Real Space Renormalization in Statistical Mechanics

1.5 MB·English

by  Efi Efrati

Real Space Renormalization in Statistical Mechanics
Efi Efrati,1, Zhe Wang,1 Amy Kolan,1,2 and Leo P. Kadanoff3,4
∗
1James Franck Institute,
The University of Chicago. 929 E. 57 st,
Chicago, IL 60637,
USA
2St. Olaf College, Northfield, MN,
USA
3James Franck Institute,
3
The university of Chicago. 929 E. 57 st,
1
0 Chicago, IL 60637,
2 USA
4The Perimeter Institute,
n
Waterloo, Ontario,
a
Canada
J

7 Thispaperdiscussesmethodsfortheconstructionofapproximaterealspacerenormal-
2 izationtransformationsinstatisticalmechanics. Inparticular,itcomparestwomethods
oftransformation: the“potential-moving” approachmostusedintheperiod1975-1980

] andthe“rewiringmethod” asithasbeendevelopedinthelastfiveyears. Thesemeth-
h
odsbothemployaparameter,calledχorD intherecentliterature,thatmeasuresthe
c
complexity of the localized stochastic variable forming the basis of the analysis. Both
e
methods are here exemplified by calculations in terms of fixed points for the smallest
m
possiblevaluesofχ. Thesecalculationsdescribethreemodelsfortwo-dimensionalsys-
- tems: The Ising model solved by Onsager, the tricritical point of that model, and the
t three-statePottsmodel.
a
The older method, often described as lower bound renormalization theory, provides a
t
s heuristic method giving reasonably accurate results for critical indices at the lowest
t. degreeofcomplexity,i.e. χ=2. Incontrast,therewiringmethod,employing“singular
a valuedecomposition”,doesnotperformaswellforlowχvaluesbutoffersanerrorthat
m apparentlydecreasesslowlytowardzeroasχisincreased. Itappearslikelythatnosuch
improvementoccursintheolderapproach.
-
d Adetailedcomparisonofthetwomethodsisperformed,withaparticulareyetodescrib-
n ingthereasonswhytheyaresodifferent. Forexample,theoldermethodisbasedonthe
o analysis of spins, simple stochastic variables located at lattice sites. The new method
uses “indices” describing linear combinations of different localized configurations. The
c
[ oldmethodquitenaturallyemployedfixedpointsforitsanalysis;thesearehardtouse
inthenewerapproach. Adiscussionisgivenofwhythefixedpointapproachprovesto

1 behardinthiscontext.
v In the new approach the calculated the thermal critical indices are satisfactory for the
3 smallest values of χ but hardly improve as χ is increased, while the magnetic critical
2 indicesdonotagreewellwiththeknowntheoreticalvalues.
3
6
.
1
0
3
1
:
v
i
X
r
a
∗ [email protected]
2
I. INTRODUCTION
A. History: the conceptual foundations
The renormalization group (1–4), provides a theoreti-
CONTENTS cal understanding of singular problems in statistical me-
chanics(5),particularlyonesinvolvingphasetransitions.
I. Introduction 2
Therearetwomainbranchesofanalysisbaseduponthis
A. History: theconceptualfoundations 2
B. StatisticalVariables 3 method, one involving work in momentum or wave vec-
C. Renormalization 3 tor space (6), the other involving so- called “real-space”
D. FixedPoint 4
methods. In this review, we follow the latter approach.
E. Criticalindices 4
Both approaches make extensive use of the following
F. Responseanalysis 4
G. Requirementsonapproximations 5 conceptual ideas, vis
H. Historyofreal-spacemethods 5
I. Comparisons 6 Scale invariance. Singularities in statistical me-
•
1. Stochasticvariables 6 chanics tend to be connected with behaviors that
2. Geometricstructure 6
are the same at different length scales. Critical
3. Calculationalstrategy 7
J. Planofpaper 7 points of phase transitions have correlations at all
length scales.
II. Therenormalizationprocess 7
A. Overview 7 Scalecovariance. Nearthephasetransition,many
1. Introducingnewstatisticalvariables 7 •
physical quantities vary as powers of characteris-
2. Tracingouttheoriginalstatisticalvariable 8
tic lengths that describe the system, or of lengths
3. Obtainingarecursionrelation 8
B. BasicStatisticalDescription 9 describing the quantities themselves, or as powers
C. Tensor-SVDRenormalization 11 of “fields” that measure the deviation of thermo-
1. SingularValueDecomposition 11
dynamic quantities from criticality. These powers
2. SVDasanapproximationmethod 11
characterize the phase transition. They are called
3. Gaugeinvarianceandinterpretationoffixed-point
tensorcomponents 12 critical indices.
D. Nullspaceoftheresponsematrix 13
1. Errors 13 Fixed point. The scale-invariance is described by
•
E. Lowerboundvariationalrenormalization 13 a Hamiltonian or free-energy-function that has el-
1. Decoration 13
ements that are independent of length scale. As
2. potentialmovingtheorem 14
3. Usingpotentialmoving 14 a result, one might expect that, for example, the
4. Spatialvectorsandtensors 15 Hamiltonian or the free energy function that de-
scribes the system will not change when the length
III. Results 15
scale changes. This unchanging behavior is de-
A. Resultsfromblockspincalculations 15
1. BlockspinrenormalizationofIsingmodels 16 scribed as “being at a fixed point”.
2. Blockspinrenormalizationforothermodels 17
B. SVDResults 17 Renormalization. Atransformationthatdescribes
•
1. GenerationofSVDfixedpoint 17 the results of changes in the length scale. Usually
C. SummaryofSVDnumericalresults 19
this transformation will not change the Hamilto-
1. Hexagonalχ=2 19
nianorfreeenergydescribingthefixedpoint. That
2. Squareχ=2 20
3. Hexagonalχ=3 20 is the reason for the name, fixed point.
4. Additionalfixedpoint(hexagonalχ=3) 20
5. Andonemore(hexagonalχ=3) 21 Universality. Near the phase transition, many dif-
•
6. Squareχ=3 21 ferent physical systems show identical behavior of
7. Squareχ=4 21
thequantitiesthatdescribecriticalbehavior. Since
8. Four-statePottsmodel 21
9. Theresponsematrixintensorrenormalization 21 these quantities are descriptive of scale-invariant
behaviorthesedescriptivequantitiesallcanbeseen
IV. Discussion 22 at large length scales.
A. Errorestimates 22
B. Worktobedone. 23
Universalityclasses. Therearemanycriticalpoints
C. Wheredowestand? 24 •
withawidevarietyofdifferentorigins. Nonetheless
Appendix: Gaugefreedomandinvarianceoftherecursion these fall into relatively few universality classes,
stepinisotropicTRG 24 each class being fully descriptive of all the details
Proof 24
of a given critical behavior.
Acknowledgments 25
Thebehaviorofdifferentcriticalsystemscanbe,inlarge
References 25 measure,classifiedbydescribingthedimensionandother
3
The tensor network representation describes the con-
topological features of the system, and then describing nectivity and interdependence of blocks and statistical
some underlying symmetry that plays a major role at variables. Because of locality the numerical value each
the critical point. Thus, the model that Lars Onsager block attains depends on a small number of statistical
solved,theIsingmodel,(7–9)ismostlydescribedbysay- variables. Every statistical variable in turn affects the
ing it is a two-dimensional system with a spin at each numerical values in a small number of different blocks.
point. The spin can point in one of two directions. The This allows the identification of a statistical variable as-
model has a symmetry under flipping the sign of a spin, suming χ different values with an index assuming the
so that it can describe a magnetic phase transition. It values 1,2, ,χ . The blocks are linked because each
{ ··· }
is equally well descriptive of a two-dimensional liquid in index appears in precisely two blocks. The blocks then
which the basic symmetry is in the interchange of high reducetotensorswhoserankisdeterminedbyhowmany
densityregionswithlowdensityones,sothatitdescribes differentindicesdeterminethevaluesassumedbyagiven
aliquid-gasphasetransition. Anymodelwiththeappro- block. Every configuration corresponds to a specific
priate symmetry and dimensionality and the right range choice of indices. It is believed, but not proven, that
of interaction strengths is likely to describe both situa- thiskindofrepresentationformsalinktothefundamen-
tions, and many others. The Ising model constitutes the tal description of the statistical problem (15). The free
simplest model of this kind. energy calculation which follows by summation over all
possibleconfigurationsofthestatisticalvariablesreduces
to a tensor product tracing out all the mutual index val-
B. Statistical Variables
ues.
There are many models and real systems that exhibit e−βF =Tri,j,k, Tijkl (1c)
critical behavior (10–13). All of those with short-range ···
(cid:89)
interactionsandspatialhomogeneityhavethesamekind The tensor indices are indirect representations of the
of characteristic behavior. One starts from a statisti- original statistical variables. Each value of a given in-
cal ensemble, that is a very large system of stochas- dex may represent a sum, with coefficients that can be
tic variables, called σr , where r defines a position in positive or negative, of the weights of statistical config-
{ }
space. The statistical calculation is defined by proba- urations in the system. Moreover, this representation
bility distribution, given as an expression of the form permitsakindofgaugeinvarianceforeachindexateach
exp( βH σr ), where β is the inverse temperature and point in space, giving the index variable a new meaning.
− { }
HistheHamiltonianforthestatisticalsystem. Onethen For example, if the index i appears in two tensors T1
ijkl
uses a sum over all the stochastic variables, defined by and T2 , then the index transformation,
ipqr
thelinearoperationdenotedastrace,todefineathermo-
dynamic quantity the free energy, F, as T1 O T1 , T2 O T2 , (2)
ijkl → i,m mjkl ipqr → i,m mpqr
m m
e−βF =Tr{σr} e−βH{σr} . (1a) for O(cid:88)O =δ , willleavet(cid:88)hepartitionfunction
(cid:16) (cid:17) m i,m j,m i,j
Eq. (1a) gives the problem formulation for statistical unchanged. This important formal property underpins
(cid:80)
physicsintroducedbyBoltzmannandGibbsanddirectly the newer statistical calculations.
used for renormalization calculations through the 1980s.
Inrecentyears, aslightlydifferentformulationhastaken
C. Renormalization
hold. SincetheHamiltonianismostoftenasumofterms,
eachcontainingafewspatially-neighboringσ values,one
r
The basic theory describing this kind of behavior was
can write the free energy as a sum of products of blocks:
derivedbyWilson(16),basedinpartuponideasderived
e−βF =trace{σr} BLOCKR, (1b) ecaornlcieerpt(1o7f–a23r)e.noTrmhealfiizrasttioenlemtraenntsfoorfmthaetiotnh.eoTryhiissitshae
R
(cid:89) changeinthedescriptionofanensembleofstatisticalsys-
each block depending on a few statistical variables. This tems, obtained by changing the length scale upon which
formulationappliesequallywelltotheolderandthenew the system is described. Such a transformation may be
formulations of the statistical mechanics. Lately, statis- applied to any statistical system, including ones which
tical scientists have realized the advantage of a particu- are or are not at a critical point. There is a whole col-
lar special form of writing the product of blocks, called lection of methods for constructing such renormalization
the tensor network representation. In this representa- transformationsanddescribingtheirproperties. Thispa-
tion, similarlytovertexmodels, thestatisticalBoltzman per will be concerned with describing one class of such
weights are associated with vertices (rather than bonds) transformations, the real-space transformations. These
(14). are ones that employ the description of the ensemble in
4
to the K . The free energy undergoes a change in value
α
ordinary space (or sometimes space-time) to construct a produced by these variations of the form:
description of the renormalization process.
Theensembleisparameterizedbyasetofcouplingcon- βδF = CαηαLd−xα (6)
−
stants, K = {Kj}. These couplings might describe the (cid:88)α
spin interactions of the early renormalization schemes,
Here,theη describeslinearcombinationsoftheh ,each
α j
with the subscripts denoting couplings to different com-
η defining a different kind of covariant scaling operator
α
binations of spin operators. Alternatively the K’s may
that describes a particular type of scaling near the criti-
be parameters that determine the tensors. The renor-
cal point. In the usual critical phenomena problems one
malization transformation increases some characteristic
suchoperatordescribesthefieldthermodynamicallycon-
distance describing the system, usually the distance be-
jugate to the order parameter whose symmetry breaking
tween neighboring lattice points on a lattice defining the
produces the phase transition, and another such field,
spatial structure, so that this distance changes accord-
conjugate to the energy density, is the deviation of tem-
ing to a(cid:48) = δL a. Correspondingly, the renormalization perature from its critical value. Still other operators,
transformation changes the coupling parameters to new
such as the stress tensors, play important roles in the
values which we denote by K(cid:48). These new couplings de- criticalbehavior,buthavesymmetriesthatpreventthem
pend upon the values of the old ones, so that
from appearing at first order in this expansion.
In Eq. (6), L is the linear dimension of the system,
K(cid:48) =R(K) (3) d is the dimensionality of the system, while Cα is a rel-
atively unimportant expansion coefficient. The crucial
Here, the function represents the effect of the renor-
R quantity in this equation is xα, the critical index defin-
malization transformation.
ingthescalingpropertiesofthescalingoperator. Forthe
usualalways-finitescalingoperatorstheexponentsxαare
D. Fixed Point positive. Operators for which the corresponding critical
exponentsliebetweenzeroandthedimensionofthesys-
The renormalization theory is particularly powerful at tem, d, arecalledrelevant operators. Theseplayamajor
thecriticalpoint. Thisapplicationofthetheoryisbased role in the thermodynamics. Operators for which the
upon the concept of a fixed point, an ensemble of statis- corresponding critical exponents are greater than d are
tical systems that describe the behavior of all individual called irrelevant operators and do not contribute to the
statistical systems within a particular universality class. to the singular behavior of the thermodynamic functions
Since the critical point is itself invariant under scaling (24; 25). In our work below, we shall compare the values
transformations,theensembleinquestionisinvariantun- of the relevant x’s as they emerge from the approximate
der a renormalization transformation. It is said to be at numerical renormalization theory with the exact values
a fixed point. The fixed point is represented by a spe- thatareoftenknownfromothernumericalworkorexact
cial set of couplings, K∗, that are invariant under the theories (26; 27).
renormalization transformation
K∗ = (K∗) (4) F. Response analysis
R
Thebehavioroftherenormalizationtransformationin
E. Critical indices the vicinity of the critical points is quantified by the re-
sponse matrix relating small changes in the couplings,
The most important physical effects are obtained by K , with the small changes they induce in the renormal-
j
studying the behavior of the renormalization transfor- ized couplings , Ki(cid:48).
mation in the vicinity of the critical points. This behav-
dK
ior is in turn best described by a set of critical indices B j = i(cid:48) . (7)
i
dK
describing the scaling of the singular part of measurable j(cid:12)K=K∗
(cid:12)
physicalquantitiesnearthecriticalpoint. Onewritesthe This matrix has right and left(cid:12)eigenfunctions defined as
(cid:12)
deviation of the Hamiltonian from its fixed point value
as B jψ α =Eαψ α and φ iB j =Eαφ j (8)
i j i α i α
j i
βH= βH∗+ hα Sα (5) (cid:88) (cid:88)
− −
α Theeigenvaluesdefinethescalingpropertiesoftheexact
(cid:88)
solution. The operators’ scaling is defined by Eq. (6).
where the components h represent the small deviations
α The eigenvalues directly determine this scaling since
ofthecouplingconstantK fromtheirfixedpointvalues,
α
whileSα aretheextensivestochasticoperatorconjugates Eα =(δL)yα =(δL)d−xα (9)
5
H. History of real-space methods
with δL being the change in length scale produced by
The first heuristic definition of a real space renor-
the renormalization. 1 Different authors describe their
malization was given in (20). After Wilson and Fisher
results in terms of E , or yα, or xα. In this paper, we
α (6; 29)demonstratedtheviabilityoftherenormalization
use the last descriptor.
approach by inventing the (cid:15) expansion, Neimeijer and
Theeigenfunctionsφandψ canbeusedtoconstructa
Van Leeuwen (30; 31) described a method for doing a
set of densities, oα(r), of operators called scaling opera- numericalcalculationoftherenormalizationfunction,R,
tors since they have simple properties under scale trans-
in terms of a small number of different couplings. These
formations. The combinations
methods were then described in one dimension (32) and
applied(33). Fromthepointofviewofthispaper,anim-
sj(r)ψ α =oα(r) (10a)
j portantadvanceoccurredwhenavariationalmethodwas
j
(cid:88) invented(34)andextensivelyemployed(33;35–42). This
define oα(r) as the densities for the scaling operator. method was described as a lower bound calculation since
These operators and their extensive counterparts Oα = itpermittedcalculationsthatgavealowerboundonpos-
oα(r) respectively scale like distances to the power sible values of the free energy. This approach permitted
xrα and yα respectively. The other coefficient in the reasonablyaccurateandextensivecalculationsofcritical
−e(cid:80)igenvalueanalysis, φa can beinterpretedbysaying that properties in two and higher dimensions. However, as
i
si generates a combination of fundamental operators ac- the 1970s came to an end, the lower bound method fell
into disuse.
cording to
In part, the disuse arose because interest turned from
si(r)= φ ioα(r) (10b) problems in classical statistical mechanics to quantum
α
problems. Although path-integral methods permit one
α
(cid:88) toconvertaquantumproblemintooneinclassicalstatis-
To make Eq. (10a) and Eq. (10b) work together, we ticalmechanics,thelowerboundmethodseemedtowork
must define the eigenvectors so that they are normalized best when the problem had the full rotational symmetry
and complete of its lattice, and hence did not apply to many quantum
problems. In 1992 White (43) invented a quantum me-
φ iψ β =δ β and φ jψ α =δ j (11) chanical real space renormalization scheme that worked
α i α α i i
i α beautifully for finding the properties of one dimensional
(cid:88) (cid:88)
quantum systems via numerical analysis. This success
started a large school of work aimed at these problems
G. Requirements on approximations and analogous problems in higher dimensions (44–47).
White’s method looked very different from the real-
The concepts of renormalization and scale invariance
space work of the 1970’s. It did, however, have an im-
lead naturally to the identification of scaling and uni-
portantprovenanceinWilson’snumericalsolutionofthe
versality and have contributed to the fundamental un-
Kondo problem (48). White studied the approximate
derstanding of critical phenomena. There is also a more
eigenstates involving long chains of correlated spins, and
practical aspect of the renormalization concepts that al-
how those long chains interacted with small blocks of
lows one to predict the location of phase transitions of
spins. Therewasalsoearlierworkthatfocusedonblocks
specificsystemsanddescribetheirnatureintermsofthe
of a small number of spins. In the course of time, con-
critical exponents. However, for most systems, carrying
nections among the different approaches began to be ap-
out the actual renormalization cannot be done exactly.
preciated. As pointed out by, for example, Cirac and
Instead, some approximation method must be used to
Verstraete (49), the correlations within wave functions
findanapproximaterenormalizationtransformation. We
were produced by summing products of correlations on
hope that the approximation method might give an in-
small blocks, producing situations described as “tensor
formativepictureofthephysicalsystem,thatitmightbe
product states”. Levin and Nave (50), Gu and Wen(51),
numericallyaccurate,andthatitmightbeimprovableso
and Vidal and coworkers(46; 47) described how an accu-
that more work can lead to better results.
rate analysis could be constructed based on the correla-
tions among statistical variables located at a very small
number of nearby lattice sites.
Ontheone-dimensionallattice,Vidal(46)andcowork-
1 If δL1 and δL2 are two rescaling factors, then the form of ers use two or three neighboring lattice sites as the basis
the eigenvectors as a function of these rescaling factors sat-
isfy E(δL1·δL2) = E(δL1)E(δL2). It follows that E(1) = 1 ofthecorrelations. InhigherdimensionsLevinandNave
and that E(cid:48)(δL1)/E(δL1)=C for some constant C, leading to useahexagonalconstructioninwhichthebasicvariables
E(δL)∝δLy (28). arethreetensorindices,eachindependentlytakingonin-
6
morerecentworkhasreplacedsummationoverspinvari-
ables by sums over tensor indices. The basic symmetries
teger values grouped around a three-legged lattice site.
are hidden in the structure of these tensors. In using
Gu and Wen correspondingly use a four index tensor de-
this tensor representation, recent workers have used uni-
scribing the intersections in a square lattice (see Figure
versality to say that they can use any problem-definition
(1)) .
that reflects a desired symmetry. They then also argue
that the proper meaning of the tensor indices will give
themdirectaccesstothedeepstructureofthestatistical
mechanics problem (52).
Eachtensorindexcantakeonχpossiblevalues,repre-
sentingthatnumberofdifferentconfigurationsofthesys-
tem. Recent workers believe, but have not proven, that
they can get perfect accuracy when χ is infinite. Conse-
quently they reach for approximation methods that per-
mitthemtoincreaseχuntilitreachesquitelargevalues.
( Note that these indices with their large number of pos-
i j i j sible values can simultaneously approximately represent
T many kinds of different variables: many-component vec-
tors, Ising spins, or continuous variables. ) In contrast
l k l k
theearlierworkersfeltthatarbitraryaccuracywouldnot
be available to them. The best that was expected was a
qualitatively accurate description of the problem.
FIG.1 ThebasictensornetworkusedherefortheSVDrenor- Weusethetermsummation variablestodescribeboth
malizationcalculations. Tensorsarerepresentedbybluesolid thespinsoftheearlierwork,andthestochasticvariables
coloredsquares. Redcirclesdenotethepositionofthetensor’s linked to the tensor indices more recently used.
indices. Every tensor has four indices. Every index assume
integer values between one and χ, and is shared between ex-
actly two tensors. Four indices determine the configuration
2. Geometric structure
ofthestatisticalvariable,andthecorrespondingtensorentry
gives the statistical weight of the configuration. Note that
the interactions represented by the tensors occupy half the Another difference can be seen in the geometric struc-
available space. The left inset shows the labeling of the in- tures used to describe the interactions among the sum-
dices;therightinsetshowsthesametensorinthestick-figure mationvariables. Inthetensorworkthesummationvari-
usually used in the literature. ables, in a similar fashion to their role in vertex models,
(e.g. see(14)),areassociatedwithbondsandtheirinter-
actionsareassociatedwithvertices. Thisconstrainseach
summationvariabletoparticipateinexactlytwointerac-
I. Comparisons
tions (connect two vertices). The interactions, however,
are less constrained and typically group together several
The main points of difference between the work of the
summation variables around a rank m vertex.
1970s and that of the last two decades include
In the earlier renormalization work, in contrast, the
summation variables are associated with vertices and
1. Stochastic variables thusmayparticipateinmorethantwointeractions. The
interactions are associated with blocks of summation
We have already mentioned that the 1975 scheme uses variables allowing more than only pairwise interactions.
spins while the recent scheme employs much more com- Thisdifferencenotonlymanifestsintheformulationof
plex spatial structures labeled by tensor indices. Both the partition function of each of the representations but
approaches need to reflect the underlying symmetry of more importantly restricts the placement of the rescaled
theproblemathand,forexamplethespinflipsymmetry summation variables and their interactions. In the ear-
of an Ising model. The early work used spin variables lier work new summation variables could be placed arbi-
that directly reflected the symmetry. 2 In contrast, the trarily provided the interactions they participate in can
be formulated in term of the old interaction blocks. In
the tensor representation the binary interaction struc-
turemustbepreservedwhenintroducingnewsummation
2 However, there were occasional uses of more complex variables.
In Burkhardt’s(35) Ising model calculation, the “spin” variable variables. Thus every introduction of a new summation
couldtakeonthreevalues: ±1andzero. Thelastvaluereflecting variableisnecessarilyassociatedwithchangingtheinter-
aholeunoccupiedbyamagneticspin. action connectivity of the old variables.
7
J. Plan of paper
The next section describes the block spin and the
rewiring methods employing singular value decomposi-
3. Calculational strategy
tion (SVD) used for renormalization by Ref.(50) and
Ref.(51). Sec. (III) outlines the results from these cal-
Theearlierworkfoundthepropertiesofcriticalpoints
culations, including some new results for both the 1975
via a method based upon the analysis of fixed points.
methodandalsotherewiringcalculations. Thefinalsec-
First, the critical system was brought to a fixed point.
tion suggests further work.
Critical indices were then calculated by looking at the
growth or decay under renormalization of small pertur-
bationsaboutthefixedpoint,usingamethodbasedupon
II. THE RENORMALIZATION PROCESS
eigenvalues (see Sec. (I.F).) The main output of the cal-
culationwereasetofcriticalindiceswhichcouldbecom-
A. Overview
pared among calculations and with theoretical results.
In contrast, tensor analysts seldom calculate fixed We now come to compare different approximate real
points.3 Instead they calculate free energies and other space renormalization schemes. The starting point for
thermodynamicquantitiesbygoingthroughalargenum- theconsideredmethodsisasystemdescribedbythesta-
berofrenormalizations, usuallyincreasingthevalueofχ tistical variables, σ , and a Hamiltonian σ . In the
{ } H{ }
as they go. (As we shall see, it is natural to square the 1975 scheme this Hamiltonian is directly used to define
value of χ in each tensor renormalization.) When they the partition function
reach a maximum convenient value of χ they employ ap-
proximationsthatenablethemtocontinuetorenormalize Z =Tr σ e−βH. (12a)
{ }
with fixed χ. These calculations then show the thermo-
Inthenewerscheme,theHamiltonianisusedtodefineto
dynamic behavior near criticality.
defineatwo-,three-,orfour-indextensoralongthelines
The non-appearance of fixed points in many of the
described in Sec. (II.B) below. The partition function
tensor calculations provides an important stylistic con-
is then defined as a statistical sum in the form of a sum
trast between that work and the studies of the 1975-era.
over indices of a product of such tensors, in the form
Thecalculationoffixedpointsforthecriticalphenomena
problemspermitsthedirectcalculationofcriticalindices two index: Z =Tr T T .....T (12b)
i,j,k,...,n ij jk ni
andthusoffersmanyinsightsintothephysicsoftheprob-
lem. The insights are obtained by keeping track of and or
understanding every coupling constant used in the anal-
four index: Z =Tr T (12c)
ysis. This is easy when there are, as in Ref.(34), sixteen i,j,k,..., ijkl
couplings. However, the more recent tensor-style work (cid:89)
In both cases, the setup of the tensor product is such
often employs indices which are summed over hundreds
that each index appears exactly twice. In this way, the
of values, each representing a sum of configurations of
systemcanmaintainitsgaugeinvarianceasaninvariance
multiple spin-like variables. All these indices are gener-
undertherotationofeachindividualindex. Wecanthen
ated and picked by the computer. The analyst does not
imagine that these partition functions may equally well
and cannot keep track of the meaning of all these vari-
bedescribedintermsofthevaluesofcouplingconstants,
ables. Therefore, even if a fixed point were generated,
K, or of the value of tensors, T.
it would not be very meaningful to the analyst. In fact,
Working from this starting point, the renormalization
theliteraturedoesnotseemtocontainmuchinformation
scheme is implemented through three steps as follows:
aboutthevaluesandconsequencesoffixedpointsforthe
new style of renormalization.
The fixed point method seems more fundamental and 1. Introducing new statistical variables
preferable, but offers major challenges when the value of
χ is large. Inthe1975-stylescheme,thenewvariablesaredefined
to be exactly similar to the old variables, σ , except
{ }
that the new variables are spaced over larger distances
than the old ones (See Figure (2).)4. A new Hamilto-
niandependingonbotholdandnewvariables,isdefined
3 Notable exceptions include the Hamiltonian work of Vidal and
coworkers(46;47),inwhichafixedpointHamiltonianisindeed
calculated. For statistical rather than quantum problems, fixed
point studies were done by Ref.(53) and Ref.(54). This fixed
pointanalyses,however,wereonlybecarriedoutforsmallvalues 4 Infact,thisidentityofoldandnewisoneofthemajorlimitations
ofχ. oftheolderscheme.
8
inwhichthatnumberisfour. Afterarearrangement,the
partition function sum in Eq. (14) may be written as
by adding to the old Hamiltonian an interaction term5.
V˜( µ , σ ). This term is defined so that the partition Z =Tr Tr U(ij,α)V(α,kl) ( T )
{ } { } αβγ... ijklmn... mnpq
function remains unchanged by the inclusion of the µ’s.
(cid:89) (cid:89)
This invariance is enforced by the condition The sum over the old tensor indices may then be per-
Tr µ e−βV˜({µ},{σ}) =1 (13a) formed, generating new tensors, Tα(cid:48)βγδ, so that
{ }
so that the partition function can be written as Z =Trαβγ... ( Tα(cid:48)βγδ) (17)
(cid:89)
Z =Tr{σ}e−βH({σ)} =Tr{σ}Tr{µ}e−βH({σ)}−βV˜({µ},{σ}). 3. Obtaining a recursion relation
(13b)
A roughly similar analysis can be used in the tensor net-
Thenewdegreesoffreedom,µ,havebeendefinedtobe
workscheme. Startingfromthedefinitionofthepartition
identicaltothevariables,σ,theonlydifferencebeingthat
functionasthetraceofaproductoftensorsinEq. (12c)
the µ’s are defined on a rescaled system. This identity
one replaces each of the rank four tensors by a product
usuallypermitstheextractionofnewcouplingconstants,
of rank three tensors, using a scheme derived from the
K(cid:48) from the new Hamiltonian. The new couplings are
singular value decomposition (SVD) theorem (See Sec.
(II.C.1) below.) as6 thenconnectedtotheoldviatherecursionrelationK(cid:48) =
(K).
R
T =Tr U V (14a) If the recursion relation is calculated exactly, the new
ijkl α ijα klα
setofcouplingswilllikelycontainmanymoretermsthan
leaving us with the old set. This proliferation of couplings reflects the
additionalinformationfromseveralblocksoftheoldsys-
Z =Trijkl... Tmnpq =Trijkl... Trαβγ... UijαVklα temthatwearetryingtocramintooneblockofthenew
(cid:89) (cid:89) (14b) one. An approximation is needed to limit the new cou-
plings. This limitation usually results in a situation in
which the possible couplings include only those that can
2. Tracing out the original statistical variable be formed from spins completely within a geometrically
defined block. Couplings which include spins from sev-
In the 1975 scheme the trace over the original statisti- eral blocks are excluded. One example of such a block is
cal variables, σ , defines a new Hamiltonian (cid:48) which shown in Figure (2).
{ } H
depends solely on the new variables µ , The tensor scheme has a different approach. In order
{ }
todorenormalizations, thenewpartitionfunctioncalcu-
e−βH(cid:48){µ} =Tr σ e−βH˜({µ},{σ}) (15) lation of Eq. (17) must have the same structure is the
{ }
old one in Eq. (12c). As we discuss in detail in Sec.
so that
(II.C.1) below, this structural identity is violated by the
Z =Tr µ e−βH(cid:48){µ} (16) exact theory in which there are many more new indices
{ } thanold. Toobtainarecursionrelation,onemustusean
One can expect that some approximation will be needed approximation to eliminate the proliferation in the sum-
in order to calculate the sum over the σ’s. mation degree, χ. As we shall discuss in Sec. (II.C.1)
A roughly analogous procedure can be applied to the below, an approximation of this kind is automatically
tensor sums in Eq. (14). If the position of the UV prod- provided by the SVD method. Using this approximation
uctsandthenewindiceshavebeendeftlychosen,theold method, one has a renormalized problem with exactly
indices will appear in a series of small islands in which the same structure as the original problem. The result
each island is only coupled to a limited number of new maybeexpressedasarecursionrelationfortherankfour
indices. Following Ref.(51), we shall work with the case tensor
T(cid:48) = (T) (18)
S
5 The˜appearsonthisV todistinguishitfromanotheruseofthe or as a recursion relation for the parameters defining
symbolV,thatistheV thatconventionallyappearsinsingular- those T’s, e.g. K(cid:48) = (K).
R
value-decompositionanalysis. There is a difficulty in using tensor components in the
6 The standard SVD scheme produces a matrix-multiplication
recursion relation of Eq. (18). Because of the gauge
product. T =UΣVTr whereΣisadiagonalmatrix. Thediago-
nalentriesinΣarenon-negativeandarecalledsingular values. invariance the components of the new tensor, T(cid:48) are not
In the notation in Eq. (14a) the Σ is absorbed into the U and uniquely defined. To ensure uniqueness, it might well be
V. better to define the tensors in terms of gauge invariant
9
decoration
potential moving
summation
FIG.2 Thesetupforapotentialmovingschemeonasquarelattice. Theoldspins(σ)aremarkedbyfilledblackcircleslocated
at the vertices of a square lattice. Note that each such spin belongs to four different squares. These squares form the “blocks”
forourcalculation. Thenewspins,µappearinonequarteroftheblocks,andaremarkedasfilledbrightredcircles. The(red)
lines emanating from these new spins denote coupling terms that link these to the old spins. Each such coupling connects a
single old spin to a new one. The potential moving places all the interactions between old spin in blue squares. The old spins
aroundeverybluesquareareconnectedonlytothemselvesandtonewspinvariables. Theycanbesummedover,givinganew
effective coupling between adjacent new spins.
parameters. While this may be done relatively easily calculations then use a Hamiltonian H( σ ) to define
{ }
for low χ values, identifying all the independent gauge the statistical weight of each configuration of the vari-
invariants for high χ value tensors may be a daunting ables. Consider a problem involving four spin variables,
task. σ1,σ2,σ3,σ4,sittingatthecornersofasquare(seeFigure
With a recursion relation at hand one may apply all (3)), each variable taking on the values 1. If this prob-
±
the tools described in the previous section and obtain a lem has the symmetry of a square, it can be described in
fixed point Hamiltonian and the corresponding critical terms of the following combinations
exponents.
In the remainder of this chapter, we describe the nuts
S =1,
0
andboltsoftherealspacerenormalizationprocess,using
S =σ +σ +σ +σ ,
asourexamplesquarelatticecalculationsbasedonIsing- 1 1 2 3 4
modelsandtheversionoftensorrenormalizationfoundin Snn =σ1σ2+σ2σ3+σ3σ4+σ4σ1,
(51). We particularly focus on understanding the differ- S =σ σ +σ σ ,
nnn 1 3 2 4
encesbetweentheolder(34)andthenewerstyles(50;51)
S =σ σ σ +σ σ σ +σ σ σ +σ σ σ ,
3 1 2 3 2 3 4 3 4 1 4 1 2
of doing renormalization work.
S =σ σ σ σ . (19)
4 1 2 3 4
The spin combination variables S form a closed alge-
i
B. Basic Statistical Description bra, i.e. any function of the spin variables of Eq. (19)
maybeexpressedasalinearsumofthesesamevariables
In Sec. (I.I.1) we pointed out that the older calcula-
with constant coefficients:
tions are based upon summations over defined stochas-
tic variables like the Ising models σr = 1. These F(S0,S1,Snn,Snnn,S3,S4)= aiSi .
±
(cid:88)
10
2 1
T =exp(K +4K +4K +2K +4K +K ),
++++ 0 1 nn nnn 3 4
T =exp(K +2K 2K K ),
+++ 0 1 3 4
{ −} − −
T =exp(K 2K +K ),
++ 0 nnn 4
3 4 { −−} −
T =exp(K 4K +2K +K ),
+ + 0 nn nnn 4
{ − −} −
T =exp(K 2K +2K K ),
+ 0 1 3 4
{ −−−} − −
T =exp(K 4K +4K +2K 4K +K ),
FIG.3 Identificationofthespinvariableslocatedatthever- −−−− 0− 1 nn nnn− 3 4
(22)
tices of a square unit cell.
where curly brackets stand for all cyclic index transfor-
mation, i.e.
T =T =T =T .
+++ ++ + + ++ +++
One important example of this set of variables, denoted { −} { − } { − } {− }
as [S] is a Hamiltonian Hsq[S] which describes the most Alternatively, one might use the index values i = [1] to
general isotropic interactions with the symmetries of a represent a sum over the statistical weights produced by
square unit block that can be formed from the set of the possible spin configuration (σ = +1) and (σ = 1)
−
σ ’s . The basic block used in the 1975 renormalization and the index [2] to represent a difference between these
i
calculation is given in terms of this Hamiltonian as two statistical weights, specifically
(+)+( ) (+) ( )
BLOCK=e−βHsq[S] with −βHsq[S]= KiSi (20) [1]= √2 − and [2]= √−2 − (23)
i
(cid:88)
Thefactorof√2isintroducedtomaketheindex-change
Here the K’s are called coupling constants and their val-
into an orthogonal transformation. Under this defini-
ues provide a numerical description of the problems at
tion the tensor-representation would also have six dis-
hand.
tinct components however their values in the different
In contrast, a whole host of new calculations replace representations change according to
the coupling constants by tensors, and use the tensor
indices as a proxy for statistical variables. To illustrate T˜ijkl =OimOjnOkoOlpTnmop, (24)
this process, we write the tensor, T , for the cases in
ijkl
where O denotes the orthogonal transformation which
which each index can take on two possible values and in
maps the indices + on the right to the new indices [1]
which there is once more the symmetry of a square. The
−
and [2] that appear on the left. For example:
tensors are situated on every other square and therefore
capture only half of the possible four spin interaction
T˜ =1 T +4T +4T
and next nearest neighbor interaction.7We use the spin 1111 4 ++++ +++− ++−−
notation to write the tensor as +(cid:0)2T+ + +4T+ +T ,
− − −−−− −−−−
T =e(cid:80)KiSi. (21) T˜{1112} =14 T+++++2T+++−−2T+−−−−(cid:1)−T−−−− .
(cid:0) (cid:1)
It is important to note that Eq. (24) gives two differ-
There is considerable flexibility in defining the indices8. ent descriptions of the very same tensor, T, in different
bases systems. The tensors remain the same, but the
For example, we could let one index-value, (+), corre-
coordinate system is varied.
spond to positive spin and the other, (-), to negative
Ofcourse,thecasedescribedhereisrathersimple. The
spin. Then the tensor components would have the fol-
renormalization transformation develops, at each step, a
lowing distinct values
succession of tensors, usually of increasing complexity,
At each step, the partition function depends upon the
tensor in question, but is independent of the particular
representation of that tensor. When applied successively
7 We note that allowing the index four possible values allows the to the redefinition of indices in each step of a long calcu-
description of every interaction in Eq. (19). However, in favor lation, theindexmethodprovidesaflexibilityandpower
ofsimplicitywerestrictourpresenttreatmenttothetwovalued
not easily available through the direct manipulation of
indextensorsonly.
8 Infactthatflexibilityisasortoffreedomundergaugetransfor- spin-like variables. We shall see this flexibility in the
mations,andthatfreedomrepresentsoneofthemainattractions specific calculations of renormalizations to be described
ofthetensorapproach. in Sec. (III) of this paper.