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Series on advances in statistical mechanics

New Problems, Methods and Techniques in Quantum Field Theory and Statistical Mechanics

Mario Rasetti

Published May 1, 1990• 5 citations
Publisher
DOI
10.1142/1095

Related papers

New Developments in Quantum Field Theory and Statistical Mechanics Cargèse 1976

Maurice M. Lévy, P. K. Mitter

Springer eBooks•Jan 1, 1977•71 citations

Introduction

N. N. Bogolubov Jr.

Despite the apparent simplicity of the formulation, the polaron problem has not yet been solved, and continues to attract much attention. It plays an important role in statistical mechanics and quantum field theory because it can be considered as the simplest example of a nonrelativistic quantum particle interacting with a quantum field. Therefore many sophisticated mathematical techniques have been tested for the first time using this problem as a model. A shining example of this is Feynman's functional integration method, which was applied first to the polaron problem, before becoming one of the main methods used in statistical mechanics and quantum field theory. Moreover, polaron theory is an expanding field of investigation in solid state physics because polarons are not only theoretical constructs but practically observable physical objects (see e.g. [6]).

CRC Press eBooks•May 17, 2000•0 citations

Studies on Quantum Field Theory and Statistical Mechanics

Shoucheng Zhang

•Jan 1, 1987•0 citations

Some aspects of quantum field theory and statistical physics

Ragnheidur Gudmundsdóttir

•Jan 1, 1986•0 citations

A New Approach to Quantum-Statistical Mechanics

Takeo Matsubara

A new method of calculating the grand partition function of many-body system is developed, adopting extensively the techniques of calculus in quantum field theory. It is shown that the grand partition function, which is a trace of the density matrix expressed in terms of field operators, can be evaluated in a way almost parallel with the evaluation of the vacuum expectation value of the S-matrix in quantum field theory, provided that appropriate modifications in notation and definitions are made. As an example, the method is applied to electron-phonon system. Further, basing on this new formalism, various non-perturbational methods are discussed.

Progress of Theoretical Physics•Oct 1, 1955•1,200 citations

Statistical mechanics of disordered quantum optimization

Chris R. Laumann

The classical statistical mechanical approach to complexity theory proceeds from the study of ensembles of computationally intractable optimization problems as a species of unusual disordered magnetic systems. Over the last thirty years, researchers have used this approach to supplement worst-case hardness results encoded by complexity theory with detailed information about thermodynamic and dynamic phase transitions in the structure of typical cases. This exchange of ideas between classical statistical mechanics and computer science enabled the development of important heuristic algorithms such as simulated annealing and survey propagation and further refined our understanding of glassiness and critical slowing in physical disordered systems. In this thesis, we map out an analogous program in the quantum context. The question is simple: what can quantum statistical mechanics reveal about the difficulty of solving hard quantum optimization problems? Or more directly, what makes those problems hard even for quantum computers? In this pursuit, we introduce the study of ensembles of optimization problems whose complexity status is intrinsically quantum mechanical (Part I) and develop techniques to study quantum spin glasses and the transverse field adiabatic algorithm applied to classically hard random optimization problems (Part II). In particular, we introduce the study of random quantum satisfiability (QSAT) and identify the coarse aspects of its phase diagram, including a new form of entanglement transition. We generalize the cavity method to the study of quantum models and use it to study the transverse field Ising glass and frustrated AKLT models on the Bethe lattice. We further apply the cavity method to extract Griffiths-McCoy singularities in a diluted (classical) ferromagnet and finally observe that there are no Goldstone bosons on the Bethe lattice.

•Jan 1, 2010•1 citation
Despite the apparent simplicity of the formulation, the polaron problem has not yet been solved, and continues to attract much attention. It plays an important role in statistical mechanics and quantum field theory because it can be considered as the simplest example of a nonrelativistic quantum particle interacting with a quantum field. Therefore many sophisticated mathematical techniques have been tested for the first time using this problem as a model. A shining example of this is Feynman's functional integration method, which was applied first to the polaron problem, before becoming one of the main methods used in statistical mechanics and quantum field theory. Moreover, polaron theory is an expanding field of investigation in solid state physics because polarons are not only theoretical constructs but practically observable physical objects (see e.g. [6]).
A new method of calculating the grand partition function of many-body system is developed, adopting extensively the techniques of calculus in quantum field theory. It is shown that the grand partition function, which is a trace of the density matrix expressed in terms of field operators, can be evaluated in a way almost parallel with the evaluation of the vacuum expectation value of the S-matrix in quantum field theory, provided that appropriate modifications in notation and definitions are made. As an example, the method is applied to electron-phonon system. Further, basing on this new formalism, various non-perturbational methods are discussed.
The classical statistical mechanical approach to complexity theory proceeds from the study of ensembles of computationally intractable optimization problems as a species of unusual disordered magnetic systems. Over the last thirty years, researchers have used this approach to supplement worst-case hardness results encoded by complexity theory with detailed information about thermodynamic and dynamic phase transitions in the structure of typical cases. This exchange of ideas between classical statistical mechanics and computer science enabled the development of important heuristic algorithms such as simulated annealing and survey propagation and further refined our understanding of glassiness and critical slowing in physical disordered systems. In this thesis, we map out an analogous program in the quantum context. The question is simple: what can quantum statistical mechanics reveal about the difficulty of solving hard quantum optimization problems? Or more directly, what makes those problems hard even for quantum computers? In this pursuit, we introduce the study of ensembles of optimization problems whose complexity status is intrinsically quantum mechanical (Part I) and develop techniques to study quantum spin glasses and the transverse field adiabatic algorithm applied to classically hard random optimization problems (Part II). In particular, we introduce the study of random quantum satisfiability (QSAT) and identify the coarse aspects of its phase diagram, including a new form of entanglement transition. We generalize the cavity method to the study of quantum models and use it to study the transverse field Ising glass and frustrated AKLT models on the Bethe lattice. We further apply the cavity method to extract Griffiths-McCoy singularities in a diluted (classical) ferromagnet and finally observe that there are no Goldstone bosons on the Bethe lattice.