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.
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.
Density functional theory (DFT) has met great success in solid state physics, quantum chemistry and in computational material sciences. In this work we show that DFT could shed light on phase transitions and entanglement at finite temperatures. Specifically, we show that the equilibrium state of an interacting quantum many-body system which is in thermal equilibrium with a heat bath at a fixed temperature is a universal functional of the first derivatives of the free energy with respect to temperature and other control parameters respectively. This insight from DFT enables us to express the average value of any physical observable and any entanglement measure as a universal functional of the first derivatives of the free energy with respect to temperature and other control parameters. Since phase transitions are marked by the nonanalytic behavior of free energy with respect to control parameters, the physical quantities and entanglement measures may present nonanalytic behavior at critical point inherited from their dependence on the first derivative of free energy. We use two solvable models to demonstrate these ideas. These results give new insights for phase transitions and provide new profound connections between entanglement and phase transitions in interacting quantum many-body physics.
Abstract When can we map a classical density profile to an external potential? In equilibrium, without time dependence, the one-body density is known to specify the external potential that is applied to the many-body system. This mapping from a density to the potential is the cornerstone of classical density functional theory (DFT). Here, we consider non-equilibrium, time-dependent many-body systems that evolve from a given initial condition. We derive explicit conditions, for example, no flux at the boundary, that ensure that the mapping from the density to a time-dependent external potential is unique. We thus prove the underlying assertion of dynamical density functional theory (DDFT) — without resorting to the so-called adiabatic approximation often used in applications. By ascertaining uniqueness for all n -body densities, we ensure that the proof — and the physical conclusions drawn from it — hold for general superadiabatic dynamics of interacting systems even in the presence of (known) non-conservative forces.
Density-functional theory (DFT) is shown to provide a conceptual and computational framework for entanglement in interacting many-body quantum systems. DFT can, in particular, shed light on the intriguing relationship between quantum phase transitions and entanglement. We use DFT concepts to express entanglement measures in terms of the first or second derivative of the ground-state energy. We illustrate the versatility of the DFT approach via a variety of analytically solvable models. As a further application we discuss entanglement and quantum phase transitions in the case of mean-field approximations for realistic models of many-body systems.
Within the framework of zero-temperature Fock-space density-functional theory (DFT), we prove that the G\^ateaux functional derivative of the universal density functional, ${\ensuremath{\delta}{F}^{\ensuremath{\lambda}}[\ensuremath{\rho}]∕\ensuremath{\delta}\ensuremath{\rho}(\mathbit{r})\ensuremath{\mid}}_{\ensuremath{\rho}={\ensuremath{\rho}}_{0}}$, at ground-state densities with arbitrary normalizations $(⟨{\ensuremath{\rho}}_{0}(\mathbit{r})⟩=n∊{\mathcal{R}}_{+})$ and an electron-electron interaction strength $\ensuremath{\lambda}$, is uniquely defined, but is discontinuous when the number of electrons
n becomes an integer, thus providing a mathematically rigorous confirmation for the ``derivative discontinuity'' initially discovered by Perdew et al. [Phys. Rev. Lett. 49, 1691 (1982)]. However, the functional derivative of the exchange-correlation functional is continuous with respect to the number of electrons in Fock space; i.e., there is no ``derivative discontinuity'' for the exchange-correlation functional at an integer electron number. For a ground-state density ${\ensuremath{\rho}}_{0,n}^{v,\ensuremath{\lambda}}(\mathbit{r})$ of an external potential $v(\mathbit{r})$, we show that ${\ensuremath{\delta}{F}^{\ensuremath{\lambda}}[\ensuremath{\rho}]∕\ensuremath{\delta}\ensuremath{\rho}(\mathbit{r})\ensuremath{\mid}}_{\ensuremath{\rho}={\ensuremath{\rho}}_{0,n}^{v,\ensuremath{\lambda}}}={\ensuremath{\mu}}_{\mathrm{SM}}^{n}\ensuremath{-}v(\mathbit{r})$, where the constant ${\ensuremath{\mu}}_{\mathrm{SM}}^{n}$ is given by the following chain of dependences: ${\ensuremath{\rho}}_{0,n}^{v,\ensuremath{\lambda}}(\mathbit{r})\ensuremath{\mapsto}[v]\ensuremath{\mapsto}{E}_{0}^{v,\ensuremath{\lambda}}(n)\ensuremath{\mapsto}{\ensuremath{\mu}}_{\mathrm{SM}}^{n}={\ensuremath{\partial}{E}_{0}^{v,\ensuremath{\lambda}}(k)∕\ensuremath{\partial}k\ensuremath{\mid}}_{k=n}$. Here
[v] is the class of the external potential $v(\mathbit{r})$ up to a real constant, and ${\ensuremath{\mu}}_{\mathrm{SM}}^{n}$ is the chemical potential defined according to statistical mechanics. At an integer electron number
N, we find that there is no freedom of adding an arbitrary constant to the value of the chemical potential ${\ensuremath{\mu}}_{\mathrm{SM}}^{N}$, whose exact value is generally not the popular preference of the negative of Mulliken's electronegativity, $\ensuremath{-}\frac{1}{2}(I+A)$, where
I and
A are the first ionization potential and the first electron affinity, respectively. In addition, for any external potential converging to the same constant at infinity in all directions, we resolve that ${\ensuremath{\mu}}_{\mathrm{SM}}^{N}=\ensuremath{-}I$. Finally, the equality ${\ensuremath{\mu}}_{\mathrm{DFT}}={\ensuremath{\mu}}_{\mathrm{SM}}^{n}$ is rigorously derived via an alternative route, where ${\ensuremath{\mu}}_{\mathrm{DFT}}$ is the Lagrangian multiplier used to constrain the normalization of the density in the traditional DFT approach.