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arXiv:1207.2755 · Journal of Physics A Mathematical and Theoretical

Signature of special behaviours of 1/r2interaction in the quantum entanglement entropy

Poulomi Sadhukhan, Somendra M. Bhattacharjee

Published Oct 8, 2012• 7 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8113/45/42/425302

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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.
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.
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 ground-state phase diagram of a spin-1 Heisenberg chain with staggered long-range (LR) interactions decaying as ∝r−α\propto r^{-α}∝r−α using a quantum Monte Carlo approach based on the split-spin representation. This formulation enables efficient large-scale simulations by mapping the spin-1 model onto spin-1/21/21/2 degrees of freedom with local projection constraints. We resolve the continuous quantum phase transition between the gapped Haldane phase at large ααα (short-range regime) and a gapless antiferromagnetically ordered Néel phase at small ααα (LR regime), where the continuous SU(2) symmetry is broken. From finite-size scaling and crossing point analyses, we determine the critical point to be at αc=2.49(1)α_c = 2.49(1)αc​=2.49(1) and extract the associated critical exponents, which indicate unconventional criticality. In particular, the transition is found to be nonconformal, characterized by a dynamical exponent z≠1z \neq 1z=1. We further analyze the scaling of entanglement entropy and bipartite fluctuations across the transition, and determine the corresponding universal scalings in both phases and at criticality.
We investigate the scaling of the bipartite entanglement entropy across Lifshitz quantum phase transitions, where the topology of the Fermi surface changes without any changes in symmetry. We present both numerical and analytical results which show that Lifshitz transitions are characterized by a well-defined set of critical exponents for the entanglement entropy near the phase transition. In one dimension, we show that the entanglement entropy exhibits a length scale that diverges as the system approaches a Lifshitz critical point. In two dimensions, the leading and subleading coefficients of the scaling of entanglement entropy show distinct power-law singularities at critical points. The effect of weak interactions is investigated using the density matrix renormalization group algorithm.