Scaling of Entanglement close to a Quantum Phase Transitions

TL;DR

It is demonstrated, for a class of one-dimensional magnetic systems, that entanglement shows scaling behaviour in the vicinity of the transition point, which connects the theory of critical phenomena with quantum information by exploring the entangling resources of a system close to its quantum critical point.

Abstract

In this Letter we discuss the entanglement near a quantum phase transition by analyzing the properties of the concurrence for a class of exactly solvable models in one dimension. We find that entanglement can be classified in the framework of scaling theory. Further, we reveal a profound difference between classical correlations and the non-local quantum correlation, entanglement: the correlation length diverges at the phase transition, whereas entanglement in general remains short ranged.

Figures (4)

• Figure 1: The change in the ground state wave-function in the critical region is analyzed considering $\partial_\lambda C(1)$ as a function of the reduced coupling strength $\lambda$. The curves correspond to different lattice sizes $N=11,41,101,251,401,\infty$. We choose $N$ odd to avoid the subtleties connected with boundary termsDMRG. On increasing the system size, the minimum gets more pronounced. also the position of the minimum changes and tends as $N^{-1.87}$ (left inset) towards the critical point $\lambda_c =1$ where for an infinite system a logarithmic divergence is present (see equation (\ref{['nn-concurrence']})). The right inset shows the behaviour of the concurrence $C(1)$ itself for an infinite system. The maximum that occurs below $\lambda_c$ is not related to the critical properties of the Ising model. As explained in the text, it is the change in the ground state and not the wavefunction itself that is a good indicator of the transition. The structure of the reduced density matrix, necessary to calculate the concurrence follows from the symmetry properties of the Hamiltonian. Reality and parity conservation of $H$ together with translational invariance fix the structure of $\rho$ to be real symmetric with $\rho_{11}$, $\rho_{22}=\rho_{33}$, $\rho_{23}$, $\rho_{14}$, $\rho_{44}$ as the only non-zero entries.
• Figure 2: The finite size scaling is performed for the case of logarithmic divergencesLIEB. The concurrence, considered as a function of the system size and the coupling constant, is a function of $N^{1/\nu}(\lambda-\lambda_m)$ only, and in the case of log divergence it behaves as $\partial_\lambda C(1)(N,\lambda)$ -- $\partial_\lambda C(1)(N,\lambda_0) \sim$$Q[N^{1/\nu}(\lambda-\lambda_m)]$ -- $Q[N^{1/\nu}(\lambda_0-\lambda_m)]$ where $\lambda_0$ is a non-critical value and $Q(x)\sim Q(\infty) \ln x$ (for large $x$). All the data from $N=41$ up to $N=2701$ collapse on a single curve. The critical exponent is $\nu=1$, as expected for the Ising model. The inset shows the divergence of the value at the minimum as the system size increases.
• Figure 3: As in the case of the nearest neighbour concurrence, data collapse is also obtained for the next-nearest-neighbour concurrence C(2). In the figure, data for system size from $N= 41$ to $N=401$ are plotted. The inset shows a peculiarity of the Ising model: $C(2)$ has its maximum precisely at the critical point for arbitrary system size (note that the maximum decreases as the system size increases). Therefore we consider the second derivative to perform the scaling analysis. It can also be seen that $C(2)$ is two orders of magnitude smaller than $C(1)$. For the smallest system sizes the concurrence is different from zero for $|i-j| = 3$ and $\lambda > 1.05$ (for $N=7$; for $N\ge 9$, $C(3)=0$ for all $\lambda$). In contrast the correlation functions are long-ranged at the critical point.
• Figure 4: The universality hypothesis for the entanglement is checked by considering the model Hamiltonian, defined in Eq.(\ref{['model']}), for a different value of $\gamma$. In this case we chose $\gamma=0.5$ and $N$ ranging from $41$ up to $401$. Data collapse, shown here for $C(1)$, is obtained for $\nu=1$, consistent with the model being in the universality class of the Ising model. In the inset is shown the divergence at the critical point for the infinite system.