In nature, everything occurs at finite temperature, and quantum phase transitions (QPTs) cannot be an exception. Nevertheless, they are still mainly discussed and formulated at zero temperature. We show that the condensation QPTs recently introduced at zero temperature can naturally be extended to finite temperature just by replacing ground-state energies with corresponding free energies. We illustrate this criterion in the paradigmatic Grover model and in a system of free fermions in a one-dimensional inhomogeneous lattice. In agreement with expected universal features, the two systems show structurally similar phase diagrams. Last, we explain how finite-temperature condensation QPTs can be used to construct quantum annealers having, at finite temperature, output probability exponentially close to 1 in the system size. As examples we consider again the Grover model and the fermionic system, the latter being well within the reach of present heterostructure technology.
In nature, everything occurs at finite temperature, and quantum phase transitions (QPTs) cannot be an exception. Nevertheless, they are still mainly discussed and formulated at zero temperature. We show that the condensation QPTs recently introduced at zero temperature can naturally be extended to finite temperature just by replacing ground-state energies with corresponding free energies. We illustrate this criterion in the paradigmatic Grover model and in a system of free fermions in a one-dimensional inhomogeneous lattice. In agreement with expected universal features, the two systems show structurally similar phase diagrams. Last, we explain how finite-temperature condensation QPTs can be used to construct quantum annealers having, at finite temperature, output probability exponentially close to 1 in the system size. As examples we consider again the Grover model and the fermionic system, the latter being well within the reach of present heterostructure technology.
Aim and Scope of this Book A plethora of systems exhibit phase transitions as the temperature or some other parameter is changed. Examples range from the ice-water phase transition observed in our daily life to the loss of ferromagnetism in iron or to the more sophisticated Mott insulator-superfluid phase transition observed in optical lattices [343]. The last five decades have witnessed a tremendous upsurge in the studies of phase transitions at finite temperature [727, 149, 333, 136, 494, 541, 556]. The success of Landau-Ginzburg theories and the concepts of spontaneous symmetry breaking and the renormalization group [27, 410, 821, 578] in explaining many of the finite temperature phase transitions occurring in nature has been spectacular. In this book, we will consider only a subclass of phase transitions called quantum phase transitions (QPTs) [154, 658, 725, 799, 185, 63, 62, 66, 141, 744] and we will discuss these mainly from the view point of recent studies of information and dynamics. QPTs are zero temperature phase transitions which are driven by quantum fluctuations and are usually associated with a non-analyticity in the ground state energy density of a quantum many-body Hamiltonian. We will focus on continuous QPTs where the order parameter vanishes continuously at the quantum critical point (QCP) at some value of the parameters which characterize the Hamiltonian. We will not discuss first order quantum phase transitions associated with an abrupt change in the order parameter. Usually, a first order phase transition is characterized by a finite discontinuity in the first derivative of the ground state energy density. A continuous QPT is similarly characterized by a finite discontinuity, or divergence, in the second derivative of the ground state energy density, assuming that the first derivative is continuous. This is of course the classical definition; we will later mention some QPTs where the ground state energy density is not necessarily singular.
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.
Phase transitions are commonly held to occur only in the thermodynamical limit of a large number of system components. Here, we exemplify at the hand of the exactly solvable Jaynes-Cummings (JC) model and its generalization to finite JC lattices that finite component systems of coupled spins and bosons may exhibit quantum phase transitions (QPTs). For the JC model we find a continuous symmetry-breaking QPT, a photonic condensate with a macroscopic occupation as the ground state, and a Goldstone mode as a low-energy excitation. For the two site JC lattice we show analytically that it undergoes a Mott-insulator to superfluid QPT. We identify as the underlying principle of the emergence of finite system QPTs the combination of increasing atomic energy and increasing interaction strength between the atom and the bosonic mode, which allows for the exploration of an increasingly large portion of the infinite dimensional Hilbert space of the bosonic mode. This suggests that finite system phase transitions will be present in a broad range of physical systems.
We extend the analysis of the class of quantum phase transitions (QPTs) that can be interpreted as condensations in state space, first introduced in [M. Ostilli and C. Presilla, J. Phys. A 54, 055005 (2021)], by generalizing the arguments of [M. Ostilli and C. Presilla, Phys. Rev. Lett. 127, 040601 (2021)] to prove the existence and determine the location (via simple bounds) of QPTs in general one-parameter lattice Hamiltonians. Unlike our original formulation, this extension also encompasses second-order QPTs, for which we provide the explicit example of the transverse-field Ising model. Our analysis suggests that, under conditions typically satisfied in physical contexts, any QPT taking place in lattice systems can be interpreted as a condensation in state space.
Introduction In the limit of zero temperature, the partition function is dominated by the ground state of the system, the free energy becomes equivalent to the ground state energy, and the central object of interest is the Hamiltonian of the system H (λ), dependent in general on one or more parameters λ. A ‘quantum’ Hamiltonian, as noted previously, is one which contains non-commuting operators. Here we concentrate on quantum spin Hamiltonians, of which the prime example is the Heisenberg model. Of particular interest is the possibility of a quantum phase transition , i.e. nonanalytic behaviour as a function of coupling λ. A quantum phase transition may exert a dominating influence on the system over a range of low temperatures above T = 0. Such phenomena have figured prominently in theoretical discussions of experiments on the cuprate superconductors, the heavy fermion materials, organic conductors, and related materials (e.g. Sachdev, 1999). In some cases, a correspondence can be found between a quantum system at zero temperature in 1 time and ( d – 1) space dimensions, and an equivalent classical system at finite temperature in d space dimensions, based on the Feynman path integral formalism – see Chapter 9 and Appendix 8 for further discussions. Under this mapping, the temperature kT in the classical system corresponds to a coupling λ in the equivalent quantum system. If a phase transition occurs, the same universal critical exponents are expected to apply in both cases.
Random quantum states drawn from the Haar ensemble with a constraint on the energy expectation value Eav=⟨ψ∣H∣ψ⟩ display \textit{eigenstate condensation}: for Eav below a critical value Ec, they develop macroscopic overlap with the ground state. We study eigenstate condensation in systems with finite-dimensional Hilbert spaces. These systems display three phases: a ground-state phase, in which energy-constrained random states have macroscopic overlap with the ground state; a high-temperature phase, in which they have exponentially small overlap with each energy eigenstate; and an anti-ground-state phase, in which they have macroscopic overlap with the most highly excited state. In local spin systems the ground-state and anti-ground-state phases approach the middle of the spectrum as 1/[system size], but -- because the condensation phase transitions have exponential, rather than polynomial, finite-size scaling -- the crossover becomes exponentially sharp in system size and the high-temperature phase is best understood as an extended phase.