The area law and real-space renormalization

0.25 MB·English

by  Andrew J. Ferris

The area lawand real-spacerenormalization
Andrew J. Ferris
De´partement de Physique, Universite´ de Sherbrooke, Que´bec, J1K 2R1, Canada
(Dated:January15,2013)
Real-spacerenormalization-grouptechniquesforquantum systemscanbedividedintotwobasiccategories
—thosecapableofrepresentingcorrelationsfollowingasimpleboundary(orarea)law,andthosewhichare
not. Idiscussthescalingoftheaccuracyofgappedsystemsinthelattercaseandanalyzetheresultantspatial
anisotropy.Itisapparentthatparticularpointsinthesystem,thataresomehow‘central’intherenormalization,
havelocalquantitiesthataremuchclosertotheexactresultsinthethermodynamiclimitthanthesystem-wide
average. Numericalresultsfromthetree-tensornetworkandtensorrenormalization-groupapproachesforthe
3
2Dtransverse-fieldIsingmodeland3DclassicalIsingmodel,respectively,clearlydemonstratethiseffect.
1
0
2 PACSnumbers:

n
a I. INTRODUCTION
J

1
Solving large, quantum mechanical systems is very chal-
1
lenging,primarilybecausethedimensionoftheHilbertspace

] describing a system with many componentsgrows exponen-
el tially with the numberof components. Direct approachesto
- suchproblems,forinstancebyexactdiagonalization,quickly
r FIG.1: (a)DepictionofanMPS(intheunitarygauge)picturedin
st becomeintractable,evenforrelativelysmall2Dand3Dquan- termsof renormalized Hilbert spaces. Eachtensor adds aphysical
tum systems. Quantum Monte Carlo (QMC) is another di-
. site(bottom)andpassestheHilbertspace(truncatedtoχ)upwards.
t rectapproach(exactuptostatisticalerror),butthesignprob-
a (b)The1DTTNfor8sites.Eachtensorcombinesandrenormalizes
m lemcausesdifficultiesformanysystemsofinterest—suchas twoneighboringblocks. (c)Asinglelayerofthe2DTTN,wherea
fermionic,frustrated,ordynamicalproblems. 2×2squareisrenormalizedintoasinglesite,firstbycombiningin
-
d Thus, in order to garner meaningful information about thexthenydirections.
n largequantumsystems,cleverapproximationsneedtobeem-
o
ployed. In this vain, many analytic and numeric techniques
c
havebeendevelopedoverthelast80years.Inthispaper,Iwill in which neighboringblocks are successively combined, de-
[
focus specifically on numerical real-space renormalization- picted in Fig. 1 (b). In this direct coarse-grainingapproach,

1 group(RG)techniques,whichcanbeappliedtobothquantum thephysicalvolumeofeachblockdoublesateachstep. This
v
andclassicalproblems. ansatz is used less frequently than MPS/DMRG because the
8
Theprocessofrenormalizationtakesadivide-and-conquer numerical cost is higher for a given amount of system en-
0
6 approach to the problem, by tackling different parts of the tanglementoraccuracy(O(χ4) vs. O(χ3)). Thetensornet-
2 system (or Hilbert space) separately or in succession, care- work can easily be extended to higher-dimensional systems
. fully simplifying or compressing the pieces at each step. In (seeFig.1(c))[4–6].
1
real-spaceRG,therenormalizationproceduregroupstogether ThemajorproblemofusingtheTTN(orMPS/DMRG[7])
0
3 spatial regionsof the system, referred to as blocks. At each in two- or higher-dimensions is that they do not respect the
1 step, twoneighboringblocksarecombined,andthensimpli- area-lawforentanglemententropywith fixed χ [4, 8]. Gen-
: fied (for instance, by truncating the Hilbert space). As the erallyspeaking,gappedphasesoflocalHamiltoniansareex-
v
i renormalizationproceeds,theblocksincludelargerandlarger pectedtohave‘local’correlations.Thus,theamountofentan-
X portionsoftheinitialsystem,untiltheentiresystemofinter- glementbetweenalarge,contiguousblockandtherestofthe
r est is encapsulated. If the RG reaches a fixed-point, we can systemshouldscaleproportionaltotheboundaryareaseparat-
a
saywehavereachedthethermodynamiclimit. ingtheregions. Ontheotherhand,wave-functionsgenerated
One of the most successful real-space RG algorithms is byMPSorTTNcontainarbitrarylargeblockswithbounded
thedensity-matrixrenormalizationgroup(DMRG)[1],which entanglement(dependingonχ),andthereforehavepoorover-
very accurately describes (quasi-)1D systems. In this ap- lapwiththetruegroundstate.
proach, a single site is added to the block at each step, and Thearealawhasmotivatedseveraltensor-networka¨nsatze
the optimal Hilbert space (limited to some dimension χ) for todescribehigherdimensionalsystems,inanattempttorepli-
the combinedsystem is determined. Working in reverse, the cate the success of DMRG. One example is the projected
procedure generates a variational wave-function known as a entangled-pairstate (PEPS) [9–16], whichcan be thoughtof
matrix-productstate (MPS) (see Fig. 1 (a)) [2]. DMRG can as a higher-dimensional generalization of MPS. The multi-
then be thoughtof as an optimization algorithm that sweeps scale entanglementrenormalizationansatz (MERA) [17–19]
over the tensors in the MPS, targeting the state with lowest addsadditionallocalentanglementtotheTTN,andinitssim-
energy. plest form (c.f. [20]) exactly replicates the area law in two-
A similar approach is the tree-tensor network (TTN) [3], or higher-dimensional systems. The drawback of these ap-
2
proacheshasbeenthelargenumericalcost, typicallyscaling
as χ10 or greater. Fortunately, one expects that as compu-
tational power increases, the accuracy of these a¨nsatze will
increasesuperpolynomiallywithχ(forgappedphases).
In the mean-time, there is immediate demand for tech-
niqueswithlowercomputationalcost. Someapproachesthat
have been tried recently include entangled plaquette states,
andperformingvariationalMonteCarlo overtensornetwork
states [21–28]. Tensor network techniques that do not obey
arealawshavebeenusedextensivelyin recentstudiesof2D
quantum systems, including DMRG in a cylindrical geom-
etry [8, 29, 30], TTN [4], and direct approximate contrac-
FIG.2:(Coloronline)Blocksofdifferentlayersinthe2DTTN.The
tions of the 3D Suzuki-Trotter decomposition using tensor
marked ‘center’ siteat coordinates (6,6) is furthest from the block
renormalization-group (TRG) and its variants [31–35]. In
boundaries. This should be the site where the Hilbert space of its
these approaches, a description of a locally-correlated state immediateenvironmentislargest,andclosesttothebulk,forallχ.
wouldrequireabond-dimensionthatgrowswithsystemsize
(e.g.exponentiallywithcylinderwidthin2DDMRG).
II. TREETENSORNETWORK
There are two possible approaches to take in these cases:
(a)useasmall,finitegeometrywhilekeepingtrackofallcor- In this section we analyze the tree tensor network on an
relations;or(b)studyalargesystemusinganansatzwithin- L × L square lattice. Here we use the 2-to-1 renormaliza-
sufficient entanglement or bond-dimension χ. The first ap- tionschemedepictedinFig.1(c),alternatingcoursegraining
proach is typically used because finite-size effects are well- along the x and y dimensions. The cost of contracting the
understood,whilefinite-χeffectsarelessclearintheseverely tensor network corresponding to the expectation value of a
undersaturated regime. The purpose of this paper is to ana- nearest-neighborHamiltonian, hΨ|Hˆ|Ψi, scales as O(χ4L),
lyze, in generality, the scaling of accuracy in approach (b). andsimilarlyforcalculatingthederivativeoftheenergywith
One reason for this to be important is that approach (b) is respecttothetensorvariables.
implicitly used in the 3D classical tensor renormalization- The wave function |Ψi generated by the TTN contains
group (or (2+1D) quantum TRG using a 3D representation blocksof size l×l (where l = 2n, forsome integer n) that
of imaginary-time evolution) — a promising technique that containboundedentanglement(Schmidtrankχ)withtherest
recently demonstrated accuracy competitive with large-scale ofthesystem. TheseblocksarehighlightedinFig.2. Intwo-
MonteCarlostudies[35]. dimensions,thisfailstosaturatethearealaw,whichdemands
that the entanglement entropy should scale linearly with the
perimeter of the block, requiring a Schmidt rank scaling as
We see that following a na¨ıve approach, the global accu-
exp(αl),forsomeconstantα.
racy of (b) scales only logarithmicallywith χ, and thus also
We now proceed to analyze the structure of the wave-
logarithmic in the numerical cost. On the other hand, pro-
functions having minimal energy. For large enough system
videdpropercare is taken, the accuracyoflocalobservables
sizeL,thebonddimensionwillbeinsufficienttodescribethe
scales polynomially with χ. This exponential improvement
entanglement of the entire system, i.e. χ < exp(αL), and
puts the approach (at least formally) on similar grounds to
thustheminimumenergywave-functionmustbedistinctfrom
finite-sizescalingofsmallsystemsusingDMRG.The‘trick’
thetruegroundstate. Ontheotherhand, χ willbesufficient
here is to realize that the anisotropic structure of the renor-
todescribetheentanglementwithinthesmallerblocks.Errors
malization meansthat notall sites are equal. Some sites are
willaccumulateprimarilybecauseofthelackofentanglement
farawayfromtheboundariesoftherenormalization,andare
betweenlargerblocks.
abletoshareampleentanglementwiththeirsurroundingenvi-
Somewherebetweenthesetwoextremestherewillbeacrit-
ronment.Inthispaper,ideallylocatedsitesarecalled‘center’
ical block-size l∗, where smaller blocks have sufficient en-
sites.
tanglement and are thus well-renormalized, whereas larger
blocksdonotpossesslargeenoughχ. Thisisthepointwhere
The paper is structured as follows. In Sec. II, the 2D
tree-tensornetworkis analyzedin-detail, with arguments for χ∼exp(αl∗). (1)
the above scaling backed up by numerical results for the
transverse-field Ising model on the square lattice. A simi- Becausetheblocksgrowexponentiallyinsizeasafunction
lar analysis for the tensor renormalization-group in higher- oflayern, andtherequiredbond-dimensionthereforegrows
dimensionsis shown to hold in Sec. III, with numericalevi- doubly-exponentiallyin n, we expect the transition between
dencefromthe(classical)3DIsingmodel(whichisrelatedto sufficiententanglementandwoefullyinadequateχtobequite
the(2+1)Dquantummodel). ThepaperconcludesinSec.IV sharp.
withanoutlookonsomepossiblefuturedirections. To simplify the analysis, we compare this situation to a
3
cluster-meanfieldtheory.Inthistheory,thefullHilbertspace largestimmediateenvironmentwhosesitesaredescribedbya
ofclustersofsizel∗×l∗ isincluded,whilenoentanglement sufficientlylargeHilbertspace,roughlyequalinalldirections.
existsbetweenneighboringclusters. Inpractice,asingleclus- We have implemented the 2D TTN to study the spin-1/2
ter is exactly-diagonalized in a self-consistent fashion with transverse-field Ising model on the square lattice, described
theirboundaryconditions,andthefixed-pointcorrespondsto byHamiltonian
thelowestenergystateintheclustermean-fieldansatz.
The errorof globalquantitiessuch as the total energycan Hˆ =− σˆzσˆz +hσˆx, (4)
i j i
be reasonablylarge when using this approach. Let’s assume <Xi,j>
thatthesystemhascorrelationlengthξ,andthattheexpecta-
where<i,j>denotesneighboringsitesandhisthestrength
tionvalueofalocalquantitydecaysexponentiallytowardsthe
of a transversemagneticfield. Because the cost of the algo-
(licmoritretchta)tblu∗lk>vaξl,useoamweafyrafcrotimonthoeftchluesstyerstbeomunwdiallribees.‘cInlotshee’ rithmcostscalesasO(χ4L),amoderatesizeL = 32isused
in orderto investigate a system that is much larger than can
totheboundaries(closerthanξ)anddisplayincorrectresults,
bedescribedexactly,whilesmallenoughtoallowarangeof
while the remaining ‘bulk’ fraction will display roughly the
χ to be used. To minimize the energy, I have employed the
correct results. In d-dimensions, the fraction of ‘error’ sites
time-dependentvariational principle [36] and used a unitary
scalesas2dξ/l∗,andthustheerrorofaglobalquantityinthe
gauge, finding a significant speed-up compared to the tradi-
2DTTNscalesas
tional,SVDapproach[4].
ξ ξα InFig.3theerrorofthemagnetizationinthexdirection(as
globalerror∝ ∼ (2)
l∗ lnχ compared to the best, center site estimate) is displayed. We
canclearlyobserveapatternofsuccessive‘windows’,where
whereEq.(1)wasusedinthesecondrelation. Theerrorofa
errors are displayed primarily on the edges of blocks. For
globalquantity,suchasthetotalenergyormagnetization,thus
largerχ,thesizeofthewindowsgrowwhiletheoverallerror
only decreases logarithmically with χ and thus computation
decreases—while thecentralsites alwaysremaininsidethe
effort.
smallestwindow.Atthecriticalpoint,closetoh=3.05,there
Although this scaling is quite poor, the ansatz takes some
is a large correlation length and the windows are somewhat
advantagefromthelocalizedentanglementinthesystemand
blurred,andtheerrorsaregreaterinmagnitudeforagiven χ.
isalreadyalargeimprovementoverexactdiagonalization[4]. We compare the predicted value of hσˆxi from the global,
On top of this, we can estimate local quantitieswith greatly
averagevalue andthe localvalue at the centersite in Fig. 4.
reduced error by understanding the structure of the ansatz.
Away from criticality, we observe that the center site value
Points inside the well-renormalized, ‘bulk’ region are sur-
converges to the quantum Monte Carlo prediction (using
roundedbyanimmediateenvironmentthatisagoodapproxi-
ALPS [37, 38]) significantly faster than the global average.
mationofthetruegroundstate(provided,again,thatl∗ >ξ).
At criticality, both values appear to be converging with a
As mentioned earlier, in a gapped phase the effect of the
slow,1/lnχscaling(extrapolationmaybeviablewhenLap-
boundaryshould decayexponentially,so in the centre of the
proachesthethermodynamiclimit).Ath=3.25thebehavior
regiontheerrorshouldscale as exp(−l∗/2ξ). Thistime, in-
isnotmonotonicbecausethewave-functionswitchesfromthe
cludingEq.(1)gives
ferromagnetictodisorderedphaseatintermediateχ.
∗ Theseresultsconfirmtheaboveanalysisandshowthatitis
−l −lnχ
localerror∼exp ∼exp ∼χ−β, (3) viabletoextractmeaningfulinformationinsystemswherethe
(cid:18) 2ξ (cid:19) (cid:18) 2ξα (cid:19)
parameterχisseverelyundersaturated.
for some β that depends on the specifics of the system (no-
tably,becomingsmallerformoreentangledsystemsorthose
withlargercorrelationlength). Thusthescalingoferrorwith III. TENSORRENORMALISATIONGROUP
computationeffortispolynomial—anexponentialimprove-
mentonthe errorofglobalquantities. If β isverylarge,the A related tensor-based, real-space renormalization tech-
methodcouldbecomecompetitivewithwell-establishedtech- nique is the tensor renormalization group (TRG). In this
niquessuch as quantumMonte Carlo (where statistical error family of methods, the tensor network corresponding to the
scalesasthesquarerootofcomputationaleffort,assumingno Markovnetworkof a classical thermalstate (whosecontrac-
sign-problem). Theperformancewilldegradesignificantlyin tion gives the partition function) is successively contracted
systemswithlargeamountsofentanglement,orclosetocriti- into larger and larger blocks. At each layer, the dimension
calpoints. ofthetensorsmustbetruncatedtopreventthedifficultyfrom
Toinvestigatetheabovenumerically,weidentifypointsin growingexponentially.TheprocessisdepictedinFig.5.
thesystemthataresomehow‘central’totherenormalization, The method is well-suited to contracting 2D tensor net-
independentofsystemparametersorχ. IntheTTN,theseare works, such asthermalstate ofa 2D classicalsystem, orthe
the pointsthatare renormalizedbest with theirenvironment, network that results from the Trotter decompositionof a 1D
favoring no particular direction. In the 2D TTN, we define quantumdensity matrix. Fornon-criticalsystems, the fixed-
these to be the points that have been successively combined pointofthisRGflowiswell-understood[32]intermsof‘cor-
withtheblockabove,totheleft,below,totheright,adinfini- ner’correlations.TheTRG,andrelated‘secondrenormaliza-
tum (see Fig. 2). For any choice of χ, such a point has the tion group’ (SRG) methods are able to encapsulate the cor-
4
FIG. 5: The contraction of the 2D Markov network (i.e. partition
function)ontheleftisapproximatedbytheTRGschemeontheright,
from[35].Inthe‘secondrenormalisationgroup’,theprojectors(tri-
angular tensors) are optimized to maximize the partition function.
The above diagram can also be considered an ansatz for the prob-
abilitydistributionitself,byopeningadditionallegsontheMarkov
networktensors(circles)correspondingtothestateatthatsite.
relations of the 2D classical system quasi-exactlywith fixed
χ.
Thistechniquehasbeensuccessfullyappliedina3Dclassi-
cal(or2Dquantum)setting,achievingimpressiveaccuracies
FIG.3:(Coloronline)Spatialvariationsinthemagnetizationinfield (see e.g. Ref. [35]). However, in these higher-dimensional
direction at various values of h, as predicted by the lowest-energy settings, the edges of the 3D blocks grow in length as the
TTNwiththedisplayedbond-dimensionsχ.Thecolorvaluesrepre- renormalizationproceeds,andstrictlyspeakingχwouldneed
sentdifferencestothepredictionatχ=112. Inthegappedregions to growexponentiallyto encapsulateallthe correlationsin a
away from the critical point (near h = 3.05), we observe a ‘win-
(non-critical) system. In general, for this kind of blocking
dowing’ pattern, withwindowsizegrowingwith χ, andrapidcon-
scheme in d dimensions, the correlations that need to be to
vergenceatthecentralpointsoftherenormalization(atx,y = 21).
beaccountedforateachleveloftherenormalisationgrowsas
Whenthecorrelationlengthislonger,theeffectisblurredsomewhat
andconvergencewasnotreachedwithavalueofχ=112. Ld−2. Thus,theTRGfailstoaccountforwhatonemightcall
a‘corner’lawforcorrelations[40,41],relatedtothequantum
arealawforentanglemententropythatmanifestsnaturally in
a(d+1)-dimensionalquantumtheory,whereonedimension
istime.
The structure of the TRG is strikingly similar to the TTN
and one mightsuppose that a similar analysis in the limit of
large 3D systems with insufficient χ mighthold. Numerical
evidenceinthe3DIsingmodelappearstosupportthisclaim.
Iimplementedthe3Dhigher-orderSRG(HOSRG)approach
fromRef. [35], usinganSVD-update[4, 39] tooptimizethe
tensorstomaximizethe(global)partitionfunction.Thecom-
putational effort scales as O(χ11lnL), and we are able to
study much largersystems, with L = 212. In Fig. 6 (a) we
see a clear indication of the ‘corner’ errors in the local en-
ergy,inasimilarfashiontothe‘window’patterninFig.3. In
Fig.6(b)theaverageenergyandcentersiteenergyarecom-
paredfordifferentvaluesof χ, andwe see even close to the
criticalpoint(atT ≈ 4.1)thelocalquantityconvergestothe
MonteCarlopredictionmuchmorerapidly.
FIG.4: (Coloronline)Predictionsforthemagnetizationinthefield
directionwithvariousmagneticfieldstrengthsh,aspredictedbythe
lowest-energyTTNwithgivenbond-dimension χ. Theredcrosses
IV. DISCUSSION
correspond to system-wide averages, while the blue points corre-
spond to the central sites of the renormalization. The latter con-
We have analyzed real-space renormalization procedures
verge much faster than the former, away from the critical value of
h≈ 3.05. ThedashedlinescorrespondtoQMCresultsforthesys- that are unable to account for local correlations in higher-
tem at temperature T = 0.005. In (a) the two lines represent the dimensionalquantum and classical systems. The anisotropy
statisticaluncertainty, whilein(b–d) theerroriscomparabletothe oftree-structureda¨nsatze,suchasTTNandTRG,canbetaken
linewidth. advantage-oftoidentifysitesthatarecentraltotherenormal-
ization, providing an (exponentially more) accurate descrip-
5
(b) −1.9 tems,whiletherehasbeenlessdiscussionontheinherentin-
abilitytoaccountforallcorrelationsinalargesystem.In-fact,
−1.91
it is possible that the central-site technique has been imple-
Energy−1.92 mentedinthesestudiesinthepast.
−1.93
From here, two possible directions to increase the effec-
−1.94 tiveness of real-space RG techniques in higher dimensions
0.4 0.6 0.8 1
(log χ)−1 become apparent. The first would be find more efficient al-
gorithms in the under-correlated regime. For example, the
FIG.6: (Coloronline)Predictionsforthe3DIsingmodelatT = 4
HOSRGwasastepinthisdirection[35],comparedtoearlier
(lessthanthecriticaltemperatureofT ≈4.1)usingtheHOSRGap-
3DTRGalgorithms. Inthepresentwork,itcouldbebenefi-
proach. (a)Thebond-energyofthez-bondsthroughacross-section
ofthex–yplane,wherethevalueatthecentersitelabeledx,y=21 cialto replacethe TTNwith a MPShavinga tree-likestruc-
has been subtracted. The 3D renormalization scheme displays er- ture,investigatedalreadyinRef.[42]. Thecostwouldreduce
rorspredominantly onthecornersof therenormalizedregions. (b) toO(χ3L2),butonewouldstillbelimitedtomoderatesystem
Theenergy(per-site)fordifferent values of χ, compared toclassi- sizes.
calMonteCarloresults. Theredcrossescorrespondtosystem-wide
Thesecondapproachwouldbe to use anansatz thattakes
averages,whilethebluepointscorrespondtothecentralsitesofthe
renormalization. Similar to the quantum TTN, the latter converge intoaccountthecorrelationstructureofthesystem. PEPSand
muchfasterthantheformer. ThedashedlinecorrespondstoMonte MERAalreadyexisttodescribehigher-dimensionalquantum
Carlosimulationsofa48×48×48lattice. systems. IntherealmofclassicalMarkovnetworks,progress
was made in Ref. [32] to use a TRG-like approach to take
account of all local correlations (for 2D classical systems).
tionoflocalquantities. Thescalingofaccuracytonumerical However,a moredirect, MERA-likeapproachto 2Dand3D
cost is expected to be polynomial(forgappedsystems), for- classicalsystemswouldrepresentamajoradvancementinthis
mallyputtingthetechniqueonasimilarfootingto2DDMRG field.
ofsmallsystemsandquantumMonteCarlo (thoughin prac- Acknowledgements
ticethealgorithmsusedheremightnotbeasefficient).
There has been some reluctance to use a severely under-
correlatedansatz fora quantumwave-function,whererecent IwouldliketothankGuifreVidalandDavidPoulinfordis-
focushasbeenonDMRGinsmallgeometriessuchasnarrow cussions. Thiswork was supportedby NSERC and FQRNT
cylinders. On the other hand, the TRG approach has shown through the network INTRIQ, as well as the visitor pro-
verypromisingresultsin3Dclassical(and2Dquantum)sys- grammeatthePerimeterInstituteforTheoreticalPhysics.
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