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Corner transfer matrix derived variational methods in lattice statistical mechanics and quantum many-body systems

Chhenh, Chunhoa

Description

The concept of the corner transfer matrix (CTM) was first discovered by Baxter in 1968, when he derived a set of variational matrix equations, which for finite matrix sizes, could be numerically solved to obtain a sequence of approximations for the statistical mechanical properties of a system of monomers and dimers on a rectangular lattice [1]. It was not until 1978, however, in a seminal paper entitled "Variational Approximations for Square Lattice Models in Statistical Mechanics" [4], that...[Show more]

dc.contributor.authorChhenh, Chunhoa
dc.date.accessioned2018-11-22T00:04:53Z
dc.date.available2018-11-22T00:04:53Z
dc.date.copyright2011
dc.identifier.otherb3005816
dc.identifier.urihttp://hdl.handle.net/1885/150093
dc.description.abstractThe concept of the corner transfer matrix (CTM) was first discovered by Baxter in 1968, when he derived a set of variational matrix equations, which for finite matrix sizes, could be numerically solved to obtain a sequence of approximations for the statistical mechanical properties of a system of monomers and dimers on a rectangular lattice [1]. It was not until 1978, however, in a seminal paper entitled "Variational Approximations for Square Lattice Models in Statistical Mechanics" [4], that Baxter outlined his CTM variational method, which brought to light the potential power of the former objects to obtain numerics and series expansions for unsolved models and to calculate the order parameter of solved ones. Subsequent numerical work led to the realisation that the method, though general, was not very efficient; and increasing efficiency required making model specific modifications, which restricted the transferability of the resulting algorithm to other models. The CTM variational method was thus not widely adopted, despite holding much promise. More recently, however, Nishino and Okunishi discovered that White's density matrix renormalisation group (DMRG) algorithm [55, 56] could be efficiently extended to study two-dimensional classical lattice models, if the density matrix was approximated by Baxters CTMs. Numerical tests of their algorithm, the corner transfer matrix renormalisation group (CTMRG) method, were met with much success. Notable among these is its implementation within the infinite projected entangled-pair states (iPEPS) algorithm of Orus and Vidal to simulate the ground state of infinite two-dimensional quantum systems. In this thesis we review CTM derived variational methods: Baxter's original CTM iterative method, and developments of the CTM concept within the DMRG algorithm by Nishino and Okunishi, which led to the CTMRG method to calculate critical phenomena of classical and quantum systems. This will begin with elucidating the theoretical formalisms of both methods in two dimensions and their generalisations to three dimensions; followed by review of important application, namely, within the iPEPS algorithm of Orus and Vidal to numerically study infinite planar quantum systems.
dc.format.extentx, 86 leaves.
dc.language.isoen_AU
dc.rightsAuthor retains copyright
dc.subject.lccQC176.8.L3 C54 2011
dc.subject.lcshStatistical mechanics
dc.subject.lcshLattice dynamics
dc.titleCorner transfer matrix derived variational methods in lattice statistical mechanics and quantum many-body systems
dc.typeThesis (PhD)
local.description.notesThesis (Ph.D.)--Australian National University
dc.date.issued2011
local.type.statusAccepted Version
local.contributor.affiliationAustralian National University.
local.identifier.doi10.25911/5d611f3b11174
dc.date.updated2018-11-20T04:31:54Z
dcterms.accessRightsOpen Access
local.mintdoimint
CollectionsOpen Access Theses

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