Non-unitary Conformal Field Theory and Logarithmic Operators for Disordered Systems

21 Pages·0.23 MB·English

by  Z. Maassarani

Non-unitary Conformal Field Theory and Logarithmic
Operators for Disordered Systems
Z. Maassarani
D. Serban
7
9
9
1 CEA-SACLAY, Service de Physique Th´eorique

n
F-91191 Gif-sur-Yvette Cedex, FRANCE
a
J

0
1
Abstract

3
v We consider the supersymmetric approach to gaussian disordered systems like
2 the random bond Ising model and Dirac model with random mass and random
6
potential. These models appeared in particular in the study of the integer quantum
0
5 Halltransition. Thesupersymmetricapproachreveals anosp(2/2)1 affinesymmetry
0 at the pure critical point. A similar symmetry should hold at other fixed points.
6
We apply methods of conformal field theory to determine the conformal weights
9
/ at all levels. These weights can generically be negative because of non-unitarity.
h
Constraints such as locality allow us to quantize the level k and the conformal
t
-
p dimensions. This provides a class of (possibly disordered) critical points in two
e spatial dimensions. Solving the Knizhnik-Zamolodchikov equations we obtain a set
h
of four-point functions which exhibit a logarithmic dependence. These functions
:
v
are related to logarithmic operators. We show how all such features have a natural
i
X setting in the superalgebra approach as long as gaussian disorder is concerned.
r
a
April 1996
SPHT-T96/040
hep-th/9605062
1 Introduction
There is growing evidence that some disordered systems at criticality share two unusual
features: The existence of logarithmic operators in the spectrum of the theory and the
existence of an infinite number of relevant operators with negative conformal dimensions.
Logarithmic operators seem to be connected to hidden continuous symmetries. In [1]
logarithmic operators were found to generate a change in a coupling constant of the
effective WZNW action for SU(2r), r 0, obtained after use of the replica trick. The
→
existence of an infinite number of conformal operators [2] means that there is an infinite
number of relevant perturbations which render the critical point unstable. Such a set of
conformal dimensions is also related to the phenomenon of multifractality.
Our aim in this paper is to show that both features follow naturally from a supersym-
metric treatment of the disorder. We consider the two-dimensional random bond Ising
model at criticality and the random Dirac model in 2 + 1 dimensions. These gaussian
models allow a supersymmetric formulation of averages over disorder [3]. One identifies
a global superalgebra symmetry of the effective action and assumes it is enhanced to an
affine symmetry at a new critical point [2, 4], as happened at the pure critical point. One
then derives the Sugawara stress-energy tensor and obtains the set of conformal dimen-
sions associated with the primary fields. Additional constraints such as locality allow to
further restrict the operator content of the theory. This provides a class of, possibly disor-
dered, critical points in two spatial dimensions. The specific structure of the superalgebra
implies that such dimensions can generically be negative. The presence of logarithmic
operators is also straightforward from the point of view of this algebra. Unlike ordinary
Lie algebras, superalgebras have indecomposable (not fully reducible) representations. It
is then possible to show on general grounds that such representations imply the existence
of logarithmic operators and logarithms in correlation functions.
This paper is organized as follows. In section 2 we briefly recall the supersymmetric
approach for the random bond Ising model and the random Dirac model. In section 3 we
write down the osp(2/2) current algebra and find the Sugawara stress-tensor. In section
k
4 we briefly review the representations of osp(2/2) and obtain the conformal weights of
primary fields. In section 5 we derive the quantization of k from locality constraints. In
section 6 we show how indecomposable representations lead to logarithmic operators and
consider an atypical representation. In section 7 we obtain some logarithm-dependent
four-point functions. We conclude in section 8.
2 The supersymmetric method for disordered sys-
tems
The supersymmetric method is applicable to models which are gaussian at fixed disorder,
but it provides a good starting point for disentangling properties of a large class of disor-
dered conformal field theories. The study of the random bond Ising model reveals that an
appropriate algebraic framework for studying gaussian disordered systems at criticality
should be based on affine Lie superalgebra with zero superdimension. In the case of the
random Ising model this algebra is osp(2N/2N). The fact that the algebra has an equal
number of bosonic and fermionic generators ensures that the Virasoro central charge of
1
the Sugawara stress-tensor vanishes. This is needed by construction for a disordered sys-
tem. The vanishing of the central charge does not imply the triviality of the theory since
it is not unitary. The relevance of affine Lie superalgebras was independently realized in
refs. [2, 4].
We now briefly recall the supersymmetric method for the study of the random bond
Ising model and the random Dirac model. These models have been analyzed using the
replica method in refs. [5, 6, 7]. The random bond Ising model has also been studied in
the context of massless scattering theories [8].
The random bond Ising model
In the scaling limit, near criticality, the Ising model is described by a massive real
Majorana fermion with mass m τ = (TC−T) were T is the critical temperature. In the
∼ TC C
presence of disorder, i.e. when the coupling constants between sites belong to a random
set, the mass becomes a function of space position. The random Ising model is defined
by the action (z = x+iy):
d2x
S[m(x)] = ψ∂ ψ +ψ∂ ψ +im(x)ψψ (1)
z¯ z
4π
Z (cid:16) (cid:17)
where ψ and ψ are grassmanian fields. The mass m(x) is chosen to be a quenched random
variable with a gaussian measure:
1 d2x
P[m] = exp (m(x) m)2 . (2)
−4g 2π − !
Z
The energy operator ǫ(x) = iψ(x)ψ(x) has dimension one. The Harris criterion tells
us that randomness in the bond interaction is marginal in the 2d Ising model. It turns out
that it is not exactly marginal but only marginally irrelevant. At criticality the disorder
only induces logarithmic corrections to the pure system.
In order to compute averages of products of correlation functions one introduces a
number of copies of fermions and of their supersymmetric partners equal to the number
ofcorrelationfunctionsintheproduct[3,4]. Onethenrewritestheseaveragesasfermionic
and bosonic path integrals. For the Ising model with m = 0 one obtains the following
effective action for the disorder average of the product of two correlation functions:
d2x g d2x
S = ψ ∂ ψ +ψ ∂ ψ +η∂ γ +η¯∂ γ¯ + Φ
eff 2π − z¯ + − z + z¯ z 8 π pert
Z (cid:16) (cid:17) Z
g d2x
= S + Φ (3)
∗ pert
8 π
Z
with
2
Φ = ψ ψ ψ ψ +η¯γ ηγ¯ . (4)
pert − + − − + −
(cid:16) (cid:17)
The ψ are complex fermions and η and γ are complex bosonic fields. This action can
be viewed as a perturbation of the (non-unitary) conformal field theory specified by the
action S . This fixes the normalization of the fields to be:
∗
1 1
ψ (z)ψ (w) , γ(z)η(w) . (5)
− +
∼ z w ∼ z w
− −
2
The central charge of the Virasoro algebra is zero. Note that since the fermions ψ have
±
dimension one half the perturbing field Φ has dimension two. It is therefore marginal.
pert
The conformal field theory specified by S is invariant under an affine supersymmetric
∗
algebra whose conserved currents are:
G (z) = η(z)ψ (z) , G (z) = γ(z)ψ (z) ,
± ± ± ±
K(z) = η2(z) , K(z) = γ2(z) ,
b
J(z) =: ψ (z)ψ (z) : , H(z) =: γ(z)η(z) : . (6)
− + c
The dots refer to fermionic and bosonic normal ordering. There are four fermionic cur-
rents, G and G , which are generators of supersymmetric transformations, and four
± ±
bosonic ones. They form a representation of the affine osp(2/2) [12, 13] current algebra
b
at level one.
The perturbing field can also be written as a bilinear in the currents,
1
Φ = 2 JJ HH + (KK +KK)+G G G G +G G G G . (7)
pert − + − + + − + −
− 2 − −
(cid:20) (cid:21)
c c b b b b
In other words the perturbation (4) is a current-current perturbation. Therefore the
action S preserves a global osp(2/2) symmetry.
eff
The random Dirac model
The random model of Dirac fermions has been introduced in connection with the
quantum Hall transition [9]. Its action is:
d2x
S = ψ ∂ ψ +ψ ∂ ψ
2π − z¯ + − z +
Z (cid:16)
m(x) V(x)
+i (ψ ψ ψ ψ )+i (ψ ψ +ψ ψ ) . (8)
2 − + − − + 2 − + − + !
The random variables m and V have a gaussian distribution with widths g and g .
M V
We denote by Φ and Φ the perturbing fields coupled to the constants g and g after
M V M V
averaging over the disorder. In the two-copy sector we can write them in terms of the
osp(2/2) currents:
Φ = 2HH 2JJ +KK +KK +2G G 2G G +2G G 2G G , (9)
V − + + − − + + −
− − −
Φ = 2JJ 2HH +KK +KK +2G G 2G G 2G G +2G G . (10)
M c c − b+ b+ − b− + + b−
− − −
We see that Φ = Φ . c c b b b b
M pert
Both Φ and Φ preserve a global osp(2/2) symmetry, whose generators are
M V
H +H , J +J , K +K , K +K ,
0 0 0 0 0 0 0 0 (11)
G +G G +G , G +G , G +G .
+0 +0 +0 +0 −0 −0 −c0 c−0
for the perturbation Φ , and b b b b
M
H +H , J +J , K K , K K ,
0 0 0 0 0 − 0 0 − 0 (12)
G +G , G G , G G , G +G .
+0 +0 +0 − +0 −0 − −0 −c0 c−0
b b b b
3
for the perturbation Φ . Here the index 0 denotes the zero modes of the currents
V
Ja(z) = Jaz−n−1. If both perturbations are present, the interaction preserves an
n n
u(1/1) symmetry generated by
P
H +H , J +J , G +G , G +G . (13)
0 0 0 0 +0 +0 −0 −0
The perturbative study of the Dirac theory with a rabndom pbotential and a random
mass is very similar to the perturbative study of the random mass Ising model. However
the crucial difference is that contrary to the randomness of the mass, the randomness of
the potential is marginally relevant. At g = 0, the one-loop beta function is given by
M
g˙ = 8g2. This means that g grows at large distances. The infrared fixed point has not
V V V
yet been determined [9]. Should the affine symmetry, present at the UV fixed point of
the pure system, be restored at the IR point, then we believe this point should belong to
the set we find in section 5.
3 The current algebra approach
The critical free theory described by the action S has a current algebra symmetry,
∗
osp(2/2) , on both its holomorphic and antiholomorphic sectors. When randomness is
1
introduced in the pure system the effective action is no longer critical, and the current
algebra symmetry is reduced to a global osp(2/2) symmetry. However at a new critical
point the conformal invariance restores the current algebra symmetry. The value of the
level k is not preserved by the renormalization flow. Therefore its value at the IR fixed
point could be different from its value at the UV point. It is possible to extract properties
at the new critical point by studying the current algebra and its associated stress-energy
tensor. In this context we consider osp(2/2) at arbitrary k and obtain the Sugawara
k
tensor.
3.1 The osp(2/2) algebra
k
In this section we write down the singular terms of the Operator Product Expansions [10]
satisfied by the currents of the affine osp(2/2) algebra at level k. The non-trivial OPE
of the currents are easily obtained from the currents (6). The level k appearing below is
equal to 1 for these currents. We find:
k k
J(z)J(w) ; H(z)H(w) −
∼ (z w)2 ∼ (z w)2
− −
1 1
J(z)G (w) ± G (w) ; J(z)G (w) ± G (w)
± ± ± ±
∼ z w ∼ z w
− −
1 1
b b
H(z)G (w) G (w) ; H(z)G (w) − G (w) (14)
± ± ± ±
∼ z w ∼ z w
− −
2 2
b b
H(z)K(w) K(w) ; H(z)K(w) − K(w)
∼ z w ∼ z w
− −
k 1
c c
G (z)G (w) + (H(w) J(w))
± ∓
∼ (z w)2 z w ±
− −
b
4
2k 4
K(z)K(w) + H(w)
∼ (z w)2 z w
− −
1c 1
G (z)G (w) K(w) ; G (z)G (w) K(w)
− + − +
∼ z w ∼ z w
− −
2 2
b b c
K(z)G (w) − G (w) ; K(z)G (w) G (w)
± ± ± ±
∼ z w ∼ z w
− −
b c b
Requiring thealgebratobeassociative constrainsthepossiblecentral termsextensions
to the ones appearing in the OPE’s (14). We can rewrite these OPE’s in a more compact
form as
κab Jc(w)
Ja(z)Jb(w) k +fab . (15)
∼ (z w)2 c z w
− −
The fab are the structure constants of the Lie superalgebra osp(2/2), and κab is propor-
c
tional to its non-degenerate Killing form.
3.2 The Sugawara stress-energy tensor
We now construct the Sugawara stress-energy tensor [11]. It is bilinear in the currents
defined in the previous section. Because of singularities which appear at coinciding points
one has to consider a regularized version where normal ordered products of currents ap-
pear. This is equivalent to a point splitting procedure where the singular parts appearing
in the OPE’s of the currents are subtracted. We take the usual definition for the normal
ordered product of two fields A(z) and B(w):
dz A(z)B(w)
: AB : (w) (16)
≡ 2πi z w
Iw −
It should be noted that this ordering prescription does not coincide in general with the
Wick prescription used in (6). The Sugawara stress-energy tensor T(z) is given by
1
T(z) = κ : Ja(z)Jb(z) : (17)
ab
κ
where κ is the inverse of κab. The constant κ is determined by requiring Ja(z) to be a
ab
primary field of conformal weight one:
Ja(w) ∂Ja(w)
T(z)Ja(w) + . (18)
∼ (z w)2 z w
− −
When calculating OPE’s one has to add a minus sign each time two odd generators are
permuted. We find
1 1
T(z) = : H(z)H(z) J(z)J(z) (K(z)K(z)+K(z)K(z))
4 2k − − 2
− (cid:18)
+ G (z)G (z) G (z)G (z)+G (z)G (z)c G (zc)G (z) : . (19)
+ − − + − + + −
− −
(cid:17)
The Sugawara tensobr also satisfies b b b
c/2 T(w) ∂T(w)
T(z)T(w) +2 + . (20)
∼ (z w)4 (z w)2 z w
− − −
5
For a general Lie superalgebra, it is easy to show that the Virasoro central charge c is
proportional to the superdimension; this is the difference between the number of even and
odd generators. Therefore c vanishes for the osp(2/2) algebra, and more generally for any
superalgebra with an equal number of even and odd generators. More precisely one has:
κ κab sdimG
ab
c = 2k = 2k = 0 . (21)
κ κ
Because c vanishes T(z) is a primary field, contrary to the case c = 0 where T is just a
6
level-two descendant of the unit operator.
4 Primary Fields
We briefly describe the representations of osp(2/2) and find the corresponding conformal
weights.
4.1 Some osp(2/2) representations
Unlike ordinary Lie algebras, the are two types of representations for most superalgebras.
ThetypicalrepresentationsareirreducibleandaresimilartothoseofordinaryLiealgebras.
Theatypicalrepresentations havenocounterpartintheordinaryLiealgebrasetting. They
can be irreducible or not fully reducible (read reducible but indecomposable).
The superalgebra osp(2/2) is isomorphic to the superalgebra spl(2/1). The repre-
sentation theory of the latter algebra was studied in [12, 13]. The quadratic Casimir of
osp(2/2) is
1 1
C = H2 (KK +KK) J2 +G G G G +G G G G . (22)
2 + − − + − + + −
2 − 2 − − −
(cid:18) (cid:19)
c c b b b b
The four even generators K,K,H,J form a su(2) u(1) subalgebra. The correspondence
⊕
with the notation of [12] is: Q = K/2, Q = K/2, Q = H/2, B = J/2, V =
+ − 3 +
G /√2, V = G /√2, W c= G /−√2, W = G /√2. −
+ − + + − − −
− − c
Let b and q be the eigenvalues of B and Q . Generically, a representation (b,q),
3 3
bb C , q = 0, 1,1, 3,..., contains fobur su(2) u(1) multiplets:
∈ 2 2 ⊕
b,q,q , q = q, q +1,...,q 1,q if q 0 , (23)
3 3
| i − − − ≥
1 1 1 3 1 1
b+ ,q ,q , q = q + ,...,q ,q if q , (24)
3 3
| 2 − 2 i − 2 − 2 − 2 ≥ 2
1 1 1 3 1 1
b ,q ,q , q = q + ,...,q ,q if q , (25)
3 3
| − 2 − 2 i − 2 − 2 − 2 ≥ 2
b,q 1,q , q = q +1,...,q 2,q 1 if q 1 . (26)
3 3
| − i − − − ≥
The action of the four even generators on these multiplets is the one implied by the
notation. The four odd generators mix the different multiplets. The vector v = b,q,q
| i | i
is a highest weight vector, i.e. it satisfies
K v = G v = G v = 0 . (27)
+ −
| i | i | i
c b b
6
The quadratic Casimir of this representation is C = 2(q2 b2).
2
−
If b = q the representation is denoted by [b,q] and is typical; the quadratic and
6 ±
cubic Casimirs do not vanish. All the vectors in the representation can be obtained
from the highest weight vector v by applying on it polynomials in the generators. The
| i
representation [b,q] has dimension 8q. The representation [0,1/2] is four-dimensional and
contains one spin 1/2 and two spin 0 multiplets.
Atypical representations
When b = q several kinds of atypical representations arise. Both Casimirs vanish,
±
andyet these representations arenot the trivialone-dimensional representation. Onekind
has dimension 4q + 1 and is denoted by [q] . To obtain [q] (resp. [q] ) one drops the
± + −
two multiplets (25) and (26) (resp. (24) and (26)). These representations are irreducible.
Atypical indecomposable representations
Generally, they are semidirect sums of atypical irreducible representations. They can
contain two, three or four terms and they arise in tensor products of irreducible represen-
tations. An interesting example, containing four irreducible representations, arises in the
tensor product of two representations [0,1/2]. The result is the direct sum of [0,1] and
an eight-dimensional representation which is the semidirect sum of [1/2] , [1/2] and two
− +
[0] representations. We call it [0, 1/2,1/2,0] (see fig. 1 b). The vector s is invariant;
−
it is annihilated by all the generators. The quadratic Casimir C vanishes on all states
2
except t: C t = 4s. We have
2
1 1 1 1 1 1 1 1 1
s = ,0 ,0 0, 0, + 0, 0, + ,0 ,0
2 |2 i⊗|− 2 i−| 2i⊗| −2i | −2i⊗| 2i |− 2 i⊗|2 i
(cid:18) (cid:19)
1 1 1 1 1 1 1 1 1
t = ,0 ,0 + 0, 0, 0, 0, + ,0 ,0
2 |2 i⊗|− 2 i | 2i⊗| −2i−| −2i⊗| 2i |− 2 i⊗|2 i
(cid:18) (cid:19)
4.2 Tensor product of osp(2/2) representations
The tensor product of two irreducible representations of a superalgebra is not necessarily
completely reducible. Ref. [12] gives a sufficient condition for a tensor product of two
osp(2/2)representations to be completely reducible, and the irreducible components. The
result is the following
[b,q] [b′,q′] = [b+b′,q +q′] [b+b′,q +q′ 1] ... [b+b′, q q′ ]
⊗ ⊕ − ⊕ ⊕ | − |
[b+b′,q+q′ 1]... [b+b′, q q′ +1]
⊕ − ⊕ | − |
[b+b′ +1/2,q+q′ 1/2]... [b+b′ +1/2, q q′ +1/2]
⊕ − ⊕ | − |
[b+b′ 1/2,q+q′ 1/2]... [b+b′ 1/2, q q′ +1/2] (28)
⊕ − − ⊕ − | − |
if b > q 1, b′ > q′ 1 .
± ≥ 2 ± ≥ 2
For the other values of b, b′, this expression still gives the correct content of su(2) u(1)
⊕
charges of the tensor product. This expression also permits to obtain some informationon
the reducibility: if two components have different Casimirs, then they belong to different
(maybeindecomposable) representations. Vanishing Casimirs are, generally, signs ofsome
pathologies.
7
b b
G ^ b+b’+1/2
+ G
+
^ t
K K q
b+b’
G ^
- G s
-
b+b’-1/2
-1 -1/2 1/2 1 q
b)
a)
Fig.1: a) The tensor product of the representations [b,1/2] and [b′,1/2]. If b > 1/2, b′ >
1/2, the result is the direct sum of [b+b′,1] (white dots), [b+b′ +1/2,1/2] (black dots) and
[b + b′ 1/2,1/2] (grey dots). The arrows represent the action of the generators. b) The
−
representation [0, 1/2,1/2,0]. The vector t (black dot) is cyclic and s (white dot) is invariant.
−
The action of the fermionic generators G , G , G and G is represented by arrows.
+ − + −
b b
4.3 osp(2N/2N) representations
When calculating the disorder average of product of correlation functions for the Ising
andDirac models, we can introduce an arbitrary number of copies of fermions and bosons.
One is led is to consider osp(2N/2N) as symmetry. The theory is consistent if the results
of the calculations do not depend on N. One step in this direction is to show that the
conformal dimensions of the primary fields are independent of N.
The main difference between the superalgebras osp(2/2) and osp(2N/2N) with N > 1
is that the even subalgebra of the first one is not simple. This is why representations
of the former are indexed by a continuous parameter, while the representation of the
latter are characterized only by discrete variables. According to the classification of the
superalgebras [14], osp(2/2) is the first of the series C(N + 1), while osp(2N/2N) is a
superalgebra of the type D(N,N).
The finite dimensional typical representation of the superalgebras were classified by
Kac [14]. For D(N,N), the typical representations are characterized by their highest
weights, Λ = N a δ + N a ǫ ; δ and ǫ , i = 1,N form an orthogonal basis with
i=1 i i i=1 N+i i i i
(ǫ ,ǫ ) = δ , (δ ,δ ) = δ , (ǫ ,δ ) = 0. The ‘numerical marks’ a satisfy the following
i j − ijP i j Pij i j i
conditions:
i) a Z , i = N,
i +
∈ 6
ii) j = a a ... a 1(a +a ) Z ,
N − N+1 − − 2N−2 − 2 2N−1 2N ∈ +
iii) a = ... = a = 0, if j N 2; a = a , if j = N 1.
N+j+1 2N 2N−1 2N
≤ − −
8
The first two conditions express the fact that Λ is a dominant weight for the even
algebra Sp(2N) O(2N).
⊗
The value of the quadratic Casimir for the representation with highest weight Λ is
given, up to a normalization, by (Λ,Λ + 2ρ), with ρ = ρ ρ and ρ is half the
0 1 0(1)
−
sum of the even (odd) positive roots. Considering the positive roots associated to the
distinguished Dynkin diagram (the Dynkin diagram with a single fermionic root), we
obtain
N
ρ = (N i)(ǫ δ ) .
i N−i+1
− −
i=1
X
We normalize the value of the quadratic Casimir of the adjoint representation (2,0,...,0)
to be 1; the Casimir of the representation (a ,...,a ) will be
1 2N
1 N
C (Λ) = (a i+1)2 (a +N i)2 .
2 i N+i
4 − − −
Xi=1h i
For the fundamental representation (1,0,...,0) we have C = 1/4, independent of N.
2
In order to obtain compatibility between the results obtained in the osp(2/2) and
osp(2N/2N)frameworks, thecontinousparameterboftheosp(2/2)representations hasto
be constrained to the discrete values compatible with the value of the discrete parameters
of the osp(2N/2N) representations.
4.4 Conformal dimensions of the primary fields
Theprimary fields φα(w)arehighest weight vectors ofaffine osp(2/2) andoftheVirasoro
k
algebra. In terms of operator products one has:
φγ(w)
Ja(z)φβ(w) (Ta)γβ , (29)
∼ z w
−
φα(w) ∂φα(w)
T(z)φα(w) ∆ + . (30)
∼ (z w)2 (z w)
− −
Using (15) and (29) one then shows that the matrices Ta form a representation of the
osp(2/2) algebra. But this applies to any algebra. Using the Sugawara form (17) of
the stress-energy tensor and (15) one obtains the conformal weights corresponding to a
particularrepresentation. Fortherepresentations [b,q]and[q] theCasimir C isdiagonal
± 2
and we obtain:
1 2(q2 b2)
∆ = κ TaTb = − . (31)
(b,q) ab
κ 2 k
−
We are also able to calculate the conformal dimensions of the primary fields for all N.
Due to the fact that the dual Coxeter number of osp(2N/2N) equals 1 for all N, these
conformal dimensions do not change if we change the number of copies to N′ > N; for
example
1
∆ =
(1,0,...,0)
4 2k
−
is the conformal dimension of the fields in the fundamental representation.
The conformal dimensions are potentially negative; this reflects the non-unitarity of
the theory we considered.
9