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arXiv:0707.4223 · Europhysics Letters (EPL)

Spatial reflection and renormalization group flow of quantum many-body systems with matrix product state representation

Li-Xiang Cen, Z. D. Wang

Published May 1, 2008• 1 citation
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Abstract

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.

DOI
10.1209/0295-5075/82/40003

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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.
The property of quantum many-body systems under spatial reflection and the relevant physics of 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.
Matrix product density operator (MPDO) provides an efficient tensor network representation of mixed states on one-dimensional quantum many-body systems. We study a real-space renormalization group transformation of MPDOs represented by a circuit of local quantum channels. We require that the renormalization group flow is exact, in the sense that it exactly preserves the correlation between the coarse-grained sites and is therefore invertible by another circuit of local quantum channels. Unlike matrix product states (MPS), which always have a well-defined isometric renormalization transformation, we show that general MPDOs do not necessarily admit a converging exact renormalization group flow. We then introduce a subclass of MPDOs with a well-defined renormalization group flow, and show the structure of the MPDOs in the subclass as a representation of a pre-bialgebra structure. As a result, such MPDOs obey generalized symmetry represented by matrix product operator algebras associated with the pre-bialgebra. We also discuss implications with the classification of mixed-state quantum phases.
We characterize the conditions under which a translationally invariant matrix product state (MPS) is invariant under local transformations. This allows us to relate the symmetry group of a given state to the symmetry group of a simple tensor. We exploit this result in order to prove and extend a version of the Lieb-Schultz-Mattis theorem, one of the basic results in many-body physics, in the context of MPS. We illustrate the results with an exhaustive search of SU(2)-invariant two-body Hamiltonians which have such MPS as exact ground states or excitations.
The density-matrix renormalization group (DMRG) method has established itself over the last decade as the leading method for the simulation of the statics and dynamics of one-dimensional strongly correlated quantum lattice systems. The DMRG is a method that shares features of a renormalization group procedure (which here generates a flow in the space of reduced density operators) and of a variational method that operates on a highly interesting class of quantum states, so-called matrix product states (MPSs). The DMRG method is presented here entirely in the MPS language. While the DMRG generally fails in larger two-dimensional systems, the MPS picture suggests a straightforward generalization to higher dimensions in the framework of tensor network states. The resulting algorithms, however, suffer from difficulties absent in one dimension, apart from a much more unfavourable efficiency, such that their ultimate success remains far from clear at the moment.
The density-matrix renormalization group (DMRG) method has established itself over the last decade as the leading method for the simulation of the statics and dynamics of one-dimensional strongly correlated quantum lattice systems. The DMRG is a method that shares features of a renormalization group procedure (which here generates a flow in the space of reduced density operators) and of a variational method that operates on a highly interesting class of quantum states, so-called matrix product states (MPSs). The DMRG method is presented here entirely in the MPS language. While the DMRG generally fails in larger two-dimensional systems, the MPS picture suggests a straightforward generalization to higher dimensions in the framework of tensor network states. The resulting algorithms, however, suffer from difficulties absent in one dimension, apart from a much more unfavourable efficiency, such that their ultimate success remains far from clear at the moment.