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arXiv:cond-mat/0612111 · Physical Review A

Quantum critical point and entanglement in a matrix-product ground state

Amit Tribedi, Indrani Bose

Published Apr 3, 2007• 15 citations
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Abstract

In this paper, we study the entanglement properties of a spin-1 model, the exact ground state of which is given by a matrix product (MP) state. The model exhibits a critical point transition at a parameter value $a=0$. The longitudinal and transverse correlation lengths are known to diverge as $a\ensuremath{\rightarrow}0$. We use three different entanglement measures $S(i)$ (the one-site von Neumann entropy), $S(i,j)$ (the two-body entanglement), and $G(2,n)$ (the generalized global entanglement) to determine the entanglement content of the MP ground state as the parameter $a$ is varied. The entanglement length, associated with $S(i,j)$, is found to diverge in the vicinity of the quantum critical point $a=0$. The first derivative of the entanglement measure $E$ $[=S(i),S(i,j)]$ with respect to the parameter $a$ also diverges. The first derivative of $G(2,n)$ with respect to $a$ does not diverge as $a\ensuremath{\rightarrow}0$ but attains a maximum value at $a=0$. At the quantum critical point itself all three entanglement measures become zero. We further show that multipartite correlations are involved in the quantum phase transitions at $a=0$.

DOI
10.1103/physreva.75.042304

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In this paper, we study the entanglement properties of a spin-1 model, the exact ground state of which is given by a matrix product (MP) state. The model exhibits a critical point transition at a parameter value a=0a=0a=0. The longitudinal and transverse correlation lengths are known to diverge as a\ensuremath→0a\ensuremath{\rightarrow}0a\ensuremath→0. We use three different entanglement measures S(i)S(i)S(i) (the one-site von Neumann entropy), S(i,j)S(i,j)S(i,j) (the two-body entanglement), and G(2,n)G(2,n)G(2,n) (the generalized global entanglement) to determine the entanglement content of the MP ground state as the parameter aaa is varied. The entanglement length, associated with S(i,j)S(i,j)S(i,j), is found to diverge in the vicinity of the quantum critical point a=0a=0a=0. The first derivative of the entanglement measure EEE [=S(i),S(i,j)][=S(i),S(i,j)][=S(i),S(i,j)] with respect to the parameter aaa also diverges. The first derivative of G(2,n)G(2,n)G(2,n) with respect to aaa does not diverge as a\ensuremath→0a\ensuremath{\rightarrow}0a\ensuremath→0 but attains a maximum value at a=0a=0a=0. At the quantum critical point itself all three entanglement measures become zero. We further show that multipartite correlations are involved in the quantum phase transitions at a=0a=0a=0.
We consider a spin ladder model which is known to have matrix product states as exact ground states with spin liquid characteristics. The model has two critical-point transitions at the parameter values u=0u=0u=0 and $\ensuremath{\infty}$. We study the variation of entanglement and fidelity measures in the ground states as a function of uuu and specially look for signatures of quantum phase transitions at u=0u=0u=0 and $\ensuremath{\infty}$. The two different entanglement measures used are S(i)S(i)S(i) (the single-site von Neumann entropy) and S(i,j)S(i,j)S(i,j) (the two-body entanglement). At the quantum critical point (QCP) $u=\ensuremath{\infty}$, the entanglement measure EEE [=S(i),S(i,j)][=S(i),S(i,j)][=S(i),S(i,j)] vanishes but remains nonzero at the other QCP u=0u=0u=0. The first and second derivatives of EEE with respect to the parameter uuu and the entanglement length associated with S(i,j)S(i,j)S(i,j) are further calculated to identify special features, if any, near the QCPs. We further determine the GS fidelity FFF and a quantity ln∣D∣\text{ln}|D|ln∣D∣ related to the second derivative of FFF and show that these quantities calculated for finite-sized systems are good indicators of QPTs occurring in the infinite system.
We consider a one-parameter family of matrix-product states of spin-1 particles on a periodic chain and study in detail the entanglement properties of such a state. In particular, we calculate exactly the entanglement of one site with the rest of the chain, and the entanglement of two distant sites with each other, and show that the derivative of both these properties diverge when the parameter ggg of the states passes through a critical point. Such a point can be called a point of quantum phase transition, since at this point the character of the matrix-product state, which is the ground state of a Hamiltonian, changes discontinuously. We also study the finite size effects and show how the entanglement depends on the size of the chain. This later part is relevant to the field of quantum computation where the problem of initial state preparation in finite arrays of qubits or qutrits is important. It is also shown that the entanglement of two sites have scaling behavior near the critical point.
The entanglement in quantum XY spin chains of arbitrary length is investigated via the geometric measure of entanglement. The emergence of entanglement is explained intuitively from the perspective of perturbations. The model is solved exactly and the energy spectrum is determined and analyzed in particular for the lowest two levels for both finite and infinite systems. The overlaps for these two levels are calculated analytically for arbitrary number of spins. The entanglement is hence obtained by maximizing over a single parameter. The corresponding ground-state entanglement surface is then determined over the entire phase diagram, and its behavior can be used to delineate the boundaries in the phase diagram. For example, the field-derivative of the entanglement becomes singular along the critical line. The form of the divergence is derived analytically and it turns out to be dictated by the universality class controlling the quantum phase transition. The behavior of the entanglement near criticality can be understood via a scaling hypothesis, analogous to that for free energies. The entanglement density vanishes along the so-called disorder line in the phase diagram, the ground space is doubly degenerate and spanned by two product states. The entanglement for the superposition of the lowest two states is also calculated. The exact value of the entanglement depends on the specific form of superposition. However, in the thermodynamic limit the entanglement density turns out to be independent of the superposition. This proves that the entanglement density is insensitive to whether the ground state is chosen to be the spontaneously Z2Z_2Z2​ symmetry broken one or not. The finite-size scaling of entanglement at critical points is also investigated from two different view points. First, the maximum in the field-derivative of the entanglement density is computed and fitted to a logarithmic dependence of the system size, thereby deducing the correlation length exponent for the Ising class using only the behavior of entanglement. Second, the entanglement density itself is shown to possess a correction term inversely proportional to the system size, with the coefficient being universal (but with different values for the ground state and the first excited state, respectively).
A useful approach to characterize and identify quantum phase transitions lies in the concept of multipartite entanglement. In this paper, we consider well-known measures of multipartite (global) entanglement, i.e., average linear entropy of one-qubit and two-qubit reduced density matrices, in order to study topological quantum phase transition (TQPT) in the Kitaev Toric code Hamiltonian with a nonlinear perturbation. We provide an exact mapping from aforementioned measures in the above model to internal energy and energy-energy correlations in the classical Ising model. Accordingly, we find that the global entanglement shows a continuous and sharp transition from a maximum value in the topological phase to zero in the magnetized phase in a sense that its first-order derivative diverges at the transition point. In this regard, we conclude that not only can the global entanglement serve as a reasonable tool to probe quantum criticality at TQPTs, but it also can reveal the highly entangled nature of topological phases. Furthermore, we also introduce a conditional version of global entanglement which becomes maximum at the critical point. Therefore, regarding a general expectation that multipartite entanglement reaches maximum value at the critical point of quantum many-body systems, our result proposes that the conditional global entanglement can be a good measure of multipartite entanglement in TQPTs.
We derive a general relation between the nonanalyticities of the ground state energy and those of a subclass of the multipartite generalized global entanglement (GGE) measure defined by de Oliveira et al. [Phys. Rev. A 73, 010305(R) (2006)] for many-particle systems. We show that GGE signals both a critical point location and the order of a quantum phase transition (QPT). We also show that GGE allows us to study the relation between multipartite entanglement and QPTs, suggesting that multipartite but not bipartite entanglement is favored at the critical point. Finally, using GGE we were able, at a second-order QPT, to define a diverging entanglement length (EL) in terms of the usual correlation length. We exemplify this with the XY spin-1/2 chain and show that the EL is half the correlation length.
The entanglement and quantum correlation measures have been investigated for the ground state of a spin chain with a Kitaev-type exchange interactions on alternating bonds, along with a transverse magnetic field. There is a macroscopic degeneracy in the ground state for zero magnetic field, implying a quantum critical point. In this model, many entanglement and correlation measures, except the macroscopic multispecies entanglement measure, do not show true singular behavior in the vicinity of the critical point, as seen in the transverse-Ising model ground state and related models. We compute the pair concurrence measure of entanglement, the pair quantum discord to track the quantum correlations, and a global entanglement measure and a multispecies entanglement measure to investigate multiparty entanglement, both analytically and numerically. The nearest-neighbor concurrence shows a peak structure as a function of magnetic field, near the critical point, for various values of the ratio of interaction strengths, but its derivative does not show a singular behavior close to the critical point. A similar behavior is shown by the quantum discord and the global entanglement. The multispecies entanglement shows a clear signature of the critical point behavior, and the first-order derivative shows a divergence near zero magnetic field, for equal interaction strengths.