4+16π 2√328+32π+π2
H = − (13)
absolute difference between the two to the datum value c 20+π+√328+32π+π2
ratio defines a quality of the datum, δ(H,T)= ∆χ/χ. −
| |
Fig.2showsthelongitudinalsusceptibilityofthetrans- Approximately, H =3.03692 and γ =1.226645. Recent
c
verse field Ising model on the square lattice calculated estimate for H is given in ref.[14]. Despite being based
c
numerically in the range 0.1 T 4 and 2 H 4.2 on poor data in this area, the quantum critical suscepti-
≤ ≤ ≤ ≤
withastep∆H =∆T =0.1. Thedatawithapoorqual- bility eq.(11) continues seamlessly onto the line T = 0.
ityδ(H,T)>3.2%arenotshown. χ0 =1/(2Hc)=0.165 Therefore, γ is the critical exponent of the susceptibility
sets an atomic scale for the susceptibility. We sort out ofthe three dimensionalIsingmodel. The quantumcrit-
our data, six hundred points in total, into five quality ical susceptibility shows two crossovers as temperature
baskets: δ 0.05% δ 0.1% δ 0.4% δ varies at T∗ = 1/π and at T∗ = π. For the regular part
1 2
≤ ≤ ≤ ≤ ≤ ≤ ≤
1.6% δ 3.2%, shown in dark blue, blue, pink, green of the susceptibility we try a polynomial. The best fit
≤ ≤
and red colors correspondingly on the Fig.2. The closer is given by the expression that depends on temperature
4
eqs.(10,11,14). The quality δ(H,T) of datum gives us
a crude estimate of how far it may potentially vary
as the maximum weight increases from w = 11 to
0.01 max
w = . Averaging all data variations over a basket
max
∞
of points we get a measure for the final potential vari-
ation. On the other hand, we can measure the current
discrepancy between the data and the fit, at w =11,
max
0.001 andaverageitoverthe samebasketofpoints. Ifthe cur-
rent discrepancy is exceeding the potential variation by
far it is unlikely that the data and the fit will converge
atw . Alternatively,whenthepotentialvariationis
0.001 0.01 →∞
exceeding the current discrepancy by far, a special pat-
tern of alternating signs is necessary for the data and
FIG. 3: Average deviation of the susceptibility data, sorted
the fit to converge that seems also unlikely. The best
outintofivequalitybaskets,from thefitvsaveragevariation
chance for the data and the fit to converge is when the
of the data as themaximum weight increases by one.
potentialvariationapproximatelyequalsthe currentdis-
crepancy. Our data and the fit present widely varying
sharplyaboveT∗ andexplicitlyvanishesatthequantum both measures shown in the Fig.3. Remarkably, these
2
critical point (H ,0): two measures are more or less equal, see Fig.3.
c
In conclusion, using the graph expansion the longitu-
H2 H2 T 9γ
χ (H,T)=a 1 +b (14) dinalsusceptibility of the transversefield Ising modelon
reg Hc2 (cid:18)Hc2 − (cid:19) (cid:18)π(cid:19) the square lattice has been calculated numerically. The
resulthasbeennon-contradictoryinterpretedintermsof
where a = 0.00313 and b = 0.000115. This fit promotes
a simple function. Such interpretation might be useful
the eq.(11) as the asymptotic susceptibility at the quan-
when searching for a corresponding settings in the Ein-
tum critical point. Probably, it also indicates a hidden
stein general relativity. It is quite possible that our in-
symmetry as though the transverse field Ising Hamilto-
terpretation is erroneous. It is also possible that eq.(13)
nian eq.(1) is the fixed point from the beginning of the
givesthe correctcriticalexponentofthe susceptibilityof
renormalization flow. The first term in eq.(14), prob-
the three dimensional Ising model.
ably, describes a tail from a crossover at H∗ 7 where
∼
therecedingquantumcriticalsusceptibilityeq.(11)trans- I am grateful to SLAC Scientific Computing Services
forms into an one-site susceptibility χ(H,T) = 1/(2H) for providing resources for numerical simulations.
at the high transverse fields. There is no evidence of the
quantum-to-classical crossover in the numerical simula-
tions for graphs with weights w 11.
≤
The quantum critical susceptibility eq.(11) has a sin-
[1] seee.g. A.M.Polyakov,’GaugeFieldsandStrings’, Har-
gularityline(H,T (H))onthephasediagramFig.1. We
Q wood (1987)
prove that this line lies above the Ising phase transi-
[2] S. Sachdev, ’Quantum Phase Transitions’, Cambridge
tion line (H,T (H)). The quantum-to-classicalcrossover
c University Press (2001)
occurs at some line (H,T∗(H)), where χQC(T) eq.(11) [3] S. Sachdev,Phys. Rev.B 55, 142 (1997)
transforms into χ (T) eq.(2). Approximating χ (T) [4] J.M.Maldacena,Adv.Theor.Math.Phys.2,231(1998)
cl QC
by a power law like eq.(2) and imposing the two conti- [Int. J. Theor. Phys.38, 1113 (1999)]
nuityconditions, χ =χ anddχ /dT =dχ /dT at [5] P. Pfeuty, Ann.Phys. (N.Y.) 57, 79 (1970)
QC cl QC cl
[6] T. Bitko, T. F. Rosenbaum, and G. Aeppli, Phys. Rev.
T = T∗(H), we find that T∗ T = (7/4γ)(T∗ T ).
− c − Q Lett. 77, 940 (1996)
Since γ <7/4, we find T (H)<T (H).
c Q [7] R. Coldea, D. A.Tennant,E. M. Wheeler, E. Wawrzyn-
One objection against the quantum critical suscepti- ska et al., Science327, 177 (2010)
bility eq.(11) is that usually the location of the critical [8] E.Barouch,B.M.McCoyandT.T.Wu,Phys.Rev.Lett.
point on the phase diagram is not universal. In the one- 31, 1409 (1973)
dimensional chain H = 1 due to the duality [10]. The [9] M. Suzuki,Prog. Theor. Phys. 56, 1454 (1976)
c
changeof the signin eq.(11) atsmall H servesas to sep- [10] E.FradkinandL.Susskind,Phys.Rev.D17,2637(1978)
[11] P. Butera and M. Comi, Phys. Rev.B 56, 8212 (1997)
arate the quantum from the classical regions whereas it
[12] R.GuidaandJ.Zinn-Justin,J.Phys.A31,8103(1998)
is problematic at high temperature. Also, crossovers in [13] M. Hasenbusch,J. Phys. A 32, 4851 (1999)
eq.(11) differs markedly from the characteristic triangu- [14] C.J. Hamer, J. Phys. A 33, 6683 (2000)
lar quantum criticallity region [2]. [15] M.P. Gelfand and R.R.P. Singh, Adv. in Phys. 49, 93
The following test demonstrates that there is no (2000)
intrinsic contradiction between our data and the fit [16] Integer sequenceA181528, www.oeis.org