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Topics in theoretical condensed matter physics

Chaitanya Murthy

Published Jan 1, 2020• 0 citations
OpenAlex

Abstract

We investigate various topics in theoretical quantum condensed matter physics.First, we show that a one-dimensional quantum wire with as few as 2 channels of interacting fermions can host metallic Luttinger liquid states of matter that are stable against all perturbations up to $q^\\text{th}$-order in fermion creation/annihilation operators for any fixed finite $q$.The leading relevant perturbations are thus complicated operators that are expected to modify the physics only at very low energies, below accessible temperatures. The stability of these Luttinger liquid fixed points is due to strong interactions between the channels, which can (but need not) be chosen to be purely repulsive. Our results might enable elementary physical realizations of these phases, and may also serve as a useful paradigm for thinking about higher-dimensional non-Fermi liquids. Separately, we present an elementary but general description of relaxation to gaussian and equilibrium generalized Gibbs states in lattice models of fermions or bosons with quadratic hamiltonians. Our analysis applies to arbitrary initial states that satisfy a mild condition on clustering of correlations.We obtain quantitive, model-independent predictions for how quickly local quantities relax in such systems.These predictions can be tested in near-term quantum gas experiments.Finally, we study chaotic many-body quantum systems that obey the eigenstate thermalization hypothesis (ETH).We show that a known bound on the growth rate of the out-of-time-order four-point correlator in such systems follows directly from the general ETH structure of operator matrix elements.This ties together two key paradigms of thermal behavior in isolated many-body quantum systems.We also consider a bipartition of the system, and study the entanglement properties of an energy eigenstate with nonzero energy density.When the two subsystems have nearly equal volumes, we find a universal correction to the entanglement entropy that is proportional to the square root of the system's heat capacity (or a sum of capacities, if there are conserved quantities in addition to energy).

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Field Theories of Condensed Matter Physics

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Presenting the physics of the most challenging problems in condensed matter using the conceptual framework of quantum field theory, this book is of great interest to physicists in condensed matter and high energy and string theorists, as well as mathematicians. Revised and updated, this second edition features new chapters on the renormalization group, the Luttinger liquid, gauge theory, topological fluids, topological insulators and quantum entanglement. The book begins with the basic concepts and tools, developing them gradually to bring readers to the issues currently faced at the frontiers of research, such as topological phases of matter, quantum and classical critical phenomena, quantum Hall effects and superconductors. Other topics covered include one-dimensional strongly correlated systems, quantum ordered and disordered phases, topological structures in condensed matter and in field theory and fractional statistics.

Cambridge University Press eBooks•Feb 28, 2013•1,185 citations
We investigate various topics in theoretical quantum condensed matter physics.First, we show that a one-dimensional quantum wire with as few as 2 channels of interacting fermions can host metallic Luttinger liquid states of matter that are stable against all perturbations up to q^\\text{th}-order in fermion creation/annihilation operators for any fixed finite qqq.The leading relevant perturbations are thus complicated operators that are expected to modify the physics only at very low energies, below accessible temperatures. The stability of these Luttinger liquid fixed points is due to strong interactions between the channels, which can (but need not) be chosen to be purely repulsive. Our results might enable elementary physical realizations of these phases, and may also serve as a useful paradigm for thinking about higher-dimensional non-Fermi liquids. Separately, we present an elementary but general description of relaxation to gaussian and equilibrium generalized Gibbs states in lattice models of fermions or bosons with quadratic hamiltonians. Our analysis applies to arbitrary initial states that satisfy a mild condition on clustering of correlations.We obtain quantitive, model-independent predictions for how quickly local quantities relax in such systems.These predictions can be tested in near-term quantum gas experiments.Finally, we study chaotic many-body quantum systems that obey the eigenstate thermalization hypothesis (ETH).We show that a known bound on the growth rate of the out-of-time-order four-point correlator in such systems follows directly from the general ETH structure of operator matrix elements.This ties together two key paradigms of thermal behavior in isolated many-body quantum systems.We also consider a bipartition of the system, and study the entanglement properties of an energy eigenstate with nonzero energy density.When the two subsystems have nearly equal volumes, we find a universal correction to the entanglement entropy that is proportional to the square root of the system's heat capacity (or a sum of capacities, if there are conserved quantities in addition to energy).
We show that a one-dimensional quantum wire with as few as two channels of interacting fermions can host metallic states of matter that are stable against all perturbations up to qth order in fermion creation or annihilation operators for any fixed finite q. Thus, the leading relevant perturbations are complicated operators that are expected to modify the physics only at very low energies, below accessible temperatures. The stability of these non-Fermi liquid fixed points is due to strong interactions between the channels, which can (but need not) be chosen to be purely repulsive. Our results might enable elementary physical realizations of these phases.
We study different aspects of integrable boundary quantum field theories, focusing mostly on the ``boundary sine-Gordon model'' and its applications to condensed matter physics. The first part of the review deals with formal problems. We analyze the classical limit and perform semi-classical quantization. We show that the non-relativistic limit corresponds to the Calogero-Moser model with a boundary potential. We construct a lattice regularization of the problem via the XXZ chain. We classify boundary bound states. We generalize the Destri de Vega method to compute the ground state energy of the theory on a finite interval. The second part deals with some applications to condensed matter physics. We show how to compute analytically time and space dependent correlations in one-dimensional quantum integrable systems with an impurity. Our approach is based on a description of these systems in terms of massless scattering of quasiparticles. Correlators follow then from matrix elements of local operators between multiparticle states -- the massless form-factors. Although, in general an infinite sum of these form-factors has to be considered, we find that for the current, spin and energy operators only a few (two or three) are necessary to obtain an accuracy of more than 1\%. Our results hold for arbitrary impurity strength, in contrary to the perturbative expansions in the coupling constants. As an example, we compute the frequency dependent condunctance, at zero temperature, in a Luttinger liquid with an impurity, and also discuss the susceptibility in the Kondo model and the time-dependent properties of the two-state problem with dissipation.
This dissertation is a study of various aspects of metals with strong interactions between electrons, with a particular emphasis on the problem of charge transport through them. We consider the physics of clean or weakly-disordered metals near some quantum critical points, and highlight novel transport regimes that could be relevant to experiments. We then develop a variety of exactly-solvable lattice models of strongly interacting non-Fermi liquid metals using novel non-perturbative techniques based on the Sachdev-Ye-Kitaev models, and relate their physics to that of the ubiquitous “strange metal” normal state of most correlated-electron superconductors, providing controlled theoretical insight into the possible mechanisms behind it. Finally, we use ideas from the field of quantum chaos to study mathematical quantities that can provide evidence for the existence of quasiparticles (or the lack thereof) in quantum many-body systems, in the context of metals with correlated electrons.
Abstract The theory of quantum critical phenomena is introduced to study some current many-body problems in condensed matter physics. Renormalization group concepts are applied to strongly correlated electronic materials which are close to a zero-temperature instability. These systems have enhanced effective masses and susceptibility. Scaling arguments yield the exponents which govern the critical behavior of these quantities in terms of the usual critical exponents associated with a zero-temperature phase transition. We show the existence of a new energy scale, related to the quantum nature of the many-body instability, which can be generally associated with the setting of Fermi-liquid behavior with decreasing temperature in three-dimensional strongly interacting electronic systems. The theory of quantum critical phenomena is used to investigate the Kondo lattice problem, which provides a model to describe heavy-fermion systems and to introduce a scaling theory of the Mott transition with special emphasis on charge fluctuation effects. However, this report is not a review on heavy fermions and Mott insulators. The microscopic theories of these systems are still controversial and present some of the most challenging and instigating problems in condensed matter physics. This state of affairs stimulated the author to review and extend the scaling approach. The scaling theory we develop provides a powerful tool, based on the notion of universality, to understand the physical properties of correlated systems beyond the mean-field level. This is illustrated by our treatment of the one-dimensional Hubbard model, where, although the Fermi-liquid fixed point does not survive the fluctuations, the scaling approach is still useful. Finally, we discuss briefly how disorder affect our results.
Abstract The theory of quantum critical phenomena is introduced to study some current many-body problems in condensed matter physics. Renormalization group concepts are applied to strongly correlated electronic materials which are close to a zero-temperature instability. These systems have enhanced effective masses and susceptibility. Scaling arguments yield the exponents which govern the critical behavior of these quantities in terms of the usual critical exponents associated with a zero-temperature phase transition. We show the existence of a new energy scale, related to the quantum nature of the many-body instability, which can be generally associated with the setting of Fermi-liquid behavior with decreasing temperature in three-dimensional strongly interacting electronic systems. The theory of quantum critical phenomena is used to investigate the Kondo lattice problem, which provides a model to describe heavy-fermion systems and to introduce a scaling theory of the Mott transition with special emphasis on charge fluctuation effects. However, this report is not a review on heavy fermions and Mott insulators. The microscopic theories of these systems are still controversial and present some of the most challenging and instigating problems in condensed matter physics. This state of affairs stimulated the author to review and extend the scaling approach. The scaling theory we develop provides a powerful tool, based on the notion of universality, to understand the physical properties of correlated systems beyond the mean-field level. This is illustrated by our treatment of the one-dimensional Hubbard model, where, although the Fermi-liquid fixed point does not survive the fluctuations, the scaling approach is still useful. Finally, we discuss briefly how disorder affect our results.
Presenting the physics of the most challenging problems in condensed matter using the conceptual framework of quantum field theory, this book is of great interest to physicists in condensed matter and high energy and string theorists, as well as mathematicians. Revised and updated, this second edition features new chapters on the renormalization group, the Luttinger liquid, gauge theory, topological fluids, topological insulators and quantum entanglement. The book begins with the basic concepts and tools, developing them gradually to bring readers to the issues currently faced at the frontiers of research, such as topological phases of matter, quantum and classical critical phenomena, quantum Hall effects and superconductors. Other topics covered include one-dimensional strongly correlated systems, quantum ordered and disordered phases, topological structures in condensed matter and in field theory and fractional statistics.