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Physical Review B

Von Neumann entropy and bipartite number fluctuation in quantum phase transitions

Myung-Hoon Chung, David Paul Landau

Published Mar 25, 2011• 12 citations
Publisher

Abstract

In order to characterize quantum phase transitions, we consider von Neumann entanglement entropy of bipartition systems. We propose a bipartition such as even-odd in a one-dimensional system or checkerboard in a two-dimensional case. The entanglement entropy successfully indicates the critical point in the spinless fermion system where the metal-insulator transition is known to exist. Furthermore, we calculate the bipartite number fluctuation which shows a shoulder near the transition.

DOI
10.1103/physrevb.83.113104

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In order to characterize quantum phase transitions, we consider von Neumann entanglement entropy of bipartition systems. We propose a bipartition such as even-odd in a one-dimensional system or checkerboard in a two-dimensional case. The entanglement entropy successfully indicates the critical point in the spinless fermion system where the metal-insulator transition is known to exist. Furthermore, we calculate the bipartite number fluctuation which shows a shoulder near the transition.
We show that the concept of bipartite fluctuations F provides a very efficient tool to detect quantum phase transitions in strongly correlated systems. Using state-of-the-art numerical techniques complemented with analytical arguments, we investigate paradigmatic examples for both quantum spins and bosons. As compared to the von Neumann entanglement entropy, we observe that F allows us to find quantum critical points with much better accuracy in one dimension. We further demonstrate that F can be successfully applied to the detection of quantum criticality in higher dimensions with no prior knowledge of the universality class of the transition. Promising approaches to experimentally access fluctuations are discussed for quantum antiferromagnets and cold gases.
The von Neumann entanglement entropy is a useful measure to characterize a quantum phase transition. We investigate the nonanalyticity of this entropy at disorder-dominated quantum phase transitions in noninteracting electronic systems. At these critical points, the von Neumann entropy is determined by the single particle wave function intensity, which exhibits complex scale invariant fluctuations. We find that the concept of multifractality is naturally suited for studying von Neumann entropy of the critical wave functions. Our numerical simulations of the three dimensional Anderson localization transition and the integer quantum Hall plateau transition show that the entanglement at these transitions is well described using multifractal analysis.
The constituents of a quantum many-body system can be inextricably linked, a phenomenon known as quantum entanglement. Entanglement can be used as a resource for quantum computing, quantum communication and detecting phase transitions, among others. The amount of entanglement can be quantified via the von Neumann and Rényi entropies, which have their origins in information theory. In this work, the quantum entanglement between subsystems of a one dimen- sional lattice model of fermions is quantified. The von Neumann and Rényi entropies were calculated for two types of subsystems. In the first study, the subsystems were treated as two subsets of particles, and in the second, as two spatial subregions. Finally, by considering particle superselection rules, the amount of entanglement that can actually be accessed as a resource was calcu- lated. In all cases, the quantum entanglement served to detect phase transitions in the model.
Extensive body of work has shown that for the model of a non-interacting electron in a random potential there is a quantum critical point for dimensions greater than two — a metal–insulator transition. This model also plays an important role in the plateau-to-plateu transition in the integer quantum Hall effect, which is also correctly captured by a scaling theory. Yet, in neither of these cases the ground state energy shows any non-analyticity as a function of a suitable tuning parameter, typically considered to be a hallmark of a quantum phase transition, similar to the non-analyticity of the free energy in a classical phase transition. Here we show that von Neumann entropy (entanglement entropy) is non-analytic at these phase transitions and can track the fundamental changes in the internal correlations of the ground state wave function. In particular, it summarizes the spatially wildly fluctuating intensities of the wave function close to the criticality of the Anderson transition. It is likely that all quantum phase transitions can be similarly described.
Extensive body of work has shown that for the model of a non-interacting electron in a random potential there is a quantum critical point for dimensions greater than two — a metal–insulator transition. This model also plays an important role in the plateau-to-plateu transition in the integer quantum Hall effect, which is also correctly captured by a scaling theory. Yet, in neither of these cases the ground state energy shows any non-analyticity as a function of a suitable tuning parameter, typically considered to be a hallmark of a quantum phase transition, similar to the non-analyticity of the free energy in a classical phase transition. Here we show that von Neumann entropy (entanglement entropy) is non-analytic at these phase transitions and can track the fundamental changes in the internal correlations of the ground state wave function. In particular, it summarizes the spatially wildly fluctuating intensities of the wave function close to the criticality of the Anderson transition. It is likely that all quantum phase transitions can be similarly described.
We study the bipartite von Neumann entropy of two particles interacting via a long-range scale-free potential V ( r ) ∼ − g / r 2 in three dimensions, close to the unbinding transition. The nature of the quantum phase transition changes from critical (−3/4 < g < 1/4) to first order ( g < −3/4) with g = −3/4 as a multicritical point. Here, we show that the entanglement entropy has different behaviours for the critical and the first-order regimes. But there still exists an interesting multicritical scaling behaviour for all g ∈ ( − 2 < g < 1/4) controlled by the g = −3/4 case.