High-gradient operators in perturbed Wess-Zumino-Witten field theories in two dimensions

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by  Shinsei Ryu

High-gradient operators in perturbed Wess-Zumino-Witten field theories in two
dimensions
S. Ryu,1 C. Mudry,2 A. W. W. Ludwig,3 and A. Furusaki4
1 Department of Physics, University of California, Berkeley, CA 94720, USA
2 Condensed matter theory group, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland
3 Department of Physics, University of California, Santa Barbara, CA 93106, USA
4 Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan
(Dated: January 31, 2010)
0
1 Manyclassesofnon-linearsigmamodels(NLσMs)areknowntocontaincompositeoperatorswith
0 anarbitrarynumber2sofderivatives(“high-gradientoperators”) whichappeartobecomestrongly
2 relevant within renormalization group (RG) calculations at one (or fixed higher) loop order, when
n thenumber2sofderivativesbecomeslarge. ThisoccursatmanyconventionalfixedpointsofNLσMs
a whichareperturbativelyaccessiblewithintheusualǫ-expansionind=2+ǫdimensions. Sincesuch
J operators are not prohibited from occurring in the action, they appear to threaten the very exis-
1 tence of such fixed points. At the same time, for NLσMs describing metal-insulator transitions of
3 Andersonlocalization inelectronicconductors,thestrongRG-relevanceoftheseoperatorshasbeen
previously related to statistical properties of the conductance of samples of large finite size (“con-

] ductancefluctuations”). Inthispaper,weanalyzethisquestion,notforperturbativeRGtreatments
ll ofNLσMs,butfortwo-dimensional Wess-Zumino-Witten(WZW)modelsat levelk,perturbatively
a inthecurrent-currentinteractionoftheNoethercurrent(“non-AbelianThirring/Gross-Neveumod-
h
els”). WZWmodelsarespecial(“PrincipalChiral”)NLσMsonaLieGroupGwithaWZWtermat
-
s level k. In these models the role of high-gradient operators is played by homogeneous polynomials
e of order 2s in the Noether currents, whose scaling dimensions we analyze. For the Lie Supergroup
m G=GL(2N|2N)andk=1,thiscorrespondstotime-reversalinvariantproblemsofAndersonlocal-
ization intheso-called chiralsymmetryclasses, andthestrengthofthecurrent-currentinteraction,
.
t ameasureofthestrengthofdisorder,isknowntobecompletelymarginal(foranyk). Wefindthat
a
m all high-gradient (polynomial) operators are, toone loop order, irrelevant or relevant dependingon
thesign of that interaction.
-
d
n
I. INTRODUCTION reparametrizationof the target manifold, is
o
c
[ 1 ddx
S = G φ(x) ∂ φi(x)∂ φj(x) (1.1)
1 Fluctuations are known to play a key role in suf- 4πtZ ad−2 ij(cid:2) (cid:3) µ µ
v ficiently low-dimensional systems, whether classical or
8 quantum, as they can preempt spontaneous symmetry where Gij[φ] is a component of the metric tensor on M,
1 breaking. Whenthesymmetryisbothglobalandcontin- t is the coupling constant, and a is the short-distance
1 uous,thetoolofchoicetoaddresstheroleoffluctuations cutoff.
0 inlow-dimensionalsystemsisthenon-linearsigmamodel The target manifold can be either compact or non-
.
2 (NLσM). However, the usefulness of NLσMs has come compact. An example of a NLσM on a compact target
0 to transcend situations in which a pattern of symmetry manifold is the O(N)/O(N −1) NLσM with 2 < N =
0 breaking is immediately obvious. For example, NLσMs 3,4,5,··· when the target manifold is the unit sphere
1 have been used with success in the context of Ander- SN−1 in N-dimensional Euclidean space. When N = 3
:
v son localization (see Ref. 1 for a review) to access the it describes spontaneous symmetry breaking in a classi-
i transitionfromametallictoaninsulatingphaseinduced cal ferromagnet. Non-compact target manifolds are of
X
byweakdisorderorto computeprobabilitydistributions relevance to the problem of Anderson localization in the
r
a of spectral,2 wavefunction,3 and transport characteris- bosonic “replica limit” N → 0 or when the manifold is
tics in chaotic metallic grains and disordered electronic generalizedtoasupermanifold.1InAndersonlocalization
systems.1 the couplingconstantt hasthe meaningofthe inverseof
the mean dimensionless conductance.5
Quitegenerally,theconstructionofagenericNLσMon Theimplicitassumptionmadeintheconstruction(1.1)
a connected Riemannian manifold M of finite dimension is that all the invariant scalars that contain 2s (1<s =
n, the “target manifold”, can proceed in the following 2,3,···)derivativesofthefieldcanbeignored. Thestan-
way.4 One assigns to any point from Euclidean space in dard justification for this assumption is that their “en-
ddimensions,specifiedbycoordinatesxµ (µ=1,··· ,d), gineering dimension” 2s is much larger than the spatial
a point in the manifold M with the coordinates φi(x) dimensiond=2+ǫ,i.e., theyareirrelevantinthe renor-
(i = 1,··· ,n). The simplest action S, which is made malization group (RG) sense, and this is expected to re-
of two derivatives of the coordinates φi, and is invari- mainsoafterrenormalizationind=2+ǫdimensionsfor
ant under both the rotations of Euclidean space and small ǫ, and thus small t.
2
This assumption was called into question in Refs. 6– The substitution t → −t does not affect the value of
16, for which the main results can be illustrated most the minimal (i.e., dominant, or “leading”) one-loopscal-
simply by the example of the O(N)/O(N −1) NLσM. ing dimensions, when the spectrum of anomalous one-
We recall that the O(N)/O(N −1) NLσM has an infra- loop dimensions17 of all high gradientoperators of order
red unstable fixed point located, to one loop order, at s is distributed symmetrically about zero. This turns
t∗ = ǫ/(N −2), from which emerges a renormalization outtobethecasewheneverm,n>1intheaboveexam-
group(RG) flow to strong and weak coupling. In Ref. 8, ples. Ontheotherhand,therearesometargetmanifolds,
a family of perturbations of the O(N)/O(N −1) NLσM the simplest examples being SN−1 = O(N)/O(N −1)
action (1.1), which we shall call high-gradientoperators, and CPN−1 = U(N)/U(N − 1)× U(1), for which the
was considered. A high-gradient operator of order s is full spectrum ofone-loopanomalousdimensions of order
a homogeneous polynomial of order 2s in the derivatives s turns out to be not symmetric about zero, in which
of the fields (all located at the same point) which is a case the substitution t → −t matters. For example,
scalar with respect to both the symmetry group of the high-gradientoperatorsaremademoreirrelevantbyone-
NLσM [i.e., O(N)] and the rotation group of Euclidean loop renormalization effects in the non-compact NLσM
space. Theminimum(i.e.,dominant,or“leading”)value onU(N−1,1)/U(N−1)×U(1). (We referthe readerto
ofthe one-loopscaling dimensions17 ofthe high-gradient AppendixAforamoredetaileddiscussionof“one-sided”
operators of order s at the fixed point t∗ is found8 to be versus“two-sided”spectraofone-loopanomalousscaling
dimensions for high-gradient operators in NLσMs.)
x(s) = 2s−s(s−1)t∗+O(ǫ2). (1.2)
Of course, one can only conclude that high-gradient
operators become relevant for sufficiently large values of
Although strongly irrelevant by power counting (i.e., in
s,ifthestrongrelevanceseenintheone-loopexpressions
the absence of fluctuation corrections, t∗ → 0), high-
fortheir scalingdimensions persistswhen allhigher loop
gradient operators of order s thus acquire a one-loop
contributions (not computed here or in other works on
scaling dimension smaller than two when the order 2s of
thissubject)havebeentakingintoaccount. Forexample,
derivatives is largeenough so that st∗ ≈sǫ/(N−2)∼2,
theone-loopexpressionsmaynotbecharacteristicinthe
and thus would appear to become relevant,based onthe
large-s limit, if the actual expansion parameter is not ǫ
one-loop result. In d = 2 dimensions, the lowest one-
butsǫ.22,23 Asanyinsightforresolvingthe natureofthe
loop scaling dimension18 for all high-gradient operators
ǫ expansion for high-gradient operators in NLσMs must
of order s is
come from outside the ǫ expansion itself, progress has
stalled since the early 1990’s.
x(s) = 2s−s(s−1)t+O(t2) (1.3)
The aim of this paper is to study the operators that
along the trajectory to strong coupling away from the play the role of the high-gradient operators in field the-
infra-red unstable fixed point t = 0. Two-loop counter- ories which are two-dimensional Wess-Zumino-Witten
partstoEqs.(1.2)and(1.3)yieldthe sameconclusion:15 (WZW)theories24−28 onaLiegroupG,perturbedbyan
high-gradient operators of sufficiently high-order s ap- interaction quadratic in the Noether currents (“current-
pear to be relevant for any given dimension d=2+ǫ at current interaction”). Such theories are often also re-
the non-trivial fixed point. ferred to as “two-dimensional non-Abelian Thirring (or
Similar results hold for the NLσMs defined on the Gross-Neveu) models”. Any WZW theory, which is a
compact target manifolds (M and N are positive inte- Principal-Chiral-Non-Linear-sigma model supplemented
gers)Sp(M+N)/Sp(M)×Sp(N),6,7 U(M+N)/U(M)× by a WZW term at level k, gives a prescription to
U(N),9,10 O(M+N)/O(M)×O(N),11andonfamiliesof construct high-gradient operators in terms of powers of
compactK¨ahler(andsuper)manifolds.16Generalizations Noether currents. Because it is possible to represent
tothenon-compacttargetmanifoldsSp(M,N)/Sp(M)× the Noether currents in WZW theories in terms of free
Sp(N), U(M,N)/U(M)×U(N), and O(M,N)/O(M)× fermions,26 one might be inclined to think that such op-
O(N) follow from the rule that the coupling t of the erators are perhaps not capable of displaying a “patho-
compact NLσM entering in one-loop anomalous dimen- logical” spectrum of scaling dimension as in Eq. (1.2).
sions must be replaced by −t in the corresponding non- However, as we demonstrate in this paper, the situation
compact NLσM. In Anderson localization, compact tar- is more interesting. Indeed, we will see that under con-
getmanifolds arisewhenusingfermionic replicasfor dis- ditions specified below, the one-loop spectra of the form
order averaging, whereas non-compact target manifolds (1.3) and (1.2) can be realized by perturbing a WZW
arise when using the bosonic replicas for disorder aver- critical point by a current-currentperturbation.
aging. If one uses supersymmetric disorder averaging, We also want to investigate if there is a difference be-
the resulting NLσM has both compactandnon-compact tweenthepropertiesofhigh-gradientoperatorsinunitary
sectors.1 The high-gradient operators in the NLσM de- and non-unitary non-Abelian Thirring models. This is
fined on AdS ×S5 (AdS is non-compact whereas S5 is importantbecause NLσMs describing the physics of An-
5 5
compact) have also been discussed in the context of the derson localization are non-unitary field theories. More-
AdS/CFT correspondence. (See, for example, Refs. 19– over,high-gradientoperatorsin these theories havebeen
21.) previously related to the statistical fluctuations of the
3
conductance of a disordered metal.29−32 In this con- The result that the spectrum of one-loop anomalous
text, an appealing physical interpretation of the spectra dimensions of high-gradient operators is strongly depen-
(1.3) and (1.2) has been proposed, attributing them to dent onthe sign ofg has importantimplications in the
M
a broad tail in the probability distribution of the con- contextofAndersonlocalizationbecause,asdiscussedin
ductance.29−32 However, given that this interpretation Ref. 33, the gl(M|M) Thirring (or Gross-Neveu)model
k
depends cruciallyonthe ability toinvertthe s→∞and at k =1 describes a disordered electronic system, where
b
ǫ → 0 limits in spectra which are analogous to those in g > 0 and g > 0 correspond to the strengths of dis-
A M
Eqs. (1.3) and (1.2), it would be useful to have an ex- order potentials. The theory with g > 0 thus offers
M
ample of a critical field theory describing a problem of an example of a critical theory for Andersonlocalization
Anderson localization for which one can study the RG- with no relevant high-gradient operator. – For example,
relevanceofhigh-gradientoperatorswithout resortingto thisfieldtheorydescribes33atight-bindingmodelofelec-
theǫ-expansion,andforwhichonecanreasonablyexpect trons on the honeycomb lattice with (real-valued) ran-
a broad distribution of the conductance. dom hopping matrix elements which are non-vanishing
We now provide an outline of the article and a sum- only between the two sublattices of the bipartite hon-
mary of our results. eycomb lattice (see also Ref. 34). Versions of the hon-
It is shown in Sec. II that high-gradient operators in eycomb tight-binding model provide the basic electronic
the (unitary) su(2) Thirring model with strength g of structure of graphene. In the classification scheme of
k
the “current-current interaction”, are made more irrele- Zirnbauer, and Altland and Zirnbauer,35−38 this model
vantbytheprebsenceoftheseinteractionswhenthelatter belongs to the “chiral-orthogonal”symmetry class (class
are (marginally) irrelevant (g < 0, in our conventions). BDI). (Another example of a problem of Anderson lo-
Ontheotherhand,alongtherenormalizationgroup(RG) calization in the same symmetry class is provided by a
flowdrivenbya(marginally)relevantcurrent-currentin- random tight-binding model on a square lattice with π-
teraction (g > 0, in our conventions), a one-loop spec- flux through every plaquette.39)
trumoftheform(1.3)isrecoveredinthe“classical”limit By contrast, when g < 0, the spectrum of one-loop
M
1/k → 0. The inverse level 1/k plays here the role of a scaling dimensions is unbounded from below for any k
“quantum” parameter. Indeed, for any finite k, we find as is the case in Eq. (1.2). The full spectrum of one-
that the quadratic growth in s in the unbounded one- loopanomalousscalingdimensionsofhighgradientoper-
loopspectrum (1.3)doesnotpersistforvaluesofslarger atorsasitappears,e.g.,intheGrassmanianNLσMswith
than k. In effect, 1/k determines the efficiency in “tam- target manifolds Sp(M +N)/Sp(M)×Sp(N), U(M +
ing” the strong RG-relevance of high-gradient operators N)/U(M)×U(N), O(M +N)/O(M)×O(N), which is
seen at one-loop order, which is related to the fact that symmetric about zero, is only recovered in the extreme
thereexistsarepresentationofthecurrentalgebraofthe “classical” limit M,k → ∞. In the context of Anderson
level-k WZW theory in terms of free fermions. localization, the case with g < 0 describes the surface
M
Section III is devoted to high-gradient operators in state of a three-dimensional topological insulator in the
what we will call the gl(M|M) Thirring (or Gross- chiral-symplecticclass(symmetryclassCII)ofAnderson
k
Neveu) model which was discussed in Ref. 33. This localization.40–42
b
is the gl(M|M) WZW theory on the Lie Supergroup After concluding in Sec. IV, we review in Appendix B
k
GL(M|M), perturbed by two current-current perturba- the realization of the gl(2N|2N) Thirring (or Gross-
k=1
tions, obne which we call g , which is exactly marginal, Neveu) model as a problem of Anderson localization in
M b
and another which we call g , which flows logarithmi- twodimensionsinsymmetryclassBDI,whichwasestab-
A
cally under the RG at a rate dependent on g . In spite lished in Ref. 33.
M
of the presence of an RG flow of the coupling g there
A
exists a sector of the theory, the so-called PSL(M|M)
sector, which is scale (conformally) invariant through- II. HIGH-GRADIENT OPERATORS AND su(2)
k
out.33 The high-gradient operators turn out to reside in WZW THEORIES b
this conformally invariant sector, and are unaffected by
the presence of the coupling g . We will show that, for The O(3)/O(2) NLσM with coupling constant t is the
A
k = 1, the spectrum of one-loop anomalous dimensions simplest example of a NLσM containing infinitely many
of high-gradient operators is fundamentally different for high-gradientoperatorsallof which wouldappear to be-
positiveandnegativevaluesofthecouplingconstantg . comerelevantbasedonone-loopresults. Thishappensat
M
In particular, when g > 0 all high-gradient operators the infra-redunstable fixed point t∗ =ǫ in d=2+ǫ>2
M
are made more irrelevant by the current-current per- dimensions within the one-loop approximation as long
turbations, whereas they are made more relevant when as the order s of these high-gradient operators is large
g < 0. We close Sec. III by comparing the anoma- enough. A precursor to this perturbative property also
M
lous scaling dimensions of high-gradientoperatorsin the occursind=2dimensionsclosetotheinfra-redunstable
gl(M|M) Thirring (or Gross-Neveu) models and those fixed point t =0 as the NLσM flows to strong coupling.
k
intheGL(2N|2N)/OSp(2N|2N)NLσMs,observingthat Along this flow, the spectrum of one-loop dimensions18
b
they behave in the same way. forthehigh-gradientoperatorsisunboundedfrombelow.
4
In two dimensions, the O(3)/O(2) NLσM, supple- plane. We shall also refer to the (anti) holomorphic sec-
mented by a topological theta-term at θ = π, flows tor of the theory as the (right-) left-moving sector.
to a critical field theory, the SU(2) WZW theory with The su(2) current algebra (2.1) has a representation
k
su(2) current algebra, su(2) WZW theory.43−45 in terms of free-fermions. More precisely, it is obtained
k=1 k=1
The strongly relevant high-gradient operators near the from thbe action46
ibnfra-red unstable fixed poibnt t = 0 must become irrel-
k
evant at the WZW critical point, because the full op- dz¯dz
S := ψc†∂¯ψ +ψ¯c†∂ψ¯ (2.2a)
erator content of the su(2) WZW theory is known ∗ Z 2πi ι cι ι cι
k=1 Xι=1 (cid:0) (cid:1)
to contain only a finite number of relevant fields (with
scaling dimensions boubnded from below and above by constructed from k-independent flavors of left (ψ)
zero and two, respectively). The purpose of this section and right (ψ¯) moving Dirac fermions, whereby each
is to perturb the su(2)k WZW theory with a current- one transforms in the fundamental representation of
current perturbation and to examine the fate of those SU(2)×SU(k), with the partition function
operators in the sub(2) WZW theory which correspond
k
to the high-gradient operators in the O(3)/O(2) NLσM.
Z := D[ψ†,ψ,ψ¯†,ψ¯] exp(−S ). (2.2b)
We will refer to tbhese operators still as “high-gradient ∗ Z ∗
operators”. We are going to argue that the spectrum
of one-loop scaling dimensions associated with all high- One has the operator product expansions (OPE)
gradient operators is bounded from below by the low-
δ δ
est one-loop scaling dimension corresponding to high- ψ (z)ψd†(0)=ψd†(z)ψ (0)∼ ιι′ cd,
gradient operators of order k. This result is very dif- cι ι′ ι′ cι z
ferent from the unbounded spectrum (1.3) of one-loop ψ¯ (z)ψ¯d†(0)=ψ¯d†(z)ψ¯ (0)∼ διι′δcd, (2.3)
scaling dimensions associated with high-gradient opera- cι ι′ ι′ cι z¯
tors in the two-dimensionalO(3)/O(2) NLσM. ψ (z)ψ¯d†(0)∼0,
cι ι′
In the following sections, we first review the su(2)
k
WZWtheoryperturbedbyacurrent-currentinteraction. forι,ι′ =1,··· ,k andc,d=1,2. Inturn,theOPE(2.3)
Second, we identify high-gradient operators of ordber s. imply that the left and right Noether currents
Finally, we compute the leading one-loop dimensions of
high-gradient operators of order s up to one loop. k (σ ) d k (σ ) d
J := ψc† α c ψ , J¯ := ψ¯c† α c ψ¯ ,
α ι 2 dι α ι 2 dι
Xι=1 Xι=1
(2.4)
A. Definitions
with α=1,2,3 obey the SU(2) current algebra (2.1).
k
The most fundamental property of the su(2) WZW
k The field theory defined by Eq. (2.2) is a free-fermion
theoryisthe existenceofapairofholomorphicandanti-
fieldtheory. Thecontentoflocaloperatorsisthusknown.
holomorphicSU(2)Noether currents, J , J b, J , andJ¯,
1 2 3 1 Itcontainsafinite numberoffieldswhosescalingdimen-
J¯,J¯,respectively,whichsatisfytheaffine(Kac-Moody)
2 3 sionsareboundedbetween0and2andarethusrelevant,
current algebra
asitshouldbeforafieldtheorydefinedonaHilbertspace
with a positive definite inner product and with a spec-
kC i
J (z)J (0)= αβ + f γJ (0)+··· , trum bounded from below which is built on the Dirac-
α β z2 z αβ γ
Fermisea,inshortaunitaryfieldtheory. Clearly,within
kC i (2.1a)
J¯ (z¯)J¯ (0)= αβ + f γJ¯ (0)+··· , the set of powers of the Noether currents (2.4) there is
α β z¯2 z¯ αβ γ thus no roomfor aninfinite family ofrelevantoperators.
J (z)J¯ (0)=0, We perturb the free-fermion field theory by a current-
α β
current interaction O of the SU(2) currents (2.4).
at level k = 1,2,3,··· , where the invariant (Casimir) I k
tensor of rank 2 in su(2) has the contravariant and co-
variant representations (in our conventions) Z := D[ψ†,ψ,ψ¯†,ψ¯] exp(−S),
Z
1
Cαβ = 2δαβ, Cαβ =2δαβ, (2.1b) S :=S +g dz¯dzO (z¯,z), (2.5)
∗ Z 2πi I
respectively, while the structure constant of su(2) is the O (z¯,z):=CαβJ (z)J¯ (z¯)≡2J (z)J¯ (z¯).
fully antisymmetric Levi-Civita tensor of rank 3, I α β α α
f γ =ǫ , α,β,γ =1,2,3. (2.1c) We take the coupling constant g to be real. The (uni-
αβ αβγ
tary) field theory (2.5) is often referred to as a non-
ThedotsinEq.(2.1a)aretermsoforderzeroandhigher Abelian Thirring (Gross-Neveu) model. Suitable non-
in powers of z (z¯) with z =x+iy (z¯=x−iy) the holo- unitary generalizations of the field theory (2.5) compute
morphic (antiholomorphic) coordinates of the Euclidean (disorder average) moments of Green’s functions in a
5
class of problems of Anderson localization in d = 2 di- Thehigh-gradientoperatorsinEq.(2.9)areclassicalin
mensions that we will investigate later on in this paper. thesensethatquantumfluctuationsencodedthroughthe
The one-loop beta function, Pauli principle (or, equivalently, through the underlying
Dirac-Fermi sea) in the free-fermion representation(2.4)
β = dg =4g2, (2.6) of the current algebra, have not yet been accounted for.
g dl To account for these quantum fluctuations, one needs to
introduce a point-splitting procedure that allows for the
encodes the change in the coupling constant caused by
proper normal ordering, i.e., the correct subtraction of
the infinitesimal rescaling a → (1 + dl)a of the short-
all short-distance singularities49
distance cutoff a. Thus, the current-current interaction
is (marginally) irrelevant (relevant) for g < 0 (g > 0) :Tα1···α¯sJ ···J¯ :(z,z¯)≡
with the free-fermion fixed point at g =0. α1 α¯s
ingTahtetShUe(n2o)kn-Ntroiveitahlefirxceudrrpeonitnst(o2f.4t)hearPertinhcoispealapCpheiarar-l zlii→mzhTα1···α¯sJα1(z1)···J¯α¯s(z¯s)
NLσMontheSU(2)groupmanifoldwithaWess-Zumino − (all short-distance singularities) .
term.25−27 This has, the well-known(Euclidean) action i
(2.10)
S = k d2xtr ∂ G−1∂ G +kΓ[G], (2.7a) Two objects of the form (2.9) that are linearly inde-
16π Z µ µ pendent classically might not survive as a pair of dis-
(cid:0) (cid:1)
tinct quantum operators of the form (2.10) after nor-
where G ∈ SU(2) is a group element, and the integral mal orderinghas been implemented. More precisely, one
Γ[G]overathree-dimensionalballB withcoordinatesrµ might anticipate that the underlying free-fermion repre-
and whose boundary ∂B is d=2-dimensional Euclidean sentation of the current algebra must manifest itself as
space, soonasthe orders becomes largerthan the number k of
fermionic flavors by changing the book-keeping relating
1
Γ[g]:= d3rǫ tr G−1∂ GG−1∂ GG−1∂ G classical expressions labeled by SU(2) tensors of rank 2s
24π Z µνλ µ ν λ
(cid:0) (cid:1) and quantum operators.
B
(2.7b) Indeed, we are going to show that this is the mecha-
is the Wess-Zumino term.24 nismthatpreventshigh-gradientoperatorsoforders>k
The Noether currents which generate the from acquiring one-loop scaling dimensions smaller than
SU(2) ×SU(2) symmetry at the critical point the smallest one-loopscaling dimensions associatedwith
left right
of the WZW theory can be fully represented by the the set of all high-gradient operators of order s ≤ k.
fermionic expressions in Eq. (2.4). In the bosonic (i.e., In other words, the smallest one-loop dimension associ-
NLσM) representation, these currents are built out of atedwiththesetofall high-gradientoperatorsisreached
first-order derivatives of the bosonic fields, within the set of all high-gradient operators of order
s ≤ k when g > 0. It is thus bounded from below when
J ∝ktr (∂G)G−1σ , J¯ ∝ktr G−1(∂¯G)σ . g >0.
α α α α
(cid:2) (cid:3) (cid:2) ((cid:3)2.8) Had we ignored the underlying free-fermion represen-
The relationship (2.8) suggests that composite opera- tation of the current algebra altogether, we would have
tors built out of monomials in the currents (2.4) in the wrongly predicted that, when g > 0, the one-loop di-
WZW theory are the counterparts of the high-gradient mensions associated with the classical objects (2.9) are
operatorsinNLσM.Forthisreason,weshallstillcallthe of a form similar to the ones in Eq. (1.2) i.e., that the
formerfamilyofcompositeoperatorshigh-gradientoper- set of one-loop dimensions of high-gradient operators is
ators. The “classical” counterparts of the high-gradient unboundedfrombelow. Ontheotherhand,thisclassical
operators of order s in the NLσM are thus the homoge- predictionisrecoveredinthelimitk →∞withs/k→0.
neous polynomials For this reasonwe shall separate the computation of the
most relevant one-loop dimension associated with high-
Tα1···αsα¯1···α¯sJ ···J J¯ ···J¯ (2.9) gradient operators of order s into the case when s ≤ k
α1 αs α¯1 α¯s
and the case when k <s.
oftheleftandrightcurrentsthatareinvariantunderthe Inthiscontext,wewouldliketoremindthereaderthat
diagonal SU(2) symmetry group of the interacting the- thesu(2) WZWtheoriesareknowntodescribequantum
k
ory.47 The generating set of classicalhigh-gradientoper- criticalpoints in the parameterspaceof quantumspin-S
atorsof orders is specified once allthe linearly indepen- antifberromagneticchainswhenk =2S. Here,we observe
dent rank 2s tensors Tαβ···γδ··· in the adjointrepresenta- that both, the number of relevant perturbations and the
tionofSU(2)thatareinvariantunderSU(2)transforma- number of independent local composite operators built
tions canbe fully enumerated. In turn, the most general out of the generators of SU(2) which are SU(2) singlets,
SU(2) invariant tensor of even rank in the adjoint rep- grows with S. [For S = 1/2 the algebra obeyed by the
resentation is the product of the Casimir tensor of rank Pauli matrices only allows one invariant SU(2) tensor of
2.48 rank 2, the 2×2 unit matrix.] On the other hand, the
6
strength of quantum fluctuations in su(2) WZW theo- upon normal ordering. To leading order in a short-
k
ries decreases with increasing k = 2S for the same rea- distance expansion, this classical expression is replaced
son as the role of quantum fluctuatiobns decreases with by
increasing S in quantum spin chains.
k
Tα1···αsα¯1···α¯s :J ···J :(z)
α1ι1 αsιs
B. Anomalous dimensions of high-gradient ι16=···6=ιs6=X¯ι16=···6=¯ιs=1
operators ×:J¯ ···J¯ :(z¯)+··· . (2.13)
α¯1¯ι1 α¯s¯ιs
Here, the terms included in the ··· arise from the OPE
As the most general SU(2) invariant tensor of even
for the product J (z)J (0) when any two flavor in-
rank in the adjoint representation is the product of the α ι α ι
i i j j
Casimir tensor of rank 2,48 we define the three diagonal dices ιi and ιj are identical. Evidently, normal ordering
SU(2) invariants out of the three current bilinears (or the Pauli principle) has a much more potent effect
when k <s, for the condition ι 6=···6= ι 6=¯ι 6=···6=
1 s 1
H :=CαβJαJ¯β, A:=CαβJαJβ, B :=CαβJ¯αJ¯β, ¯ιs can then never be met so that the leading order term
aboveisabsent. Theoperatorcontentswithandwithout
(2.11a)
normal ordering thus look very different. When k < s,
together with the SU(2) invariant some operators in the set (2.12) completely disappear to
leading order because of Fermi statistics. This will be
C :=AB. (2.11b) demonstrated explicitly for the case of k = 1 [see Eqs.
(2.28) and (2.29) below], for which we will show, after
Thespaceofthehigh-gradientoperatorsisthenspanned correctly taking into account normal ordering, that all
by the family high-gradient operators which would be relevant classi-
cally (when g >0) disappear from the operator content.
Hs,Hs−2C,··· ,H2C[s/2]−1,C[s/2] (2.12)
n o
1. RG equation for un-regularized high-gradient operators
madeof[s/2]+1“classical”operators.50Wecalltheseop-
erators“classical”becausewehavenotyettakenintoac-
To compute the leading one-loop scaling dimensions
countthe shortdistancesingularitiesassociatedwiththe
for the high-gradientoperators (2.12), we start from the
definition of composite operators (i.e., the “Pauli princi-
field theory (2.5) in which we substitute the action by
ple” discussed above). As announced below Eq. (2.10),
these singularities need to be subtracted from the “clas- dz¯dz
sical”expressions(2.12)uponnormalordering. We shall S :=S∗+gZ 2πi OI
nevertheless ignore the issue of normal ordering at first
2m+n=s (2.14)
and compute the one-loop RG equation for these “un- dz¯dz
− Z(s) a2s−2 CmHn.
regularized” (or un-normal-ordered) operators, a step of m,n Z 2πi
mX,n=0
no consequence in the (“classical”) limit s/k → 0. We
shall then contrast this un-regularized calculation with To determine the one-loop dimensions of the couplings
thefullquantumcalculationforthespecialcaseofk =1, {Z(s) |2m+n=s},wedonotneedthe fullone-loopRG
m,n
i.e.,whenthepropernormalorderingprocedurehasbeen
flows, i.e., the RG equations for the coupling constants
accounted for. (s) (s)
up to and including order Z Z , but only the linear
We shall see that the calculation without normal or- m,n p,q
(s)
inZ contributionstotheone-loopRGflows. Thus,all
dering gives an infinite tower of high-gradient operators m,n
we need are the OPE of CmHn(z,z¯) with O (0), where
thatareallrelevanttoone-looporderforsufficientlylarge I
the integers m and n satisfy 1 < 2m+n = s ≤ k. Fur-
s and for g > 0. The one-loop spectrum of anomalous
thermore, we shall introduce the short-hand notation
dimensions isidenticalto the one inthe O(N)/O(N−1)
NLσM when N = 3. Indeed, once the normal ordering 1
procedure is ignored, the high-gradient operators (2.12) A×B =C ⇐⇒A(z,z¯)B(0)= C(z,z¯)+··· (2.15)
zz¯
areanalogoustothosediscussedinRef.15,andthecalcu-
lations of the anomalousdimensions in the sˆu(2) WZW for the OPE relating the operators A, B, and C. Here,
k
model and in the O(N)/O(N−1) NLσM run along par- the dots are meant to contain not only regular terms of
allel tracks. zeroth and higher order in z or z¯ but also second and
Theeffectofnormalorderingisweakerthesmallers/k higher order poles in z or z¯.
is, i.e., the closer proximity to the semi-classical limit of As an intermediary step, one verifies that the OPE
the WZW theory. To see this, consider the case when (2.15) between the building blocks H and C with OI
k >s. The“classical”expressionTα1···α¯sJα1···J¯α¯s(z,z¯) (observe that OI =H) are
for the composite operator made of a local product of
holomorphic and antiholomorphic currents is modified H×OI = −4H, C×OI =0. (2.16)
7
Here, we introduced yet another short-hand notation current monomials with repeating flavor indices. The
linearized RG flows (2.21) have a lower triangular struc-
A···B or A···B, by which we mean that one current
ture, i.e., there is no feedback effect on the flow of a
in A and one current in B are contracted with the rule
high-gradient operator of order s from lower-order high-
C i gradient operators. Thus, we conclude that the leading
J (z)J (0)=δ αβ+ f γJ (0)+··· ,
αι βι′ ιι′(cid:18) z2 z αβ γι (cid:19) [s/2]+1one-loopscalingdimensionsassociatedwiththe
family of high-gradient operators (2.12) when k ≥ s =
C i (2.17)
J¯ (z¯)J¯ (0)=δ αβ+ f γJ¯ (0)+··· , 2m+n are given by
αι βι′ ιι′(cid:18) z¯2 z¯ αβ γι (cid:19)
J (z)J¯ (0)=0, x(ms),n = 2(2m+n)−4gn(n+1). (2.22)
αι βι′
Observe that the spectrum of anomalous dimensions17
for any α,β = 1,2,3 and ι,ι′ = 1,··· ,k at the free-
fermion fixed point g =0, where γ(s) :=−4gn(n+1), 2m+n=s (2.23)
m,n
J :=ψa†(σα)abψ , J¯ :=ψ¯a†(σα)abψ¯ . (2.18) isone sidedwith respectto 0. Wheng ≤0these anoma-
αι ι 2 bι αι ι 2 bι lous dimensions are positive, i.e., the scaling dimensions
are larger than their engineering value. The opposite
When A and B consist of more than one J or J¯ , and
α α happens when g ≥ 0, i.e., when the current-current per-
whentherearemanypossibleWickcontractionsbetween
turbation is (marginally) relevant. When g > 0 and for
A and B, the short-hand notations A···B, A···B and a given 1 < s ≤ k, the smallest one-loop anomalous di-
mension occurs for the pair (m,n)=(0,s),
A···Bmeantheresultingoperatorobtainedbytakingall
possible such Wick contractions. One also verifies that γ(s) := min γ(s) =−4gs(s+1). (2.24)
min m,n
the OPE (2.15) between the building blocks HH, CH, 2m+n=s
and CC with OI are For g > 0, the quadratic dependence on s can overcome
the linear dependence on s in the one-loop dimension
HH×OI =−4H2+4C, x(msi)n := 2s+γm(si)n. If the order s (1 < s ≤ k) is allowed
(2.19) to be sufficiently large, the one-loop dimension x(s) de-
CH×O = CC×O =0. min
I I creasespastthevalue2andeventuallybecomesnegative.
The quadratic dependence on s is reminiscent of that
Wetheninferthat,foranypair(m,n)ofpositiveinteger
for the one-loopdimensions (1.2) in the O(N)/O(N−1)
that satisfies 1<2m+n=s≤k,
NLσM. However, in contrast to the (2+ǫ)-dimensional
O(N)/O(N−1)NLσMatitsnon-trivialfixedpointt∗,a
CmHn×O = C×O ×mCm−1Hn
I I value smaller than 2 for the one-loop dimensions x(s)
min
+ H×O ×nCmHn−1 is not a threat to the internal stability of the WZW
I fixed point g = 0 since it occurs along a flow to strong
+CH×O ×mnCm−1Hn−1 coupling. Moreover, it is known that in d = 2 dimen-
I sions the O(3)/O(2) NLσM with theta term at θ = π
m(m−1) (2.20) flows in the infrared into the level k = 1 SU(2) WZW
+CC×O × Cm−2Hn
I 2 fixed point. While the spectrum of one-loop dimensions
n(n−1) of high-gradient operators at the WZW fixed point is
+HH×O × CmHn−2
I 2 bounded from below (as we will recall below), the spec-
=−2n(n+1)CmHn trum of these operators is unbounded from below in
theweaklycoupled2-dimensionalO(3)/O(2)NLσM(the
+2n(n−1)Cm+1Hn−2.
presence of the theta term does not affect this result).
The contributions to the RG equations obeyed by the
couplings Zm(s,)n where 1 < 2m+n = s ≤ k needed to 2. Normal ordering revisited
extract the spectrum of one-loop dimensions are
dZ(s) WeshallillustratetheeffectsoftheFermistatisticsfor
m,n = 2−2s Z(s) +4gn(n+1)Z(s) the family of high-gradient operators (2.12) when s = 2
dl m,n m,n (2.21)
(cid:0) (cid:1) for the case of a (marginally) relevant (g > 0) current-
−4g(n+2)(n+1)Z(s) +··· . current interaction. We shall then show for the special
m−1,n+2
case of k = 1 and s = 2 that the two one-loop dimen-
Here,the dots include non-linearcontributionsofsecond sions associated with the family of high-gradient opera-
order in g or Zm(s,)n. tors (2.12) are unchanged, to one loop order, i.e.,
The linearized RG flows (2.21) are closed. This is a
justification a posteriori for neglecting the RG effects of x(s) =x(s) =4. (2.25)
0,2 1,0
8
We start from the family of high-gradient opera- Z(2) associated with :O :, we find that their one-loop
0,2 2
tors (2.12) with s = 2. For clarity of presentation, we dimensionsremainequaltotheirengineeringdimensions,
rename the two members of this family,
x(2) =x(2) =4. (2.31)
1,0 0,2
O ≡CαβJ J CγδJ¯ J¯, O ≡CαβJ J¯ CγδJ J¯.
1 α β γ δ 2 α β γ δ The lesson that we draw from the example s = 2
(2.26)
and k = 1 is that it is necessary to use normal order-
As implied by Eq. (2.10) these are two classical expres-
ing to properly define composite operators. Had we not
sions. The two quantum expressions involve point split-
used normal ordering, we would have incorrectly pre-
ting and normal ordering as in Eq. (2.10).
dicted that there are infinitely many high-gradientoper-
Without loss of generality, we consider only the left
atorswhichbecome relevant,atone-looporder,for large
current sector. Normal ordering of
enough s and for g > 0. We believe that for a generic
value of k, there is no infinity of one-loop relevant high-
kC i i
J (z)J (0)= αβ + ǫ J (0)+ ǫ ∂J (0) gradientoperators. Onlyafinitenumberofhigh-gradient
α β z2 z αβγ γ 2 αβγ γ
operators become relevant, at one-loop order, for large
δ k enough s and for g >0 when k >1.
+ αβ : ψa†∂ψ −∂ψa†ψ :(0)
4 ι aι ι aι Inthenextsection,weturnattentiontoanon-unitary
Xι=1 (cid:0) (cid:1) WZW model of relevance to the problem of Anderson
k (σ ) b (σ ) d localization to investigate whether the loss of unitarity
+ :ψa† α a ψ ψc† β c ψ :(0)
ι 2 bι ι′ 2 dι′ opens the door to an infinity of relevant high-gradient
ι,Xι′=1 operators.
+···
(2.27)
III. HIGH-GRADIENT OPERATORS AND
amounts to the subtraction from Eq. (2.27) of the terms gbl(M|M)k WZW THEORIES
singular in the limit z →0,
An interesting example of a problem of Anderson
k i localization in two dimensions which possesses a spe-
:JαJβ :(0)= JαιJβι′(0)+ 2ǫαβγ∂Jγ(0) cial so-called sublattice (or chiral) symmetry (SLS) and
ι6=Xι′=1 TRS (thus belonging to the “chiral-orthogonal”symme-
k try class BDI in the classification scheme of Zirnbauer,
δ
+ αβ : ∂ψa†ψ −ψa†∂ψ −ψa†ψ ψb†ψ :(0) and Altland and Zirnbauer36−38) is as follows. Consider
4 ι aι ι aι ι aι ι bι
Xι=1 (cid:0) (cid:1) a tight-binding model of fermions on a honeycomb lat-
(2.28) tice with random real-valued hopping matrix elements
of non-vanishing mean, which do not connect the same
for α,β = 1,2,3. The proper quantum interpretation of sublattice (so that SLS is preserved). [A related real-
the classical currents (2.26) is then ization of the same problem of Anderson localization is
provided by a random tight-binding model on a square
3
lattice with flux-π through every plaquette, introduced
:O :(z,z¯)=4 :J J :(z):J¯ J¯ :(z¯),
1 α α β β in Ref. 39.] In the absence of disorder this band struc-
αX,β=1 ture is known to exhibit the energy-momentum disper-
(2.29)
3 sion law of two species of (relativistic) Dirac fermions at
:O2 :(z,z¯)=4 :JαJβ :(z):J¯αJ¯β :(z¯). two points in the Brillouin zone at low energy near the
αX,β=1 Fermilevel(atzeroenergy). ItwasshowninRef. 33that
the SLS-preserving disorder discussed above leads to a
theory for the disorder averageswhich, in the supersym-
3. High-gradient operators when k=1 metric formulation,1 is a GL(2N|2N) Thirring (Gross-
Neveu) model. In other words, the problem of two-
When k = 1, the summation over unequal flavors dis- dimensionalAndersonlocalizationonthehoneycomblat-
appears in Eq. (2.28). (Observe in passing that : O : tice preserving SLS and TRS, is described by a set of
1
is then proportional to one component of the energy- Diracfermions(andSUSYbosonpartners)perturbedby
momentum stress tensor.) One then verifies the OPE a current-current interaction of the Noether currents of
itsunderlyingGL(2N|2N)(super)symmetry. Theinter-
action strength corresponds to the strength of the disor-
:O1:×OI =3:O1:−9:O2:, (2.30) der. ThesystemoffreeDiracfermions(andSUSYboson
:O :×O =:O :−3:O :. partners) is well known26,28 to be described by a WZW
2 I 1 2
model on the supergroup GL(2N|2N) with gl(2N|2N)
k
If we diagonalize the linearized one-loop RG flows for conformalKac-Moodycurrentalgebrasymmetryatlevel
the coupling Z(2) associated with :O : and the coupling k =1. b
1,0 1
9
This section is devoted to the one-loop RG analysis where
of high-gradient operators in the perturbed gl(M|M)
WZWtheory. Themainresultofthis sectionandofthiks cBACD :=(−)B+1δADδCB, (3.3b)
b
article applies to the case of level k = 1 of relevance for
and
the randomtight-bindingmodelsdiscussedabove. Inor-
der to state this result, we first need to recall from Ref. dB =−(−)BCδB, eBD =(−)BC+D(B+C)δD, (3.3c)
C C AC A
33 that the GL(2N|2N) Thirring (Gross-Neveu) models
possess two coupling constants; one, g , which does not withtheindicesA,B,C,D =1,··· ,M+N,whereδB de-
M C
flowunder the renormalizationgroup(RG)andanother, notes the Kroneckerdelta. The capitalizedindices A, B,
g , whichflows logarithmicallyunder the RG anda rate C, and D also carry a grade which is either 0 for M out
A
dependentong . Ourmainresultthensuggeststhatall oftheM+N valuesthattheytakeor1fortheremaining
M
higher-ordergradientoperatorsaremoreirrelevantinthe N values. It is the gradeofthe indices A andB thaten-
presence of the current-current interaction with g > 0 tersexpressionssuchas(−)A or(−)AB. The grade0 (1)
M
than at zero coupling g = 0. A positive g can be will shortly be associated with bosons (fermions). The
M M
interpreted33 as the variance of the disorder strength in positiveintegerk isthelevelofthecurrentalgebra(3.3).
the random tight-binding model in symmetry class BDI. The current algebra (3.3) is associated with the Lie su-
For the opposite sign of the coupling constant g < 0, peralgebra gl(M|N) defined by the structure constants
M
on the other hand, higher-order gradient operators have Eq.(3.3c) for A,B,C,D =1,··· ,M+N. When N =0,
a spectrum of one-loop dimensions that is unbounded the structure constants (3.3c) reduce to
from below very much as in Eq. (1.2). In the context
dB =−δB, eBD =+δD, (3.4)
of Anderson localization, the case with g <0 describes C C AC A
M
the surface state ofa three-dimensionaltopologicalinsu-
for A,B,C,D = 1,··· ,M. These define the Lie algebra
lator in the chiral-symplectic class (symmetry class CII)
gl(M)ofthenon-compactLiegroupGL(M). WhenM =
of Anderson localization.40,41
0, the structure constants (3.3c) reduce to
AsinSec.II,wearegoingtodistinguishtwolimits. In
the first (classical) limit, dB =+δB, eBD =−δD, (3.5)
C C AC A
M →∞, k →∞, (3.1) for A,B,C,D = 1,··· ,N. These define the Lie algebra
u(N) of the compact Lie group U(N).
OPEs between the high-gradient operators can be ob- There exists a free-fermion and free-boson realization
tained without any reference to the composite nature of of the current algebra (3.3) defined by the action46
the currents. One then recovers a spectrum of one-loop
scalingdimensionsforhigh-gradientoperatorsthatmim- k dz¯dz
ics closely that of the NLσMs discussed above. The sec- S∗ := Z 2πi ψA†ι∂¯ψAι+ψ¯A†ι∂ψ¯Aι (3.6a)
ond limit, Xι=1 (cid:0) (cid:1)
with the partition function
M =1,2,3,··· , k =1, (3.2)
is the opposite extreme to the first one in that the nor- Z := D[ψ†,ψ,ψ¯†,ψ¯]exp(−S ), (3.6b)
∗ Z ∗
malorderingofthecurrentsandthusofthehigh-gradient
operatorsis essentialandchangesdramaticallythe spec- where it is understood that ψ and ψ¯ are complex-
Aι Aι
trum of one-loopscaling dimensions from the “classical” valued integration variables for the M values of A with
limit (3.1). grade0whileψ andψ¯ areGrassmann-valuedintegra-
Aι Aι
tionvariablesfortheN valuesofAwithgrade1,regard-
less of the value taken by the flavor index ι = 1,··· ,k.
A. Definitions The current algebra (3.3) is then realized by the repre-
sentation
Ourstartingpointisatwo-dimensionalconformalfield
k k
theory characterizedby the current algebra33 JB := :ψ ψB†:, J¯B := :ψ¯ ψ¯B†:, (3.7)
A Aι ι A Aι ι
kcBD 1 Xι=1 Xι=1
JB(z)JD(0)= AC + dBJD(0)+eBDJB(0)
A C z2 z C A AC C as follows from the OPE
(cid:2) (cid:3)
+··· ,
δ δB
J¯B(z¯)J¯D(0)=kcBACD + 1 dBJ¯D(0)+eBDJ¯B(0) ψAι(z)ψιB′†(0)=(−1)AB+1ψιB′†(z)ψAι(0)= ιιz′ A,
A C z¯2 z¯ C A AC C δ δB
(cid:2) (cid:3) ψ¯ (z¯)ψ¯B†(0)=(−1)AB+1ψ¯B†(z¯)ψ¯ (0)= ιι′ A,
+··· , Aι ι′ ι′ Aι z¯
JB(z)J¯D(0)=0, ψ (z)ψ¯B†(0)=0,
A C Aι ι′
(3.3a) (3.8)
10
with A,B =1,··· ,M +N and ι=1,··· ,k. of high-gradient operators shares the same structure as
The expressions in Eq. (3.7) form a representation of that in Eq. (1.2). It will become clear by comparison to
the gl(M|N) current algebra in terms of free fermions. thecaseofk =1thatthelimitM,k→∞istheextreme
k
There are two Casimir invariants of rank 2 in gl(M|N) “classical” limit whereas the limit k = 1 is the extreme
thatbwe use to perturb the free field theory (3.6) with “quantum” limit.
two types of current-current interactions, both of which We restrict the family of “classical” high-gradient op-
are invariant under the global GL(M|N) symmetry,33 erators to objects of the form
Z := D[ψ†,ψ,ψ¯†,ψ¯]exp(−S), str JJ¯JJJ¯J str JJ¯J¯ ··· , (3.12)
Z (cid:0) (cid:1) (cid:0) (cid:1)
i.e., to diagonal GL(M|M)-invariant monomials of or-
dz¯dz g g
S :=S∗+Z 2πi (cid:16)2AπOA+ 2MπOM(cid:17), (3.9) dreenrtss.inFobroatnhythgeivhenoloomrdoerrpsh,icthaenednagnintieheorilnomg dorimpheincsciounrs-
OA :=−JAA(−1)AJ¯BB(−1)B ≡−strJstrJ¯, are all equal and given by 2s. This degeneracy is lifted
OM :=−JABJ¯BA(−1)A ≡−str JJ¯ . to first order in the coupling constant gM. The task of
enumeratingalllinearly-independenthigh-gradientoper-
(cid:0) (cid:1)
Formally, one may allow the coupling constants gA and ators (3.12) of order s is greatly simplified by the as-
gM to take on any real (i.e., positive or negative) values. sumption M,k → ∞. We can rule out the scenario by
However, to make connection with the above mentioned which it is a finite set of independent Casimir operators
two-dimensionaltight-binding models in symmetry class of gl(M|M) that fixes all the linearly independent clas-
BDI of Anderson localization, we must demand that gA sical high-gradient operators of order s once the limit
and gM be positive. (See Appendix B.) M → ∞ has been taken. We can also rule out the sce-
The “classical” counterparts to the high-gradient op- nariobywhichnormalorderingchangesthebook-keeping
eratorsoforders inEq.(2.9)arethehomogeneouspoly- betweenclassicalandquantumhigh-gradientoperatorsof
nomials order s once the limit k →∞ has been taken.
TA1···AsA¯1···A¯sJB1···JBsJ¯B¯1···J¯B¯s (3.10) For high-gradient operators of type Eq. (3.10) or
B1···BsB¯1···B¯s A1 As A¯1 A¯s (3.12),thecouplinggA doesnotrenormalizetheirscaling
oftheleftandrightcurrentsthatareinvariantunderthe dimensions, since gA (or OA) can be removed from the
diagonal GL(M|N) symmetry group of the interacting action (3.9) by chiral transformation. All that therefore
theory.47 The set of “classical” high-gradient operators is needed to compute their one-loop scaling dimensions
of order s is specified once all the linearly independent are their OPE with the quadratic Casimir operator OM.
rank 2s invariant tensors TA1···AsA¯1···A¯s in the adjoint We will write the following expressions for the general
B1···BsB¯1···B¯s case of GL(M|N), and will set M = N (i.e., the case
representation of GL(M|N) which are invariant under
of interest) only in Eqs. (3.19), (3.20), and (3.21). The
GL(M|N) transformations have been enumerated. At
required OPEs follow from (a) the intra-trace formula
the quantum level, normalorderingdefines the quantum
high-gradient operators of order s as in Eq. (2.10).
str JMJ¯N ×str JJ¯
We are now going to specialize to the case M = N
where the beta function for the coupling constant g =(cid:2) str (JN(cid:3))str (cid:2)MJ¯(cid:3) −str (M)str JJ¯N
M
vanishes identically, an exact result.33 (As already men- −str JMJ¯(cid:0)str (N(cid:1) )+str (JM)(cid:0)str J¯N(cid:1)
tioned, the other coupling constant g flows logarithmi-
A (cid:0) (cid:1) (cid:0) (3(cid:1).13a)
cally at a rate set by g .) The sector which we loosely
M
denote by
and (b) the inter-trace formula
PSL(M|M)∼GL(M|M)/U(1)×U(1) (3.11)
str JM str J¯N ×str JJ¯
remainsscale(conformally)invariantforanyvalueofg .
More specifically, PSL(M|M) is obtained by first factoMr- =(cid:2) str(cid:3)JN(cid:2)J¯M(cid:3) −str(cid:2) JJ(cid:3)¯NM (3.13b)
ing out the U(1) subgroup thereby obtaining the sub- −s(cid:0)tr JMN(cid:1)J¯ +s(cid:0)tr JMJ¯(cid:1)N
groupSL(M|M) ofGL(M|M), followedin a secondstep
(cid:0) (cid:1) (cid:0) (cid:1)
by the “gauging away” of the states carrying the U(1) with M and N arbitrary operators. Here we have used
charges under j := JAA and ¯j := J¯AA.33,51,52 This turns the short-hand notation of Eq. (2.15).
out to realize a line of RG fixed points (and conformal To proceed we also need to distinguish linearly inde-
field theories) labeled by the coupling constant gM.33 pendent high-gradient operators of order s. To this end,
a “quantum number”, the number of switches, is intro-
duced.9−11 The number of switches of type n and of
B. High-gradient operators when M,k→∞ ↑
type n in a single trace are defined as follows. Consider
↓
the trace
We are going to show that, when k and M are very
large, the spectrum for the one-loop scaling dimensions str J J J ···J (3.14a)
µ1 µ2 µ3 µ2n
(cid:0) (cid:1)