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Journal of Physics Conference Series

Exact results for non-integrable systems

Zsolt Gulácsi

Published Feb 8, 2013• 1 citation
Publisher

Abstract

Quantum mechanical many-body systems described by an arbitrary Hamiltonian ( ) are analyzed. It is shown how positive semidefinite operator ( ) properties are able to lead in this case to exact results related to the ground state and the low lying part of the excitation spectrum. This is done independent on dimensionality and integrability. The technique first casts the Hamiltonian in a positive semidefinite form in exact terms ( = + C , where C is a scalar). Second, the ground state is deduced by constructing the most general Hilbert space vector, on which applying , one obtains zero as a result. It is underlined, that the procedure, if applied for variable total number of particles N, allows to obtain information also related to the low lying part of the excitation spectrum. The uniqueness of the ground states can be demonstrated via the study of the kernel of ' = − C . The physical properties of the obtained phases are deduced based on ground state expectation values calculated in terms of the constructed ground states. Since for a fixed structure of , usually the transformation = + C is exact only in a restricted parameter space domain ( ), the deduced ground states are present only in . A global view on the phase diagram is obtained by different transformations of in positive semidefinite form.

DOI
10.1088/1742-6596/410/1/012011

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Quantum mechanical many-body systems described by an arbitrary Hamiltonian ( ) are analyzed. It is shown how positive semidefinite operator ( ) properties are able to lead in this case to exact results related to the ground state and the low lying part of the excitation spectrum. This is done independent on dimensionality and integrability. The technique first casts the Hamiltonian in a positive semidefinite form in exact terms ( = + C , where C is a scalar). Second, the ground state is deduced by constructing the most general Hilbert space vector, on which applying , one obtains zero as a result. It is underlined, that the procedure, if applied for variable total number of particles N, allows to obtain information also related to the low lying part of the excitation spectrum. The uniqueness of the ground states can be demonstrated via the study of the kernel of ' = − C . The physical properties of the obtained phases are deduced based on ground state expectation values calculated in terms of the constructed ground states. Since for a fixed structure of , usually the transformation = + C is exact only in a restricted parameter space domain ( ), the deduced ground states are present only in . A global view on the phase diagram is obtained by different transformations of in positive semidefinite form.
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The spectrum of elementary excitations in one-dimensional quantum liquids is generically linear at low momenta. It is characterized by the sound velocity that can be related to the ground-state energy. Here we study the spectrum at higher momenta in Galilean-invariant integrable models. Somewhat surprisingly, we show that the spectrum at arbitrary momentum is fully determined by the properties of the ground state. We find general exact relations for the coefficients of several terms in the expansion of the excitation energy at low momenta and arbitrary interaction and express them in terms of the Luttinger liquid parameter. We apply the obtained formulas to the Lieb-Liniger model and obtain several new results.
A technique based on the transformation in positive semidefinite form of the Hamiltonian (Ĥ) is developed to allow the study of the enlargement possibilities of the emergence domain of a given state. For this reason the calculation of exact ground states for the polyphenylene type of non-integrable chains is performed. Using different exact transcriptions of Ĥ in positive semidefinite form and different solutions of the matching equations, the same type of ground state is deduced in different regions of the parameter space, hence a more global view of its emergence possibilities is obtained. With this procedure an enlarged view of the appearance in the phase diagram of different studied phases can be achieved.
A class of quantum many-body models of arbitrary dimension and arbitrary statistics of particles, for which exact eigenstates may be obtained, is found. It is assumed that: (i) models contain two (or 2m) kinds of particles with 'symmetric' matrix elements of pairwise interaction (all potentials coincide with each other to within a sign and wavefunctions of free particles of two components coincide to within a phase factor; pairwise interactions are otherwise arbitrary); (ii) there exists the degeneracy of (the sum) of free-particle spectra. Exact many-body eigenstates correspond to a condensation of non-interacting composite particles ('excitons') which are not exactly bosons, into a single quantum state, and to excitations over the condensate. The origin of the possibility of exact solution is the symmetry under the continuous rotations in the isospin space of two components, to which Bogolubov canonical transformations with parameters u, nu independent of momentum correspond. The class of such models comprises, in particular, two-dimensional electron-hole systems in a strong magnetic field.
The aim of this work is to develop a technique for identifying quantum phase transitions which does not rely on the existence of a thermodynamic limit, for studies in finite systems. The approach we adopt exploits an exact mapping of the spectrum of a many-body integrable system, which admits an exact Bethe ansatz solution, into the quasi-exactly solvable spectrum of a one-particle Schrodinger operator. Bifurcations of the minima for the potential of the Schrodinger operator determine critical ground-state couplings. By considering the behaviour of certain ground-state correlation functions, these may be identified as quantum phase transitions in the many-body integrable system with finite particle number. We study two particular examples of bosonic Hamiltonians which admit second-order transitions, and discuss further applications.