Geometric Method for Solvable Lattice Spin Systems

概要

Summary of Thesis: Geometric Method for Solvable Lattice Spin
Systems

Masahiro Ogura
The term ``exactly solvable models" mean models whose physical quantity
(e.g., partition functions, ground states and so on) can be calculable without
any approximations. Exactly solvable models have played important roles in
not only physics but also mathematics. For example, by solving the 2D Ising
model exactly, Onsager indicated ferromagnetic phase transition in lattice
spin systems [1].
An important solvable class has a Hamiltonian which can be transformed
into a free fermion system such as the transverse field Ising model [2], the XY
model and the Kitaev honeycomb lattice model [3]. If a given Hamiltonian is
converted into a free fermion system, or equivalently, a Majorana quadratic
form (MQF), we can obtain the spectrum of the system. Moreover, recent
attempts have explored the generalized fermionization in higher-dimensional
system. The important difference from low-dimensional ones is that
additional ℤ2 gauge fields emerge in it. The Kitaev honeycomb lattice model
is one example of such model. However, systematic approach to this class has
never been studied.
In this thesis we construct a simple class of lattice systems which can be
transformed into MQFs [4]. This class includes well-known examples such as
the transverse field Ising model, the XY model, the Kitaev honeycomb lattice
model and a lot of other examples. We can also construct many new examples
including higher-dimensional models, fractal-like models and so on.
Our method is based on algebras, graph theory and simplicial homology
theory. For a lattice system with an algebra with a graphical representation,
we prove that the null space of the adjacency matrix of the graph provides
conserved quantities of the system. Moreover, we discover a class whose
Hamiltonians can be transformed into MQFs. When the graph of a
Hamiltonian belongs to a class of simplicial complexes, called SPSC class, we
can visually describe a transformation of the Hamiltonian.
This thesis is organized as follows.
In chapter 1, we explain the importance of exactly solvable models.
In chapter 2, we explain the famous examples including the transverse

field Ising model, the XY model, the Kitaev honeycomb lattice model and
related examples. We also describe the fundamental properties of
mathematical notions, such as algebras, combinatorics, graph theory, and
simplicial homology theory.
Chapter 3 is the theoretical part of [4]. At first, we define single-pointconnected simplicial complexes (SPSCs) as a simplicial complex with a special
condition, and calculate their homology groups. Next, we define the bond
algebra and the commutativity graph of a Hamiltonian and discuss its
conserved quantities. In particular, we discover that conserved quantities are
determined by the adjacency matrix of the commutativity graph. Finally, we
construct the method of transformation of lattice systems into a MQF. If the
commutativity graph corresponds to some SPSC, the Hamiltonian is
transformed into a MQF. Moreover, we reveal the relations between algebraic,
geometric and physical structure of Hamiltonians.
Chapter 4 is the practice part of [4]. We re-discuss the transverse field
Ising model, the XY model, the Kitaev honeycomb lattice model and related
examples. Moreover, we also show novel and interesting examples such as
higher-dimensional systems and fractal-like models.
In chapter 5 we demonstrate junction models [5]. We construct three
patterns of 1D tri-junction models and examine their zero energy boundary
modes. From this calculation we obtain various patterns of boundary modes.
We also illustrate 2D trijunction models and unveil that same boundary
modes happen as 1D ones.
Finally, in chapter 6, we discuss some new directions which should be
considered.

References
[1] L. Onsager, Phys. Rev. 65, 117 (1944).
[2] K. Minami. J. Phys. Soc. Jpn. 85, 024003 (2016).
[3] A. Kitaev, Ann. Phys. (Amsterdam,Neth.) 321, 2 (2006).
[4] M. Ogura et. al., Phys. Rev. B 102, 245118 (2020).
[5] M. Ogura and M. Sato, arXiv:2212.08891. ...