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Density Functional Theory and Information-Theoretic Diagnostics of Quantum Phase Transitions

E. Romera, Ágnes Nagy

Published Feb 1, 2026• 0 citations

Abstract

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.

DOI: 10.3390/e28020170

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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.

\n\nIt is shown that there is a linear relation between the relative Rényi entropy and the density functional fidelity susceptibility. The derivative of the relative Rényi entropy with respect to the Rényi parameter α is also proportional to the density functional fidelity susceptibility. The benefit of these relations is illustrated for the Dicke model (considering a numerical study and a variational analytical approximation). These relations are useful to detect the second-order QPT in this model.\n

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.

The fidelity, linear entropy and entanglement entropy, concepts of information theory, are used to determine the localization of quantum phase transitions of the Lipkin–Meshkov–Glick (LMG) and the Dicke models for a finite number of particles. Another concept, the so-called fidelity susceptibility, gives us information of the scaling behavior of the quantum phase transitions of the models. In particular universal curves are presented for the specific susceptibility. It is shown that the control parameters of both models have a similar dependence in the number of particles when the systems approach to the thermodynamic limit.

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.