#  Lattice fermions as spectral graphs

Jun Yumoto  d8521007(at)s.akita-u.ac.jp Department of Mathematical Science,
Akita University, Akita 010-8502, Japan  Tatsuhiro Misumi
misumi(at)phys.kindai.ac.jp Department of Physics, Kindai University, Osaka
577-8502, Japan  Research and Education Center for Natural Sciences, Keio
University, Kanagawa 223-8521, Japan

###### Abstract

We study lattice fermions from the viewpoint of spectral graph theory (SGT).
We find that a fermion defined on a certain lattice is identified as a
spectral graph. SGT helps us investigate the number of zero eigenvalues of
lattice Dirac operators even on the non-torus and non-regular lattice, leading
to understanding of the number of fermion species (doublers) on lattices with
arbitrary topologies. The procedure of application of SGT to lattice fermions
is summarized as follows: (1) One investigates a spectral graph corresponding
to a lattice fermion. (2) One obtains a matrix corresponding to the graph. (3)
One finds zero eigenvalues of the matrix by use of the discrete Fourier
transformation (DFT). (4) By taking an infinite-volume and continuum limits,
one finds the number of species. We apply this procedure to the known lattice
fermion formulations including Naive fermions, Wilson fermions and Domain-wall
fermions, and reproduce the known fact on the number of species. We also apply
it to the lattice fermion on the discretized four-dimensional hyperball and
discuss the number of fermion species on the bulk. In the end of the paper, we
discuss the application of the analysis to lattice fermions on generic
lattices with arbitrary topologies, which could lead to constructing a new
theorem regarding the number of species.

###### Contents

  1. I Introduction
  2. II Spectral graph theory
  3. III Naive fermion
    1. III.1 Dirac matrix of naive fermion
    2. III.2 Graphs corresponding to naive fermion
    3. III.3 Diagonalization of Dirac matrix $\mathcal{D}$
  4. IV Wilson fermion
    1. IV.1 Dirac matrix of Wilson fermion
    2. IV.2 Graphs corresponding to Wilson fermion
    3. IV.3 Diagonalization of Dirac matrix $\mathcal{D}_{W}$
  5. V Domain-wall fermion
    1. V.1 Dirac matrix and mass matrix of domain-wall fermion
    2. V.2 Graphs corresponding to domain-wall fermion
    3. V.3 Diagonalization of Dirac matrix $\mathcal{D}_{DW}$ and mass matrix $\mathcal{M}_{DW}$
  6. VI Lattice fermions on four dimensional hyperball
    1. VI.1 Graphs corresponding to the Dirac matrix on hyperball
    2. VI.2 Lattice action on four-dimensional hyperball
    3. VI.3 Number of species on four-dimensional hyperball
  7. VII Summary and Discussion

##  I Introduction

Non-perturbative physics of quantum field theories has been investigated by
lattice gauge theory Wilson:1974sk ; Creutz:1980zw . Despite of the
accomplishments obtained by the lattice Monte-Carlo simulation, there are
still troublesome problems in lattice fermion formulations Karsten:1980wd ;
Nielsen:1980rz ; Nielsen:1981xu ; Nielsen:1981hk , including the reconcilement
of chiral symmetry and numerical efficiency, the realization of a single Weyl
fermion, and the sign problem of the quark determinant. The known lattice
fermions, including Wilson, staggered and domain-wall (overlap) fermions, have
their own shortcomings Wilson:1975id ; Kaplan:1992bt ; Shamir:1993zy ;
Furman:1994ky ; Neuberger:1998wv ; Ginsparg:1981bj ; Kogut:1974ag ;
Susskind:1976jm ; Kawamoto:1981hw ; Sharatchandra:1981si ; Golterman:1984cy ;
Golterman:1985dz ; Kilcup:1986dg . Other lattice fermion formulations have
been investigated, but they also have their own drawbacks: The generalized
Wilson fermions based on flavored-mass terms Bietenholz:1999km ; Creutz:2010bm
; Durr:2010ch ; Durr:2012dw ; Misumi:2012eh ; Cho:2013yha ; Cho:2015ffa ;
Durr:2017wfi  and the staggered-Wilson fermions Golterman:1984cy ;
Adams:2009eb ; Adams:2010gx ; Hoelbling:2010jw ; deForcrand:2011ak ;
Creutz:2011cd ; Misumi:2011su ; Follana:2011kh ; deForcrand:2012bm ;
Misumi:2012sp ; Misumi:2012eh ; Durr:2013gp ; Hoelbling:2016qfv ;
Zielinski:2017pko  can reduce numerical costs and discretizartion errors.
However, their implementation to lattice QCD is complicated and eventually
leads to costs. The minimally doubled fermion Karsten:1981gd ; Wilczek:1987kw
; Creutz:2007af ; Borici:2007kz ; Bedaque:2008xs ; Bedaque:2008jm ;
Capitani:2009yn ; Kimura:2009qe ; Kimura:2009di ; Creutz:2010cz ;
Capitani:2010nn ; Tiburzi:2010bm ; Kamata:2011jn ; Misumi:2012uu ;
Misumi:2012ky ; Capitani:2013zta ; Capitani:2013iha ; Misumi:2013maa ;
Weber:2013tfa ; Weber:2017eds ; Durr:2020yqa  with $U(1)$ chiral symmetry
yields only two species. The shortcoming of this setup is the explicit
breaking of part of the lattice discrete symmetry. The central-branch Wilson
fermion Kimura:2011ik ; Creutz:2011cd ; Misumi:2012eh ; Chowdhury:2013ux ;
Misumi:2020eyx  has the extra $U(1)_{\overline{V}}$ symmetry prohibiting
additive mass renormalization, but its two-flavor version loses part of
hypercubic symmetry.

One of new avenues in the study of lattice fermions is to study fermions on
non-torus lattices with different topologies. However, we are so far
unfamiliar with the number of species on such lattices since the Nielsen-
Ninomiya’s no-go theorem can be only applied to the lattice fermions defined
on the torus111Some may state that the Nielsen-Ninomiya’s no-go theorem can be
applied to fermions on non-torus lattices. However, this no-go theorem cannot
explain the empirical fact that the number of fermion species is two on the
discretized two-sphere.. Toward understanding the number of species of
fermions on lattices with arbitrary topologies, we need to develop the way how
to figure out the number of zero eigenvalues of lattice Dirac operators
without using the momentum representation since the non-torus lattices do not
have discrete translational symmetry.

In this paper, we discuss lattice fermions in terms of spectral graph theory
(SGT) west2001introduction ; bondy1976graph ; mieghem_2010 ; Watts1998  and
obtain a new insight. The relation between supersymmetric lattice gauge
theories (matrix models) and graph theories has been recently discussed in
Ohta:2020ygi ; Ohta:2021xty , and the relation among Dirac-Kaehler fermions,
Euler characteristics of background spaces and quantum anomalies has been
discussed in Catterall:2018lkj ; Butt:2021brl . We now focus on the spectral
graph theory (SGT) and the discrete Fourier transformation (DFT), which tell
us the relation among a graph, the associated matrix and its eigenvalues. We
find that a Dirac operator of fermions defined on a certain lattice is
identified as a matrix of the spectral graph corresponding to the lattice
fermion. SGT and DFT clarifies the number of zero eigenvalues of lattice Dirac
operators, leading to understanding of the number of doublers on lattices with
arbitrary topologies. Our arguments are summarized as follows; First of all, a
lattice fermion on a certain lattice corresponds to one spectral graph.
Secondly, the Dirac operator of the lattice fermion can be treated as a matrix
corresponding to the graph. Thus, once we have a certain lattice and a Dirac
operator defined on the lattice, we can derive the corresponding matrix and
find the number of zero eigenvalues by use of the discrete Fourier
transformation (DFT). Finally, by taking an infinite-volume and continuum
limits, we can study the number of doublers. We apply this procedure to the
known fermion formulations including Naive fermion, Wilson fermion and Domain-
wall fermion defined on the regular lattices, although the advantage of our
technique is its use for non-torus lattices. We then apply it to the lattice
fermion on the discretized four-dimensional hyperball and study the number of
doublers on the bulk. We also discuss its application to hypersphere and
discuss the possibility of a new theorem regarding the number of doublers on
the lattice with arbitrary topologies.

The main purpose of this work is to introduce the importance of spectral graph
theory to the lattice field theory community. Although most of the results
obtained in this paper are reproductions of the known results, the method we
use here is new and it could be applied to the nontrivial cases in future.

This paper is constructed as follows: In Sec. II we introduce the spectral
graph theory. In Sec. III we discuss the naive fermion in terms of SGT and
reproduce the known results. In Sec. IV and Sec. V we discuss Wilson and
domain-wall fermions in terms of SGT and reproduce the known results. In Sec.
VI, as the first non-trivial application, we apply SGT analysis to the lattice
fermions on four-dimensional hyperball. Sec. VII is devoted to the summary and
discussion.

##  II Spectral graph theory

In the section, we introduce basic concepts of Spectral Graph Theory (SGT) for
the purpose of discussing lattice fermions from a new viewpoint. We will also
present some examples to show the importance of the concept. Let us start with
the definition of a graph west2001introduction ; bondy1976graph ; mieghem_2010
; Watts1998 .

######  Definition. 1 (graph).

A graph $G$ is a pair $G=(V,E)$, where $V$ is a set of vertices of the graph
and $E$ is a set of edges of the graph.

As examples, we exhibit two graphs in Fig. 1 with $V=\left\\{1,2,3,4\right\\}$
and $E=\left\\{\\{1,2\\},\\{1,3\\},\\{1,4\\},\\{3,4\\}\right\\}$. Here,
$\\{i,j\\}$ stands for an edge from $i$ to $j$. Note that two graphs in Fig. 1
have the common pair $G=(V,E)$ and they are the identical graphs although
their appearances are different. They have no directed edges, which will be
discussed soon, thus any two vertex in any element of $E$ in the examples can
be commutated, so we find $\\{i,j\\}=\\{j,i\\}$.

Figure 1: These examples are graphs having a pair $G=(V,E)$ with
$V=\left\\{1,2,3,4\right\\}$ and
$E=\left\\{\\{1,2\\},\\{1,3\\},\\{1,4\\},\\{3,4\\}\right\\}$.

######  Definition. 2 (directed graph or digraph).

A directed graph (or digraph) is a pair $(V,E)$ of sets of vertices and edges
together with two maps $\mathrm{init}:E\to V$ and $\mathrm{ter}:E\to V$. The
two maps are assigned to every edge $e_{ij}$ with an initial vertex
$\mathrm{init}(e_{ij})=v_{i}\in V$ and a terminal vertex
$\mathrm{ter}(e_{ij})=v_{j}\in V$. The edge $e_{ij}$ is said to be directed
from $\mathrm{init}(e_{ij})$ to $\mathrm{ter}(e_{ij})$. If
$\mathrm{init}(e_{ij})=\mathrm{ter}(e_{ij})$, the edge $e_{ij}$ is called a
loop.

As examples, two graphs in Fig. 2 are digraphs with
$V=\left\\{1,2,3,4\right\\}$ and
$E=\left\\{\\{1,2\\},\\{1,3\\},\\{1,4\\},\\{3,4\\}\right\\}$. Then initial
vertex and terminal vertex are assigned as
$\mathrm{init}(\\{i,j\\})=i,\mathrm{ter}(\\{i,j\\})=j$.

Figure 2: These examples are graphs having a pair $(V,E)$ with
$V=\left\\{1,2,3,4\right\\}$ and
$E=\left\\{\\{1,2\\},\\{1,3\\},\\{1,4\\},\\{3,4\\}\right\\}$. The initial
vertices of edges are
$\mathrm{init}(\\{1,2\\})=1,\mathrm{init}(\\{1,3\\})=1,\mathrm{init}(\\{1,4\\})=1,\mathrm{init}(\\{3,4\\})=3$,
while the terminal vertices are
$\mathrm{init}(\\{1,2\\})=2,\mathrm{init}(\\{1,3\\})=3,\mathrm{init}(\\{1,4\\})=4,\mathrm{init}(\\{3,4\\})=4$.

As another example we show Fig. 3, which is a digraph with loop. Note that
vertices in any loop can be commutated since they are the identical vertex. In
general, two vertices in a directed edge can not be commutated
$\\{i,j\\}\neq\\{j,i\\}$.

Figure 3: A digraph with two loops.

######  Definition. 3 (weighted graph).

The weighted graph has a value (the weight) for each edge in a graph or a
digraph.

As an example, we show a digraph in Fig. 4. It is a weighted graph, each of
whose edge has a weight.

Figure 4: This digraph is a weighted graph. Blue edges in the graph are those
with positive weights, while red edges are those with negative weights.

######  Definition. 4 (adjacency matrix).

The adjacency matrix $A$ of a graph is the $|V|\times|V|$ matrix given by

$A_{ij}=\begin{cases}w_{ij}&\mbox{if there is a edge from $i$ to $j$}\\\
0&\mbox{otherwise}\end{cases}\,,$ (1)

where $w_{ij}$ is the weight of an edge from $i$ to $j$.

As an example we exhibit an adjacency matrix $A$ of a graph in Fig. 4

$A=\begin{pmatrix}0&1&0&-1\\\ -4&0&2&0\\\ 0&0&0&0\\\ 3&0&-2&0\end{pmatrix}\,.$
(2)

This matrix is asymmetric. In general, the adjacency matrix of a directed
graph is asymmetric since the existence of an edge from $i$ to $j$ does not
necessarily imply that there is also an edge from $j$ to $i$.

The lattice fermion has common properties with the spectral graph we have
introduced here. In the next sections, we will show that the lattice fermion
can be identified as the spectral graph.

##  III Naive fermion

In this section we discuss the Dirac operator matrix of Naive fermion in terms
of SGT. Through this example, we will find a Dirac operator of lattice
fermions is identified as a matrix of the spectral graph corresponding to the
lattice fermion. We also show how to find the number of zero modes (the number
of fermion species) by use of discrete Fourier transformation (DFT). This
technique can be applied to any kind of matrices arising from the non-regular
lattices. By use of DFT, we will correctly derive the sixteen fermion species
in the naive fermion.

###  III.1 Dirac matrix of naive fermion

The lattice naive fermion action in four dimensions is

$S=\sum_{n}\sum_{\mu=1}^{4}\bar{\psi}_{n}\gamma_{\mu}D_{\mu}\psi_{n}\,,$ (3)

where $D_{\mu}\equiv(T_{+\mu}-T_{-\mu})/2$ with
$T_{\pm\mu}\psi_{n}=U_{n,\pm\hat{\mu}}\psi_{n\pm\hat{\mu}}$ and $\hat{\mu}$ is
a unit vector. In a free theory, we just set $U_{n,\pm\hat{\mu}}=\bm{1}$. The
sum $\sum_{n}$ is the summation over lattice site
$n=(n_{1},n_{2},n_{3},n_{4})$ and those intervals are $1\leq n_{\mu}\leq N$.
Note that the spacetime where the fermion is defined is a four-dimensional
torus because we usually impose periodic boundary conditions in each
direction. To derive the corresponding matrix of the lattice Dirac operator in
a free theory, we introduce a vector of fermion fields. Namely, a vector
$\psi$ is defined $\psi=\sum_{n}\psi_{n}\bm{e}_{n}$ where $\bm{e}_{n}$ are
standard basis which satisfy orthonormal
$\bm{e}_{n^{\prime}}\bm{e}_{n}=\delta_{n^{\prime}n}\equiv\prod_{i=1}^{4}\delta_{n^{\prime}_{i}n_{i}}$.
$\delta_{kl}$ is the Kronecker delta. Here we specify that the order of
components $\psi_{n}$ in the vector is
$(1,1,1,1)\to\cdots\to(N,1,1,1)\to(1,2,1,1)\to\cdots\to(N,N,1,1)\to(1,1,2,1)\to\cdots\to(N,N,N,N)$
in descending order. Namely,

$\psi=\begin{pmatrix}\psi_{(1,1,1,1)}\\\ \vdots\\\ \psi_{(N,1,1,1)}\\\
\psi_{(1,2,1,1,1)}\\\ \vdots\\\ \psi_{(N,N,1,1)}\\\ \psi_{(1,1,2,1)}\\\
\vdots\\\ \psi_{(N,N,N,N)}\\\ \end{pmatrix}$ (4)

in term of the vector. Thus, the action of naive fermion can be rewritten as
$S=\bar{\psi}\mathcal{D}\psi=\sum_{m}\sum_{n}\bar{\psi}_{m}\mathcal{D}_{mn}\psi_{n}$
where $\mathcal{D}$ is Dirac matrix having $\mathcal{D}_{mn}$ as $(m,n)$
component. For later use, we now introduce the tensor-product representation:
By the tensor product (or Kronecker product), this matrix can be represented
as

$\begin{split}\mathcal{D}&=\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes
P_{N}\otimes\gamma_{1}\\\ &+\bm{1}_{N}\otimes\bm{1}_{N}\otimes
P_{N}\otimes\bm{1}_{N}\otimes\gamma_{2}\\\ &+\bm{1}_{N}\otimes
P_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\gamma_{3}\\\
&+P_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\gamma_{4}\end{split}$
(5)

where $\bm{1}_{N}$ is an identity matrix of order $N$ and $P_{N}$ is a square
matrix of order $N$ Kaveh2005 ; Sabidussi1959 ; Aurenhammer1992 . The
components of the matrix $P_{N}$ is defined as
$(P_{N})_{ab}\equiv(E_{ab}-E_{ab}^{-1})/2$ where
$E_{ab}\equiv\sum_{i=1}^{N-1}\delta_{ai}\delta_{i+1\,b}+\delta_{aN}\delta_{1b}$
and
$E_{ab}^{-1}\equiv\delta_{a1}\delta_{Nb}+\sum_{i=2}^{N}\delta_{ai}\delta_{i-1\,b}$
for $a,b=1,2,\cdots,N$. $P_{N}$ is explicitly written as

$P_{N}=\frac{1}{2}\begin{pmatrix}0&1&0&&0&0&-1\\\ -1&0&1&\cdots&0&0&0\\\
0&-1&0&&0&0&0\\\ &\vdots&&\ddots&&\vdots&\\\ 0&0&0&&0&1&0\\\
0&0&0&\cdots&-1&0&1\\\ 1&0&0&&0&-1&0\end{pmatrix}$ (6)

and $E$ and $E^{\dagger}$ are similarly represented as

$E=\begin{pmatrix}0&1&0&&0&0&0\\\ 0&0&1&\cdots&0&0&0\\\ 0&0&0&&0&0&0\\\
&\vdots&&\ddots&&\vdots&\\\ 0&0&0&&0&1&0\\\ 0&0&0&\cdots&0&0&1\\\
1&0&0&&0&0&0\end{pmatrix},\quad E^{\dagger}=\begin{pmatrix}0&0&0&&0&0&1\\\
1&0&0&\cdots&0&0&0\\\ 0&1&0&&0&0&0\\\ &\vdots&&\ddots&&\vdots&\\\
0&0&0&&0&0&0\\\ 0&0&0&\cdots&1&0&0\\\ 0&0&0&&0&1&0\end{pmatrix}.$ (7)

Note that the matrix $P_{N}$ is the matrix representing the periodic boundary
condition of one direction. This matrix is the circulant matrix.

###  III.2 Graphs corresponding to naive fermion

In this subsection, based on the matrix representations we have obtained in
the previous subsection, we see how the spectral graphs correspond to the
naive lattice fermions on low- and high-dimensional lattices. Although in the
cases of non-torus lattice fermions we obtain the Dirac matrix from the
spectral graph, we now derive “the graph from the matrix" conversely.

Firstly, we consider the one dimensional lattice. In this case, the Dirac
matrix of the free naive fermion is

$\mathcal{D}^{\mathrm{1d}}=P_{N}\otimes\gamma_{1}.$ (8)

Now, the graph corresponding to this Dirac matrix is depicted in Fig. 5. This
graph schematically represents a circle $S^{1}$, where we impose a periodic
boundary condition. This is called a circulant graph in graph theory.

Figure 5: This graph corresponds to the Dirac matrix of one dimensional naive
fermion with periodic boundary condition. This graph schematically shows a
circle $S^{1}$.

In the case of two-dimensional naive fermion, the Dirac matrix is

$D^{\mathrm{2d}}=\bm{1}_{N}\otimes
P_{N}\otimes\gamma_{1}+P_{N}\otimes\bm{1}_{N}\otimes\gamma_{2}\,,$ (9)

and the graph, who schematically represents two-dimensional torus $T^{2}$, is
expressed as Fig. 6, where each direction is independent, leading to
$S^{1}\times S^{1}\sim T^{2}$. There are both an edge with the positive weight
and an edge with the negative weight between each vertex. It is notable that
the weight for edges from $n$ to $n+\hat{\mu}$ is $+\gamma_{\mu}$ and the one
for edges from $n$ to $n-\hat{\mu}$ is $-\gamma_{\mu}$.

Figure 6:  This graph corresponds to the Dirac matrix of two-dimensional naive
fermion with periodic boundary condition. This graph schematically represents
the two-dimensional torus $T^{2}$. The weight for each edge from $n$ to
$n+\hat{\mu}$ is $+\gamma_{\mu}/2$ and the one for each edge from $n$ to
$n-\hat{\mu}$ is $-\gamma_{\mu}/2$.

In three and four dimensions, the procedures are parallel. The corresponding
graphs schematically represent $S^{1}\times S^{1}\times S^{1}\sim T^{3}$ and
$S^{1}\times S^{1}\times S^{1}\times S^{1}\sim T^{4}$, respecctively. For the
cases of non-torus lattice fermions, we conversely obtain the Dirac matrix
from the spectral graph.

###  III.3 Diagonalization of Dirac matrix $\mathcal{D}$

In this subsection, we discuss how to diagonalize Dirac matrix $\mathcal{D}$
and find the number of zero eigenvalues for a free four-dimensional naive
fermion. Firstly, we will diagonalize $E$ in order to diagonalize the whole
Dirac matrix. We introduce the discrete Fourier transform (DFT) matrix $X$,
which is consisted of $X_{jk}\equiv\xi^{(k-1)(j-1)}/\sqrt{N}$ with
$\xi=e^{-\frac{2\pi i}{N}}$ for $j,k=1,2,\cdots,N$. It is clear that DFT
matrix is unitary. Then, $E$ and $E^{\dagger}$ satisfy $EX=X\Lambda$ and
$E^{\dagger}X=\Lambda^{\dagger}X$, where $\Lambda$ is a diagonal matrix as
$\Lambda=\mathrm{Diag}\left[1,\xi,\xi^{2},\cdots,\xi^{N-1}\right]$. Since
$P_{N}=E-E^{\dagger}$, the diagonalized $P_{N}$ is given as

$P_{N}X=i\,\mathrm{Diag}\left[0,\,\sin\frac{2\pi}{N},\,\sin\frac{4\pi}{N},\,\cdots,\,\sin\frac{2(N-1)\pi}{N}\right]X\equiv\Lambda_{P_{N}}X.$
(10)

There exist two zero eigenvalues in $\Lambda_{P_{N}}$ at most when $N$ is an
even number. Now, the Dirac matrix $\mathcal{D}$ can be diagonalized as

$\begin{split}\mathcal{U}^{\dagger}\mathcal{D}\,\mathcal{U}&=\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\Lambda_{P_{N}}\otimes\gamma_{1}\\\
&+\bm{1}_{N}\otimes\bm{1}_{N}\otimes\Lambda_{P_{N}}\otimes\bm{1}_{N}\otimes\gamma_{2}\\\
&+\bm{1}_{N}\otimes\Lambda_{P_{N}}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\gamma_{3}\\\
&+\Lambda_{P_{N}}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\gamma_{4}\end{split}$
(11)

where $\mathcal{U}$ is a unitary matrix defined as
$\mathcal{U}=\bigotimes_{\mu=1}^{4}X\otimes\bm{1}_{4}$. It is easy to see the
number zero eigenvalues of the Dirac matrix by introducing eigenvectors of
$E$. We denote eigenvectors of $E$ as $\ket{k}$ for $k=1,2,\cdots,N$. They
satisfy $E\ket{k}=\xi^{k-1}\ket{k}$ and
$\braket{k^{\prime}}{k}=\delta_{k^{\prime}k}$. Then, the unitary matrix
$\mathcal{U}$ is written as
$\mathcal{U}=\sum_{k_{4},k_{3},k_{2},k_{1}}\ket{k_{4},k_{3},k_{2},k_{1}}\bra{k_{4},k_{3},k_{2},k_{1}}\otimes\bm{1}_{4}$,
where eigenvectors $\ket{k_{4},k_{3},k_{2},k_{1}}$ mean
$\ket{k_{4}}\otimes\ket{k_{3}}\otimes\ket{k_{2}}\otimes\ket{k_{1}}$. Thus the
diagonalization of the Dirac matrix is expressed as

$\mathcal{U}^{\dagger}\mathcal{D}\,\mathcal{U}=\sum_{k_{4},k_{3},k_{2},k_{1}}\left[i\sum_{\mu=1}^{4}\sin\left(\frac{2\pi(k_{\mu}-1)}{N}\right)\gamma_{\mu}\right]\ket{k_{4},k_{3},k_{2},k_{1}}\bra{k_{4},k_{3},k_{2},k_{1}}.$
(12)

If $\mathcal{U}^{\dagger}\mathcal{D}\,\mathcal{U}$ has zero eigenvalues, Eq.
(12) must satisfy an equation below

$\sum_{\mu=1}^{4}\sin\left(\frac{2\pi(k_{\mu}-1)}{N}\right)\gamma_{\mu}=\bm{0}.$
(13)

However, since $\gamma$ matrices are linearly independent, the coefficient of
each $\gamma$ matrices must be zero. Finally, the conditions for the
diagonalized Dirac matrix to have zero eigenvalues are

$\sin\left(\frac{2\pi(k_{\mu}-1)}{N}\right)=0.$ (14)

The solutions of Eq. (14) are $k_{\mu}=1,\frac{N}{2}+1$. In this case, we do
not need to take a continuum and a thermodynamic limit as long as we take an
even number as $N$. Therefore, the number of zero eigenvalues are $2^{4}$
since the number of zero eigenvalues is equal to the number of combination of
the solutions in four dimensions. This result is consistent with the well-
known number of doublers in four-dimensional naive fermion.

In the end of this section, we comment on the meaning of $k_{\mu}$. This
$k_{\mu}$ can be interpreted as a “shifted momentum" in this case. However, it
is not necessarily equivalent to the momentum since it can be defined also on
the lattice in which the momentum cannot be defined. Thus, $k_{\mu}$ should be
simply interpreted as a variable for the diagonalization in general.

##  IV Wilson fermion

###  IV.1 Dirac matrix of Wilson fermion

In this section we discuss the Dirac operator matrix of Wilson fermion in
terms of SGT. The lattice Wilson fermion action in four dimensions is
expressed as

$S_{W}=\sum_{n}\sum_{\mu}\bar{\psi}_{n}\gamma_{\mu}D_{\mu}\psi_{n}+m\sum_{n}\bar{\psi}_{n}\psi_{n}\\\
+r\sum_{n}\sum_{\mu}\bar{\psi}_{n}\left(1-C_{\mu}\right)\psi_{n}\,,$ (15)

with $C_{\mu}\equiv(T_{+\mu}+T_{-\mu})/2$. $m$ is a mass parameter and $r$ is
a Wilson-fermion parameter. The periodic boundary condition is imposed on each
direction and those intervals are given as $1\leq n_{\mu}\leq N$. By
investigating the action in a manner similar to Sec. III, it is written as
$S_{W}=\bar{\psi}\mathcal{D}_{W}\psi\equiv\bar{\psi}\left(\mathcal{D}+\mathcal{M}_{W}\right)\psi$,
where $\mathcal{D}_{W}$ is the Dirac matrix of Wilson fermion and
$\mathcal{M}_{W}$ is mass matrix consisting of the Wilson term. The mass
matrix $\mathcal{M}_{W}$ is represented by use of tensor products as

$\begin{split}\mathcal{M}_{W}&=m\cdot\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{4}\\\
&+r\cdot\bigl{\\{}\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes
M_{W}\otimes\bm{1}_{4}+\bm{1}_{N}\otimes\bm{1}_{N}\otimes
M_{W}\otimes\bm{1}_{N}\otimes\bm{1}_{4}\\\ &+\bm{1}_{N}\otimes
M_{W}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{4}+M_{W}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{4}\bigr{\\}}\end{split}$
(16)

where $M_{W}=\bm{1}_{N}-(E+E^{\dagger})/2$ and the product $\cdot$ stands for
scalar product.

###  IV.2 Graphs corresponding to Wilson fermion

As with the case in Sec. III.2, we begin with the one-dimensional lattice. For
one-dimensional Wilson fermion, the Dirac matrix is given by

$D_{W}^{\mathrm{1d}}=P_{N}\otimes\gamma_{1}+m\bm{1}_{N}\otimes\bm{1}_{4}+rM_{W}\otimes\bm{1}_{4},$
(17)

and the graph corresponding to this matrix is depicted in Fig. 7. This graph
again represents the circle $S^{1}$. However, the difference from the case of
naive fermions in Fig. 6 is that Fig. 7 has a loop with the weight $M=m+r$ for
each vertex.

Figure 7:  This digraph corresponds to the Dirac matrix of one-dimensional
Wilson fermion. The weight of the loop is $M=m+r$. The weight $\pm$ for each
edge stand for $\pm(\gamma_{1}\mp r\bm{1}_{4})/2$.

In two dimensions, the Dirac matrix is

$\begin{split}D_{W}^{\mathrm{2d}}&=\bm{1}_{N}\otimes
P_{N}\otimes\gamma_{1}+P_{N}\otimes\bm{1}_{N}\otimes\gamma_{2}\\\
&+m\left(\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{4}\right)+r\left(\bm{1}_{N}\otimes
M_{W}+M_{W}\otimes\bm{1}_{N}\right)\otimes\bm{1}_{4}\end{split}$ (18)

and the graph corresponding to this matrix is given by Fig. 8. The shape of
the graph is topologically a two-dimensional torus $T^{2}$ however there is a
loop with the weight $M=m+\sum_{\mu}r$ for each vertex. Furthermore, the
weight for each edge from $n$ to $n+\hat{1}$ is $+(\gamma_{1}-r\bm{1}_{4})/2$
and the one for each edge from $n$ to $n-\hat{1}$ is
$-(\gamma_{1}+r\bm{1}_{4})/2$. In other direction $\mu=2$, we just replace
$\gamma_{1}$ by $\gamma_{2}$.

Figure 8:  This digraph corresponds to the Dirac matrix in two-dimensional
Wilson fermion. The shape of this graph is topologically equivalent to two-
dimensional torus $T^{2}$. The weight for each edge from $n$ to $n+\hat{\mu}$
is $+(\gamma_{\mu}-r\bm{1}_{4})/2$ and the one for each edge from $n$ to
$n-\hat{\mu}$ is $-(\gamma_{\mu}+r\bm{1}_{4})/2$. The weight for each loop is
$M=m+\sum_{\mu}r$.

The shape of the graph corresponding to Wilson fermion in four dimensions is
again topologically a four-dimensional torus $T^{4}$. Furthermore, there is a
loop for each vertex, which has the weight $M=m+\sum_{\mu}r$. The weight for
each edge from $n$ to $n+\hat{\mu}$ is $+(\gamma_{\mu}-r\bm{1}_{4})/2$ while
the wight for each edge from $n$ to $n-\hat{\mu}$ is
$-(\gamma_{\mu}+r\bm{1}_{4})/2$.

###  IV.3 Diagonalization of Dirac matrix $\mathcal{D}_{W}$

Firstly, we will diagonalize the mass matrix $\mathcal{M}_{W}$ so as to
diagonalize the whole Dirac matrix $\mathcal{D}_{W}$. As with the discussion
of naive fermion, we use the DFT matrix $X$. By using the matrix $X$ we can
diagonalize $M_{W}$ as

$M_{W}X=\mathrm{Diag}\left[0,\,1-\cos\frac{2\pi}{N},\,1-\cos\frac{4\pi}{N},\,\cdots,\,1-\cos\frac{2(N-1)\pi}{N}\right]X\equiv\Lambda_{M_{W}}X.$
(19)

There is only a single zero eigenvalue in $\Lambda_{M_{W}}$ at most if $N$ is
even number. As a result the diagonalization is expressed as

$\begin{split}\mathcal{U}^{\dagger}\mathcal{M}_{W}\,\mathcal{U}&=m\cdot\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{4}\\\
&+r\cdot\bigl{\\{}\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\Lambda_{M_{W}}\otimes\bm{1}_{4}+\bm{1}_{N}\otimes\bm{1}_{N}\otimes\Lambda_{M_{W}}\otimes\bm{1}_{N}\otimes\bm{1}_{4}\\\
&+\bm{1}_{N}\otimes\Lambda_{M_{W}}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{4}+\Lambda_{M_{W}}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{4}\bigr{\\}}\end{split}$
(20)

where $\mathcal{U}=\bigotimes_{\mu=1}^{4}X\otimes\bm{1}_{4}$. Since the Dirac
matrix of naive fermion $\mathcal{D}$ can be diagonalized with the unitary
matrix $U$, the Dirac matrix of Wilson fermion can be diagonalized as
$\mathcal{U}^{\dagger}\mathcal{D}_{W}\,\mathcal{U}=\mathcal{U}^{\dagger}\mathcal{D}\,\mathcal{U}+\mathcal{U}^{\dagger}\mathcal{M}_{W}\,\mathcal{U}$.
We introduce the eigenvectors $\ket{k}$ of $E$ to see the number of zero
eigenvalues in the diagonalized Dirac matrix
$\mathcal{U}^{\dagger}\mathcal{D}_{W}\,\mathcal{U}$. The diagonalized mass
matrix $\mathcal{U}^{\dagger}\mathcal{M}_{W}\,\mathcal{U}$ is written as

$\begin{split}\mathcal{U}^{\dagger}\mathcal{M}_{W}\,\mathcal{U}=&\sum_{k_{4},k_{3},k_{2},k_{1}}\left[m\,\bm{1}_{4}+r\sum_{\mu=1}^{4}\left\\{1-\cos\left(\frac{2\pi(k_{\mu}-1)}{N}\right)\right\\}\bm{1}_{4}\right]\\\
&\times\ket{k_{4},k_{3},k_{2},k_{1}}\bra{k_{4},k_{3},k_{2},k_{1}}.\end{split}$
(21)

Thus the diagonalized Dirac matrix
$\mathcal{U}^{\dagger}\mathcal{D}_{W}\,\mathcal{U}$ is

$\begin{split}\mathcal{U}^{\dagger}\mathcal{D}_{W}\,\mathcal{U}=&\sum_{k_{4},k_{3},k_{2},k_{1}}\left[i\sum_{\mu}\sin\left(\frac{2\pi(k_{\mu}-1)}{N}\right)\gamma_{\mu}+r\sum_{\mu}\left\\{\frac{m}{4r}+1-\cos\left(\frac{2\pi(k_{\mu}-1)}{N}\right)\right\\}\bm{1}_{4}\right]\\\
&\times\ket{k_{4},k_{3},k_{2},k_{1}}\bra{k_{4},k_{3},k_{2},k_{1}}.\end{split}$
(22)

If the diagonalized Dirac matrix
$\mathcal{U}^{\dagger}\mathcal{D}_{W}\,\mathcal{U}$ has zero eigenvalues, Eq.
(22) must satisfy a equation

$i\sum_{\mu}\sin\left(\frac{2\pi(k_{\mu}-1)}{N}\right)\gamma_{\mu}+r\sum_{\mu}\left\\{\frac{m}{4r}+1-\cos\left(\frac{2\pi(k_{\mu}-1)}{N}\right)\right\\}\bm{1}_{4}=\bm{0}$
(23)

Since $\gamma$ matrices and $\bm{1}_{4}$ are linearly independent, the
conditions for the diagonalized Dirac matrix to have zero eigenvalues are
given as

$\displaystyle\sin\left(\frac{2\pi(k_{\mu}-1)}{N}\right)=0$ (24)
$\displaystyle
m+4r-r\sum_{\mu}\cos\left(\frac{2\pi(k_{\mu}-1)}{N}\right)=0\,.$ (25)

The solutions of Eq. (24) are $k_{\mu}=1,\frac{N}{2}+1$. These solutions are
classified by Eq. (25) as

$M(k)=\begin{cases}m&\mbox{any $k_{\mu}=1$ in $k$.}\\\ m+2r&\mbox{one
$k_{\mu}=1+N/2$ otherwise $k_{\mu}=1$ in $k$.}\\\ m+4r&\mbox{two
$k_{\mu}=1+N/2$ otherwise $k_{\mu}=1$ in $k$.}\\\ m+6r&\mbox{three
$k_{\mu}=1+N/2$ otherwise $k_{\mu}=1$ in $k$.}\\\ m+8r&\mbox{any
$k_{\mu}=1+N/2$ in $k$.}\end{cases}$ (26)

where we define $M(k)\equiv
m+4r-r\sum_{\mu}\cos\left(\frac{2\pi(k_{\mu}-1)}{N}\right)$ for
$k=(k_{1},k_{2},k_{3},k_{4})$ in Eq. (25). Hence, the number of zero
eigenvalues of Dirac matrix $\mathcal{D}_{W}$ depends on mass parameter $m$
and Wilson-fermion parameter $r$. If we set $m=0$ and $r=1$, the number of
zero eigenvalues is just one. In this case, we do not need to take a continuum
and a thermodynamic limit as long as we take an even number as $N$. This
result is consistent with the known result of Wilson fermion.

##  V Domain-wall fermion

###  V.1 Dirac matrix and mass matrix of domain-wall fermion

In this section we discuss the Dirac operator matrix of domain-wall fermions
in terms of SGT. The domain-wall fermion action in five dimensions (4+1
dimensions) is

$\begin{split}S_{DW}=&\sum_{n,s}\bar{\psi}_{n,s}\left[\sum_{\mu}\gamma_{\mu}D_{\mu}\psi_{n,s}+\gamma_{5}D_{s}\psi_{n,s}\right]-M_{0}\sum_{n,s}\bar{\psi}_{n,s}\psi_{n,s}\\\
&+\sum_{n,s}\bar{\psi}_{n,s}\left[\sum_{\mu}\left(1-C_{\mu}\right)\psi_{n,s}+\left(1-C_{s}\right)\psi_{n,s}\right]\end{split}$
(27)

where $D_{s}\psi_{n,s}\equiv(\psi_{n,s+1}-\psi_{n,s-1})/2,\
C_{s}\psi_{n,s}\equiv(\psi_{n,s+1}+\psi_{n,s-1})/2$ and $M_{0}$ is a mass
parameter. The sum $\sum_{n,s}$ is the summation over five-dimensional lattice
sites $n,s=(n_{1},n_{2},n_{3},n_{4};s)$. Note that there are two differences
regarding boundary conditions and dynamical (or non-dynamical) variables
between the fifth direction and other directions:

  * •

On the fifth direction $s$, we impose the Dirichlet boundary condition,
$\psi_{n,0}=\psi_{n,N_{s}+1}=0$ , and its interval is $1\leq s\leq N_{s}$. We
impose a periodic boundary condition on other directions $n$ and those
intervals are $1\leq n_{\mu}\leq N$.

  * •

The link variables in the fifth direction are not dynamical while those in
other directions can be dynamical.

Based on a similar discussion in Sec. III, the action of a free domain-wall
fermion can be rewritten as
$S_{DW}=\sum_{s,t}\bar{\psi}_{s}\left[\mathcal{D}_{DW}+\mathcal{M}_{DW}\right]_{st}\psi_{t}$
for $s,t=1,2,\cdots,N_{s}$, where $\mathcal{D}_{DW}$ is the Dirac matrix of
domain-wall fermion and $\mathcal{M}_{DW}$ is a mass matrix of domain-wall
fermion from four-dimensional viewpoints. The Dirac matrix $\mathcal{D}_{DW}$
is

$\begin{split}\left[\mathcal{D}_{DW}\right]_{st}=&\delta_{st}\cdot\bigl{(}\,\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes
P_{N}\otimes\gamma_{1}+\bm{1}_{N}\otimes\bm{1}_{N}\otimes
P_{N}\otimes\bm{1}_{N}\otimes\gamma_{2}\\\ &+\bm{1}_{N}\otimes
P_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\gamma_{3}+P_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\gamma_{4}\,\bigr{)}\end{split}$
(28)

and the mass matrix $\mathcal{M}_{DW}$ is

$\begin{split}\left[\mathcal{M}_{DW}\right]_{st}=&\frac{1}{2}\Delta_{st}^{(-)}\cdot\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\gamma_{5}\\\
&+\delta_{st}\bigl{(}\,-M_{0}\cdot\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{4}\\\
&+\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes
M_{W}\otimes\bm{1}_{4}+\bm{1}_{N}\otimes\bm{1}_{N}\otimes
M_{W}\otimes\bm{1}_{N}\otimes\bm{1}_{4}\\\ &+\bm{1}_{N}\otimes
M_{W}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{4}+M_{W}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{4}\,\bigr{)}\\\
&+\frac{1}{2}\left(2\delta_{st}-\Delta_{st}^{(+)}\right)\cdot\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{4}\end{split}$
(29)

where
$\Delta_{st}^{(\pm)}=\sum_{i=1}^{N-1}\delta_{si}\delta_{i+1\,t}\pm\sum_{l=2}^{N}\delta_{sl}\delta_{l-1\,t}$.
Note that we specify the order of components $(\psi_{n})_{s}$ in the vector
$\psi_{s}$ as
$(1,1,1,1;s)\to\cdots\to(N,1,1,1;s)\to(1,2,1,1;s)\to\cdots\to(N,N,1,1;s)\to(1,1,2,1;s)\to\cdots\to(N,N,N,N;s)$
in a descending order. For simplicity, we introduce the chiral projection
$\mathcal{P}_{R}$ and $\mathcal{P}_{L}$ for the mass matrix
$\mathcal{M}_{DW}$. The matrix $\mathcal{M}_{DW}$ can then be rewritten as

$\displaystyle\left[\mathcal{M}_{DW}\right]_{st}=\mathcal{M}_{st}^{(L)}\otimes\mathcal{P}_{L}+\mathcal{M}_{st}^{(R)}\otimes\mathcal{P}_{R}$
(30)
$\displaystyle\begin{split}\mathcal{M}_{st}^{(L)}=&\delta_{st}\bigl{\\{}\,\left(1-M_{0}\right)\cdot\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\\\
&+\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes
M_{W}+\bm{1}_{N}\otimes\bm{1}_{N}\otimes M_{W}\otimes\bm{1}_{N}\\\
&+\bm{1}_{N}\otimes
M_{W}\otimes\bm{1}_{N}\otimes\bm{1}_{N}+M_{W}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\,\bigr{\\}}\\\
&-\sum_{i=1}^{N-1}\delta_{si}\delta_{i+1\,t}\cdot\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\end{split}$
(31)
$\displaystyle\begin{split}\mathcal{M}_{st}^{(R)}=&\delta_{st}\bigl{\\{}\,\left(1-M_{0}\right)\cdot\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\\\
&+\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes
M_{W}+\bm{1}_{N}\otimes\bm{1}_{N}\otimes M_{W}\otimes\bm{1}_{N}\\\
&+\bm{1}_{N}\otimes
M_{W}\otimes\bm{1}_{N}\otimes\bm{1}_{N}+M_{W}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\,\bigr{\\}}\\\
&-\sum_{l=2}^{N}\delta_{sl}\delta_{l-1\,t}\cdot\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\end{split}$
(32)

where $\mathcal{P}_{R}\equiv(\bm{1}_{4}+\gamma_{5})/2$ is a projection matrix
into right-handed components and
$\mathcal{P}_{L}\equiv(\bm{1}_{4}-\gamma_{5})/2$ is into left-handed
components.

###  V.2 Graphs corresponding to domain-wall fermion

We now discuss a graph corresponding to domain-wall fermion in two dimensions
($1+1$ dimensions). For two-dimensional domain-wall fermion, the Dirac matrix
is given as

$\left[\mathcal{D}_{DW}^{\mathrm{1d}}\right]_{st}=\delta_{st}P_{N}\otimes\gamma_{1}$
(33)

and the mass matrix is

$\begin{split}\left[\mathcal{M}_{DW}^{\mathrm{1d}}\right]_{st}&=\frac{1}{2}\Delta_{st}^{(-)}\cdot\bm{1}_{N}\otimes\gamma_{3}+\delta_{st}\left(-M_{0}\bm{1}_{N}\otimes\bm{1}_{4}+M_{W}\otimes\bm{1}_{4}\right)+\frac{1}{2}\left(2\delta_{st}-\Delta_{st}^{(+)}\right)\bm{1}_{N}\otimes\bm{1}_{4}\\\
&={\mathcal{M}_{st}^{\mathrm{1d}}}^{(L)}\otimes\mathcal{P}_{L}^{\mathrm{1d}}+{\mathcal{M}_{st}^{\mathrm{1d}}}^{(R)}\otimes\mathcal{P}_{R}^{\mathrm{1d}}\end{split}$
(34)

where

$\displaystyle{\mathcal{M}_{st}^{\mathrm{1d}}}^{(L)}=\delta_{st}\left\\{\left(1-M_{0}\right)\bm{1}_{N}+M_{W}\right\\}-\sum_{i=1}^{N-1}\delta_{si}\delta_{i+1\,t}\bm{1}_{N}$
(35)
$\displaystyle{\mathcal{M}_{st}^{\mathrm{1d}}}^{(R)}=\delta_{st}\left\\{\left(1-M_{0}\right)\bm{1}_{N}+M_{W}\right\\}-\sum_{l=2}^{N}\delta_{sl}\delta_{l-1\,t}\bm{1}_{N}$
(36)
$\displaystyle\mathcal{P}_{L}^{\mathrm{1d}}=\frac{1}{2}\left(\bm{1}_{4}-\gamma_{3}\right),\quad\mathcal{P}_{R}^{\mathrm{1d}}=\frac{1}{2}\left(\bm{1}_{4}+\gamma_{3}\right)\,.$
(37)

We adopt $\gamma_{3}$ as the extra dimensional gamma matrix. A graph
corresponding to the Dirac matrices is Fig. 9. The schematicl shape of this
graph is paetially similar to the graph in one-dimensional Wilson fermion.
There is also a loop with the weight $M=2-M_{0}$ for each vertex. Each edge
from $n$ to $n+\hat{1}$ has the weight $+(\gamma_{1}-\bm{1}_{4})/2$ and the
one from $n$ to $n-\hat{1}$ has the weight $-(\gamma_{1}+\bm{1}_{4})/2$.
However, there is another direction $s$ and each edge from $s$ to $s\pm 1$ has
the weight $\pm(\gamma_{5}\mp\bm{1}_{4})/2$. The two graphs corresponding to
the direction $\mu=1$ and in the extra direction $s$ is producted by Cartesian
product $\Box$. It is notable that a graph in the direction $\mu=1$ is
equivalent to a graph in one-dimensional Wilson fermion.

Figure 9: This digraph corresponds to the Dirac matrix and mass matrix in the
two-dimensional domain-wall fermion. The left graph is equivalent to one-
dimensional Wilson fermion. On the other hand, the right graph corresponds to
the extra direction $s$. The weight $+_{s}$ means $+(\gamma_{3}-\bm{1}_{4})/2$
and $-_{s}$ means $-(\gamma_{3}+\bm{1}_{4})/2$. $\Box$ stands for the
Cartesian product.

For the five-dimensional domain-wall fermion, a graph also consists of the two
graphs, a graph corresponding to the four dimensional Wilson fermion and a
graph in the extra direction $s$, which are unified by Cartesian product.

###  V.3 Diagonalization of Dirac matrix $\mathcal{D}_{DW}$ and mass matrix
$\mathcal{M}_{DW}$

Firstly, We diagonalize the Dirac matrix $\mathcal{D}_{DW}$ in a parallel way
to Sec. III. By using the unitary matrix $\mathcal{U}$, the diagonalized Dirac
matrix is given as

$\bigl{[}\mathcal{U}^{\dagger}\mathcal{D}_{DW}\,\mathcal{U}\bigr{]}_{st}=\delta_{st}\sum_{k_{4},k_{3},k_{2},k_{1}}\left[i\sum_{\mu=1}^{4}\sin\left(\frac{2\pi(k_{\mu}-1)}{N}\right)\gamma_{\mu}\right]\ket{k_{4},k_{3},k_{2},k_{1}}\bra{k_{4},k_{3},k_{2},k_{1}}$
(38)

for $k_{\mu}=1,2,\cdots,N$. As a result, the solutions of $k_{\mu}$ for the
Dirac matrix $\mathcal{D}_{DW}$ to have zero eigenvalues are given by
$k_{\mu}=1,1+N/2$.

Secondly, we will diagonalize the mass matrix $\mathcal{M}_{DW}$. The
diagonalized mass matrix is

$\begin{split}\bigl{[}\,\mathcal{U}^{\dagger}\mathcal{M}_{DW}\,\mathcal{U}\,\bigr{]}_{st}=\
&\mathcal{U}^{\dagger}\left(\mathcal{M}_{st}^{(L)}\otimes\mathcal{P}_{L}\right)\mathcal{U}+\mathcal{U}^{\dagger}\left(\mathcal{M}_{st}^{(R)}\otimes\mathcal{P}_{R}\right)\mathcal{U}\\\
=&\sum_{k_{4},k_{3},k_{2},k_{1}}\left[\left\\{W(k)\delta_{st}-\sum_{i=1}^{N-1}\delta_{si}\delta_{i+1\,t}\right\\}\mathcal{P}_{L}+\left\\{W(k)\delta_{st}-\sum_{l=2}^{N}\delta_{sl}\delta_{l-1\,t}\right\\}\mathcal{P}_{R}\right]\\\
&\times\ket{k_{4},k_{3},k_{2},k_{1}}\bra{k_{4},k_{3},k_{2},k_{1}}\end{split}$
(39)

where $W(k)\equiv
1-M_{0}+\sum_{\mu}\left\\{1-\cos\left(\frac{2\pi(k_{\mu}-1)}{N}\right)\right\\}$.
We now impose conditions, $|W(k)|<1$and $N_{s}=\infty$ so that mass matrix
$\mathcal{M}_{DW}$ has zero eigenvalues. We can determine the range of mass
parameter $M_{0}$ from $|W(k)|<1$.Using the solutions of $k_{\mu}$, $W(k)$ can
be classified as

$W(k)=\begin{cases}1-M_{0}&\mbox{any $k_{\mu}=1$ in $k$.}\\\ 3-M_{0}&\mbox{one
$k_{\mu}=1+N/2$ otherwise $k_{\mu}=1$ in $k$.}\\\ 5-M_{0}&\mbox{two
$k_{\mu}=1+N/2$ otherwise $k_{\mu}=1$ in $k$.}\\\ 7-M_{0}&\mbox{three
$k_{\mu}=1+N/2$ otherwise $k_{\mu}=1$ in $k$.}\\\ 9-M_{0}&\mbox{any
$k_{\mu}=1+N/2$ in $k$.}\end{cases}$ (40)

Therefore, the relation between the solutions of $k_{\mu}$, the range of mass
parameter $M_{0}$, and the number of zero eigenvalues are given in Table. 1.
The result is consistent with the known results of the domain-wall fermion. It
indicates that our method can be applied to the lattice fermion with Dirichlet
boundary conditions.

Table 1: Classification of the number of zero eigenvalues in DW fermion the solutions of $k_{\mu}$ | the range of $M_{0}$ | the number of zero eigenvalues
---|---|---
any $k_{\mu}=1$ | $0<M_{0}<2$ | 1
one $k_{\mu}=1+N/2$ otherwise $k_{\mu}=1$ | $2<M_{0}<4$ | 4
two $k_{\mu}=1+N/2$ otherwise $k_{\mu}=1$ | $4<M_{0}<6$ | 6
three $k_{\mu}=1+N/2$ otherwise $k_{\mu}=1$ | $6<M_{0}<8$ | 4
any $k_{\mu}=1+N/2$ | $8<M_{0}<10$ | 1

##  VI Lattice fermions on four dimensional hyperball

In Sec. III,IV,V, we show that there are the spectral digraphs corresponding
to the Dirac matrices of the naive fermion, Wilson fermion, and domain-wall
fermion. Then, we find out the number of fermion species based on the discrete
Fourier transformation.

Our question is whether we can obtain the Dirac matrix when the spectral
digraph corresponding to a lattice fermion is given. In this section, we will
construct the Dirac matrix from a digraph corresponding to a certain lattice
fermion and derive the number of fermion species by use of DFT. Through this
procedure, we can construct the Dirac matrix for a lattice fermion defined on
complicated lattices, where the momentum cannot be defined. Then, we can
derive the number of the species for fermions on such lattices.

From the next subsection we consider the discretized four-dimensional
hyperball $B^{4}$. This space is obtaining by imposing the Dirichlet boundary
condition on each direction of four dimensions.

###  VI.1 Graphs corresponding to the Dirac matrix on hyperball

We discuss the graph corresponding to the fermion defined on the four-
dimensional hyperball. For simplicity, we first study a graph corresponding to
lattice fermion on the discretized one-dimensional ball $B^{1}$ (or line
segment). The digraph corresponding to the discretized $B^{1}$ is given as
Fig. 10.

Figure 10: This digraph corresponds to the lattice fermion on the discretized
$B^{1}$.

Note that the hopping between two ends in position space is prohibited. The
Dirac matrix corresponding to this graph is given by

$\mathcal{D}_{B^{1}}=Q_{N}\otimes\gamma_{1}$ (41)

where
$(Q_{N})_{ab}\equiv\left(\sum_{i=1}^{N-1}\delta_{ai}\delta_{i+1\,b}-\sum_{l=2}^{N}\delta_{al}\delta_{l-1\,b}\right)/2$.
This matrix $Q_{N}$ is written as

$Q_{N}=\frac{1}{2}\begin{pmatrix}0&1&0&&0&0&0\\\ -1&0&1&\cdots&0&0&0\\\
0&-1&0&&0&0&0\\\ &\vdots&&\ddots&&\vdots&\\\ 0&0&0&&0&1&0\\\
0&0&0&\cdots&-1&0&1\\\ 0&0&0&&0&-1&0\end{pmatrix}$ (42)

The prohibition of hopping in the two ends is represented by the two elements
$(1,N)$ and $(N,1)$ which are zero in this case.

For the discretized two-dimensional ball $B^{2}$ (or disk), the digraph is
given as Fig. 11.

Figure 11: This digraph corresponds to fermions on the discretized $B^{2}$. As
with the one-dimensional case, the weight of blue edge is $+\gamma_{\mu}/2$
and the one of the red edge is $-\gamma_{\mu}/2$.

The Dirac matrix in two dimensions is given by

$\mathcal{D}_{B^{2}}=\bm{1}_{N}\otimes
Q_{N}\otimes\gamma_{1}+Q_{N}\otimes\bm{1}_{N}\otimes\gamma_{2}.$ (43)

As seen from Figs. 10 and 11, the translational symmetry is partially broken
since there is a boundary.

Through a parallel discussion, we find the shape of the digraph corresponding
to fermions on the discretized three dimensional ball $B^{3}$ become a box,
where the sites in this space is expressed by $(n_{1},n_{2},n_{3})$ whith
$1\leq n_{i}\leq N$. The Dirac matrix on the discretized $B^{3}$ is given as

$\mathcal{D}_{B^{3}}=\bm{1}_{N}\otimes\bm{1}_{N}\otimes
Q_{N}\otimes\gamma_{1}+\bm{1}_{N}\otimes
Q_{N}\otimes\bm{1}_{N}\otimes\gamma_{2}+Q_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\gamma_{3}.$
(44)

In a similar manner we can construct the digraph and the corresponding Dirac
matrix on the discretized four dimensional hyperball $B^{4}$. In the next
subsection, we will investigate the lattice action on the discretized $B^{4}$.

###  VI.2 Lattice action on four-dimensional hyperball

The free naive fermion action on four-dimensional discretized hyperball
$B^{4}$ is

$S_{B^{4}}=\sum_{n}\sum_{\mu}\bar{\psi}_{n}\gamma_{\mu}D_{\mu}\psi_{n}\,,$
(45)

where the lattice sites are $n=(n_{1},n_{2},n_{3},n_{4})$ for $1\leq n_{i}\leq
N$ and Dirichlet boundary conditions are imposed on each direction. This
action is also expressed as $S_{B^{4}}=\bar{\psi}\mathcal{D}_{B^{4}}\psi$. The
Dirac matrix $\mathcal{D}_{B^{4}}$ is

$\begin{split}\mathcal{D}_{B^{4}}&=\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes
Q_{N}\otimes\gamma_{1}\\\ &+\bm{1}_{N}\otimes\bm{1}_{N}\otimes
Q_{N}\otimes\bm{1}_{N}\otimes\gamma_{2}\\\ &+\bm{1}_{N}\otimes
Q_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\gamma_{3}\\\
&+Q_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\gamma_{4}.\end{split}$
(46)

###  VI.3 Number of species on four-dimensional hyperball

Firstly, we will discuss diagonalization of the matrix $Q_{N}$ so as to derive
the number of species on $B^{4}$. This matrix can be diagonalized by use of a
unitary matrix $Y$ defined as $(Y)_{ab}=\alpha
i^{b}\sin\left(\frac{ab\pi}{N+1}\right)$, where $\alpha$ is a normalization
coefficient Noschese2013 ; Gover1994 . By using the unitary matrix, the
diagonalized matrix is given as

$Q_{N}Y=i\mathrm{Diag}\left[\cos\left(\frac{\pi}{N+1}\right),\,\cos\left(\frac{2\pi}{N+1}\right),\,\cdots,\,\cos\left(\frac{N\pi}{N+1}\right)\right]\equiv\Lambda_{Q_{N}}X\,.$
(47)

Note that there is a single zero eigenvalue in $\Lambda_{Q_{N}}$ at most when
the number of sites in each direction $N$ is odd number. The Dirac matrix
$\mathcal{D}_{B^{4}}$ is diagonalized as

$\begin{split}\mathcal{V}^{\dagger}\,\mathcal{D}_{B^{4}}\,\mathcal{V}&=\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\Lambda_{Q_{N}}\otimes\gamma_{1}\\\
&+\bm{1}_{N}\otimes\bm{1}_{N}\otimes\Lambda_{Q_{N}}\otimes\bm{1}_{N}\otimes\gamma_{2}\\\
&+\bm{1}_{N}\otimes\Lambda_{Q_{N}}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\gamma_{3}\\\
&+\Lambda_{Q_{N}}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\gamma_{4}\end{split}$
(48)

where $\mathcal{V}$ is a unitary matrix defined as
$\mathcal{V}\equiv\bigotimes_{\mu=1}^{4}Y\otimes\bm{1}_{4}$. To study
eigenvalues in the Dirac matrix $\mathcal{D}_{B^{4}}$, we introduce
eigenvector $\ket{\lambda}$ satisfying
$Q_{N}\ket{\lambda}=i\cos\left(\frac{\lambda\pi}{N+1}\right)\ket{\lambda}$ for
$\lambda=1,2,\cdots,N$. The diagonalized Dirac matrix is then written

$\mathcal{V}^{\dagger}\,\mathcal{D}_{B^{4}}\,\mathcal{V}=\sum_{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}}\left[i\sum_{\mu=1}^{4}\cos\left(\frac{\lambda_{\mu}\pi}{N+1}\right)\gamma_{\mu}\right]\ket{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}}\bra{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}}$
(49)

where
$\ket{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}}=\ket{\lambda_{1}}\otimes\ket{\lambda_{2}}\otimes\ket{\lambda_{3}}\otimes\ket{\lambda_{4}}$.
For the diagonal matrix
$\mathcal{V}^{\dagger}\,\mathcal{D}_{B^{4}}\,\mathcal{V}$ to have zero
eigenvalues, the following equation must be satisfied,

$\sum_{\mu}\cos\left(\frac{\lambda_{\mu}\pi}{N+1}\right)\gamma_{\mu}=0\,.$
(50)

Furthermore, since $\gamma$ matrices is linearly independent, this equation is
rewritten as

$\cos\left(\frac{\lambda_{\mu}\pi}{N+1}\right)=0.$ (51)

The solution of Eq. 51 is $\lambda_{\mu}=\frac{N+1}{2}$. If the number of
sites in the each direction is even, there is no zero eigenvalues in the Dirac
matrix. As a result, there are no solutions $\lambda_{\mu}$ both satisfying
Eq. 51 and belonging to the set of integer, $\\{1,2,\cdots,N\\}$. On the other
hand, when $N$ is the odd number, there is a single zero eigenvalue.
Therefore, there is one physical pole on the bulk of four-dimensional
hyperball when the number of sites in each direction is the odd number. If we
take a thermodynamical limit for even $N$, one of the non-zero eigenvalue
approaches to zero. This result is true in any dimensional lattice since
$\gamma$ matrices in any dimension is linearly independent. Thus, lattice
fermions on the finite-volume lattice of $d$-dimensional hyperball $B^{d}$
have one physical pole on the bulk.

As is well-known, the lattice fermion defined on a lattice with boundaries can
have edge modes on the boundaries. Therefore, the edge mode works to cancel
the gauge anomaly at the boundary when we introduce gauge fields or link
variables. The existence of a single Dirac mode on the bulk in the present
lattice fermion with boundaries is reasonable as with the case of the domain-
wall fermion.

The results we have obtained in this section are again not so novel. However,
we have shown that we can easily find the number of zero eigenvalues or the
number of species by obtaining the spectral digraph corresponding to the
lattice fermions and taking the procedure of discrete Fourier transformation.

##  VII Summary and Discussion

In this note, we have introduced a new insight into lattice fermions based on
spectral graph theory (SGT). By use of SGT and discrete Fourier transformation
(DFT), we can figure out the number of zero eigenvalues of lattice Dirac
operators without considering the momentum space. Then, we can understand the
number of doublers on non-torus lattices with arbitrary topologies. The most
important fact in this analysis is that a lattice fermion on a certain lattice
corresponds to one spectral graph. Then, Dirac operators of lattice fermions
can be treated as matrices corresponding to graphs. Once we have a certain
lattice and a Dirac operator defined on the lattice, we can derive the
corresponding matrix and find the number of zero eigenvalues by use of the
discrete Fourier transformation (DFT), leading to understanding of the number
of doublers. We have applied this procedure to the known fermion formulations
including Naive fermion, Wilson fermion and Domain-wall fermion, and discuss
the number of doublers in terms of SGT. As the first non-trivial application,
we have studied lattice fermions on the discretized hyperball and discussed
the number of fermion species on the bulk.

We now discuss further application of SGT to lattice fermions;

  1. 1.

Application to discretized sphere:

It is possible to apply SGT to lattice fermions on discretized spheres via the
method in Sec. VI. The $d$ dimensional sphere can be realized by eliminating
the bulk of $d+1$ dimensional hyperball $B^{d+1}$. By using SGT, the Dirac
matrix in this situation is simply obtained since what we have to do is to
eliminate the matrix elements corresponding to the bulk. We will discuss this
case in the future work soon.

  2. 2.

A new theorem on the lattice with arbitrary Euler characteristics:

The method based on SGT can be applied to the lattices with arbitrary Euler
characteristics since we simply construct the graphs corresponding discretized
manifolds with arbitrary topology. As a result, we may be able to conjecture a
new theorem which tells us the number of species on more generic lattices by
studying lattice fermions based on SGT. One of keys for such a theorem is the
topology of the graph itself. We will also discuss this point in the future
work soon.

  3. 3.

Novel lattice fermions:

We can propose novel lattice fermions by translating a matrix with desirable
properties (minimal zero modes, hermiticity or chirality) to a spectral graph,
which corresponds to the lattice fermion. A fermion obtained by this procedure
may correspond to a lattice fermion defined on non-torus lattice.

###### Acknowledgements.

T. M. is grateful to K. Ohta, S. Matsuura and S. Kamata for the fruitful
discussion on the relation between graph theory and lattice field theory. This
work of T. M. is supported by the Japan Society for the Promotion of Science
(JSPS) Grant-in-Aid for Scientific Research (KAKENHI) Grant Numbers 19K03817
and 18H01217.

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# Emergent Chirality and Current Generation

Satyanu Bhadra  Department of Condensed Matter Physics and Materials Science,
Tata Institute of Fundamental Research, Mumbai 400005, India  Shankar Ghosh
Department of Condensed Matter Physics and Materials Science,
Tata Institute of Fundamental Research, Mumbai 400005, India  Shamik Gupta
Department of Physics, Ramakrishna Mission Vivekananda Educational and
Research Institute, Belur Math, Howrah 711202, India,
Regular Associate, Quantitative Life Sciences Section, ICTP - The Abdus Salam
International Centre for Theoretical Physics, Strada Costiera 11, 34151
Trieste, Italy

(August 27, 2024)

###### Abstract

We investigate the phenomenon of producing vibration-induced rotational motion
in a cylinder filled with achiral rods and being vibrated by an external
drive. The arrangement of the rods develops chirality as a result of the
interaction of gravity and steric hindrance, which is responsible for the
induced motion. A two-rod arrangement is sufficient to generate a persistent
motion. The average angular velocity $\langle\omega\rangle$ of the rods at
long times varies non-monotonically with the packing fraction $\phi$ for a
given drive strength $\Gamma$. Though the precise nature of the variation of
$\langle\omega\rangle$ with $\phi$ depends on the details of the interaction
between individual rods, its general characteristics hold true for rods of
various materials and geometries. A stochastic model based on the asymmetric
simple exclusion process helps in understanding the key features of our
experiments.

###### pacs:

Valid PACS appear here

It is common knowledge that when a system is subjected to an external field
that causes a current to flow in it, it behaves differently from when it is
not. Presence of current in the system implies that the underlying dynamics no
longer satisfies time-reversal symmetry. The external field induces biased
transitions between configurations by effectively rendering a subset of
configurations in the energy landscape to be higher in energy than the others.
The scenario described above is best visualized in a system of interacting
particles, which in the thermodynamic limit develops an effective single-
particle potential energy landscape that is tilted due to the application of
an external field to have two minima separated by an energy barrier. An
alternative method of generating current would be to use extended objects that
break structural symmetry, so that when a field is applied, these objects move
in a preferred direction. Chiral objects are an example of the aforementioned
type. These objects convert linear momentum to angular momentum and thus twist
and turn as they attempt to move under the influence of an external field [1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. A question of pertinent interest
is then: Can an external field drive current in a system of achiral particles
interacting only through steric hindrance, and that too not in the
thermodynamic limit but for a system of a few particles?

The aforesaid question is answered in the affirmative in this work: In a
system of rods confined to a cylindrical container, we show that vertical
shaking of the cylinder can spontaneously generate a chirality in the system,
leading to a rotational current whose direction is determined by the
handedness of the developed chirality. The experimental setup is depicted in
Fig. 1. The radius of the cylinder is $R$, while its height is $H$. This
cylinder is filled with $N$ identical rods of length $h$ and width $2r$ that
are achiral. To impart regulated vibrations to the rods, we employ an
electromagnetic shaker that vibrates the cylinder in the vertical (i.e., in
the $z$) direction with frequency $\Omega_{s}$ and amplitude $A$. The
parameter $\Gamma\equiv A\Omega_{s}^{2}/g$ sets the strength of this external
drive. We find that a rotational current is generated in the system so long as
we have $r<R$, $h>2R$, and the parameter $\Gamma$ is less than a critical
value. We performed experiments with objects of varying $h$ and found that
experimental results varied weakly. If $h>>H$, the object can topple over the
container when shaken. When subject to the drive, and when one has a few rods
in the cylinder, the system settles into a chiral arrangement with the ends of
the rods resting on the top and the bottom rim of the cylindrical container.
Despite the fact that the Euclidean geometry of the rods is non-chiral,
presence of gravity lowers the symmetry group associated with the geometry of
the rods. Chirality arises from the fact that the reduced group has fewer
connected components.

We now discuss the coordinate description of the problem, for the
representative two-rod case where the rods are in contact. We label the rod on
top at the point of contact as $a$ and the one below as $b$. In terms of
cylindrical polar coordinates, the end points of rod $a$ are
$[(R,\vartheta_{(a,L)},0),(R,\vartheta_{(a,U)},H)]$ and the end points of rod
$b$ are $[(R,\vartheta_{(b,L)},0),(R,\vartheta_{(b,U)},H)]$. Here, the $L$ and
$U$ indices in the subscript of $\vartheta$ describe the lower and upper end
points of the rods. The lower end of the rods rests on the bottom rim of the
container, while the upper end of the rods rests on the upper rim. If rod $a$
needs to be turned in a clockwise (respectively, anticlockwise) direction so
as to make it parallel to rod $b$, we identify the structure with negative
(respectively, positive) chirality. Zero chirality represents configurations
where the rods are parallel to each other.

In the following, we discuss in turn our experimental observations made with
one, two, and more than two rods in our setup. To study the dynamics of the
rods in time, we track the motion of a single tagged rod. From the top view,
the motion of the top of this tagged rod appears to correspond to the movement
of a point on a disk whose center lies along the axis of the container. It is
thus natural to characterize the motion of this rod in terms of polar
coordinates $\mathcal{S}(t),\theta(t)$. The angular coordinate $\theta(t)$
describes the instantaneous angular position of this tagged rod on this disk
(see Fig. 1, panels (d),(e)). From the time series $\theta(t)$, we construct
the cumulative angular distance $\Theta(t)$, where $\theta(t)={\rm
mod}|\Theta(t),2\pi|$. The tangential component $e_{H}$ of the axial vector in
the direction of $\hat{\theta}$ changes as the container fills up with more
rods, and is correlated with the angular velocity of the rods; This is
discussed in detail later in the paper. The tangential component can be easily
computed from images such as Fig. 1, panels (d) and (f).

Figure 1:  (a) A typical experimental scenario in which two pencils are placed
in a container. These pencils can be made to move in one direction by tapping
the container. The relative arrangement of the pencils determines the
direction. (b) A schematic representation of the experiment, including all
relevant coordinates. (c) A cylinder of earbuds generates a doubly-ruled
hyperbolic surface. Take note of the profile of the earbuds’ top-ends; the
rods on the inside are straighter than the ones on the outside. (d) A typical
snapshot of the system of rods being driven. From the top view, we can track
the top of the tagged particle, colored red. The unit vector $\hat{e}_{a}$
lies along the rod, with the tangential vector being $e_{H}\hat{\theta}$. (e)
A schematic showing the mapping of the $x-y$ coordinates at time $t$ of the
tagged particle top in Fig 1(d) onto a disk geometry, with relevant
coordinates $\mathcal{S}(t)$ and $\theta(t)$ (f) The component $e_{H}$ can be
measured by the angle made by a rod with the wall of the container. A single
toothpick and corresponding $e_{H}$ is highlighted. Figure 2: (a) Time trace
of the angular distance $\Theta(t)$ of a tagged rod in a system of $N$ rods,
for $N=2$ and various values of $\Gamma$. The case $N=1$ is also shown. (b)
Long-time average of the angular velocity, denoted by $\langle\omega\rangle$,
increases with $\Gamma$ until $\Gamma_{c}$, after which continuous chirality
toggling causes $\langle\omega\rangle$ to plummet. Inset: The variation of
$\Gamma_{c}$ vs $r/A$ for different materials of varying thickness. The red
line represents $\Gamma_{c}=r/A$. (c) The toothpicks move noisily, with the
amount of noise decreasing as $N$ increases. This is seen in the time trace of
the angular velocity $\omega$. (d) The variation of $\langle\omega\rangle$
with packing fraction $\phi$ for toothpicks ($r=2.2$mm, $h=60$mm, $R=45$mm,
$H=75$ mm). Note that the error bars reduce with increase in $\phi$. The
panels (e) and (f) show the variation of $MSD$ vs lag time $\Delta t$ for
$\Gamma<\Gamma_{c}$ and $\Gamma\geq\Gamma_{c}$ respectively.

##  Observations for $N=1$:

Chirality is an emergent property that arises from the arrangement of two or
more rods. The rods used here are not by themselves chiral. On being driven, a
single rod does not exhibit any persistent angular motion (see Fig. 2(a),
$N=1$ ).

##  Observations for $N=2$:

In terms of $\Theta(t)$, the trajectory of a tagged rod for various values of
$\Gamma$ is shown in Fig. 2(a). Occurrence of a persistent rotational current
necessitates the use of at least two rods. The direction of rotation of two
rods is determined by the instantaneous chirality in the system. Constant
toggling between states with two different signs of chirality results in a
continuous current reversal and a concomitant change in sign of the slope of
$\Theta(t)$ versus $t$. The rate of toggling increases as $\Gamma$ increases,
and there exists a $\Gamma_{c}$ beyond which the long-time average
$\langle\omega\rangle$ of the angular velocity starts to drop, as seen in Fig.
2(b). By measuring the mean square displacement $MSD$ as a function of the lag
time $\Delta t$, we see that for $\Gamma<\Gamma_{c}$, the motion is completely
ballistic ($MSD\propto(\Delta t)^{2}$) as seen in Fig 2(e). For
$\Gamma\geq\Gamma_{c}$, the motion is ballistic at shorter timescales, and
makes a crossover into diffusive motion ($MSD\propto\Delta t$) for longer
timescales, as seen in 2(f).

Figure 3: (a) Snapshots of rod packing with increase in number $N$ of rods. As
we load more rods, the tendency to form clusters rather than equidistant
structures increases. Well-defined shells of rods are observed beyond a
certain number of rods. (b) Variation of long-time average
$\langle\omega\rangle$ of the angular velocity with radial distance
$\mathcal{S}$, measured along the radius of the container, for various packing
fractions $\phi$, highlighting the presence of a velocity gradient as one
travels from the container’s center to the edge. This gradient decreases with
increasing $\phi$. (c) The variation of the autocorrelation function
$C(\mathcal{R},\Delta t)$ for various $\phi$ and respective exponential fits
$g(t)=\exp(-t/\tau)$. Inset: Variation of $\tau$ with $\phi$. (d) Variation of
$e_{H}/R$ and $\langle\omega\rangle$ with $\phi$. All experiments were
performed using toothpicks ($r=2.2$mm, $h=60$mm, $H=75$mm, $R=37$mm).

## Observations for $N>2$:

As the number of rods in the container increases beyond two, the arrangement
of the rods evolves from equidistant regular structures (see the case $N=2$ in
Fig. 3(a)) to a single shell that resembles a doubly-ruled hyperbolic surface
(see the case $N=30$ in Fig. 3(a) and Fig. 1(c)). When the number is further
increased, multiple concentric shells are formed. For low values of the
packing fraction $\phi\equiv N(r/R)^{2}$, the rods move from one shell to the
next as these shells shear against each other. The presence of shearing
between shells is demonstrated by a gradient in $\langle\omega\rangle$ as a
function of the radial distance $\mathcal{S}$ measured along the radius of the
container (Fig. 3(b)). As the packing fraction $\phi$ increases,
$\langle\omega\rangle$ becomes increasingly independent of $\mathcal{S}$. The
motion of the rods appears to become more coherent with increase in $\phi$;
$\phi_{\rm rigid}$ refers to the value of $\phi$ at which the shearing between
layers completely disappears. In order to quantify the onset of coherence in
the motion, one may consider studying an appropriate autocorrelation function.
To this end, we map the top of two tagged rods onto two points on a disc whose
coordinates are $(\mathcal{S}_{1}(t),\theta_{1}(t))$ and
$(\mathcal{S}_{2}(t),\theta_{2}(t))$ and measure the Pythagorean distance
between the points. Denoting this distance at time $t$ by $\mathcal{R}(t)$, we
have
$\mathcal{R}^{2}(t)=\mathcal{S}_{1}^{2}(t)+\mathcal{S}_{2}^{2}(t)-2\mathcal{S}_{1}\mathcal{S}_{2}\cos\left(\theta_{1}(t)-\theta_{2}(t)\right).$
The autocorrelation function is defined as $C(\mathcal{R},\Delta
t)\equiv\langle\mathcal{R}(t)\mathcal{R}(t+\Delta t)\rangle$ for time lag
$\Delta t$. For various values of $\phi$, the behavior of the correlation
function is presented in Fig. 3(c). Because the two tagged rods are rotating
within the cylinder, the distance between them is an oscillating function. As
a result, the correlation function as a function of $\Delta t$ shows a decay
with superimposed small oscillations. Fitting the decaying profile to an
exponential yields the correlation time $\tau$. Exponential fit
$g(t)=\exp(-t/\tau)$ to the different correlation functions are represented by
the dashed-dotted lines in Fig. 3(c). As $\phi\rightarrow\phi_{\rm rigid}$,
the system makes a transition from being floppy, where the rods can move
freely with respect to one another, to being rigid, where all the rods move
coherently, and correspondingly, the timescale $\tau$ becomes large as seen in
Fig. 3(c) inset. Similar rigidity transitions have been observed in other
experimental systems [15, 16].

Apart from the linear drift in $\Theta(t)$, there exist oscillations about the
linear function, corresponding to the rattling motion of the rods observed in
experiments. For low densities, each rod can access a large section of the
configuration space. On being driven by the shaker, the injected energy does
not fully contribute to the global motion of the rods, but is used by
individual rods to explore nearby configurations in its allowed configuration
space. This leads to a noisy, rattling motion of the rods at low densities,
making the motion of the individual rods to be essentially random on short
timescales (Fig. 2(c)). The rattling of the rods decreases as the number of
rods in the container increases since the available configuration space for
each rod decreases; the system eventually becomes rigid (Fig. 2(c), (d)).

Each rod is tilted at an agle with respect to the vertical along the wall of
the container. We quantify this tilt as the as tangential component $e_{H}$ of
the axial vector lying along $\hat{\theta}$, where $\hat{\theta}$ is the
normal vector to the radial vector from the centre of the container. $e_{H}$
of the rods changes as the container fills up with more rods. For the rod $a$
shown in Fig.1(b) $e_{H}=R\sin(\vartheta_{(a,U)}-\vartheta_{(a,L)})$. The
quantities $\langle\omega\rangle$ and $e_{H}$ of a tagged rod are depicted as
a function of $\phi$ in Fig. 3(d). Since the rods travel with different speeds
in different layers, we measure the $\langle\omega\rangle$ of a tagged rod
placed in the outermost shell for all experiments. Both $e_{H}$ and
$\langle\omega\rangle$ have similar non-monotonic trends as a function of
packing fraction $\phi$. It is tempting to attribute the increase in
$\langle\omega\rangle$ to the corresponding increase in $e_{H}$. The rods in
the inner shell, on the other hand, are inclined at a lower angle than those
at the edge (Fig. 1(c)). As one advances radially inwards, $e_{H}$ drops,
whereas investigations show that inner shells move faster (Fig. 3(b)). As a
result, $e_{H}$ cannot be the only element influencing particle speed.

The frictional dissipation in the system increases with $\phi$. In the inset
of Fig. 4(a), we present a schematic representation of our experimental set-up
that measures the force of friction between the rods and the wall of the
cylinder. The probe, a toothpick attached to the load cell, is lowered
vertically into the container that already has requisite number of toothpicks
to make the packing fraction $\phi$. To measure the force of friction between
the rods, we insert the probe in the middle of the container, while to measure
the frictional force between the rods and the wall, we ensure that the probe
continuously brushes against the wall of the container. The process of
lowering of the probe is continued till the lower end of the probe comes
within 2 mm of the base of the container. The highest frictional force
recorded between the probe and the rods is $F_{f,b}$, and the force between
the rods and the wall is $F_{f,w}$. Both these measurements are made at the
deepest point of insertion, which is 2 mm from the base of the container. The
experiment was repeated with different $\phi$ values; for $\phi>0.66$ it
becomes increasingly difficult to insert rods into the container, and the
value of both $F_{f,b}$ and $F_{f,w}$ rise steeply (Fig. 4(a)). The rise in
both $F_{f,b}$ and $F_{f,w}$ and the drop in $\langle\omega\rangle$ occur at
values of $\phi$ close to each other. It is to be noted the these two values
$\langle\omega\rangle$ and the forces of friction are measured from different
experiments. As discussed above, while measuring the force of friction, the
system was static. These observations indicate that there are two roles of the
frictional force in the system. The inter-particle friction provides
constraints and hence is closely associated to the rigidity transition of the
system (Fig 3(c)). On the other hand the friction between the rod and the wall
provides resistance to the particle current. At high packing fraction $\phi$,
the friction between wall and the rods exceeds the driving force associated
with the current, hence the current goes to zero (Fig 3(d)).

Figure 4: (a) Variation of the long-time average $\langle\omega\rangle$ of the
angular velocity ($\Gamma=2$) and the frictional force $F_{f}$ with packing
fraction $\phi$ for toothpicks ($r=2.2$mm, $h=60$mm, $H=75$mm, $R=37$mm).
Inset shows a schematic of the setup used to measure $F_{f}$, where a probe
attached to a load cell is inserted into the container full of $N$ rods. (b)
On plotting $\langle\omega\rangle/\omega_{\rm max}$ vs $\phi/\phi_{c}$, a good
scaling relationship for the rise is observed. (c,d) $\omega_{\rm max}$ is
dependent on both the material and the diameter, whereas $\phi_{c}$ is not
affected by the material but by only the diameter.

Experiments with different materials of varying sizes and shapes reveal that
the nature of the $\langle\omega\rangle$ vs. $\phi$ relationship is always the
same. The following materials were used in our setup: Steel rods (SR, $2r$ =
6mm), Toothpicks (TP, $2r$ = 2.2mm), Bamboo skewers with tips (ST, $2r$ =
2.5mm), Bamboo skewers without tips (S/T, $2r$ = 2.5mm), Wood blocks (WB, $2r$
= 6mm), Glass rods (GR, $2r$ = 3.5, 6, 8 mm), Wood blocks (WB, $2r$ = 6mm),
Wood blocks (WB, $2r$ = 6mm), Glass rods (GR, $2r$ = 3.5, 6, 8 mm). The
ordinate is scaled as $\langle\omega\rangle/\omega_{\rm max}$ and the abscissa
is scaled as $\phi/\phi_{c}$ to compare results from different experiments
(Fig. 4(b)). For the region $\phi/\phi_{c}<1$, the figure exhibits good
scaling collapse of the data. It may be seen from Fig. 4(b) that $\omega_{\rm
max}$ is controlled by both the material and the diameter, but $\phi_{c}$ is
only influenced by the diameter and not by the material.

Figure 5:  The graph depicts the energy $E$ of two rods in a cylinder as a
function of $\vartheta_{(b,L)}-\vartheta_{(a,L)}$ and
$\vartheta_{(b,U)}-\vartheta_{(a,U)}$ for all possible allowed configurations.
The red (negative chirality) and black (positive chirality) markers indicate
configurations that satisfy the condition $d=2r$, with $d$ and $r$ being the
shortest distance at point of contact of two rods, and the radius of the rods,
respectively. Refer to Fig. 1(b) for the definition of the $\vartheta_{i}$’s.

## Energetics for the case $N=2$:

The energy of each rod is characterized in terms of the tilt angle, and the
coordinates of the two ends of a rods are used as its geometrical descriptor.
The coordinate description of the problem is depicted in detail in Fig. 1(b).
In these coordinates, the potential energy $E$ of two rods $a$ and $b$ with
tilt angles $\psi_{1}$ and $\psi_{2}$ respectively is
$(mgh/2)(\sin(\psi_{1})+\sin(\psi_{2}))$. The sine function of the tilt angle
$\psi_{1}$ is
$\sin(\psi_{1})=\left(1+\left(\frac{2R}{H}\right)^{2}\sin^{2}\left(\frac{\vartheta_{(a,U)}-\vartheta_{(a,L)}}{2}\right)\right)^{-\frac{1}{2}}$.

In this case, the tilt angles can be expressed in terms of the coordinates of
the end points of the rod. When the angular gap between the top and the bottom
end of a rod is maximized, the energy for one rod is at its minimum. When a
second rod is introduced, the overall energy must be minimized while also
considering the steric repulsion of the two rods in contact. This restriction
is written as $d\geq 2r$, where $d$ is the shortest distance at point of
contact between two rods. The scatter plots in Fig. 5 illustrate the possible
configuration for $R=1$, $H=2$, and $r=0.05$, with the restriction $d\geq 2r$.

The color code represents the total energy $E$ of the two rods. The black and
red dots in the figure show the configurations where the rods come into
contact with $d=2r$. The black and red dots represent positive and negative
chirality, respectively. (i) There are two global minima to be found. Each one
has a chirality of its own. As a result, the configuration space corresponding
to distinct chiralities is divided by an energy barrier. (ii) Permissible
configuration pathways connect these minima. (iii) Because of the energy
introduced by the shaker, the rods can explore neighboring configurations.

The configurations of the rods are constrained in the energy diagram for
smaller values of $\Gamma$ to a specific chirality. As a result, the rods
continue to move in the same direction. The energy supplied by the shaker, on
the other hand, is sufficient to toggle the chirality of the arrangement for
$\Gamma>\Gamma_{c}$, and crossing this energy barrier results in current
reversal. The quantity $\langle\omega\rangle$ would consequently be zero. The
relevant energy barrier $\Delta E$ that divides the chiral states and the
energy $2mgA\Gamma_{c}$ imparted by the shaker may be used to calculate
$\Gamma_{c}$. The motion of one rod ascending over the other describes a
simple route in the configuration space that permits the chirality to toggle.
This corresponds to the situation where we have $\Delta E=2mgr$ and
$\Gamma_{c}=\frac{\Delta E}{2Amg}=\frac{r}{A}$. For $A=0.4$mm, $r=1.1$mm, we
have $\Gamma_{c}=2.75$, which is consistent with experimental findings for the
data plotted in Fig. 2(b). We have seen that this relation holds for various
objects of different materials and diameters as seen in the inset of Fig.
2(b), as long as the height of the object isn’t so high that it topples out of
the container on being shaken.

## Simulations for $N>2$:

The geometric model discussed in the foregoing is of use mainly in the two-rod
scenario, and its extension to analyze the case of multiple rods is not easy.
To proceed, we note that the motion of individual rods on the timescale of our
experiments appears stochastic, as mentioned earlier while discussing the
rattling motion of rods. This warrants a stochastic dynamical model to explain
qualitatively our experimental findings. A few reasonable simplifications are
in order, all of which are motivated by experimental observations. When viewed
from above, the rods appear as discs moving preferentially in one direction on
circular tracks. There are several concentric tracks available for the rods to
move in, and when allowed, the rods can move from one track to another.
Moreover, manufacturing defects in the cylindrical container lead to the rods
speeding up and slowing down at different random regions of the container,
leading to a quenched disorder in the system. Steric hindrance forbids two
discs to be on top of one another. All these aspects may be accounted for by
considering a lattice model of hard-core discs, wherein the separation between
neighboring lattice sites is given by the diameter of a disc and hard-core
constraint allows only one disc to be placed centered on a lattice site.

The lattice constituting our model consists of multiple lanes placed adjacent
to one another, and with periodic boundary conditions only in the longitudinal
direction. Thus, referring to Fig.6(a) – (e), periodic boundary condition
applies as one moves from left to right along the lattice, while no periodic
condition applies as one wraps around the lattice from top to bottom. We
specifically consider $N$ hard-core discs on a two-dimensional lattice of
$L_{1}$ rows and $L_{2}$ columns, with filling fraction $\phi=N/L$ and
$L=L_{1}L_{2}$. The dynamics of our model involves stochastic hopping of discs
to vacant nearest-neighbor sites, either in the same or in the adjacent lane,
with the preferential motion of rods accounted for by considering the hopping
to be biased along one of the two longitudinal directions. We will here
consider the extreme case of biased hopping in which hopping is possible only
to the right, see Fig. 6(a)–(e). The results are expected to remain
qualitatively unaffected on considering the motion to be possible both to the
right and to the left, with a bias, say, to the right. A schematic diagram
showing the rules of the simulation is shown in Fig. 6 (a-e). The model has
spatially-distributed disorder in that every site has an associated hopping
rate $\alpha$ for a disc that occupies the site, which we consider to be a
random variable sampled from a bimodal distribution:
$P(\alpha=q)=f,P(\alpha=1)=1-f$, with $0<f<1$. The lattice is generated with a
fixed probability $f$ of having slow sites , and $1-f$ for fast sites. Slow
sites are marked in gray in Fig. 6 (a,e). Referring to Fig.6, panels (a) and
(b), the particle marked white and occupying a site with associated hopping
rate $1$, panel (a), or, $q$, panel (b), is shown to move to the right
neighboring site. Note that the particle move is allowed in the two cases
since the destination site was empty before the move. If however the
destination site happens to be occupied, as is the case in panel (c), then the
white particle has the possibility to change lanes. A situation may arise when
the destination sites in the two adjacent lanes are empty, in which case, the
particle moves to either of the two sites with equal probability $q/2$
(respectively, $1/2$) if the particle before the move was on a slow site
(respectively, a fast site). The former case is shown in Fig.6(c). Another
situation may arise when only one of the two possible destination sites in the
adjacent lanes are empty; in this case, the particle moves to this empty site
with probability $1$ or $q$, depending respectively on whether the particle
before making the move was on a fast or a slow site, see Fig.6(d). Figure 6(e)
represents the case when all possible destination sites are occupied, and
hence the white particle cannot make any move whatsoever. In our experiments,
we have observed that a rod makes a move provided another rod in its vicinity
collides with it and pushes it forward. The probability to find another rod in
the neighborhood grows linearly with the filling fraction of rods. This aspect
is taken care of in our stochastic model by stipulating that the parameter $q$
is a linear function of $\phi$. The stochastic model proposed here is a
nontrivial generalization of the paradigmatic asymmetric simple exclusion
process [17]. The lattice is periodic in x direction, and the average current
$\langle\omega\rangle$ is measured is measured in this direction.

We studied our proposed model by performing Monte Carlo simulations of the
dynamics, whereby evolution in discrete time steps involves sampling a
particle at random and attempting to move it to a nearest-neighbor vacant site
according to the rules mentioned in the preceding paragraph. These simulations
were carried out for $L=100,200,500$, and different values of $\phi$. There
were no appreciable differences on changing lattice size. The system was
allowed to settle into a stationary state, after which measurements of average
velocity $\langle\omega\rangle$ of individual discs were made (because of
periodic boundary conditions, the velocity that one measures is actually the
angular velocity). Our obtained results for $\langle\omega\rangle$ versus
$\phi$ show the same characteristic rise and fall as in our experimental data;
The position and the height of the peak depends on both $f$ and $q$ (Fig. 6,
panels (f) and (g)). We have also performed alternative simulations where
particles were demarcated with probability $f$ to be slow with slow speed $q$,
obtaining similar numerical results. This simulation scheme mirrors the
experimental feature of imperfections between particles.

Figure 6:  (a) $\ldots$ (e) shows the schematic for the simulation rules. Here
the particle marked white is chosen to move, while black particles remain
stationary. Slow sites are marked gray, while fast sites are marked in light
blue. For this proposed theoretical model of hard-core discs hopping between
the sites of a two-dimensional lattice, (f) shows the average particle
velocity $\langle\omega\rangle$ vs. packing fraction $\phi$ for different
values of the parameter $f$ characterizing the hop rate distribution, with hop
rate parameter $q=2\phi$, see text. (g) By changing the parameter $k$ in
$q=k\phi$, we can change the position and the height of peaks in the
$\langle\omega\rangle$ versus $\phi$ curve.

The different translational and rotational degrees of freedom of an Euclidean
object may get coupled in an external field, resulting in a dynamics that
violates time-reversal symmetry. Toys such as the rattle back or the tippe top
are single-particle examples of this phenomenon. The present study extends in
a non-trivial and hitherto unexplored manner this domain of work to a multi-
particle setup of interacting achiral objects, wherein chirality that drives a
current in the system is an emergent property.

Shamik Gupta acknowledges support from the Science and Engineering Research
Board (SERB), India under SERB-TARE scheme Grant No. TAR/2018/000023, SERB-
MATRICS scheme Grant No. MTR/2019/000560, and SERB-CRG scheme Grant No.
CRG/2020/000596. SG acknowledges support of the Department of Atomic Energy,
Government of India, under Project No. 12-R $\&$DTFR-5.10-0100.

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