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arXiv:1612.03795 · Journal of Statistical Mechanics Theory and Experiment

Solvable quantum Grassmann matrices

Dionysios Anninos, G A Silva

Published Apr 1, 2017• 19 citations
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Abstract

Abstract We explore systems with a large number of fermionic degrees of freedom subject to non-local interactions. We study both vector and matrix-like models with quartic interactions. The exact thermal partition function is expressed in terms of an ordinary bosonic integral, which has an eigenvalue repulsion term in the matrix case. We calculate real time correlations at finite temperature and analyze the thermal phase structure. When possible, calculations are performed in both the original Hilbert space as well as the bosonic picture, and the exact map between the two is explained. At large N , there is a phase transition to a highly entropic high temperature phase from a low temperature low entropy phase. Thermal two-point functions decay in time in the high temperature phase.

DOI
10.1088/1742-5468/aa668f

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Abstract We explore systems with a large number of fermionic degrees of freedom subject to non-local interactions. We study both vector and matrix-like models with quartic interactions. The exact thermal partition function is expressed in terms of an ordinary bosonic integral, which has an eigenvalue repulsion term in the matrix case. We calculate real time correlations at finite temperature and analyze the thermal phase structure. When possible, calculations are performed in both the original Hilbert space as well as the bosonic picture, and the exact map between the two is explained. At large N , there is a phase transition to a highly entropic high temperature phase from a low temperature low entropy phase. Thermal two-point functions decay in time in the high temperature phase.
We explore systems with a large number of fermionic degrees of freedomsubject to non-local interactions. We study both vector and matrix-like modelswith quartic interactions. The exact thermal partition function is expressed interms of an ordinary bosonic integral, which has an eigenvalue repulsion term inthe matrix case. We calculate real time correlations at finite temperature andanalyze the thermal phase structure. When possible, calculations are performedin both the original Hilbert space as well as the bosonic picture, and the exactmap between the two is explained. At large N, there is a phase transition toa highly entropic high temperature phase from a low temperature low entropyphase. Thermal two-point functions decay in time in the high temperature phase.
We explore different limits of exactly solvable vector and matrix fermionic quantum mechanical models with quartic interactions at finite temperature. The models preserve a $U(1)\ifmmode\times\else\texttimes\fi{}SU(N)\ifmmode\times\else\texttimes\fi{}SU(L)$ symmetry at the classical level and we analyze them through bosonization techniques introducing scalar (singlet) and matrix (nonsinglet) bosonic fields. The bosonic path integral representations in the vector limits (N,1)(N,1)(N,1) and (1,L)(1,L)(1,L) are matched to fermionic Fock space Hamiltonians expressed in terms of quadratic Casimirs and some additional terms involving the Cartan subalgebra, which makes explicit the symmetries preserved by scalar and matrix bosonizations at the quantum level. For the case of nonsinglet bosonization we find an equivalence between the vector model and the Polychronakos-Frahm spin model. Using this relation, we compute the free energy. Finally, we compute the eigenvalue distribution in the large NNN, LLL limit with $\ensuremath{\alpha}=\frac{L}{N}$ fixed. The model displays a third order phase transition as we vary the temperature which, in the $\ensuremath{\alpha}\ensuremath{\gg}1$ limit, can be characterized analytically. We conclude finding the critical curve in the parameter space were the eigenvalue distribution transitions from single to double cut.
Abstract The functional integral representation of the density matrix at thermal equilibrium in non-relativistic quantum mechanics (QM) with many degrees of freedom, in the grand canonical formulation is introduced. In QM, Hamiltonians H(p,q) can be also expressed in terms of creation and annihilation operators, a method adapted to the study of perturbed harmonic oscillators. In the holomorphic formalism, quantum operators act by multiplication and differentiation on a vector space of analytic functions. Alternatively, they can also be represented by kernels, functions of complex variables that correspond in the classical limit to a complex parametrization of phase space. The formalism is adapted to the description of many-body boson systems. To this formalism corresponds a path integral representation of the density matrix at thermal equilibrium, where paths belong to complex spaces, instead of the more usual position–momentum phase space. A parallel formalism can be set up to describe systems with many fermion degrees of freedom, with Grassmann variables replacing complex variables. Both formalisms can be generalized to quantum gases of Bose and Fermi particles in the grand canonical formulation. Field integral representations of the corresponding quantum partition functions are derived.
Applying Feynman diagrammatics to non-fermionic strongly correlated models with local constraints might seem generically impossible for two separate reasons: (i) the necessity to have a Gaussian (non-interacting) limit on top of which the perturbative diagrammatic expansion is generated by Wick’s theorem, and (ii) Dyson’s collapse argument implying that the expansion in powers of coupling constant is divergent. We show that for arbitrary classical lattice models both problems can be solved/circumvented by reformulating the high-temperature expansion (more generally, any discrete representation of the model) in terms of Grassmann integrals. Discrete variables residing on either links, plaquettes, or sites of the lattice are associated with the Grassmann variables in such a way that the partition function (as well as all correlation functions) of the original system and its Grassmann-field counterpart are identical. The expansion of the latter around its Gaussian point generates Feynman diagrams. Our work paves the way for studying lattice gauge theories by treating bosonic and fermionic degrees of freedom on equal footing.
Quantum Mechanics was established as the theory of the microscopic world, which allowed to understand processes in atoms and molecules. Its emergence led to a new scientific paradigm that quickly spread to different research fields. Two relevant examples are Quantum Thermodynamics and Quantum Many-Body Theory, where the former aims to characterize thermodynamic processes in quantum systems and the latter intends to understand the properties of quantum many-body systems. In this thesis, we tackle some of the questions in the overlap between these disciplines, focusing on the concepts of temperature and correlations. Specifically, it contains results on the following topics: locality of temperature, correlations in long-range interacting systems and thermometry at low temperature. The problem of locality of temperature is considered for a system at thermal equilibrium and consists in studying whether it is possible to assign a temperature to any of the subsystems of the global system such that both local and global temperatures are equal. We tackle this problem in two different settings, for generic one-dimensional spin chains and for a bosonic system with a phase transition at non-zero temperature. In the first case, we consider generic one-dimensional translation-invariant spin systems with short-range interactions and prove that it is always possible to assign a local temperature equal to the global one for any temperature, including at criticality. For the second case, we consider a three-dimensional discretized version of the Bose-Einstein model at the grand canonical ensemble for some temperature and particle density, and characterize its non-zero-temperature phase transition. Then, we show that temperature is locally well-defined at any temperature and at any particle density, including at the phase transition. Additionally, we observe a qualitative relation between correlations and locality of temperature in the system. Moving to correlations, we consider fermionic two-site long-range interacting systems at thermal equilibrium. We show that correlations between anti-commutative operators at non-zero temperature are upper bounded by a function that decays polynomially with the distance and with an exponent that is equal to the interaction exponent, which characterizes the interactions in the Hamiltonian. Moreover, we show that our bound is asymptotically tight and that the results extend to density-density correlations as well as other types of correlations for quadratic and fermionic Hamiltonians with long-range interactions. Regarding the results on thermometry, we consider a bosonic model and prove that strong coupling between the probe and the system can boost the thermal sensitivity for low temperature. Furthermore, we provide a feasible measurement scheme capable of producing optimal estimates at the considered regime.