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a Matrix Model Representation of the Integrable Xxz Heisenberg Chain on Random Surfaces

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A matrix model representation of the integrable XXZHeisenberg chain on random surfaces
Jan Ambjorn
1
,Niels Bohr Institute, Blegdamsvej 17, 2100, Copenhagen, Denmark,
Ara Sedrakyan
2
Yerevan Physics Institute, Alikhanian Br. str. 2, Yerevan 36, Armenia
Abstract
We consider integrable models, i.e. models deﬁned by
R
-matrices, on randomManhattan lattices (RML). The set of random Manhattan lattices is deﬁned asthe set dual to the lattice random surfaces embedded on a regular d-dimensionallattice. As an example we formulate a random matrix model where the partitionfunction reproduces annealed average of the XXZ Heisenberg chain over all RML.A technique is presented which reduces the random matrix integration in partitionfunction to an integration over their eigenvalues.
1
e-mail:
ambjorn@nbi.dk
2
e-mail:
sedrak@nbi.dk
1
1 Introduction
One of the major goals of non-critical string theory was to describe the non-perturbative physics of non-Abelian gauge ﬁelds in two, three and four dimen-sions. The asymptotic freedom of these theories allowed us to understand thescattering observed at high energies [1, 2], but it also made the long distance, lowenergy sector of the theories non-perturbative and indicated a non-trivial struc-ture of the vacuum [3]. It became necessary to develop non-perturbative toolswhich would allow us to study phenomena associated with e.g. conﬁnement. Onepossibility which attracted a lot of attention was the attempt by Polyakov to re-formulate the non-Abelian gauge theories as a string theory. This line of researchled Polyakov to his seminal work on non-critical string theory [4]. He showed thatthe presence of the conformal anomaly forces us to include the conformal factor inthe string path integral, and that the action associated with the conformal factoris the Liouville action. In his approach the study of non-critical string theorybecomes equivalent to the study of two-dimensional quantum gravity (governedby the Liouville theory) coupled to certain conformal matter ﬁelds.Attempts to understand and deﬁne rigorously the quantum Liouville theorytriggered the lattice formulation, presenting the two-dimensional random sur-faces appearing in the string path integral as a sum over triangulated piecewiselinear surfaces [5, 6, 8]. This sum over “random triangulations” (or “dynamicaltriangulations” (DT)) could be represented by matrix integrals and in this waycertain matrix integrals became almost synonymous to non-critical string theory.Somewhat surprisingly many of the lattice models were exactly solvable and atthe same time it was possible to solve the continuum quantum Liouville modelvia conformal bootstrap, and whenever results of the two non-pertubative meth-ods could be compared, agreement was found. However, the solutions only madesense for matter ﬁelds with a central charge
c
≤
1 [11]. Thus the understandingof non-critical string theory in three and four dimensions, where the non-Abeliangauge theories are non-trivial, is still missing. It is thus of great interest to try tostudy new classes of random surface models which might allow us to penetratethe
c
= 1 barrier. This is one of the main motivations of this paper. We pro-pose to consider a new class of random lattices, the so-called random Manhattanlattices, which we were led to via the 3d Ising model [12, 13] and via the studyof the Chalker-Coddington network model [15]. The study of the latter modelled to the idea that an
R
-matrix could be associated to a random Manhattanlattice, and we will consider how to couple in general a matter system deﬁnedby an
R
-matrix to a random lattice. By summing over the random lattices (i.e.taking the annealed average) we thus introduce a coupling between the integrablemodel and two-dimensional quantum gravity.More precisely we start with an integrable model on a 2d square lattice, as-2
suming we know the R-matrix. We then show that the same R-matrix can beused on a so-called Random Manhattan Lattice (RML) (see Fig. 1), which isa lattice where the links have ﬁxed arrows which indicate the allowed fermionhopping. No hopping is allowed in directions opposite to arrows. The summa-tion over the RMLs can be performed by a certain matrix integral related to theR-matrix. This matrix integral is somewhat diﬀerent from the the conventionalmatrix integrals used to describe conformal ﬁeld theories with
c <
1 coupled to2d quantum gravity, and thus there is hope than one can penetrate to
c
= 1barrier. Below we describe the construction in detail.
2 The model
As mentioned one arrives in a natural way to a RML comes from the study of the 3d Ising model on a regular cubic lattice. The high temperature expansionof the Ising model can be expressed as a sum over random lattice surfaces of thekind shown in Fig. 1, and on these two-dimensional lattice surfaces one constructsa kind of dual lattice by the following procedure: the lattice surface consists of plaquettes. consider the mid-points of the links on the plaquettes as sites of thedual lattice, and consider arrows on the links as shown in Fig. 2.Figure 1: Dual lattice surface3
(a) (b)
Figure 2: Assignment of arrows to dual latticeThe allover orientation of arrows on the plaquettes should be such that the ﬂowto neighbouring plaquettes is continuous as illustrated in Fig. 3. This type of duallattice with arrows will be a ﬁnite Manhattan lattice corresponding to a particularplaquette lattice surface on the regular three-dimensional lattice. There is a oneto one correspondence between the plaquette surfaces on the regular lattice andthe ﬁnite Manhattan lattices described above.
R RRRRRRRRRRRRRRRRRRRRRRR
Figure 3: Random Manhattan latticeA second way of obtaining a RML is by starting from oriented double linegraphs, like the ones introduced by ’t Hooft, and then modify the double linepropagator like shown in Fig. 4.We will now attach an
R
-matrix of an integrable model to the squares of the RML with the index assignment shown in Fig. 5. Two neighbouring squareswill share one of indices, and the same is thus the case for the corresponding
R
-matrices, and a summation over values of the indices are understood, resultingin a matrix-like multiplication of
R
-matrices. To a RML Ω we now associate the4
(b)(a)
Figure 4: Manhattan lattices and double line graphs
βα
α β
αβ
R
βα
Figure 5: Index assignment of the
R
-matrixpartition function
Z
(Ω) =
R
∈
Ω
ˇ
R,
(1)where the summation over indices is dictated by the lattice. Our ﬁnal partitionfunction is deﬁned by summing over all possible (connected) lattices Ω:
Z
=
∑
Ω
Z
(Ω) (2)The summation over the elements in Ω, i.e. the summation over a certain setof random 2d lattices, is a regularized version of the sum over 2d geometriesprecisely in the same way as in ordinary DT. It is known the sum over randompolygons (triangles, squares, pentagons etc) with positive weights under quitegeneral conditions leads to the correct continuum limit, i.e. the functional integralover 2d geometries, when the link length goes to zero. Thus it is natural toassume that the sum over RML will also represent in correct way the sum over2d geometries when the link length goes to zero. Under this assumption we havecoupled a given two-dimensional model, integrable on a regular lattice, to two-dimensional quantum gravity. In the next Section we will, in order to make thediscussion mor eexplicit, consider the
XXZ
Heisenberg model, where the R-matrixis known.5

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