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arXiv:hep-th/9401094 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Discrete phase transitions associated with topological lattice field theories inD≥2 dimensions

Naoki Sasakura

Published Jan 15, 1995• 1 citation
arXivHTMLPDF

Abstract

We investigate the neighborhood of topological lattice field theories (TLFT's) in the parameter space of general lattice field theories in D\ensuremath{\ge}2 dimensions, and discuss the phase structures associated with them. We first discuss the decomposition of a TLFT into a direct sum of irreducible TLFT's, which cannot be decomposed anymore. Using this decomposed form, we discuss the phase structure and the renormalization group flow near a TLFT in a certain restricted parameter space. We find that a TLFT is on a multiple first-order phase transition point as well as on the fixed point of the flow. The phase structure is controlled by the physical states on a (D-1)-sphere of the TLFT. The flow agrees with the Nienhuis-Nauenberg criterion. We also discuss the neighborhood of a TLFT in general directions by a perturbative method, the so-called cluster expansion. We investigate especially the ${\mathit{Z}}_{\mathit{p}}$ analogue of the Turaev-Viro model, and find that the TLFT is in general on a higher order discrete phase transition point. The phase structures depend on the topology of the base manifold and are controlled by the physical states on topologically nontrivial surfaces. We also discuss the correlation lengths of local fluctuations, and find some long-range modes propagating along topological defects. Thus various discrete phase transitions are associated with TLFT's.

DOI
10.1103/physrevd.51.703

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We investigate the neighborhood of topological lattice field theories (TLFT's) in the parameter space of general lattice field theories in D\ensuremath{\ge}2 dimensions, and discuss the phase structures associated with them. We first discuss the decomposition of a TLFT into a direct sum of irreducible TLFT's, which cannot be decomposed anymore. Using this decomposed form, we discuss the phase structure and the renormalization group flow near a TLFT in a certain restricted parameter space. We find that a TLFT is on a multiple first-order phase transition point as well as on the fixed point of the flow. The phase structure is controlled by the physical states on a (D-1)-sphere of the TLFT. The flow agrees with the Nienhuis-Nauenberg criterion. We also discuss the neighborhood of a TLFT in general directions by a perturbative method, the so-called cluster expansion. We investigate especially the Zp{\mathit{Z}}_{\mathit{p}}Zp​ analogue of the Turaev-Viro model, and find that the TLFT is in general on a higher order discrete phase transition point. The phase structures depend on the topology of the base manifold and are controlled by the physical states on topologically nontrivial surfaces. We also discuss the correlation lengths of local fluctuations, and find some long-range modes propagating along topological defects. Thus various discrete phase transitions are associated with TLFT's.
We consider fixed-point models for topological phases of matter formulated as discrete path integrals in the language of tensor networks. Such zero-correlation length models with an exact notion of topological invariance are known in the mathematical community as state-sum constructions or lattice topological quantum field theories. All of the established ansatzes for fixed-point models imply the existence of a gapped boundary as well as a commuting-projector Hamiltonian. Thus, they fail to capture topological phases without a gapped boundary or commuting-projector Hamiltonian, most notably chiral topological phases in 2+12+12+1 dimensions. In this paper, we present a more general fixed-point ansatz not affected by the aforementioned restrictions. Thus, our formalism opens up a possible way forward towards a microscopic fixed-point description of chiral phases and we present several strategies that may lead to concrete examples. Furthermore, we argue that our more general ansatz constitutes a universal form of topological fixed-point models, whereas established ansatzes are universal only for fixed-points of phases, which admit topological boundaries.
I discuss aspects of higher-order topological field theory of crystalline insulators with no other symmetries. I show how the topology and geometry of the crystalline lattice is organized in terms of so-called elasticity tetrads which are ground state degrees of freedom labelling translational lattice topological charges, higher-form conservation laws and responses on sub-dimensional manifolds of the bulk insulator. The quasitopological responses obtained in this way depend on the lattice and its embedding in space, as expected for weak topology. In a topological crystalline insulator, they classify higher-order responses and global symmetries in a transparent fashion in generic dimensions. This hierarchy coincides with the dimensional hierarchy of topological terms, the multipole expansion, and anomaly inflow, related to a mixed number of elasticity tetrads and electromagnetic gauge fields. In the continuum limit of the elasticity tetrads, the semi-classical expansion in momentum space can be used to derive the higher-order or subdimensional topological responses to local U(1) symmetries, such as electromagnetic gauge fields, with explicit formulas for the higher-order quasi-topological invariants in terms of the elasticity tetrads and Green’s functions. The topological responses in arbitrary dimensions are readily generalized to parameter space to allow for e.g. multipole pumping. The simple results further bridge the recently appreciated connections between topological field theory, higher-form symmetries and gauge fields and their relation to fractonic excitations and topological defects with restricted mobility in the elasticity of crystalline insulators.
Topological defects in two-dimensional conformal field theories (CFTs) are canonical examples of generalized symmetries in quantum field theories. They not only encode key features of the underlying theory, but their study has also enhanced our understanding of renormalization group (RG) flows in theories that contain them. Lattice models exist which discretize the states, energy-momentum tensor, and primary fields of many 2D CFTs, raising the natural question of whether topological defects can be similarly discretized. In this thesis, we demonstrate that in many cases all topological defects of minimal model CFTs can be discretized, enabling analytical and ab-initio numerical computations of their properties. In some of these constructions, we make essential use of the integrable structure of the underlying lattice models. The lattice realization of these operators also permits the study of defect RG flows between CFTs with different topological defects inserted. These topological defects and defect RG flows are realized on the lattice not only for Hermitian but also for non-Hermitian lattice Hamiltonians. Studying topological defects in minimal model CFTs on the lattice has also opened the door to a deeper understanding of topological defects in CFTs where they remain poorly understood.
Higher gauge theory is a higher order version of gauge theory that makes possible the definition of 2-dimensional holonomy along surfaces embedded in a manifold where a gauge 2-connection is present. In this paper, we study Hamiltonian models for discrete higher gauge theory on a lattice decomposition of a manifold. We show that a construction for higher lattice gauge theory is well-defined, including in particular a Hamiltonian for topological phases of matter in 3+1 dimensions. Our construction builds upon the Kitaev quantum double model, replacing the finite gauge connection with a finite gauge 2-group 2-connection. Our Hamiltonian higher lattice gauge theory model is defined on spatial manifolds of arbitrary dimension presented by slightly combinatorialized CW-decompositions (2-lattice decompositions), whose 1-cells and 2-cells carry discrete 1-dimensional and 2-dimensional holonomy data. We prove that the ground-state degeneracy of Hamiltonian higher lattice gauge theory is a topological invariant of manifolds, coinciding with the number of homotopy classes of maps from the manifold to the classifying space of the underlying gauge 2-group.\n\n\n\nThe operators of our Hamiltonian model are closely related to discrete 2-dimensional holonomy operators for discretized 2-connections on manifolds with a 2-lattice decomposition. We therefore address the definition of discrete 2-dimensional holonomy for surfaces embedded in 2-lattices. Several results concerning the well-definedness of discrete 2-dimensional holonomy, and its construction in a combinatorial and algebraic topological setting are presented.
Topological defects play a fundamental role in the investigation of symmetries in quantum field theories. For conformal field theories in two space-time dimensions, it is possible to construct these defects using lattice models allowing ab-initio analytical and numerical computations of their characteristics. In this work, topological defects are investigated in non-unitary conformal field theories using appropriate variations of the restricted solid-on-solid models. The relevant impurity models and the corresponding defect operators are constructed for the lattice system. Numerical computations are performed for the energy spectrum, eigenvalues of the defect operators as well as thermodynamic characteristics and compared with analytical predictions. Finally, renormalization group flows between the different fixed points are analyzed using numerical methods.
We describe a family of phase transitions connecting phases of differing nontrivial topological order by explicitly constructing Hamiltonians of the Levin-Wen [Levin and Wen, Phys. Rev. B 71, 045110 (2005)] type which can be tuned between two solvable points, each of which realizes a different topologically ordered phase. We show that the low-energy degrees of freedom near the phase transition can be mapped onto those of a Potts model, and we discuss the stability of the resulting phase diagram to small perturbations about the model. We further explain how the excitations in the condensed phase are formed from those in the original topological theory, some of which are split into multiple components by condensation, and we discuss the implications of our results for understanding the nature of general achiral topological phases in 2 +++ 1 dimensions in terms of doubled Chern-Simons theories.