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arXiv:0710.1687 · Journal of Physics A Mathematical and Theoretical

Statistically interacting quasiparticles in Ising chains

Ping Lu, Jared Vanasse, Christopher G. Piecuch, Michael Karbach, Gerhard H. Muller

Published Jun 11, 2008• 14 citations
arXivHTMLPDF

Abstract

The exclusion statistics of two complementary sets of quasiparticles, generated from opposite ends of the spectrum, are identified for Ising chains with spin s=1/2,1. In the s=1/2 case the two sets are antiferromagnetic domain walls (solitons) and ferromagnetic domains (strings). In the s=1 case they are soliton pairs and nested strings, respectively. The Ising model is equivalent to a system of two species of solitons for s=1/2 and to a system of six species of soliton pairs for s=1. Solitons exist on single bonds but soliton pairs may be spread across many bonds. The thermodynamics of a system of domains spanning up to $M$ lattice sites is amenable to exact analysis and shown to become equivalent, in the limit M -> infinity, to the thermodynamics of the s=1/2 Ising chain. A relation is presented between the solitons in the Ising limit and the spinons in the XX limit of the s=1/2 XXZ chain.

DOI
10.1088/1751-8113/41/26/265003

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The exclusion statistics of two complementary sets of quasiparticles, generated from opposite ends of the spectrum, are identified for Ising chains with spin s=1/2,1. In the s=1/2 case the two sets are antiferromagnetic domain walls (solitons) and ferromagnetic domains (strings). In the s=1 case they are soliton pairs and nested strings, respectively. The Ising model is equivalent to a system of two species of solitons for s=1/2 and to a system of six species of soliton pairs for s=1. Solitons exist on single bonds but soliton pairs may be spread across many bonds. The thermodynamics of a system of domains spanning up to MMM lattice sites is amenable to exact analysis and shown to become equivalent, in the limit M -> infinity, to the thermodynamics of the s=1/2 Ising chain. A relation is presented between the solitons in the Ising limit and the spinons in the XX limit of the s=1/2 XXZ chain.
The quasiparticle composition of the eigenstates of quantum spin chains with nearest-neighbor exchange coupling subject to strong axial anisotropy is the focus of this dissertation. Complementary sets of quasiparticles are generated from opposite ends of the spectrum for the systematic generation of all eigenstates. The exclusion statistics of all quasiparticles thus introduced are determined exactly and used, in selected cases, for the exact thermodynamic analysis of the underlying spin chain model. This project starts from the Ising model with spin s = 1/2 and then branches out to the spin s = 1/2 XXZ model in the axial regime and to the spin s = 1 Ising model. In the s = 1/2 Ising case, the two complementary sets of quasiparticles are antiferromagnetic domain walls (solitons) and ferromagnetic domains (strings). In the s = 1 Ising case, they are soliton pairs and nested strings, respectively. The string quasiparticles of the s = 1/2 Ising chains are identified as limiting cases of Bethe ansatz solutions of the axial XXZ model. The thermodynamics of the Ising chain for s 1/2 is shown to be equivalent to that of a system of two species of solitons and for s = 1 to that of a system of six species of soliton pairs. Solitons exist on single bonds but soliton pairs may be spread across many bonds. The thermodynamics of a system of domains spanning up to M lattice sites is amenable to exact analysis and shown to become equivalent, in the limit M → ∞, to the thermodynamics of the s = 1/2 Ising chain. A relation is presented between the solitons in the Ising limit and the spinons in the XX limit of the s = 1/2 XXZ chain. The exclusion statistics is semionic for both spinons and solitons but the spinon vacuum is unique whereas the soliton vacuum is twofold with broken symmetry.
The coordinate Bethe ansatz solutions of the XXZ model for a one-dimensional spin-1/2 chain are analyzed with focus on the statistical properties of the constituent quasiparticles.Emphasis is given to the special cases known as XX, XXX, and Ising models, where considerable simplifications occur.The XXZ spectrum can be generated from separate pseudovacua as configurations of sets of quasiparticles with different exclusion statistics.These sets are complementary in the sense that the pseudovacuum of one set contains the maximum number of particles from the other set.The Bethe ansatz string solutions of the XXX model evolve differently in the planar and axial regimes.In the Ising limit they become ferromagnetic domains with integer-valued exclusion statistics.In the XX limit they brake apart into hard-core bosons with (effectively) fermionic statistics.Two sets of quasiparticles with spin 1/2 and fractional statistics are distinguished, where one set (spinons) generates the XXZ spectrum from the unique, critical ground state realized in the planar regime, and the other set (solitons) generates the same spectrum from the twofold, antiferromagnetically ordered ground state realized in the axial regime.In the Ising limit, the solitons become antiferromagnetic domain walls.
The coordinate Bethe ansatz solutions of the XXZ model for a one-dimensional spin-1/2 chain are analyzed with focus on the statistical properties of the constituent quasiparticles. Emphasis is given to the special cases known as XX, XXX, and Ising models, where considerable simplifications occur. The XXZ spectrum can be generated from separate pseudovacua as configurations of sets of quasiparticles with different exclusion statistics. These sets are complementary in the sense that the pseudovacuum of one set contains the maximum number of particles from the other set. The Bethe ansatz string solutions of the XXX model evolve differently in the planar and axial regimes. In the Ising limit they become ferromagnetic domains with integer-valued exclusion statistics. In the XX limit they brake apart into hard-core bosons with (effectively) fermionic statistics. Two sets of quasiparticles with spin 1/2 and fractional statistics are distinguished, where one set (spinons) generates the XXZ spectrum from the unique, critical ground state realized in the planar regime, and the other set (solitons) generates the same spectrum from the twofold, antiferromagnetically ordered ground state realized in the axial regime. In the Ising limit, the solitons become antiferromagnetic domain walls.
The mapping between the fermion and spinon compositions of eigenstates in the one-dimensional spin-1/2 XX model on a lattice with N sites is used to describe the spinon interaction from two different perspectives: (i) For finite N the energy of all eigenstates is expressed as a function of spinon momenta and spinon spins, which, in turn, are solutions of a set of Bethe ansatz equations. The latter are the basis of an exact thermodynamic analysis in the spinon representation of the XX model. (ii) For N -> infinity the energy per site of spinon configurations involving any number of spinon orbitals is expressed as a function of reduced variables representing momentum, filling, and magnetization of each orbital. The spins of spinons in a single orbital are found to be coupled in a manner well described by an Ising-like equivalent-neighbor interaction, switching from ferromagnetic to antiferromagnetic as the filling exceeds a critical level. Comparisons are made with results for the Haldane-Shastry model.
Chapter 1: The statistical mechanics of particles with shapes on a one-dimensional lattice is investigated in the context of the s = 1 Ising chain with uniform nearest-neighbor coupling, quadratic single-site potential, and magnetic field, which supports four distinct ground states: : ↑↓↑↓ ···⟩, : ∘ ∘ ···⟩, : ↑↑ ··· ⟩, : ↑ ∘ ↑ ∘ ··· ⟩. The complete spectrum is generated from each ground state by particles from a different set of six or seven species. Particles and elements of pseudo-vacuum are characterized by motifs (patterns of several consecutive site variables). Particles are floating objects that can be placed into open slots on the lattice. Open slots are recognized as permissible links between motifs. The energy of a particle varies between species but is independent of where it is placed. Placement of one particle changes the open-slot configuration for particles of all species. This statistical interaction is encoded in a generalized Pauli principle, from which the multiplicity of states for a given particle combination is determined and used for the exact statistical mechanical analysis. Particles from all species belong to one of four categories: compacts, hosts, tags, or hybrids. Compacts and hosts find open slots in segments of pseudo-vacuum. Tags find open slots inside hosts. Hybrids are tags with hosting capability. In the taxonomy of particles proposed here, 'species' is indicative of structure and 'category' indicative of function. The hosting function splits the Pauli principle into exclusion and accommodation parts. Near phase boundaries, the state of the Ising chain at low temperature is akin to that of miscible or immiscible liquids with particles from one species acting as surfactant molecules. Chapter 2: The s = 1/2 Ising chain with uniform nearest-neighbor and next-nearest-neighbor coupling is used to construct a system of floating particles characterized by motifs of up to six consecutive local spins. The spin couplings cause the assembly of particles which, in turn, remain free of interaction energies even at high density. All microstates are configurations of particles from one of three different sets, excited from pseudo-vacua associated with ground states of periodicities one, two, and four. The motifs of particles and elements of pseudo-vacuum interlink in two shared site variables. The statistical interaction between particles is encoded in a generalized Pauli principle, describing how the placement of one particle modifies the options for placing further particles. In the statistical mechanical analysis arbitrary energies can be assigned to all particle species. The entropy is a function of the particle populations. The statistical interaction specifications are transparently built into that expression. The energies and structures of the particles alone govern the ordering at low temperature. Under special circumstances the particles can be replaced by more fundamental particles with shorter motifs that interlink in only one shared site variable. Structures emerge from interactions on two levels: particles with shapes from coupled spins and long-range ordering tendencies from statistically interacting particles with shapes. Chapter 3: The s = [special characters omitted] Ising spin chain with uniform nearest-neighbor coupling, quadratic single-site potential, and magnetic field is shown to be equivalent to a system of 17 species of particles with internal structure. The same set of particles (with different energies) is shown to generate the spectrum of the s = ½ Ising chain with dimerized nearest-neighbor coupling. The particles are free of interaction energies even at high densities. The mutual exclusion statistics of particles from all species is determined by their internal structure and encoded in a generalized Pauli principle. The exact statistical mechanical analysis can be performed for thermodynamically open or closed systems and with arbitrary energies assigned to all particle species. Special circumstances make it possible to merge two or more species into a single species. All traits that distinguish the original species become ignorable. The particles from the merged species are effectively indistinguishable and obey modified exclusion statistics. Different mergers may yield the same end-product, implying that the inverse process (splitting any species into subspecies) is not unique. In a macroscopic system of two merged species at thermal equilibrium, the concentrations of the original species satisfy a functional relation governed by their mutual statistical interaction. That relation is derivable from an extremum principle. In the Ising context the system is open and the particle energies depend on the Hamiltonian parameters. Simple models of polymerization and solitonic paramagnetism each represent a closed system of two species that can transform into each other. Here they represent distinguishable traits with different energies of the same physical particle.
The statistical mechanics of particles with shapes on a one-dimensional lattice is investigated in the context of the s=1s=1s=1 Ising chain with uniform nearest-neighbor coupling, quadratic single-site potential, and a magnetic field, which supports four distinct ground states: $|\ensuremath{\uparrow}\ensuremath{\downarrow}\ensuremath{\uparrow}\ensuremath{\downarrow}\ensuremath{\cdots}\ensuremath{\rangle}$, $|\ensuremath{\circ}\ensuremath{\circ}\ensuremath{\cdots}\ensuremath{\rangle}$, $|\ensuremath{\uparrow}\ensuremath{\uparrow}\ensuremath{\cdots}\ensuremath{\rangle}$, $|\ensuremath{\uparrow}\ensuremath{\circ}\ensuremath{\uparrow}\ensuremath{\circ}\ensuremath{\cdots}\ensuremath{\rangle}$. The complete spectrum is generated from each ground state by particles from a different set of six or seven species. Particles and elements of the pseudovacuum are characterized by motifs (patterns of several consecutive site variables). Particles are floating objects that can be placed into open slots on the lattice. Open slots are recognized as permissible links between motifs. The energy of a particle varies between species but is independent of where it is placed. Placement of one particle changes the open-slot configuration for particles of all species. This statistical interaction is encoded in a generalized Pauli principle, from which the multiplicity of states for a given particle combination is determined and used for the exact statistical mechanical analysis. Particles from all species belong to one of four categories: compacts, hosts, tags, or hybrids. Compacts and hosts find open slots in segments of pseudovacuum. Tags find open slots inside hosts. Hybrids are tags with hosting capability. In the taxonomy of particles proposed here, ``species'' is indicative of structure and ``category'' indicative of function. The hosting function splits the Pauli principle into exclusion and accommodation parts. Near phase boundaries, the state of the Ising chain at low temperature is akin to that of miscible or immiscible liquids with particles from one species acting as surfactant molecules.