scholarly journals Extended Bloch theorem for topological lattice models with open boundaries

2019 ◽  
Vol 99 (8) ◽  
Author(s):  
Flore K. Kunst ◽  
Guido van Miert ◽  
Emil J. Bergholtz
Keyword(s):  
Lattice Models ◽  
Open Boundaries ◽  
Bloch Theorem ◽  
Topological Lattice

scholarly journals A Proof of the Bloch Theorem for Lattice Models

2019 ◽  
Vol 177 (4) ◽  
pp. 717-726 ◽  
Author(s):  
Haruki Watanabe
Keyword(s):  
Boundary Condition ◽  
Tight Binding ◽  
Lattice Models ◽  
Quantum Systems ◽  
Open Boundary ◽  
Current Operator ◽  
Open Boundary Condition ◽  
Expectation Value ◽  
Bloch Theorem ◽  
Large Quantum

Abstract The Bloch theorem is a powerful theorem stating that the expectation value of the U(1) current operator averaged over the entire space vanishes in large quantum systems. The theorem applies to the ground state and to the thermal equilibrium at a finite temperature, irrespective of the details of the Hamiltonian as far as all terms in the Hamiltonian are finite ranged. In this work we present a simple yet rigorous proof for general lattice models. For large but finite systems, we find that both the discussion and the conclusion are sensitive to the boundary condition one assumes: under the periodic boundary condition, one can only prove that the current expectation value is inversely proportional to the linear dimension of the system, while the current expectation value completely vanishes before taking the thermodynamic limit when the open boundary condition is imposed. We also provide simple tight-binding models that clarify the limitation of the theorem in dimensions higher than one.

scholarly journals TOPOLOGICAL LATTICE MODELS IN FOUR DIMENSIONS

1992 ◽  
Vol 07 (30) ◽  
pp. 2799-2810 ◽  
Author(s):  
HIROSI OOGURI
Keyword(s):  
Partition Function ◽  
Piecewise Linear ◽  
Lattice Models ◽  
Closed Surface ◽  
Statistical Weight ◽  
Four Dimensions ◽  
Angular Momenta ◽  
Topological Lattice ◽  
Wilson Operator ◽  
Dimensional Version

We define a lattice statistical model on a triangulated manifold in four dimensions associated to a group G. When G= SU (2), the statistical weight is constructed from the 15j-symbol as well as the 6j-symbol for recombination of angular momenta, and the model may be regarded as the four-dimensional version of the Ponzano-Regge model. We show that the partition function of the model is invariant under the Alexander moves of the simplicial complex, thus it depends only on the piecewise linear topology of the manifold. For an orientable manifold, the model is related to the so-called BF model. The q-analog of the model is also constructed, and it is argued that its partition function is invariant under the Alexander moves. It is discussed how to realize the 't Hooft operator in these models associated to a closed surface in four dimensions as well as the Wilson operator associated to a closed loop. Correlation functions of these operators in the q-deformed version of the model would define a new type of invariants of knots and links in four dimensions.

scholarly journals Phase transitions in topological lattice models via topological symmetry breaking

2012 ◽  
Vol 14 (1) ◽  
pp. 015004 ◽  
Author(s):  
F J Burnell ◽  
Steven H Simon ◽  
J K Slingerland
Keyword(s):  
Phase Transitions ◽  
Symmetry Breaking ◽  
Lattice Models ◽  
Topological Lattice

scholarly journals Three-dimensional topological lattice models with surface anyons

2013 ◽  
Vol 87 (4) ◽  
Author(s):  
C. W. von Keyserlingk ◽  
F. J. Burnell ◽  
S. H. Simon
Keyword(s):  
Three Dimensional ◽  
Lattice Models ◽  
Topological Lattice

scholarly journals Parafermionization, bosonization, and critical parafermionic theories

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Yuan Yao ◽  
Akira Furusaki
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Degrees Of Freedom ◽  
Lattice Models ◽  
Partition Functions ◽  
Minimal Models ◽  
Fermionic Model ◽  
Conformal Field ◽  
Critical Theories ◽  
Wigner Transformation

AbstractWe formulate a ℤk-parafermionization/bosonization scheme for one-dimensional lattice models and field theories on a torus, starting from a generalized Jordan-Wigner transformation on a lattice, which extends the Majorana-Ising duality atk= 2. The ℤk-parafermionization enables us to investigate the critical theories of parafermionic chains whose fundamental degrees of freedom are parafermionic, and we find that their criticality cannot be described by any existing conformal field theory. The modular transformations of these parafermionic low-energy critical theories as general consistency conditions are found to be unconventional in that their partition functions on a torus transform differently from any conformal field theory whenk >2. Explicit forms of partition functions are obtained by the developed parafermionization for a large class of critical ℤk-parafermionic chains, whose operator contents are intrinsically distinct from any bosonic or fermionic model in terms of conformal spins and statistics. We also use the parafermionization to exhaust all the ℤk-parafermionic minimal models, complementing earlier works on fermionic cases.

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