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The universal behavior of the quantum correlation of Ising model in a transverse field

Meng Qin, Zhongzhou Ren, Xin Zhang

Published Sep 16, 2015• 0 citations
OpenAlex

Abstract

We study the ground state quantum correlation of Ising model in a transverse field by implementing the quantum renormalization group (QRG) theory. It is shown that various quantum correlation measures and the Clauser-Horne-Shimony-Holt inequality will highlight the critical point related with quantum phase transitions, and demonstrate nonanalytic phenomenon and scaling behavior when the size of the system becomes large. Our results also indicate a universal behavior of the critical exponent of Ising model under QRG that the critical exponent will not change with the variation of different quantities, even when the quantities vary from entanglement measures to quantum correlation measures. This means that the two kinds of quantum correlation criterion, including the entanglement/separability paradigm and the information-theoretic paradigm, have some connections between them. These remarkable behaviors may have important implications on condensed matter physics because the critical exponent directly associates with the correlation length exponent.

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We study the ground state quantum correlation of Ising model in a transverse field by implementing the quantum renormalization group (QRG) theory. It is shown that various quantum correlation measures and the Clauser-Horne-Shimony-Holt inequality will highlight the critical point related with quantum phase transitions, and demonstrate nonanalytic phenomenon and scaling behavior when the size of the system becomes large. Our results also indicate a universal behavior of the critical exponent of Ising model under QRG that the critical exponent will not change with the variation of different quantities, even when the quantities vary from entanglement measures to quantum correlation measures. This means that the two kinds of quantum correlation criterion, including the entanglement/separability paradigm and the information-theoretic paradigm, have some connections between them. These remarkable behaviors may have important implications on condensed matter physics because the critical exponent directly associates with the correlation length exponent.
We study the ground state quantum correlation of Ising model in a transverse field (ITF) by implementing the quantum renormalization group (QRG) theory. It is shown that various quantum correlation measures and the Clauser-Horne-Shimony-Holt inequality will highlight the critical point related with quantum phase transitions, and demonstrate nonanalytic phenomena and scaling behavior when the size of the systems becomes large. Our results also indicate a universal behavior of the critical exponent of ITF under QRG theory that the critical exponent of different measures is identical, even when the quantities vary from entanglement measures to quantum correlation measures. This means that the two kinds of quantum correlation criterion including the entanglement-separability paradigm and the information-theoretic paradigm have some connections between them. These remarkable behaviors may have important implications on condensed matter physics because the critical exponent directly associates with the correlation length exponent.
We investigate the behavior of quantum coherence of the ground states of 2D Heisenberg XY model and 2D Ising model with transverse field on square lattices, by using the method of Quantum Renormalization Group (QRG). We show that the non-analytic behavior of quantum coherence near the critical point, can detect quantum phase transition (QPT) of these models. We also use the scaling behavior of maximum derivative of quantum coherence, with system size, to find the critical exponent of coherence for both models and also the length exponent of the Ising model. The results are in close agreement with the ones obtained from entanglement analysis, that is while quantum coherence needs less computational calculations in comparison to entanglement approaches.
We analyze the fidelity per lattice site for two different ground states of the one-dimensional quantum Ising model in a transverse field near the critical point. It is found that, in the thermodynamic limit, the fidelity per lattice site is singular, and the derivative of its logarithmic function with respect to the transverse field strength is logarithmically divergent at the critical point. The scaling behavior is confirmed numerically by performing a finite-size scaling analysis for systems of different sizes, consistent with the conformal invariance at the critical point. This allows us to extract the correlation length critical exponent, which turns out to be universal in the sense that the correlation length critical exponent does not depend on either the anisotropic parameter or the transverse field strength.
Abstract We investigate the quantum coherence behavior of the ground states of 2D Heisenberg XY model and 2D Ising model with a transverse field on square lattices using the Quantum Renormalization Group (QRG) method. Our analysis focused on the ground state density matrix and its marginal states, revealing non-analytic behavior of quantum coherence (especially two-site coherence) near the critical point. This behavior allowed us to detect quantum phase transitions (QPT) in these models. By examining the scaling behavior of the maximum derivative of quantum coherence with system size, we determined the critical exponent of coherence for both models and the length exponent of the Ising model. Additionally, we investigated the time evolution of coherence in both models. Our results closely align with those obtained from entanglement analysis, that is while quantum coherence requires fewer computational calculations compared to discord and entanglement approaches.
In this paper we consider the quantum phase transition in the Ising model in the presence of a transverse field in one, two, and three dimensions from a multipartite entanglement point of view. Using exact numerical solutions, we are able to study such systems up to 25 qubits. The Meyer-Wallach measure of global entanglement is used to study the critical behavior of this model. The transition we consider is between a symmetric Greenberger-Horne-Zeilinger--like state to a paramagnetic product state. We find that global entanglement serves as a good indicator of quantum phase transition with interesting scaling behavior. We use finite-size scaling to extract the critical point as well as some critical exponents for the one- and two-dimensional models. Our results indicate that such multipartite measure of global entanglement shows universal features regardless of dimension ddd. Our results also provide evidence that multipartite entanglement is better suited for the study of quantum phase transitions than the much-studied bipartite measures.
We use the quantum renormalization-group (QRG) method to study the entanglement and quantum phase transition (QPT) in the one-dimensional spin-1/2 Heisenberg—Ising model [Lieb E, Schultz T and Mattis D 1961 Ann. Phys. (N.Y.) 16 407]. We find the quantum phase boundary of this model by investigating the evolution of concurrence in terms of QRG iterations. We also investigate the scaling behavior of the system close to the quantum critical point, which shows that the minimum value of the first derivative of concurrence and the position of the minimum scale with an exponent of the system size. Also, the first derivative of concurrence between two blocks diverges at the quantum critical point, which is directly associated with the divergence of the correlation length.