390 research outputs found
Diagonalizing transfer matrices and matrix product operators: a medley of exact and computational methods
Transfer matrices and matrix product operators play an ubiquitous role in the
field of many body physics. This paper gives an ideosyncratic overview of
applications, exact results and computational aspects of diagonalizing transfer
matrices and matrix product operators. The results in this paper are a mixture
of classic results, presented from the point of view of tensor networks, and of
new results. Topics discussed are exact solutions of transfer matrices in
equilibrium and non-equilibrium statistical physics, tensor network states,
matrix product operator algebras, and numerical matrix product state methods
for finding extremal eigenvectors of matrix product operators.Comment: Lecture notes from a course at Vienna Universit
Quantum mechanics of null polygonal Wilson loops
Scattering amplitudes in maximally supersymmetric gauge theory are dual to
super-Wilson loops on null polygonal contours. The operator product expansion
for the latter revealed that their dynamics is governed by the evolution of
multiparticle GKP excitations. They were shown to emerge from the spectral
problem of an underlying open spin chain. In this work we solve this model with
the help of the Baxter Q-operator and Sklyanin's Separation of Variables
methods. We provide an explicit construction for eigenfunctions and eigenvalues
of GKP excitations. We demonstrate how the former define the so-called
multiparticle hexagon transitions in super-Wison loops and prove their
factorized form suggested earlier.Comment: 51 pages, 15 figure
Form-factors in the Baxter-Bazhanov-Stroganov model I: Norms and matrix elements
We continue our investigation of the Z_N-Baxter-Bazhanov-Stroganov model
using the method of separation of variables [nlin/0603028]. In this paper we
calculate the norms and matrix elements of a local Z_N-spin operator between
eigenvectors of the auxiliary problem. For the norm the multiple sums over the
intermediate states are performed explicitly. In the case N=2 we solve the
Baxter equation and obtain form-factors of the spin operator of the periodic
Ising model on a finite lattice.Comment: 24 page
Topological Defects on the Lattice I: The Ising model
In this paper and its sequel, we construct topologically invariant defects in
two-dimensional classical lattice models and quantum spin chains. We show how
defect lines commute with the transfer matrix/Hamiltonian when they obey the
defect commutation relations, cousins of the Yang-Baxter equation. These
relations and their solutions can be extended to allow defect lines to branch
and fuse, again with properties depending only on topology. In this part I, we
focus on the simplest example, the Ising model. We define lattice spin-flip and
duality defects and their branching, and prove they are topological. One useful
consequence is a simple implementation of Kramers-Wannier duality on the torus
and higher genus surfaces by using the fusion of duality defects. We use these
topological defects to do simple calculations that yield exact properties of
the conformal field theory describing the continuum limit. For example, the
shift in momentum quantization with duality-twisted boundary conditions yields
the conformal spin 1/16 of the chiral spin field. Even more strikingly, we
derive the modular transformation matrices explicitly and exactly.Comment: 45 pages, 9 figure
Fine structure of anomalous dimensions in N=4 super Yang-Mills theory
Anomalous dimensions of high-twist Wilson operators in generic gauge theories
occupy a band of width growing logarithmically with their conformal spin. We
perform a systematic study of its fine structure in the autonomous SL(2)
subsector of the dilatation operator of planar N=4 super Yang-Mills theory
which is believed to be integrable to all orders in 't Hooft coupling. We
resort in our study on the framework of the Baxter equation to unravel the
properties of the ground state trajectory and the excited trajectories in the
spectrum. We use two complimentary approaches in our analysis based on the
asymptotic solution of the Baxter equation and on the semiclassical expansion
to work out the leading asymptotic expression for the trajectories in the upper
and lower part of the band and to find how they are modified by the
perturbative corrections.Comment: 34 pages; v2: minor corrections, reference adde
BFKL Spectrum of N=4 SYM: non-Zero Conformal Spin
We developed a general non-perturbative framework for the BFKL spectrum of
planar N=4 SYM, based on the Quantum Spectral Curve (QSC). It allows one to
study the spectrum in the whole generality, extending previously known methods
to arbitrary values of conformal spin . We show how to apply our approach to
reproduce all known perturbative results for the Balitsky-Fadin-Kuraev-Lipatov
(BFKL) Pomeron eigenvalue and get new predictions. In particular, we re-derived
the Faddeev-Korchemsky Baxter equation for the Lipatov spin chain with non-zero
conformal spin reproducing the corresponding BFKL kernel eigenvalue. We also
get new non-perturbative analytic results for the Pomeron eigenvalue in the
vicinity of point and we obtained an explicit formula for
the BFKL intercept function for arbitrary conformal spin up to the 3-loop order
in the small coupling expansion and partial result at the 4-loop order. In
addition, we implemented the numerical algorithm of arXiv:1504.06640 as an
auxiliary file to this arXiv submission. From the numerical result we managed
to deduce an analytic formula for the strong coupling expansion of the
intercept function for arbitrary conformal spin.Comment: 70 pages, 5 figures, 1 txt, 2 nb and 2 mx files; v2: references
added, typos fixed and nb file with Mathematica stylesheet attached; v3: more
typos fixed; v4: the text edited according to the report of the refere
Quantum spectral curve as a tool for a perturbative quantum field theory
An iterative procedure perturbatively solving the quantum spectral curve of
planar N=4 SYM for any operator in the sl(2) sector is presented. A Mathematica
notebook executing this procedure is enclosed. The obtained results include
10-loop computations of the conformal dimensions of more than ten different
operators.
We prove that the conformal dimensions are always expressed, at any loop
order, in terms of multiple zeta-values with coefficients from an algebraic
number field determined by the one-loop Baxter equation. We observe that all
the perturbative results that were computed explicitly are given in terms of a
smaller algebra: single-valued multiple zeta-values times the algebraic
numbers.Comment: 36 pages plus tables; v2: minor changes, references added, ancillary
files with mathematica notebooks adde
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