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arXiv:cond-mat/0607236

Stability and properties of striped phases in systems of interacting fermions or hard-core bosons

Volodymyr Derzhko

Published Jul 10, 2006• 0 citations
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Abstract

In this thesis we deal with the specific collective phenomena in condensed matter - striped-structures formation. Such structures are observed in different branches of condensed matter physics, like surface physics or physics of high-temperature superconductors. These quasi-one-dimensional objects appear in theoretical analyses as well as in computer simulations of different theoretical models. Here, the main topic of interest is the stability of striped structures in certain quantum models, where a tendency towards crystallization competes with a tendency towards phase separation, and some basic properties of these structures.

DOI
10.48550/arxiv.cond-mat/0607236

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In this thesis we deal with the specific collective phenomena in condensed matter - striped-structures formation. Such structures are observed in different branches of condensed matter physics, like surface physics or physics of high-temperature superconductors. These quasi-one-dimensional objects appear in theoretical analyses as well as in computer simulations of different theoretical models. Here, the main topic of interest is the stability of striped structures in certain quantum models, where a tendency towards crystallization competes with a tendency towards phase separation, and some basic properties of these structures.
This Ph.D. thesis in mathematical physics concerns systems of interacting fermions with strong correlations. For these systems the physical properties can only be described in terms of the collective behavior of the fermions. Moreover, they are often characterized by a close competition between fermion localization versus delocalization, which can result in complex and exotic physical phenomena. Strongly correlated fermion systems are usually modelled by many-body Hamiltonians for which the kinetic- and interaction energy have the same order of magnitude. This makes them challenging to study as the application of conventional computational methods, like mean field- or perturbation theory, often gives unreliable results. Of particular interest are Hubbard-type models, which provide minimal descriptions of strongly correlated fermions. The research of this thesis focuses on such models defined on two-dimensional square lattices. One motivation for this is the so-called high-Tc problem of the cuprate superconductors. A main hypothesis is that there exists an underlying Fermi surface with nearly flat parts, i.e. regions where the surface is straight. It is shown that a particular continuum limit of the lattice system leads to an effective model amenable to computations. This limit is partial in that it only involves fermion degrees of freedom near the flat parts. The result is an effective quantum field theory that is analyzed using constructive bosonization methods. Various exactly solvable models of interacting fermions in two spatial dimensions are also derived and studied.
This Ph.D. thesis in mathematical physics concerns systems of interacting fermions with strong correlations. For these systems the physical properties can only be described in terms of the collective behavior of the fermions. Moreover, they are often characterized by a close competition between fermion localization versus delocalization, which can result in complex and exotic physical phenomena. Strongly correlated fermion systems are usually modelled by many-body Hamiltonians for which the kinetic- and interaction energy have the same order of magnitude. This makes them challenging to study as the application of conventional computational methods, like mean field- or perturbation theory, often gives unreliable results. Of particular interest are Hubbard-type models, which provide minimal descriptions of strongly correlated fermions. The research of this thesis focuses on such models defined on two-dimensional square lattices. One motivation for this is the so-called high-Tc problem of the cuprate superconductors. A main hypothesis is that there exists an underlying Fermi surface with nearly flat parts, i.e. regions where the surface is straight. It is shown that a particular continuum limit of the lattice system leads to an effective model amenable to computations. This limit is partial in that it only involves fermion degrees of freedom near the flat parts. The result is an effective quantum field theory that is analyzed using constructive bosonization methods. Various exactly solvable models of interacting fermions in two spatial dimensions are also derived and studied.
In recent years, quantum phase transitions have attracted the interest of both theorists and experimentalists in condensed matter physics. These transitions, which are accessed at zero temperature by variation of a non-thermal control parameter, can influence the behaviour of electronic systems over a wide range of the phase diagram. Quantum phase transitions occur as a result of competing ground state phases. The cuprate superconductors which can be tuned from a Mott insulating to a d -wave superconducting phase by carrier doping are a paradigmatic example. This review introduces important concepts of phase transitions and discusses the interplay of quantum and classical fluctuations near criticality. The main part of the article is devoted to bulk quantum phase transitions in condensed matter systems. Several classes of transitions will be briefly reviewed, pointing out, e.g., conceptual differences between ordering transitions in metallic and insulating systems. An interesting separate class of transitions is boundary phase transitions where only degrees of freedom of a subsystem become critical; this will be illustrated in a few examples. The article is aimed at bridging the gap between high-level theoretical presentations and research papers specialized in certain classes of materials. It will give an overview on a variety of different quantum transitions, critically discuss open theoretical questions, and frequently make contact with recent experiments in condensed matter physics.
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).
The first part of the thesis deals with so called numerical renormalization group methods, and in particular the density-matrix renormalization group (DMRG). This method performs extremely well for large one-dimensional quantum systems like spin-chains and Hubbard models. We discuss general features of the quantum states obtained by using this method for large systems. One finds that the method works better, i.e. faster convergence and higher accuracy, when applied to systems having an energy gap between the ground state and the first excited state. An investigation of the behavior of the DMRG for a simple gapless system is done, mainly focusing on the properties of correlation functions. The second part of the thesis deals with topics related to superconductivity. It starts with a discussion of magnetic Josephson junctions and the physical processes governing the behavior of such a system. We also introduce and discuss models of high-temperature superconductors, in particular the t-J model and its relation to stripes. Starting from the t-J model on a two dimensional square lattice, we discuss an approximation where the spin-configuration is assumed static. We discuss the properties of the fictitious magnetic field experienced by the electrons due to the static spin-configuration. Configurations being close to ferromagnetic and antiferromagnetic turn out to have very different properties. Furthermore, we derive an effective model for this system and look for different phases of this model. In particular we investigate whether the system takes advantage of a spin-generated fictitious flux. Uniform phases are discussed in a Hartree-Fock approximation and we also consider stripes between antiferromagnetically ordered regions, these being modeled in a self-consistent Hartree approximation. Finally, and inspired by the fictitious fluxes, there is a chapter discussing the statistics of particles living on a circle. It is shown that there are two free parameters in the quantization of such a system and we also sort out the relation between fermions and hard-core bosons in one dimension.