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arXiv:0911.2844 · Physical Review Letters

Phase Transitions in the Distribution of Bipartite Entanglement of a Random Pure State

Céline Nadal, Satya N. Majumdar, Massimo Vergassola

Published Mar 19, 2010• 115 citations
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Abstract

Using a Coulomb gas method, we compute analytically the probability distribution of the Renyi entropies (a standard measure of entanglement) for a random pure state of a large bipartite quantum system. We show that, for any order q>1 of the Renyi entropy, there are two critical values at which the entropy's probability distribution changes shape. These critical points correspond to two different transitions in the corresponding charge density of the Coulomb gas: the disappearance of an integrable singularity at the origin and the detachment of a single-charge drop from the continuum sea of all the other charges. These transitions, respectively, control the left and right tails of the entropy's probability distribution, as verified also by Monte Carlo numerical simulations of the Coulomb gas equilibrium dynamics.

DOI
10.1103/physrevlett.104.110501

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Using a Coulomb gas method, we compute analytically the probability distribution of the Renyi entropies (a standard measure of entanglement) for a random pure state of a large bipartite quantum system. We show that, for any order q>1 of the Renyi entropy, there are two critical values at which the entropy's probability distribution changes shape. These critical points correspond to two different transitions in the corresponding charge density of the Coulomb gas: the disappearance of an integrable singularity at the origin and the detachment of a single-charge drop from the continuum sea of all the other charges. These transitions, respectively, control the left and right tails of the entropy's probability distribution, as verified also by Monte Carlo numerical simulations of the Coulomb gas equilibrium dynamics.
analytically the statistics of the Renyi and von Neumann entropies (standard measures of entanglement), for a random pure state in a large bipartite quantum system. The full probability distribution is computed by first mapping the problem to a random matrix model and then using a Coulomb gas method. We identify three different regimes in the entropy distribution, which correspond to two phase transitions in the asso-ciated Coulomb gas. The two critical points correspond to sudden changes in the shape of the Coulomb charge density: the appearance of an integrable singularity at the origin for the first critical point, and the detachment of the rightmost charge (largest eigenvalue) from the sea of the other charges at the second critical point. Analytical results are verified by Monte Carlo numerical simulations. A short account of part of these results appeared recently in Nadal et al. (Phys. Rev. Lett. 104:110501, 2010).
We study the probability distribution of entanglement in the quantum symmetric simple exclusion process, a model of fermions hopping with random Brownian amplitudes between neighboring sites. We consider a protocol where the system is initialized in a pure product state of M particles, and we focus on the late-time distribution of Rényi-q entropies for a subsystem of size ℓ. By means of a Coulomb gas approach from random matrix theory, we compute analytically the large-deviation function of the entropy in the thermodynamic limit. For q>1, we show that, depending on the value of the ratio ℓ/M, the entropy distribution displays either two or three distinct regimes, ranging from low to high entanglement. These are connected by points where the probability density features singularities in its third derivative, which can be understood in terms of a transition in the corresponding charge density of the Coulomb gas. Our analytic results are supported by numerical Monte Carlo simulations.
The probability of large deviations of the smallest Schmidt eigenvalue for random pure states of bipartite systems, denoted as A and B, is computed analytically using a Coulomb gas method. It is shown that this probability, for large N, goes as exp[-βN^{2}Φ(ζ)], where the parameter β is the Dyson index of the ensemble, ζ is the large deviation parameter, while the rate function Φ(ζ) is calculated exactly. Corresponding equilibrium Coulomb charge density is derived for its large deviations. Effects of the large deviations of the extreme (largest and smallest) Schmidt eigenvalues on the bipartite entanglement are studied using the von Neumann entropy. Effect of these deviations is also studied on the entanglement between subsystems 1 and 2, obtained by further partitioning the subsystem A, using the properties of the density matrix's partial transpose ρ_{12}^{Γ}. The density of states of ρ_{12}^{Γ} is found to be close to the Wigner's semicircle law with these large deviations. The entanglement properties are captured very well by a simple random matrix model for the partial transpose. The model predicts the entanglement transition across a critical large deviation parameter ζ. Log negativity is used to quantify the entanglement between subsystems 1 and 2. Analytical formulas for it are derived using the simple model. Numerical simulations are in excellent agreement with the analytical results.
The R\'enyi-Shannon entropy allows extraction of some universal information about many-body wave functions. For a critical spin chain with central charge c=1c=1c=1, we show that it exhibits a phase transition at some value nc{n}_{c}nc​ of the R\'enyi parameter nnn which depends on the Luttinger parameter RRR. A replica-free formulation establishes a connection to boundary entropies in conformal field theory and reveals that the transition is triggered by a vertex operator which becomes relevant at the boundary. Our numerical results (XXZ and ${J}_{1}\ensuremath{-}{J}_{2}$ spin chains) match the continuum limit prediction, confirming its universal character. The replica approach used in previous works turns out to be correct only for n<ncn<{n}_{c}n<nc​. From the point of view of two-dimensional Rokhsar-Kivelson states, this transition reveals a singularity in the entanglement spectra.
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.