Quantum Groups, Correlation Functions and Infrared Divergences

7 Pages·0.1 MB·English

by  Haye Hinrichsen

QUANTUM GROUPS, CORRELATION FUNCTIONS AND
INFRARED DIVERGENCES
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9 Haye Hinrichsen and Vladimir Rittenberg
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Universit¨at Bonn, Physikalisches Institut
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Nussallee 12, D-5300 Bonn 1, Germany
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Abstract
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We show in two simple examples that for one-dimensional quantum chains with quantum group
:
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symmetries, the correlationfunctions oflocaloperatorsare, ingeneral, infrareddivergent. If one
i
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considers, however, correlation functions invariant under the quantum group, the divergences
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a cancel out.
BONN HE-93-02
hep-th/9301056
January 1993
In this paper we consider the anisotropic XY chain in a magnetic field which is defined by the
Hamiltonian
1 L−1
H(q,η) = − η σxσx + η−1σyσy + qσz + q−1σz , (1)
2 j j+1 j j+1 j j+1
jX=1 (cid:16) (cid:17)
where q and η are complex parameters and σx,y,z are Pauli matrices placed on site j. Up to
j
boundary terms, which will play a crucial role in our discussion, H can be rewritten as
1 L L
H = − η σxσx + η−1σyσy − h σz (2)
2 j j+1 j j+1 j
jX=1 (cid:16) (cid:17) jX=1
with h = q+q−1. This Hamiltonian has a long history [1] and provides a good model for Helium
2
adsorbed on metallic surfaces (η real and q on the unit circle). It also gives the master equation
of the kinetic Ising model [2] (q = 1 and η real). The correlation functions for this model given
x,y,z x,y,z
by Eq. (2) with periodic boundary conditions (σ = σ ) have been computed for η and
L+1 1
h real [3]. The calculations are relatively simple since the Hamiltonian (2) can be diagonalized
in terms of free fermions. It is the aim of this paper to compute the correlation functions of
the Hamiltonian given by Eq. (1) in the massless regime. We will be concerned with two
cases: η = 1 and q on the unit circle (notice that in this case H given by Eq. (1) is not
hermitian whereas H in Eq. (2) is) and q = 1, η on the unit circle (where H in Eqs. (1) and (2)
are nonhermitian). Notice that the correlation functions in the latter case have not yet been
computed with any boundary conditions. Our interest in these calculations goes beyond the
particular physics of the model. As opposed to H defined by Eq. (2), the Hamiltonian defined
by Eq. (1) is invariant under quantum group deformations, and we want to use the fact that H
can be easily diagonalized and the correlation functions can be computed in order to find out if
there are unexpected phenomena in the massless regime (it turns out that there are). For other
quantum chains with higher rank quantum symmetries or supersymmetries, the calculations are
much more involved.
In order to see the symmetry properties [4] we first perform a Jordan-Wigner transformation
and introduce new matrices τ1 and τ2 by
j j
τ1 = σz σx , τ2 = σz σy (j = 1...L) (3)
j i j j i j
(cid:16)iY<j (cid:17) (cid:16)iY<j (cid:17)
which obey the Clifford algebra
{τµ,τν} = 2 δµ,ν (i,j = 1...L; µ,ν = 1,2) (4)
i j i,j
and get
i L−1
H(q,η) = ητ2τ1 − η−1τ1τ2 + qτ1τ2 + q−1τ1 τ2 . (5)
2 j j+1 j j+1 j j j+1 j+1
jX=1(cid:16) (cid:17)
This Hamiltonian is invariant under the two-parameter (α and β) quantum algebra
{T1,T1} = 2[E] {T1,T2} = 0 (6)
0 0 α 0 0
{T2,T2} = 2[E] [E,T1] = [E,T2] = 0
0 0 β 0 0
together with a suitable coproduct [4], where
q αE −α−E
α = , β = qη, [E] = . (7)
α
η α−α−1
1
In the representation of the quantum chain (5), the operators T1, T2, and E are obtained from
0 0
the coproduct rules [4] and read
L L
T1 = α−L+21 αjτ1, T2 = β−L+21 βjτ2, E = L, (8)
0 j 0 j
j=1 j=1
X X
where L is the length of the chain. For generic α and β the nontrivial irreducible representations
of the algebra (6) are two-dimensional. Because of the branching rules given by the representa-
tion theory of the algebra (6) all energy levels of H are twice degenerated.
If η = 1, the total magnetization operator N = 1 L σz = i L τ2τ1 also commutes with
2 j=1 j 2 j=1 j j
the Hamiltonian. In this case the algebra (6) can be enlarged by the commutation relations
P P
[N,T1] = iT2 [N,T2] = −iT1 [N,E] = 0 . (9)
0 0 0 0
The coproduct for the operator N can be read off from its definition. This is the quantum
superalgebra U SU(1|1) [5]. The Hamiltonian (1) is also obtained if one asks for SU(2) sym-
q q
metry for q = i and one uses unrestricted representations [6].
Before computing the correlation functions, we give the diagonalized form of H [4]
L−1
H(q,η) = Λ iT2T1 , (10)
k k k
k=1
X
where
1
Λk = 1 (α−eiπLk)(α−e−iπLk)(β −eiπLk)(β −e−iπLk) 2 (11)
4αβ !
are fermionic excitation energies. The operators Tγ are certain linear combinations of the τµ
k j
and obey the commutation relations
{Tµ,Tν} = 2δµ,ν , {Tµ,Tν} = 0 . (k,l = 1...L−1; µ,ν = 1,2) (12)
k l k,l 0 k
This implies that the operators iT2T1 have the eigenvalues ±1 and commute for different k.
k k
For complex parameters q and η the operators Tµ are not hermitian. Notice that T1 and T2 do
k 0 0
not appear in H. Observe also (see Eq. (11)), that the spectrum of H(q,η) coincides with the
spectrum of H(η,q) (this is a generalized duality property [4]).
As mentioned above the ground state is twofold degenerate. The corresponding subspace is
spanned by the basis vectors |0+i and |0−i which are defined by
iT2T1|0±i = −|0±i (13)
k k
iT2T1|0±i = ± [L] [L] |0±i
0 0 α β
q
where 1 ≤ k ≤ L−1. The corresponding bra-vectors are defined by
h0±|iT2T1 = −h0±| (14)
k k
h0±|iT2T1 = ± [L] [L] h0±|
0 0 α β
q
Although in general the Hamiltonian (1) is non-hermitian, it is symmetric, i.e. H = HT. Thus
going from the bras to the kets we take the transpose only (no complex conjugation).
Since Tµ|0±i =6 0, it is impossible to define a linear combination of |0+i and |0−i as a
0
scalar with respect to the quantum algebra. In the special case η = 1, where the algebra can
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be enlarged by the relations (9), the states |0+i and |0−i are also eigenstates of the additional
operator N. Thus we have to choose either |0+i or |0−i as the ground state of the system.
Now we consider the correlation function
Gzz(R,S,L) = h0±|σzσz |0±i = h0±|iτ2τ1|0±ih0±|iτ2τ1|0±i−h0±|iτ2τ1|0±ih0±|iτ2τ1|0±i,
l m l l m m l m m l
(15)
where we have used the Wick theorem and taken R = m−l and S = m+l. One can show that
qS−L−1ηR 1 L−1
h0±|iτ2τ1|0±i = ± + Λ−1 × (16)
m l [L]α[L]β 4Lq k=X−L+1 k
k6=0
q
(β −e−iπLk)(α−eiπLk)eiπLkR − (β −e−iπLk)(α−e−iπLk)eiπLkS .
h i
We want to compute the limit Gzz(R,S) = lim Gzz(R,S,L) with R and S fixed. If the
L→∞
spectrum is massive, the factor Λ−1 in Eq. (16) is bounded and we can replace the sum by an
k
integral. This integral is finite and thus the limit L → ∞ exists in the massive case. We are
interested however in the massless case where the situation is different.
By Eq. (11) the spectrum is massless if at least one of the parameters α or β lies on the
unit circle. Hence in terms of q and η, there are three massless situations: (q ∈ S1,η = 1),
(q = 1,η ∈ S1) and (q ∈ S1,η ∈ S1). In the last case the spectrum contains imaginary
excitations and the states |0±i defined in Eq. (13) loose their physical meaning as states of
minimal energy. This case will not be considered (The Ising line q = η, also massless, is not
interesting for the present investigation).
Letusconsider thefirstcasewhereη = 1 and q = eiπφ. Wewouldexpect thethermodynamic
limit L → ∞ of correlation functions to exist even for non-periodic boundary conditions in both
the massive and the massless case. For example the correlation function for a conformally
invariant system with free boundary conditions on a strip [7] has the form
1
lim Gstrip(R,S,L) = Gstrip(R,S) = F(S2/R2), (17)
L→∞ R2x
ρ→∞
where x is the bulk exponent. Here F is a function which satisfies F(ρ) = 1 giving the
correlation function in the bulk
1
Gstrip(R) = . (18)
R2x
On the other hand F(1 + δ) δ=→0 δxs−x, where x is the surface exponent. This gives the
s
correlation function when the first point (in the notation of Eq. (15) at position l) is kept near
the boundary and the second is taken far away:
(4l)xs−x
Gstrip(l,R) = . (19)
Rxs+x
For the boundary conditions defined in Eq. (1), where H is non-hermitian in the first and
the last position of the chain, we will find however divergent results as we will show in the
following. If q = eiπφ = eiπrs and L is a multiple of s, Eq. (16) is not defined since one of the
Λ vanishes. Hence in order to perform a thermodynamic limit, we have to use a sequence in
k
L given by L = L′s+t, where 0 < t < s and L′ are integers. But even then one can show that
Eq. (16) diverges like logL when L′ → ∞:
4e2iπφ(S−1)sin2πφ
Gzz(R,S,L) = logL 2logS −log(S +R)−log(S −R) (20)
π2
(cid:20) (cid:16) (cid:17)
2eiπφ(S−1)sinπφ 4sinπφR
+ +(2−4φ)cosπφR + ...
π πR
(cid:16) (cid:17)(cid:21)
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This divergence is a consequence of the boundary conditions in Eq. (1) since for periodic
boundary conditions (no U SU(1|1) symmetry !), one obtains the non-divergent result [3]
q
4sin2πφR
Gzz(R) = m2 − (21)
z π2R2
where m2 is the magnetization per site whose expression is given in Ref. [3]. The occurence of
z
oscillatory terms reflects the fact that we are here in an incommensurate phase. Notice that for
q = η = 1 the divergence in Eq. (20) disappears. Similar divergences occur if we take η = 1
and q = eiπφ −ε with small ε or η = 1+ε, q = eiπφ and ε → 0.
In the second case, q = 1 and η = eiπφ, we also obtain a divergent result:
8sinπφ
Gzz(R,S,L) = logL logR+log(2sinπφ)+C sinπφ
π2
(cid:20)(cid:16) (cid:17)
2sin(πφ(S −1)) sinπφR
+(1−2φ)cosπφ+ (22)
S −1
sin(πφ(S +R−1)) sin(πφ(S −R−1))
− − + ...
S +R−1 S −R−1
(cid:21)
HereC istheEulerconstant. Inthiscaseasimilarcalculationwithperiodicboundaryconditions
(the Hamiltonian is non-hermitian) gives again an infrared divergent result.
The main message of this paper is that there is a way to find correlation functions with a
well defined thermodynamic limit in the presence of a quantum group. Instead of computing
the correlation functions of two arbitrary local operators, let us consider the invariant quantity:
c = qm−liτ2τ1 + ql−miτ2τ1 − ηm−liτ2τ1 − ηl−miτ2τ1 (23)
l,m l l m m l m m l
This operatorisacombination oftwo localoperatorsiτ2τ1 = σz andtwo dislocationlines (iτ2τ1
l l l l m
and iτ2τ1, see Ref. [1]). It is symmetric (c = c ) and commutes with the generators of the
m l l,m m,l
quantum group:
[c ,T1] = [c ,T2] = 0 (24)
l,m 0 l,m 0
Moreover c is the unique combination which is invariant (there is only one Casimir operator
l,m
for the algebra (6)). For η = 1 and q = eiπφ we derive from Eq. (16) that the corresponding
correlation function is given by
1 L−1 k
Gc(R,S,L) = h0±|c |0±i = sgn(|φ|−| |) × (25)
l,m
L L
k=−L+1
X
k6=0
sin π(k +φ) sin2 π(k −φ)R
2 2 L 2 L eiπLk(S−1)
(cid:16) sin π(cid:17)(k −(cid:16)φ) (cid:17)
(cid:20)
2 L
k (cid:16) (cid:17)
+ cos(π R) − cos(πφR) .
L
(cid:21)
Since c is invariant under the quantum group, the correlation function (25) does not depend
l,m
on the choice of the ground state. As one sees the correlation function is infrared finite. Because
of the special combination in Eq. (23) the zero of the denominator in Eq. (25) (which comes
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from the zero in the dispersion relation) is cancelled by a zero in the numerator. In the limit
L → ∞ we obtain
2 |R|−1 sin πφ(S −j −1) sin πφ(S +j −1)
Gc(R,S) = 2isinπφ eiπφj −e−iπφj
π  (cid:16) S −j −1 (cid:17) (cid:16) S +j −1 (cid:17)
(cid:20) j=1
X
 
sin πφ(S −R−1) sin πφ(S +R−1)
− eiπφ(R−1) − eiπφ(1−R) (26)
(cid:16) S −R−1 (cid:17) (cid:16) S +R−1 (cid:17)
sin πφ(S −1) 2sinπφR
+ 2cosπφ + + (2−4φ)cosπφR ,
(cid:16) S −1 (cid:17) R
(cid:21)
which is an exact result. We can now derive the ”bulk” and ”surface” behaviour (see Eqs. (18)
and (19))
4sinπφR
Gc(R) = (2−4φ)cosπφR+ +0(R−2) (27)
πR
2 |R|
Gc(l,R) ≈ (2−4φ)cosπφR+ sinπφeiπφ(R+2l−1) log . (28)
π 4l
We note that the chain given by Eq. (1) is the simplest one of a series of quantum chains, the so-
called Perk-Schultz ones [8], which have as symmetry the U SU(M|N) quantum superalgebras
q
[9], and we expect to find similar results in all cases. We also suspect that our results will have
consequences on the calculations of correlation functions performed directly in the L → ∞ limit
using the quantum affine symmetry [10].
In the second case where q = 1 and η = eiπφ we are led to
1 L−1 k
Gc(R,S,L) = sgn(|φ|−| |) ×
L L
k=−L+1
X
k6=0
sin π(k +φ) sin2 π(k −φ)R
2 L 2 L
2
(cid:16) sin (cid:17)π(k −(cid:16)φ) (cid:17)
(cid:20)
2 L
+ eiπLk(S−1) c(cid:16)os(πφR) (cid:17)− cos(πkR) .
L
(cid:16) (cid:17) (cid:21)
Here we get the infrared finite result
2 |R|−1 2j +1 |R|−1 sin πφ(2j +1)
Gc(R,S) = 2+ sinπφ − (29)
π j(j +1) (cid:16)j(j +1) (cid:17)
(cid:20) (cid:16) jX=1 (cid:17) jX=1
1 1
+ − sin πφ(S +R−1)
S −1 S +R−1
(cid:16) (cid:17) (cid:16) (cid:17)
1 1
+ − sin πφ(S −R−1) + (2−4φ) cosπφ
S −1 S −R−1
(cid:16) (cid:17) (cid:16) (cid:17) (cid:21)
which is also an exact expression. The ”bulk” and the ”surface” limit turn out to be given by
2
Gc(R) ≈ (2−4φ) cosπφ+ sinπφ log(4sin2πφ)+logR2 +2C (30)
π
(cid:16) (cid:17)
2 2sinπφ(2l−1)
Gc(l,R) ≈ (2−4φ) cosπφ+ sinπφ log(4sin2πφ)+logR2 +2C −
π π(2l−1)
(cid:16) (cid:17)
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where C is the Euler constant.
We do not yet have an explanation for the ”miraculous” cancellations of divergent quantities
which occur when considering invariant quantities. Probably these quantities ”know” to pick up
only the right states of the Hilbert space, and in this subspace a scalar product can be defined in
spite of the nonhermiticity of the Hamiltonian. A similar but completely unrelated phenomenon
occurs in σ-models and their generalizations [11], where again only group (not quantum group!)
invariant quantities give infrared finite correlation functions.
One of us (V.R.) would like to thank the University of Geneva and the Einstein Center of the
Weizmann Institute for their hospitality. We would also like to thank R. Flume, A. Ganchev,
C. Gomez, A. Schwimmer and E. Sokatchev for useful discussions.
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