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
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
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.
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.
What is a quantum phase transition? Consider a Hamiltonian, H ( g ), whose degrees of freedom reside on the sites of a lattice, and which varies as a function of a dimensionless coupling, g . Let us follow the evolution of the ground state energy of H ( g ) as a function of g . For the case of a finite lattice, this ground state energy will generically be a smooth, analytic function of g . The main possibility of an exception comes from the case when g couples only to a conserved quantity (i.e. H ( g ) = H 0 + gH 1 , where H 0 and H 1 commute). This means that H 0 and H 1 can be simultaneously diagonalized and so the eigenfunctions are independent of g even though the eigenvalues vary with g ; then there can be a level-crossing where an excited level becomes the ground state at g = gc (say), creating a point of nonanalyticity of the ground state energy as a function of g (see Fig. 1.1). The possibilities for an infinite lattice are richer. An avoided level-crossing between the ground and an excited state in a finite lattice could become progressively sharper as the lattice size increases, leading to a nonanalyticity at g = gc in the infinite lattice limit. We shall identify any point of nonanalyticity in the ground state energy of the infinite lattice system as a quantum phase transition: The nonanalyticity could be either the limiting case of an avoided level-crossing or an actual level-crossing.
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.
Quantum phase transitions manifest as an abrupt change in the ground state of a many-body system; yet it is an open question whether this sudden change necessarily precludes a continuous dynamical connection between the two phases. We introduce a classification of quantum phase transitions based on this geometric aspect of the ground-state manifold, that differs from known classifications. By leveraging the framework of counterdiabatic driving, we explicitly construct schedules that dynamically connect one phase to another. This strategy allows us to uncover a large class of quantum phase transitions, where the states on both sides are separated only by a finite geometric distance in the thermodynamic limit. We term such transitions traversable, since exact counterdiabatic driving links the two phases via a finite dynamical protocol in the thermodynamic limit. We show that multiple known transitions fall into this class -- e.g., symmetry-breaking transitions obeying hyperscaling and discontinuous transitions with an enhanced continuous symmetry. We further show the existence of quantum phase transitions that cannot be crossed dynamically even with the help of nonlocal counterdiabatic driving, as they would require divergent amplitudes and frequencies. Geometrically, these nontraversable transitions correspond to an infinite distance separating the two phases of matter; we show that the class comprises continuous transitions exhibiting mean-field universality, and discontinuous transitions arising from the competition between metastable minima. Our geometric classification goes beyond the known taxonomy, is independent of local order parameters and renormalization group fixed points, and has direct implications for the complexity of state preparation and adiabatic quantum computation.
What Is a Quantum Phase Transition? Consider a Hamiltonian, H(g) , whose degrees of freedom reside on the sites of a lattice, and which varies as a function of a dimensionless coupling g . Let us follow the evolution of the ground state energy of H(g) as a function of g . For the case of a finite lattice, this ground state energy will generically be a smooth, analytic function of g . The main possibility of an exception comes from the case when g couples only to a conserved quantity (i.e., H(g) = H 0 + gH 1 , where H 0 and H 1 commute). This means that H 0 and H 1 can be simultaneously diagonalized and so the eigenfunctions are independent of g even though the eigenvalues vary with g ; then there can be a level-crossing where an excited level becomes the ground state at g = g c (say), creating a point of nonanalyticity of the ground state energy as a function of g (see Fig. 1.1). The possibilities for an infinite lattice are richer. An avoided level-crossing between the ground and an excited state in a finite lattice could become progressively sharper as the lattice size increases, leading to a nonanalyticity at g = g c in the infinite lattice limit. We shall identify any point of nonanalyticity in the ground state energy of the infinite lattice system as a quantum phase transition: The nonanalyticity could be either the limiting case of an avoided level-crossing or an actual level-crossing.