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arXiv:quant-ph/0512031 · Physical Review A

Linking entanglement and quantum phase transitions via density-functional theory

Lian-Ao Wu, M. S. Sarandy, Daniel A. Lidar, Lu Jeu Sham

Published Nov 27, 2006• 111 citations
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Abstract

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.

DOI
10.1103/physreva.74.052335

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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.
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.
We develop a general theory of the relation between quantum phase transitions (QPTs) characterized by nonanalyticities in the energy and bipartite entanglement. We derive a functional relation between the matrix elements of two-particle reduced density matrices and the eigenvalues of general two-body Hamiltonians of d-level systems. The ground state energy eigenvalue and its derivatives, whose nonanalyticity characterizes a QPT, are directly tied to bipartite entanglement measures. We show that first-order QPTs are signaled by density matrix elements themselves and second-order QPTs by the first derivative of density matrix elements. Our general conclusions are illustrated via several quantum spin models.
The maximum von Neumann entropy principle subject to given constraints of mean values of some physical observables determines the density matrix. Similarly the stationary action principle in the case of time-dependent (dissipative) situations under similar constraints yields the density matrix. The free energy and measures of entanglement are expressed in terms of such a density matrix and thus define respective functionals of the mean values. In the light of several model calculations, it is found that the density matrix contains information about both quantum entanglement and phase transitions even though there may not be any direct relationship implied by one on the other.
In this work, we establish a general theory of phase transitions and quantum entanglement in the equilibrium state at arbitrary temperatures. First, we derive a set of universal functional relations between the matrix elements of a two-body reduced density matrix of the canonical density matrix and the Helmholtz free energy of the equilibrium state, which implies that the Helmholtz free energy and its derivatives are directly related to entanglement measures because any entanglement measures are defined as a function of the reduced density matrix. Then we show that the first-order phase transitions are signaled by the matrix elements of the reduced density matrix while the second-order phase transitions are witnessed by the first derivatives of the reduced density matrix elements. Near the second-order phase-transition point, we show that the first derivative of the reduced-density-matrix elements present universal scaling behaviors. Finally, we establish a theorem which connects the phase transitions and bipartite entanglement at arbitrary temperatures. Our general results are demonstrated in an experimentally relevant many-body spin model.
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.
In this chapter we discuss methods to calculate the entanglement of a system using density-functional theory. We firstly introduce density-functional theory and the local-density approximation (LDA). We then discuss the concept of the 'interacting LDA system'. This is characterised by an interacting many-body Hamiltonian which reproduces, uniquely and exactly, the ground state density obtained from the single-particle Kohn-Sham equations of density-functional theory when the local-density approximation is used. We motivate why this idea can be useful for appraising the local-density approximation in many-body physics particularly with regards to entanglement and related quantum information applications. Using an iterative scheme, we find the Hamiltonian characterising the interacting LDA system in relation to the test systems of Hooke's atom and helium-like atoms. The interacting LDA system ground state wavefunction is then used to calculate the spatial entanglement and the results are compared and contrasted with the exact entanglement for the two test systems. For Hooke's atom we also compare the entanglement to our previous estimates of an LDA entanglement. These were obtained using a combination of evolutionary algorithm and gradient descent, and using an LDA-based perturbative approach. We finally discuss if the position-space information entropy of the density—which can be obtained directly from the system density and hence easily from density-functional theory methods—can be considered as a proxy measure for the spatial entanglement for the test systems.