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Quantum Field Theory

Giuseppe Mussardo

Published Aug 27, 2009• 0 citations
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Abstract

Abstract The statistical models we have analyzed so far are defined on a lattice and they have a microscopic length-scale given by the lattice spacing a. In all these models there is, however, another length-scale provided by the correlation length : this is a function of the coupling constants and can be varied by varying the external parameters of the systems.

DOI
10.1093/oso/9780199547586.003.0007

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Abstract The statistical models we have analyzed so far are defined on a lattice and they have a microscopic length-scale given by the lattice spacing a. In all these models there is, however, another length-scale provided by the correlation length : this is a function of the coupling constants and can be varied by varying the external parameters of the systems.
We study critical spin systems and field theories using matrix product states, and formulate a scaling hypothesis in terms of operators, eigenvalues of the transfer matrix, and lattice spacing in the case of field theories. The critical point, exponents, and central charge are determined by optimizing them to obtain a data collapse. We benchmark this method by studying critical Ising and Potts models, where we also obtain a scaling Ansatz for the correlation length and entanglement entropy. The formulation of those scaling functions turns out to be crucial for studying critical quantum field theories on the lattice. For the case of $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ with mass parameter ${\ensuremath{\mu}}^{2}$ and lattice spacing aaa, we demonstrate a double data collapse for the correlation length $\ensuremath{\delta}\ensuremath{\xi}(\ensuremath{\mu},\ensuremath{\lambda},D)=\stackrel{\texttildelow{}}{\ensuremath{\xi}}((\ensuremath{\alpha}\ensuremath{-}{\ensuremath{\alpha}}_{c})(\ensuremath{\delta}/a{)}^{\ensuremath{-}1/\ensuremath{\nu}})$ with DDD the bond dimension, $\ensuremath{\delta}$ the gap between eigenvalues of the transfer matrix, and ${\ensuremath{\alpha}}_{c}={\ensuremath{\mu}}_{R}^{2}/\ensuremath{\lambda}$ the parameter which fixes the critical quantum field theory.
The explicit dependence of correlation length as function of temperature and external field is found for the Ising model on a simple cubic lattice near the phase transition point. , where and h c are the renormalized values of the external field h and reduced temperature τ. The critical amplitude ξ0 is proportional to the lattice constant through the coefficient, which is determined by the parameter of the interaction potential. The size of the critical region is established by the temperature and field. It is shown that the essential increasing of the system correlation length takes place at () with h→0 and at h′<h* (, h′=βh) with τ→0.
We have investigated the statistical properties of an ensemble of disordered 1D spatial spin chains (SSCs) of finite length, placed in an external field, with consideration of relaxation effects. The short-range interaction complex-classical Hamiltonian was first used for solving this problem. A system of recurrent equations is obtained on the nodes of the spin-chain lattice. An efficient mathematical algorithm is developed on the basis of these equations with consideration of advanced Sylvester conditions which allows one to step by step construct a huge number of stable spin chains in parallel. The distribution functions of different parameters of spin glass system are constructed from first principles by analyzing the calculation results of the 1D SSCs ensemble. It is shown that the behaviors of different distributions parameters are quite different even at weak external fields. The ensemble energy and constants of spin-spin interactions are being changed smoothly depending on the external field in the limit of statistical equilibrium, while some of them such as the mean value of polarizations of the ensemble and parameters of its orderings are frustrated. We have also studied some critical properties of the ensemble such as catastrophes in the Clausius-Mossotti equation depending on the value of the external field. We have shown that the generalized complex-classical approach excludes these catastrophes, which allows one to organize continuous parallel computing on the whole region of values of the external field including critical points. A new representation of the partition function is suggested based on these investigations. Being opposite to the usual definition, it is a complex function and its derivatives are everywhere defined, including at critical points.