In the thermodynamic limit, the equilibrium state of a many-body system can be characterized by three pairs of conjugate thermodynamic variables:
E/T,V/P,N/μ. In this limit, the thermodynamic properties in all ensembles are equivalent up to the leading order of
E,V,N. However, for systems of finite size, this ensemble equivalence is no longer exact, and the thermodynamic properties may differ substantially among ensembles. To quantify these finite-size effects rigorously, it is desirable to develop a universal ensemble theory applicable to systems of arbitrary size, providing exact expressions for entropy and, thereby, giving rise to the precise value of all equilibrium thermodynamic quantities. In this work, we propose a theory that determines the exact entropy expressions for classical non-interacting many-body systems of arbitrary size across all statistical ensembles, based on only two postulates: \textbf{stationarity}, requiring that the physical laws be invariant under time translation, and \textbf{unbiasedness}, requiring that the equilibrium mixed state maximize the entropy subject to the prescribed constraints. Moreover, we show that the entropy expressions obtained in different ensembles converge to the common asymptotic form
S≍ln((3N4πmeE)3N/2⋅(NV)N)+N, consistent with the predictions of the large deviation theory.
Ground-state properties are central to our understanding of quantum many-body systems.At first glance, it seems natural and essential to obtain the ground state before analyzing its properties; however, its exponentially large Hilbert space has made such studies costly, if not prohibitive, on sufficiently large system sizes.Here, we propose an alternative strategy based upon the expectation values of an ensemble of operators and the elusive yet vital quantum constraints between them where the search for ground-state properties simply equates to classical constrained minimization.These quantum constraints are generally obtainable via sampling and then machine learning on a large number of systematically consistent quantum many-body states.We showcase our perspective on one-dimensional fermion chains and spin chains for applicability, effectiveness, caveats, and unique advantages especially for strongly correlated systems, thermodynamic-limit systems, property designs, etc.
The fractional quantum Hall (FQH) effect plays a prominent role in the study of topological phases of matter and of strongly correlated electron systems in general. FQH systems have been demonstrated to show many interesting novel properties such as fractional charges, and are believed to harbor even more intriguing phenomena such as fractional statistics. However, there remain many interesting questions to be addressed in this regime. The work reported in this thesis aims to push the envelope of our understanding of the low-energy properties of FQH states using microscopic principles. In the first part of the thesis, we present a systematic perturbative approach to study excitations in the thin cylinder/torus limit of the quantum Hall states. The approach is applied to the Haldane-Rezayi and Gaffnian quantum Hall states, which are both expected to have gapless excitations in the usual two-dimensional thermodynamic limit. For the Haldane-Rezayi state, we confirm that gapless excitations are present also in the ``one-dimensional'' thermodynamic limit of an infinite thin cylinder, in agreement with earlier considerations based on the wave functions alone. In contrast, we identify the lowest excitations of the Gaffnian state in the thin cylinder limit, and conclude that they are gapped, using a combination of perturbative and numerical means. We discuss possible scenarios for the cross-over between the two-dimensional and the one-dimensional thermodynamic limit in this case. In the second part of the thesis, we study the low energy spectral properties of positive center-of-mass conserving two-body Hamiltonians as they arise in models of FQH states. Starting from the observation that positive many-body Hamiltonians must have ground state energies that increase monotonously in particle number, we explore what general additional constraints can be obtained for two-body interactions with "center-of-mass conservation" symmetry, both in the presence and absence of particle-hole symmetry. We find general bounds that constrain the evolution of the ground state energy with particle number, and in particular constrain the chemical potential at T=0. Special attention is given to Hamiltonians with zero modes, in which case similar bounds on the first excited state are also obtained, using a duality property. In this case, in particular an upper bound on the charge gap is also obtained. We further comment on center-of-mass and relative-decomposition in disk geometry within the framework of second quantization.