By using a recently proposed probabilistic approach, we determine the exact ground state of a class of matrix Hamiltonian models characterized by the fact that in the thermodynamic limit the multiplicities of the potential values assumed by the system during its evolution are distributed according to a multinomial probability density. The class includes (i) the uniformly fully connected models, namely a collection of states all connected with equal hopping coefficients and in the presence of a potential operator with arbitrary levels and degeneracies, and (ii) the random potential systems, in which the hopping operator is generic and arbitrary potential levels are assigned randomly to the states with arbitrary probabilities. For this class of models we find a universal thermodynamic limit characterized only by the levels of the potential, rescaled by the ground-state energy of the system for zero potential, and by the corresponding degeneracies (probabilities). If the degeneracy (probability) of the lowest potential level tends to zero, the ground state of the system undergoes a quantum phase transition between a normal phase and a frozen phase with zero hopping energy. In the frozen phase the ground state condenses into the subspace spanned by the states of the system associated with the lowest potential level.
By using a recently proposed probabilistic approach, we determine the exact ground state of a class of matrix Hamiltonian models characterized by the fact that in the thermodynamic limit the multiplicities of the potential values assumed by the system during its evolution are distributed according to a multinomial probability density. The class includes (i) the uniformly fully connected models, namely a collection of states all connected with equal hopping coefficients and in the presence of a potential operator with arbitrary levels and degeneracies, and (ii) the random potential systems, in which the hopping operator is generic and arbitrary potential levels are assigned randomly to the states with arbitrary probabilities. For this class of models we find a universal thermodynamic limit characterized only by the levels of the potential, rescaled by the ground-state energy of the system for zero potential, and by the corresponding degeneracies (probabilities). If the degeneracy (probability) of the lowest potential level tends to zero, the ground state of the system undergoes a quantum phase transition between a normal phase and a frozen phase with zero hopping energy. In the frozen phase the ground state condenses into the subspace spanned by the states of the system associated with the lowest potential level.
We prove that generic quantum local Hamiltonians are gapless. In fact, we prove that there is a continuous density of states above the ground state. The Hamiltonian can be on a lattice in any spatial dimension or on a graph with a bounded maximum vertex degree. The type of interactions allowed for include translational invariance in a disorder (i.e., probabilistic) sense with some assumptions on the local distributions. Examples include many-body localization and random spin models. We calculate the scaling of the gap with the system's size when the local terms are distributed according to a Gaussian β orthogonal random matrix ensemble. As a corollary, there exist finite size partitions with respect to which the ground state is arbitrarily close to a product state. When the local eigenvalue distribution is discrete, in addition to the lack of an energy gap in the limit, we prove that the ground state has finite size degeneracies. The proofs are simple and constructive. This work excludes the important class of truly translationally invariant Hamiltonians where the local terms are all equal.
Abstract We formalize and prove the extension to finite temperature of a class of quantum phase transitions, acting as condensations in the space of states, recently introduced and discussed at zero temperature (Ostilli and Presilla 2021 J. Phys. A: Math. Theor. 54 055005). In details, we find that if, for a quantum system at canonical thermal equilibrium, one can find a partition of its Hilbert space into two subspaces, and , such that, in the thermodynamic limit, dim ℋ cond / dim ℋ → 0 and the free energies of the system restricted to these subspaces cross each other for some value of the Hamiltonian parameters, then, the system undergoes a first-order quantum phase transition driven by those parameters. The proof is based on an exact probabilistic representation of quantum dynamics at an imaginary time identified with the inverse temperature of the system. We also show that the critical surface has universal features at high and low temperatures.
Abstract We demonstrate that a large class of first-order quantum phase transitions, namely, transitions in which the ground state energy per particle is continuous but its first order derivative has a jump discontinuity, can be described as a condensation in the space of states. Given a system having Hamiltonian H = K + gV , where K and V are two non commuting operators acting on the space of states F , we may always write F = F cond ⊕ F norm where F cond is the subspace spanned by the eigenstates of V with minimal eigenvalue and F norm = F cond ⊥ . If, in the thermodynamic limit, M cond / M → 0, where M and M cond are, respectively, the dimensions of F and F cond , the above decomposition of F becomes effective, in the sense that the ground state energy per particle of the system, ϵ , coincides with the smaller between ϵ cond and ϵ norm , the ground state energies per particle of the system restricted to the subspaces F cond and F norm , respectively: ϵ = min{ ϵ cond , ϵ norm }. It may then happen that, as a function of the parameter g , the energies ϵ cond and ϵ norm cross at g = g c . In this case, a first-order quantum phase transition takes place between a condensed phase (system restricted to the small subspace F
A new microcanonical equilibrium state is introduced for quantum systems with finite-dimensional state spaces. Equilibrium is characterized by a uniform distribution on a level surface of the expectation value of the Hamiltonian. The distinguishing feature of the proposed equilibrium state is that the corresponding density of states is a continuous function of the energy, and hence thermodynamic functions are well defined for finite quantum systems. The density of states, however, is not in general an analytic function. It is demonstrated that generic quantum systems therefore exhibit second-order phase transitions at finite temperatures.
The general structure of thermodynamic equilibrium states for a class of quantum mechanical (multi-lattice) systems is elaborated, combining quantum statistical and thermodynamical methods. The quantum statistical formulation is performed in terms of recent operator algebraic concepts emphasizing the role of the permutation symmetry due to homogeneous coarse graining and employing the internal symmetries. The variational principle of the free energy functional is derived, which determines together with the symmetries the general form of the limiting Gibbs states in terms of their central decomposition. The limiting minimal free energy density and its possible equilibrium states are analyzed on various levels of the description by means of convex analysis, where the Fenchel transforms of the free energies provide entropy like potentials. On the thermodynamic level a modified entropy surface is obtained, which specifies only in combination with its concave envelope the regions of pure and mixed phase states. The symmetry properties of a certain model allow to specify the (non-) differentiability of the minimal free energy density. A characterization and classification of phase transitions in terms of quantum statistical equilibrium states is proposed, and the connection to the Landau theory is established demonstrating that the latter implies a (continuous) deformation of the sets of equilibrium states along a canonically given curve.
Nonrelativistic Hamiltonians with large, even infinite, ground-state degeneracy are studied by connecting the degeneracy to the property of a Dirac operator. We then identify a special class of Hamiltonians, for which the full space of degenerate ground states in any spatial dimension can be exhibited explicitly. The two-dimensional version of the latter coincides with the Pauli Hamiltonian, and recently-discussed models leading to higher-dimensional Landau levels are obtained as special cases of the higher-dimensional version of this Hamiltonian. But, in our framework, it is only the asymptotic behavior of the background 'potential' that matters for the ground-state degeneracy. We work out in detail the ground states of the three-dimensional model in the presence of a uniform magnetic field and such potential. In the latter case one can see degenerate stacking of all 2d Landau levels along the magnetic field axis.