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Communications in Theoretical Physics

Properties of Parity Non-conserved and Parity Conserved States in Matrix Product Systems

Jing-Min Zhu

Published Aug 1, 2010• 11 citations
Publisher

Abstract

In terms of reflection transformation of a matrix product state (MPS), the parity of the MPS is defined. Based on the reflective parity non-conserved MPS pair we construct the even-parity state |Φ e ⟩ and the odd-parity state |Φ o ⟩. It is interesting to find that the parity non-conserved reflective MPS pair have no long-range correlations; instead the even-parity state |Φ e ⟩ and the odd-parity state |Φ o ⟩ constructed from them have the same long-range correlations for the parity non-conserved block operators. Moreover, the entanglement between a block of n contiguous spins and the rest of the spin chain for the states |Φ e ⟩ and |Φ o ⟩ is larger than that for the reflective MPS pair except for n = 1, and the difference of them approaches 1 monotonically and asymptotically from 0 as n increases from 1. These characteristics indicate that MPS parity as a conserved physical quantity represents a kind of coherent collective quantum mode, and that the parity conserved MPSs contain more correlation, coherence, and entanglement than the parity non-conserved ones.

DOI
10.1088/0253-6102/54/2/31

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In terms of reflection transformation of a matrix product state (MPS), the parity of the MPS is defined. Based on the reflective parity non-conserved MPS pair we construct the even-parity state |Φ e ⟩ and the odd-parity state |Φ o ⟩. It is interesting to find that the parity non-conserved reflective MPS pair have no long-range correlations; instead the even-parity state |Φ e ⟩ and the odd-parity state |Φ o ⟩ constructed from them have the same long-range correlations for the parity non-conserved block operators. Moreover, the entanglement between a block of n contiguous spins and the rest of the spin chain for the states |Φ e ⟩ and |Φ o ⟩ is larger than that for the reflective MPS pair except for n = 1, and the difference of them approaches 1 monotonically and asymptotically from 0 as n increases from 1. These characteristics indicate that MPS parity as a conserved physical quantity represents a kind of coherent collective quantum mode, and that the parity conserved MPSs contain more correlation, coherence, and entanglement than the parity non-conserved ones.
In terms of reflection transformation of a matrix product state (MPS),the parity of the MPS is defined. Based on the reflective parity non-conserved MPS pair we construct the even-parity state |Ψ_e and the odd-parity state |Ψ_o.It is interesting to find that the parity non-conserved reflective MPS pair have no long-range correlations;instead the even-parity state |Ψ_e and the odd-parity state |Ψ_o constructed from them have the same long-range correlations for the parity non-conserved block operators.Moreover,the entanglement between a block of n contiguous spins and the rest of the spin chain for the states |Ψ_e and |Ψ_o is larger than that for the reflective MPS pair except for n = 1,and the difference of them approaches 1 monotonically and asymptotically from 0 as n increases from 1.These characteristics indicate that MPS parity as a conserved physical quantity represents a kind of coherent collective quantum mode,and that the parity conserved MPSs contain more correlation,coherence,and entanglement than the parity non-conserved ones.
Abstract For our proposed composite parity-conserved matrix product state (MPS), if only a spin block length is larger than 1, any two such spin blocks have correlation including classical correlation and quantum correlation. Both the total correlation and the classical correlation become larger than that in any subcomponent; while the quantum correlations of the two nearest-neighbor spin blocks and the two next-nearest-neighbor spin blocks become smaller and for other conditions the quantum correlation becomes larger, i.e., the increase or the production of the long-range quantum correlation is at the cost of reducing the short-range quantum correlation, which deserves to be investigated in the future; and the ration of the quantum correlation to the total correlation monotonically decreases to a steady value as the spacing spin length increasing.
We present a new general and much simpler scheme to construct various quantum phase transitions (QPTs) in spin chain systems with matrix product ground states. By use of the scheme we take into account one kind of matrix product state (MPS) QPT and provide a concrete model. We also study the properties of the concrete example and show that a kind of QPT appears, accompanied by the appearance of the discontinuity of the parity absent block physical observable, diverging correlation length only for the parity absent block operator, and other properties which are that the fixed point of the transition point is an isolated intermediate-coupling fixed point of renormalization flow and the entanglement entropy of a half-infinite chain is discontinuous.
According to our scheme to construct quantum phase transitions (QPTs) in spin chain systems with matrix product ground states, we first successfully combine matrix product state (MPS) QPTs with spontaneous symmetry breaking. For a concrete model, we take into account a kind of MPS QPTs accompanied by spontaneous parity breaking, though for either side of the critical point the GS is typically unique, and show that the kind of MPS QPTs occur only in the thermodynamic limit and are accompanied by the appearance of singularities, diverging correlation length, vanishing energy gap and the entanglement entropy of a half-infinite chain not only staying finite but also whose first derivative discontinuous.
The property of quantum many-body systems under spatial reflection and the relevant physics of the renormalization group (RG) procedure are revealed. By virtue of the matrix product state (MPS) representation, various attributes for translational invariant systems associated with spatial reflection are manifested. We demonstrate subsequently a conservation rule of the conjugative relation for reflectional MPS pairs under RG transformations and illustrate further the property of the fixed points of RG flows. Finally, we show that a similar rule exists with respect to the target states in the density matrix renormalization group algorithm.
For the matrix product system of a one-dimensional spin-½ chain, we present a new model of quantum phase transitions and find that in the thermodynamic limit, both sides of the critical point are respectively described by phases |Ψ a 〉 = |1...1〉 representing all particles spin up and |Ψ b 〉 = |0...0〉 representing all particles spin down, while the phase transition point is an isolated intermediate-coupling point where the two phases coexist equally, which is described by the so-called N -qubit maximally entangled GHZ state . At the critical point, the physical quantities including the entanglement are not discontinuous and the matrix product system has long-range correlation and N -qubit maximal entanglement. We believe that our work is helpful for having a comprehensive understanding of quantum phase transitions in matrix product states of one-dimensional spin chains and of potential directive significance to the preparation and control of one-dimensional spin lattice models with stable coherence and N -qubit maximal entanglement.