264 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 discrete Dubrovin equations
The discrete equations of motion for the quantum mappings of KdV type are
given in terms of the Sklyanin variables (which are also known as quantum
separated variables). Both temporal (discrete-time) evolutions and spatial
(along the lattice at a constant time-level) evolutions are considered. In the
classical limit, the temporal equations reduce to the (classical) discrete
Dubrovin equations as given in a previous publication. The reconstruction of
the original dynamical variables in terms of the Sklyanin variables is also
achieved.Comment: 25 page
New techniques for integrable spin chains and their application to gauge theories
In this thesis we study integrable systems known as spin chains and their applications to the study of the AdS/CFT duality, and in particular to N “ 4 supersymmetric Yang-Mills theory (SYM) in four dimensions.First, we introduce the necessary tools for the study of integrable periodic spin chains, which are based on algebraic and functional relations. From these tools, we derive in detail a technique that can be used to compute all the observables in these spin chains, known as Functional Separation of Variables. Then, we generalise our methods and results to a class of integrable spin chains with more general boundary conditions, known as open integrable spin chains.In the second part, we study a cusped Maldacena-Wilson line in N “ 4 SYM with insertions of scalar fields at the cusp, in a simplifying limit called the ladders limit. We derive a rigorous duality between this observable and an open integrable spin chain, the open Fishchain. We solve the Baxter TQ relation for the spin chain to obtain the exact spectrum of scaling dimensions of this observable involving cusped Maldacena-Wilson line.The open Fishchain and the application of Functional Separation of Variables to it form a very promising road for the study of the three-point functions of non-local operators in N “ 4 SYM via integrability
Integrable Matrix Models in Discrete Space-Time
We introduce a class of integrable dynamical systems of interacting classical
matrix-valued fields propagating on a discrete space-time lattice, realized as
many-body circuits built from elementary symplectic two-body maps. The models
provide an efficient integrable Trotterization of non-relativistic
-models with complex Grassmannian manifolds as target spaces,
including, as special cases, the higher-rank analogues of the Landau-Lifshitz
field theory on complex projective spaces. As an application, we study
transport of Noether charges in canonical local equilibrium states. We find a
clear signature of superdiffusive behavior in the Kardar-Parisi-Zhang
universality class, irrespectively of the chosen underlying global unitary
symmetry group and the quotient structure of the compact phase space, providing
a strong indication of superuniversal physics.Comment: v2, 60 pages, 10 figures, 1 tabl
Quantizations of D=3 Lorentz symmetry
Using the isomorphism
we develop a new
simple algebraic technique for complete classification of quantum deformations
(the classical -matrices) for real forms and
of the complex Lie algebra in
terms of real forms of : ,
and . We prove that the
Lorentz symmetry
has three different Hopf-algebraic quantum deformations which are expressed in
the simplest way by two standard and
-analogs and by simple Jordanian
twist deformations. These quantizations are
presented in terms of the quantum Cartan-Weyl generators for the quantized
algebras and as well as in
terms of quantum Cartesian generators for the quantized algebra
. Finaly, some applications of the deformed Lorentz
symmetry are mentioned.Comment: 22 pages, V2: First and final sections (Sect. 1, Sect. 6) has been
partialy rewritten and extended, in Sect. 2-4 only minor corrections, in
Sect. 5 notational changes and the clarifications of some formulas; 13 new
references adde
Space-like dynamics in a reversible cellular automaton
In this paper we study the space evolution in the Rule 54 reversible cellular
automaton, which is a paradigmatic example of a deterministic interacting
lattice gas. We show that the spatial translation of time configurations of the
automaton is given in terms of local deterministic maps with the support that
is small but bigger than that of the time evolution. The model is thus an
example of space-time dual reversible cellular automaton, i.e. its dual is also
(in general different) reversible cellular automaton. We provide two equivalent
interpretations of the result; the first one relies on the dynamics of
quasi-particles and follows from an exhaustive check of all the relevant time
configurations, while the second one relies on purely algebraic considerations
based on the circuit representation of the dynamics. Additionally, we use the
properties of the local space evolution maps to provide an alternative
derivation of the matrix product representation of multi-time correlation
functions of local observables positioned at the same spatial coordinate.Comment: 28 pages, 3 figure
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