According to the recently proposed density-functional analog of quantum phase transitions (QPTs) the ``density'' determines the ``control parameter'' (corresponding to the density-functional theory ``external potential''). It was also proved that in the nondegenerate case there is a one-to-one map between the ground-state wave function and the control parameter. It is now shown that there is a one-to-one map between the ground-state subspace and the control parameter. It is pointed out that there is a one-to one map between the subspace density and the control parameter. The R\'enyi entropies are proved to be strictly monotonic functions of the control parameter in the neighborhood of the transition point. These properties are illustrated with the Dicke model which exhibits a QPT from a normal phase (nondegenerate) to the superradiant case (degenerate) in the thermodynamic limit.
According to the recently proposed density-functional analog of quantum phase transitions (QPTs) the density'' determines the control parameter'' (corresponding to the density-functional theory ``external potential''). It was also proved that in the nondegenerate case there is a one-to-one map between the ground-state wave function and the control parameter. It is now shown that there is a one-to-one map between the ground-state subspace and the control parameter. It is pointed out that there is a one-to one map between the subspace density and the control parameter. The R'enyi entropies are proved to be strictly monotonic functions of the control parameter in the neighborhood of the transition point. These properties are illustrated with the Dicke model which exhibits a QPT from a normal phase (nondegenerate) to the superradiant case (degenerate) in the thermodynamic limit.
It is shown that there is a one-to-one map between the ground state wave function and the control parameter. Wu et al. [Phys. Rev. A2006, 74, 052335] introduced an analogue of the Density Functional Theory (DFT) density and showed that the "density" determines the "control parameter" (corresponding to the DFT "external potential"). We prove that any strictly monotonous function can be used to obtain a new density with a different control parameter (or DFT "external potential"), and the new density determines the new control parameter. Moreover, there is also a one-to-one map between the Rényi entropy of a given order and the control parameter. Therefore, the Rényi entropy can be used as a control parameter.
Within density functional theory (DFT), where the density is the fundamental variable, quantum phase transitions (QPTs) can be formulated through a Hamiltonian H^=H^0+∑iξiA^i, such that the control parameters {ξi} are in bijective correspondence (in the nondegenerate case) with the “densities” ai=⟨A^i⟩, and the functional Q({ai}) acts as the Legendre transform of the energy; this structure even permits the use of Rényi entropy (for a given order) as an alternative control parameter, while degeneracy can be handled via a subspace density. On this foundation, information-theoretic measures provide sensitive diagnostics of criticality: fidelity and its susceptibility χ, Fisher information, relative Rényi entropy, and the Kullback–Leibler divergence are locally linked by Rq≈qIKL≈2qχ(δλ)2, revealing their proportionality in the small-parameter-shift regime. Applied to the Dicke model, numerical analyses show that fidelity exhibits pronounced curvature or divergence near λc=ωω0/2 and that the response sharpens with increasing j, corroborating that these information measures capture QPTs with precision within the DFT framework.
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.
We systematically analyze the various phase transitions of the anisotropic Dicke model that is endowed with both rotating and counterrotating light-matter couplings. In addition to the ground-state quantum phase transition (QPT) from the normal to the superradiant phase, the anisotropic Dicke model also exhibits other transitions, namely, the excited state quantum phase transition (ESQPT), ergodic to nonergodic transition (ENET), and the temperature-dependent phase transition. We show that these phase transitions are profitably studied not only with the standard consecutive level spacing ratio, but also with the aid of various eigenvector quantities such as von Neumann entanglement entropy, the participation ratio, multifractal dimension, and mutual information. For ENET, both the statics and dynamics of the participation ratio offer a consistent and useful picture. An exciting finding from our work is that the ESQPT and the ENET are closely related to each other. We show this with the aid of two characteristic energies in the spectrum corresponding to jumps in von Neumann entropy.
We investigate quantum phase transitions (QPTs) in spin chain systems characterized by local Hamiltonians with matrix product ground states. We show how to theoretically engineer such QPT points between states with predetermined properties. While some of the characteristics of these transitions are familiar, like the appearance of singularities in the thermodynamic limit, diverging correlation length, and vanishing energy gap, others differ from the standard paradigm: In particular, the ground state energy remains analytic, and the entanglement entropy of a half-chain stays finite. Examples demonstrate that these kinds of transitions can occur at the triple point of "conventional" QPTs.
Quantum phase transitions occur in non-Hermitian systems. In this work we show that density functional theory, for the first time, uncovers universal critical behaviors for quantum phase transitions and quantum entanglement in non-Hermitian many-body systems. To be specific, we first prove that the non-degenerate steady state of a non-Hermitian quantum many body system is a universal function of the first derivative of the steady state energy with respect to the control parameter. This finding has far-reaching consequences for non-Hermitian systems. First, it bridges the non-analytic behavior of physical observable and no-analytic behavior of steady state energy, which explains why the quantum phase transitions in non-Hermitian systems occur for finite systems. Second, it predicts universal scaling behaviors of any physical observable at non-Hermitian phase transition point with scaling exponent being (1 - 1/p) with p being the number of coalesced states at the exceptional point. Third, it reveals that quantum entanglement in non-Hermitian phase transition point presents universal scaling behaviors with critical exponents being (1 - 1/p). These results uncover universal critical behaviors in non-Hermitian phase transitions and provide profound connections between entanglement and phase transition in non-Hermitian quantum many-body physics.