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# A note on confluence
in typed probabilistic lambda calculi††thanks: This paper is part of the first
author’s masters thesis [9]. Partially founded by PICT-2019-1272 and 21STIC10
Qapla’.

Rafael Romero  CONICET-Universidad de Buenos Aires.
Instituto de Ciencias de la Computación.
Buenos Aires, Argentina<EMAIL_ADDRESS>CONICET-Universidad de Buenos
Aires.
Instituto de Ciencias de la Computación.
Buenos Aires, Argentina.Departamento de Ciencia y Tecnología.
Universidad Nacional de Quilmes.
Bernal, Buenos Aires, Argentina. Alejandro Díaz-Caro CONICET-Universidad de
Buenos Aires.
Instituto de Ciencias de la Computación.
Buenos Aires, Argentina.Departamento de Ciencia y Tecnología.
Universidad Nacional de Quilmes.
Bernal, Buenos Aires, Argentina<EMAIL_ADDRESS>
###### Abstract

On the topic of probabilistic rewriting, there are several works studying both
termination and confluence of different systems. While working with a lambda
calculus modelling quantum computation, we found a system with probabilistic
rewriting rules and strongly normalizing terms. We examine the effect of small
modifications in probabilistic rewriting, affine variables, and strategies on
the overall confluence in this strongly normalizing probabilistic calculus.

##  1 Introduction

When dealing with probabilistic lambda calculus, we can find two different
sources of divergence.

  * •

A single redex may reduce in two different ways via a probabilistic reduction.

  * •

A term with multiple redexes and no strategy, could be reduced in different
ways.

For example, we can consider a lambda calculus extended with a coin
$\fullmoon$ reducing to $0$ or $1$ with probability $\frac{1}{2}$ each. Then,
taking just the coin, we are in the first case of divergence. While taking,
for example, $(\lambda x.\lambda y.yxx)\fullmoon$, we are in the second case,
since we can either beta reduce, or reduce the coin.

There is no point in trying to achieve confluence in the first case: the coin
is non-confluent by design. However, we can analyse the branching paths and
verify that the probability of reducing to a particular term stays the same,
regardless of the reduction sequence. This is what we call probabilistic
confluence.

To study this kind of cases, Bournez and Kirchner developed the notion of PARS
[4], later refined in [3]. Using the techniques described in [7, 8] we can
define the rewriting rules over the distributions.

If we denote by $[(p_{\\_}1,t_{\\_}1),\dots,(p_{\\_}n,t_{\\_}n)]$ the
probability distribution where $t_{\\_}i$ has probability $p_{\\_}i$, the
possible reductions from the previous example are depicted in Figure 1. The
resulting distributions are not only different, but also divergent, since in
the left branch, the probability to arrive, for example, to $\lambda y.y01$ is
$0$, while it is one of the possible results in the right branch.

${{[({\frac{1}{2}},(\lambda x.\lambda y.yxx)0),({\frac{1}{2}},(\lambda
x.\lambda y.yxx)1)]}}$${(\lambda x.\lambda y.yxx)\fullmoon}$${{[(1,\lambda
y.y\fullmoon\fullmoon)]}}$${{[({\frac{1}{2}},\lambda
y.y00),({\frac{1}{2}},\lambda y.y11)]}}$${{[({\frac{1}{4}},\lambda
y.y00),({\frac{1}{4}},\lambda y.y01),({\frac{1}{4}},\lambda
y.y10),({\frac{1}{4}},\lambda y.y11)]}}$$\scriptstyle{*}$ Figure 1: Counter
example of confluence

In this short paper we consider a simply typed lambda calculus extended with a
coin, and show different possibilities for achieving some sort of confluence,
without giving preference to any of them.

In Section 2 we introduce the calculus to be studied, without any restrictions
either in the rewriting rules, or in the typing rules. As we argued above,
this naive definition is not confluent (cf. Figure 1), unless a strategy is
defined (in which case it becomes trivially probabilistically confluent, as
will be discussed in Section 3.1). In Section 3.2 we show that we can achieve
confluence by internalising the probabilistic reductions in the terms. In
Section 3.3 we show that we can achieve probabilistic confluence by taking an
affine-linear type system. Then, in Section 4, we show that we can relax the
type system in an if-then-else branching, obtaining a probabilistic confluence
result modulo a computational equivalence.

##  2 The $\lambda_{\circ}$lambda coin calculus

In this section we present $\lambda_{\circ}$ (read “lambda coin”), which is
the simply typed lambda calculus extended with booleans ($1$ and $0$), an if-
then-else construction, and a coin $\fullmoon$. Terms are inductively defined
by

$t:=x\mid\lambda x.t\mid tt\mid 1\mid
0\mid\mathsf{if~{}}t\mathsf{~{}then~{}}t\mathsf{~{}else~{}}t\mid\fullmoon$

The rewrite system is given in Table 1. The rules $t\rightarrow_{p}r$ mean
that $t$ reduces with probability $p$ in one step to $r$, where the sum of
probabilities of reducing one redex is $1$. In particular, every non-
contextual rule has probability $1$, since there is only one rule per redex,
except for the coin, which reduces with probability $\frac{1}{2}$ to $0$ and
probability $\frac{1}{2}$ to $1$. If
$t\rightarrow_{p_{\\_}1}r_{\\_}1\dots\rightarrow_{p_{\\_}n}r_{\\_}n$ we may
write $t\rightarrow_{\prod_{\\_}{i=1}^{n}p_{\\_}i}^{*}r_{\\_}n$.

The reduction rules are intentionally as permissive as possible (even the
branches of an if-then-else can be reduced) in order to analyse its (lack of)
confluence. The type system is given in Table 2.

$\begin{array}[]{r@{\ }l@{\qquad}r@{\ }l@{\qquad}r@{\ }l}(\lambda
x.t)r&\rightarrow_{1}t[r/x]&\mathsf{if~{}}1\mathsf{~{}then~{}}t\mathsf{~{}else~{}}u&\rightarrow_{1}t&\fullmoon&\rightarrow_{\frac{1}{2}}1\\\
&&\mathsf{if~{}}0\mathsf{~{}then~{}}t\mathsf{~{}else~{}}u&\rightarrow_{1}u&\fullmoon&\rightarrow_{\frac{1}{2}}0\end{array}$

$\lambda x.t\rightarrow_{p}\lambda x.rt\rightarrow_{p}r\qquad
ts\rightarrow_{p}rst\rightarrow_{p}r\qquad
st\rightarrow_{p}srt\rightarrow_{p}r\qquad\mathsf{if~{}}t\mathsf{~{}then~{}}u\mathsf{~{}else~{}}v\rightarrow_{p}\mathsf{if~{}}r\mathsf{~{}then~{}}u\mathsf{~{}else~{}}vt\rightarrow_{p}r$

$\mathsf{if~{}}t\mathsf{~{}then~{}}u\mathsf{~{}else~{}}v\rightarrow_{p}\mathsf{if~{}}t\mathsf{~{}then~{}}r\mathsf{~{}else~{}}vu\rightarrow_{p}r\qquad\mathsf{if~{}}t\mathsf{~{}then~{}}u\mathsf{~{}else~{}}v\rightarrow_{p}\mathsf{if~{}}t\mathsf{~{}then~{}}u\mathsf{~{}else~{}}rv\rightarrow_{p}r$

Table 1: Rewrite system for $\lambda_{\circ}$.

$A:=\mathbb{B}\mid A\rightarrow A$

$\Gamma,x:A\vdash x:A\qquad\Gamma\vdash\lambda x.t:A\rightarrow
B\Gamma,x:A\vdash t:B\qquad\Gamma\vdash tr:B\lx@proof@logical@and\Gamma\vdash
t:A\rightarrow B\Gamma\vdash r:A$

$\Gamma\vdash 1:\mathbb{B}\qquad\Gamma\vdash
0:\mathbb{B}\qquad\Gamma\vdash\mathsf{if~{}}t\mathsf{~{}then~{}}u\mathsf{~{}else~{}}v:A\lx@proof@logical@and\Gamma\vdash
t:\mathbb{B}\Gamma\vdash u:A\Gamma\vdash
v:A\qquad\Gamma\vdash\fullmoon:\mathbb{B}$

Table 2: Type system for $\lambda_{\circ}$.

Strong normalization for this calculus follows trivially from the proof for
the simply typed $\lambda$-calculus with booleans. The only reduction added is
the probabilistic coin toss and it takes at most one step for each operator.
Hence, using the rewriting over probabilistic distributions techniques from
[7], we only need to show local confluence in order to achieve global
confluence altogether. This is an adaptation of Newman’s lemma for
probabilistic calculi (see [7] for a longer discussion about probabilistic
confluence).

Clearly, $\lambda_{\circ}$ is not confluent, as already seen in the
introduction (see Figure 1). It is easy to see that these distributions
represent different results.

The divergence stems from three characteristics of the calculus: (1) Lack of a
reduction strategy. (2) Probabilistic reductions. (3) Duplication of
variables.

Removing just one of these elements renders the system confluent, however each
modification comes with its own trade-off. We will examine each case, one by
one, in the following section.

##  3 Removing the divergence sources

###  3.1 Defining a strategy

The definition of a strategy is the easiest modification. Choosing a reduction
strategy makes all critical pairs disappear, since there is only one possible
reduction rule to be applied for each term distribution. For example, in
Figure 1 a call-by-name strategy would take the right path, where call-by-
value would take the left one.

Ultimately the choice lies in how to interpret the duplication of variables.
Reducing via call-by-name means that the probabilistic event is duplicated.
Whereas, a call-by-value strategy duplicates the outcome of said event (see
[5] for a discussion on this choice).

###  3.2 Internalising the probabilities

Following [6], we can modify the reduction on $\lambda_{\circ}$ to internalise
the entire distribution of a term. In this particular case, every reduction
has probability $1$, and the coin toss deterministically reduces to its
probability distribution. We write $t\oplus_{\\_}{p}r$ for the probability
distribution $[(p,t),(1-p,r)]$, $(t\oplus_{\\_}pr)\oplus_{\\_}qs$ for the
probability distribution $[(qp,t),(q(1-p)r),(1-q,s)]$, etc. Then, we can
consider the rewrite rule $\fullmoon\to 0\oplus_{\\_}{\frac{1}{2}}1$. This
idea is common in non-probabilistic settings as well, e.g., [2].

Following this approach brings confluence to the calculus, since every
repetition of a probabilistic event rewrites to the same result, its
distribution. For example, the two branches of Figure 1 become $(\lambda
x.\lambda y.yxx)(0\oplus_{\\_}{\frac{1}{2}}1)$ and $\lambda
y.y\fullmoon\fullmoon$, both converging to $\lambda
y.y(0\oplus_{\\_}{\frac{1}{2}}1)(0\oplus_{\\_}{\frac{1}{2}}1)$.

Although this is a valid solution, it forces us to consider every possible
state of a program at the same time along with its probability of occurrence,
making the management of the system more complex. Here we are dealing with a
simple coin, but more involved calculi might have several different
reductions, each with its own distribution. If not designed correctly, a
language that holds every possible state in the probability distribution can
easily become too cumbersome to be effective.

###  3.3 Affine variables

The last reasonable solution is to restrict duplication. One way to do this is
by controlling the appearance of variables at the type system level, with an
affine type system, see Table 3.

$A:=\mathbb{B}\mid A\rightarrow A$

$\Gamma,x:A\vdash x:A\qquad\Gamma\vdash\lambda x.t:A\rightarrow
B\Gamma,x:A\vdash t:B\qquad\Gamma,\Delta\vdash
tr:B\lx@proof@logical@and\Gamma\vdash t:A\rightarrow B\Delta\vdash r:A$

$\Gamma\vdash 1:\mathbb{B}\qquad\Gamma\vdash
0:\mathbb{B}\qquad\Gamma,\Delta_{\\_}1,\Delta_{\\_}2\vdash\mathsf{if~{}}t\mathsf{~{}then~{}}u\mathsf{~{}else~{}}v:A\lx@proof@logical@and\Gamma\vdash
t:\mathbb{B}\Delta_{\\_}1\vdash u:A\Delta_{\\_}2\vdash
v:A\qquad\Gamma\vdash\fullmoon:\mathbb{B}$

Table 3: Affine type system for $\lambda_{\circ}$ (different contexts are
considered to be disjoint).

This type system solves the counterexample from Figure 1, since the considered
term has no type in this system. In particular, we can prove the following
property:

######  Lemma 3.1.

If $r\rightarrow_{p}s$ then $t[r/x]\rightarrow_{p}t[s/x]$.

Notice that this property is not true in the unrestricted $\lambda_{\circ}$.
For example, while $\fullmoon\rightarrow_{\frac{1}{2}}1$, we have $(\lambda
y.yxx)[\fullmoon/x]=\lambda
y.y\fullmoon\fullmoon\rightarrow_{\frac{1}{4}}^{*}\lambda y.y11$.

The drawback in this approach is clear, there is a loss in expressivity. Of
course, in some cases, this restriction is a desirable quality. For example,
in quantum computing it may serve to avoid cloning qubits, a forbidden
operation in quantum mechanics.

##  4 Computational confluence with a sub-affine type system

The solution considered in Section 3.3 seems quite extreme. In particular,
using the same variable in different branches of an if-then-else construction
does not actually duplicate it, since only one of those branches will remain.
However, changing the rule $\mathsf{if}$ from Table 3 to $\mathsf{if}_{\\_}s$
given by

$\Gamma,\Delta\vdash\mathsf{if~{}}t\mathsf{~{}then~{}}u\mathsf{~{}else~{}}v:A\lx@proof@logical@and\Gamma\vdash
t:\mathbb{B}\Delta\vdash u:A\Delta\vdash v:A$

breaks confluence anyway. We call this calculus “sub-affine”. Consider the
following example. Let $\mathsf{not}=\lambda
x.\mathsf{if~{}}x\mathsf{~{}then~{}}0\mathsf{~{}else~{}}1$, then

${(\lambda x.\lambda
y.\mathsf{if~{}}y\mathsf{~{}then~{}}x\mathsf{~{}else~{}}\mathsf{not}~{}x)\fullmoon}$${\left[\begin{tabular}[]{c}$(1/2,(\lambda
x.\lambda
y.\mathsf{if~{}}y\mathsf{~{}then~{}}x\mathsf{~{}else~{}}\mathsf{not}~{}x)0);$\\\
$(1/2,(\lambda x.\lambda
y.\mathsf{if~{}}y\mathsf{~{}then~{}}x\mathsf{~{}else~{}}\mathsf{not}~{}x)1)$\end{tabular}\right]}$${\left[(1,(\lambda
y.\mathsf{if~{}}y\mathsf{~{}then~{}}\fullmoon\mathsf{~{}else~{}}\mathsf{not}~{}\fullmoon))\right]}$${\left[\begin{array}[c]{l}(\nicefrac{{1}}{{2}},\lambda
y.\mathsf{if~{}}y\mathsf{~{}then~{}}0\mathsf{~{}else~{}}1);\\\
(\nicefrac{{1}}{{2}},\lambda
y.\mathsf{if~{}}y\mathsf{~{}then~{}}1\mathsf{~{}else~{}}0)\end{array}\right]}$${\left[\begin{array}[c]{l}(\nicefrac{{1}}{{4}},\lambda
y.\mathsf{if~{}}y\mathsf{~{}then~{}}0\mathsf{~{}else~{}}0);\\\
(\nicefrac{{1}}{{4}},\lambda
y.\mathsf{if~{}}y\mathsf{~{}then~{}}0\mathsf{~{}else~{}}1);\\\
(\nicefrac{{1}}{{4}},\lambda
y.\mathsf{if~{}}y\mathsf{~{}then~{}}1\mathsf{~{}else~{}}0);\\\
(\nicefrac{{1}}{{4}},\lambda
y.\mathsf{if~{}}y\mathsf{~{}then~{}}1\mathsf{~{}else~{}}1)\end{array}\right]}$

Note that terms in both distributions are in normal form. The two paths are
syntactically divergent, however the resulting programs share the same
behaviour under the same inputs. If we were to apply the resulting
abstractions to $0$ or to $1$, both paths would yield
$[(\nicefrac{{1}}{{2}},0);(\nicefrac{{1}}{{2}},1)]$. Therefore, these
distributions are semantically confluent, they are not the same terms but they
represent the same function.

We can formalise this notion for the sub-affine calculus as follows. Let

$C:=\lozenge\mid Cv$

be an elimination context, where $\lozenge$ is called “placeholder” and $v$ is
a normal closed term. We write $C\langle t\rangle$ for $C[t/\lozenge]$. Notice
that $C\langle t\rangle$ is a term. We say that $C$ is an elimination context
of $A$, written $C^{A}$, if for all $\vdash t:A$, we have $\vdash C\langle
t\rangle:\mathbb{B}$. That is, it applies $t$ until it reaches the basic type
$\mathbb{B}$.

The computational equivalence is defined as follows.

######  Definition 4.1.

Let $D_{\\_}1=[(p_{\\_}i,t_{\\_}i)]_{\\_}i$ and
$D_{\\_}2=[(q_{\\_}j,r_{\\_}j)]_{\\_}j$ be two distributions of terms, all
closed of type $A$. Then, we say that these distributions are computationally
equivalent (notation $D_{\\_}1\equiv D_{\\_}2$) if for all $C^{A}$ we have
$C\langle t_{\\_}i\rangle\rightarrow_{u_{\\_}{ik}}^{*}b_{\\_}{ik}$ and
$C\langle r_{\\_}j\rangle\rightarrow_{s_{\\_}{jh}}^{*}c_{\\_}{jh}$ such that
the $b_{\\_}{ik}$ and $c_{\\_}{jh}$ are in normal form, and
$[(p_{\\_}iu_{\\_}{ik},b_{\\_}{ik})]_{\\_}{ik}\sim[(q_{\\_}js_{\\_}{jh},c_{\\_}{jh})]_{\\_}{jh}$
, where $\sim$ denotes the equality on distributions.

The previous definition means that two distributions are computationally
equivalent if by applying the resulting terms to all the possible inputs, they
produce the same probability distribution of results. Notice that the
definition is not assuming confluence.

Then, we can prove that the confluence modulo computational equivalence of the
sub-affine calculus (Theorem 4.4). We need the following two lemmas, which
follow by straightforward induction.

######  Lemma 4.2.

Let $y\not\in\mathsf{FV}(r)$, then $t[q/y][r/x]=t[r/x][q[r/x]/y]$. ∎

######  Lemma 4.3.

Let $t$ reduce to the distribution $D$, then $t[r/x]$ reduces to $D[r/x]$. ∎

######  Theorem 4.4 (Computational confluence).

Let $\vdash t:A$ in the sub-affine calculus. If $t$ reduces to the
distribution $D_{\\_}1$ and to the distribution $D_{\\_}2$, then
$D_{\\_}1\equiv D_{\\_}2$.

###### Proof.

We only consider the six critical pairs, four derived from the if-then-else
construction and two from the classical lambda calculus. Non-critical pairs
are trivially probabilistically confluent.

${\mathsf{if~{}}1\mathsf{~{}then~{}}r\mathsf{~{}else~{}}s}$${{[(p_{\\_}i,\mathsf{if~{}}1\mathsf{~{}then~{}}r_{\\_}i\mathsf{~{}else~{}}s)]_{\\_}i}}$${{[(1,r)]}}$${{[(p_{\\_}i,r_{\\_}i)]_{\\_}i}}$${\mathsf{if~{}}1\mathsf{~{}then~{}}s\mathsf{~{}else~{}}r}$${{[(p_{\\_}i,\mathsf{if~{}}1\mathsf{~{}then~{}}s\mathsf{~{}else~{}}r_{\\_}i)]_{\\_}i}}$${{[(1,s)]}}$${{[(p_{\\_}i,s)]_{\\_}i}\sim[(1,s)]}$

The symmetrical cases with $0$ close in a similar way. The fifth critical pair
closes by Lemma 4.3.

${{[(p_{\\_}i,(\lambda x.t_{\\_}i)r)]_{\\_}i}}$${(\lambda
x.t)r}$${{[(1,t[r/x])]}}$${{[(p_{\\_}i,t_{\\_}i[r/x])]_{\\_}i}}$

The last critical pair requires a more thorough analysis.

${{[(p_{\\_}i,(\lambda x.t)r_{\\_}i)]_{\\_}i}}$${(\lambda
x.t)r}$${{[(1,t[r/x])]}}$${{[(p_{\\_}i,t[r_{\\_}i/x])]_{\\_}i}}$

We must prove that for all $C^{A}$, if $C\langle
t[r_{\\_}i/x]\rangle\rightarrow_{q_{\\_}{ij}}^{*}b_{\\_}{ij}$ and $C\langle
t[r/x]\rangle\rightarrow_{s_{\\_}{k}}^{*}c_{\\_}k$, then
$[p_{\\_}iq_{\\_}{ij},b_{\\_}{ij}]_{\\_}{ij}\sim[s_{\\_}k,c_{\\_}k]_{\\_}k$.
It is enough to take only one path to
$[p_{\\_}iq_{\\_}{ij},b_{\\_}{ij}]_{\\_}{ij}$ and to
$[s_{\\_}k,c_{\\_}k]_{\\_}k$ according to the definition of $\equiv$. If
$x\notin\mathsf{FV}(t)$, then $C\langle t[r/x]\rangle=C\langle
t\rangle=C\langle t[r_{\\_}i/x]\rangle$. If $x$ appears once in $t$, $C\langle
t[r/x]\rangle\rightarrow_{p_{\\_}i}C\langle t[r_{\\_}i/x]\rangle$. If $x$
appears more than once in $t$, it appears at most $2^{n}$ times (where $n$ is
the number of if-then-else constructions), so we can proceed by induction on
$n$.

  * •

If $n=1$, then there is one if-then-else, say
$\mathsf{if~{}}c\mathsf{~{}then~{}}s_{\\_}1\mathsf{~{}else~{}}s_{\\_}2$ and
$x$ appears both in $s_{\\_}1$ and in $s_{\\_}2$. Therefore, there are three
cases:

    * –

$c$ is $0$ or $1$, then, for $j=0$ or $j=1$, we have $C\langle
t[r/x]\rangle\rightarrow_{1}^{*}C\langle
s_{\\_}j[r/x]\rangle\rightarrow_{p_{\\_}i}^{*}C\langle
s_{\\_}j[r_{\\_}i/x]\rangle$. Notice that $C\langle
t[r_{\\_}i/x]\rangle\rightarrow_{1}C\langle s_{\\_}j[r_{\\_}i/x]\rangle$.

    * –

$c\rightarrow_{q_{\\_}j}^{*}j$ with $j=0,1$, so we have $C\langle
t[r/x]\rangle\rightarrow_{q_{\\_}j}^{*}C\langle
s_{\\_}j[r/x]\rangle\rightarrow_{p_{\\_}i}^{*}C\langle
s_{\\_}j[r_{\\_}i/x]\rangle$. That is, we arrived to the distribution
$[(q_{\\_}jp_{\\_}i,s_{\\_}j[r_{\\_}i/x])]_{\\_}{ij}$. Notice that since
$C\langle t[r_{\\_}i/x]\rangle\rightarrow_{q_{\\_}j}C\langle
s_{\\_}j[r_{\\_}i/x]\rangle$, we arrive to the same distribution in the right
branch of this critical pair.

    * –

$c$ is an open term, hence not reducing to a constant $0$ or $1$. In such a
case, since $t[r/x]$ is a closed term, it means that the if-then-else
construction in $t$ is under a lambda-abstraction, therefore, we must beta-
reduce first, either by with the argument in $t$, or with an external argument
given by the context $C$. We can repeat the same process until we get one of
the previous treated cases (that is, at some point, the if-then-else becomes a
redex).

  * •

If $n>1$, we proceed as the previous case, reducing the if-then-else first,
which reduces $n$ and so the induction hypothesis applies.∎

##  5 Conclusion

In this paper we have analysed the different possibilities to transform a
simple probabilistic calculus into a (probabilistically / computationally)
confluent calculus. The main contribution on this note is Theorem 4.4, which
proves that we can relax the affinity restriction on “non-interfering” paths.
This technique has been considered in the first author master’s thesis [9] to
prove the computational confluence of the quantum lambda calculus
$\lambda_{\\_}\rho$ [6].

## References

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aainstitutetext: Theory Center, IPNS, High Energy Accelerator Research
Organization (KEK), Tsukuba, Ibaraki 305-0810, Japanbbinstitutetext: School of
High Energy Accelerator Science, The Graduate University for Advanced Studies
(Sokendai), Tsukuba 305-0810, Japan

# Anderson Localization in high temperature QCD: background configuration
properties and Dirac eigenmodes

Guido Cossu<EMAIL_ADDRESS>a,b Shoji Hashimoto<EMAIL_ADDRESS>
###### Abstract

We investigate the properties of the background gauge field configurations
that act as disorder for the Anderson localization mechanism in the Dirac
spectrum of QCD at high temperatures. We compute the eigenmodes of the Möbius
domain-wall fermion operator on configurations generated for the $SU(3)$ gauge
theory with two flavors of fermions, in the temperature range
$[0.9,1.9]T_{c}$. We identify the source of localization of the eigenmodes
with gauge configurations that are self-dual and support negative fluctuations
of the Polyakov loop $P_{L}$, in the high temperature sea of $P_{L}\sim 1$.
The dependence of these observations on the boundary conditions of the valence
operator is studied. We also investigate the spatial overlap of the left-
handed and right-handed projected eigenmodes in correlation with the
localization and the corresponding eigenvalue. We discuss an interpretation of
the results in terms of monopole-instanton structures.

###### Keywords:

Lattice QCD, Anderson localisation, Monopole-instantons

##  1 Introduction

Quantum chromodynamics (QCD) is an inherently non-perturbative description of
the strong force that governs the interaction of quarks. It has proven to be
valid at zero temperature and to predict the phase transition to the quark-
gluon plasma phase that occurs at high temperature. This phase transition
separates the regime of confining and spontaneously broken chiral vacuum from
the deconfining and chirally symmetric one. In the case of fermions in the
fundamental representation, the deconfinement temperature $T_{d}$ and the
chiral restoration temperature $T_{c}$ are remarkably close to each other,
suggesting a deeper connection between two otherwise unrelated properties of
the vacuum. There is still no widely accepted explanation for the correlation
between the two transitions.111In the case of physical fermion masses for 2+1
flavors the two transitions at the crossover are actually separated
Aoki:2006we , but nevertheless they are very close.

The eigenvalues and eigenvectors of the QCD Dirac operator are important tools
to probe the properties of the vacuum. They control the low energy physics and
chiral restoration. A significant example of their relevance is the well-known
Banks-Casher relation Banks:1979yr  that relates the spectral density
$\rho(\lambda)$ at near-zero virtuality $\lambda\sim 0$ to the chiral
condensate $\Sigma$, i.e. $\Sigma=\pi\rho(\lambda=0)$, in which the
thermodynamical and massless limit are assumed. The spectral densities near
the origin on the two sides of the chiral phase transition are radically
different. Therefore, it is expected that the properties of the related
eigenmodes reflect this change. In this study we examine the relation between
the lowest eigenmodes of the Dirac operator and their supporting background
gauge configurations in order to shed some light on the possible connection
between confinement and chiral symmetry.

Another set of important characteristics of the Dirac spectrum are the local
fluctuations of the eigenvalues in the bulk and the localisation of the
corresponding eigenmodes. The local fluctuations in the Dirac spectrum at high
temperatures are quite different from their low temperature counterparts. The
high temperature Dirac spectrum experiences a transition between a low-energy
region, where the eigenvalues are uncorrelated, and the higher part of the
spectrum that exhibits non-trivial Random Matrix Theory (RMT) type of
correlations (see PhysRevLett.105.192001 ; PhysRevD.86.114515 ;
Giordano:2013taa  and Section 2). The critical energy at which the quantum
transition occurs, $\lambda_{c}$, is called the mobility edge. The
distribution of the low eigenvalues for the low temperature region is expected
to agree with RMT models by the presence of a chiral condensate. The actual
symmetry class in RMT depends on the Dirac operator discretization, the gauge
group and the dimensionality of the space Verbaarschot:2000dy .

There is a tight connection between the local spectral correlations and the
localisation of the modes. It has been proven using staggered and overlap
fermions in PhysRevLett.105.192001 ; PhysRevD.86.114515 ; Bruckmann:2011cc ;
Giordano:2013taa  (and we provide a confirmation using domain-wall fermions in
Section 2) that the lowest modes of the high temperature spectrum are
localised below the threshold $\lambda_{c}$. Above the critical eigenvalue the
modes are delocalised, occupying the whole available volume and scaling
accordingly. This behaviour is parallel to the Anderson localisation mechanism
Anderson:1958vr  that explains the transition from a metallic phase
(delocalised electron wave functions) to an insulator (localised modes) by the
increasing density of impurities in a crystal. In the localised phase the wave
functions are no more extended waves (Bloch waves in the limit of zero
disorder) but are described by an exponential decay with a typical scale
called the localisation length. Thus the QCD vacuum at high temperature is
acting as an insulator for the lowest modes. The transition between the two
regimes is known as Anderson transition and it is a quantum transition of
second order for more than two dimensions Abrahams:AndersonLocalization . It
is driven by the amount of impurities (disorder) in the crystal.

The authors of Giordano:2013taa  have shown that the scaling critical
exponents for the QCD spectrum match the expectation from the three
dimensional Anderson Hamiltonian model, confirming that there is an Anderson-
like mechanism in action in high temperature QCD. The critical point depends
on the temperature Pittler:2014qea , an indication that the disorder is
depending on the temperature too. Moreover, the low temperature phase
eigenmodes exhibit a metallic behaviour, being delocalised at all energies.
There is no critical $\lambda_{c}$ in the spectrum in this regime. The
interesting conjecture is whether the opening of a localised region in the
spectrum is connected to the chiral phase transition GarciaGarcia:2006gr ;
GarciaGarcia:2005vj ; Giordano:2013taa ; Pittler:2014qea ; Giordano:2016cjs
.222 Anderson localisation has been considered in lattice QCD also in the
context of construction of chiral fermions, domain-wall and overlap, regarding
the localisation of the kernel eigenmodes and its effect on the chirality of
the discretised action (see for example Aoki:2001su ; Golterman:2003qe ).

The attractive question, once we assume Anderson localisation in QCD, is what
are the impurities, or the disorder, in the background gauge configurations
that trigger the localisation of the eigenmodes. How is the diffusion of
fermions affected by the impurities? Are these related to the chiral
transition and/or deconfinement? This is the central topic of this study.
Concerning this question, in recent years Bruckmann:2011cc  it has been shown
for $SU(2)$ pure gauge theories and argued using spin models Giordano:2015vla
that the Polyakov loop is acting as a source of localisation. In Section 3 we
show numerical evidences for several temperatures and masses that the lowest
modes cluster where the real part of the Polyakov loop is negative and this
correlation increases with the temperature. Higher modes, delocalised, show no
preference regarding the value of Polyakov loop. Furthermore, we study the
relation of the eigenvalues with the local action $F_{\mu\nu}F_{\mu\nu}(x)$,
the local topology $F_{\mu\nu}\tilde{F}_{\mu\nu}(x)$ and the variation of the
boundary conditions for the fermion field. A dependence on the boundary
condition of the spectrum at high temperature has been analytically proven
(see for example Bilgici:2009tx ) and it is another useful way of probing the
properties of the background configurations. In Section 3.3 we examine the
left and right projected components of the non-zero eigenmodes and their
relation with the corresponding eigenvalues.

In the last section we propose an interpretation of the results on the gauge-
impurities based on Bogomolny-Prasad-Sommerfield (BPS) monopole-instanton
objects Prasad:1975kr ; Bogomolny:1975de . We argue that all the properties
observed are explained by molecules of self-dual monopoles, sometimes called
dyons in the literature.

We then draw conclusions and perspectives for future work to understand how
strong is the relation between the deconfinement and the chiral phase
transition.

##  2 Dirac spectrum and localisation properties

###  2.1 Numerical setup

We study the properties of the eigenmodes of the Möbius domain-wall fermion
operator Brower:2005qw ; Brower:2012vk  on configurations around the phase
transition temperature. The gauge configurations are generated with two
degenerate flavors of Möbius domain-wall dynamical fermions and tree-level
improved Symanzik gauge action. Stout smearing Morningstar:2003gk  is applied
to the gauge links inserted in the fermionic action. A detailed account on the
choice of parameters is found in Hashimoto:2014gta . We simulated several
ensembles in the temperature range $[0.9,1.9]T_{d}$ where $T_{d}$ is the
deconfinement temperature ($T_{d}\simeq T_{c}$), and several masses. The
deconfinement transition temperature is estimated to be $T_{d}=175(5)$ MeV
from the measurement of the average Polyakov loop. The range of bare quark
masses is $[2,40]$ MeV depending on the ensemble. We have three type of
lattice sizes $16^{3}\times 8$, $32^{3}\times 8$ and $32^{3}\times 12$ to
control the dependence on the volume and lattice spacing. Table 1 lists the
ensembles presented in this work.

We calculated, using the implicitly restarted Lanczos algorithm saad-eig , the
eigenmodes of the massive hermitian Dirac operator
$H(m)=\gamma_{5}D^{4d}_{DW}(m)\equiv\gamma_{5}D$ (which also defines the
notation for $D$ from now on). Here the four-dimensional effective Dirac
operator of the five-dimensional domain-wall fermion is constructed by
combining it with the Pauli-Villars operator Brower:2012vk . We employed the
IroIro++ codeset Cossu:2013ola  to generate and analyse the configurations.

For our study of the left and right projected modes we need the eigenmodes of
the non-hermitian operator, that guarantees that the expectation values of
$\gamma_{5}$ are zero for non-zero eigenmodes. In order to obtain the
eigenmodes of the non-hermitian operator we assume that $D$ satisfies the
Ginsparg-Wilson relation. This assumption is checked numerically and it is
satisfied with enough precision for our purposes. The depth of the fifth
dimension $L_{s}$ is chosen such that this requirement is satisfied. We use
only ensembles with lattice spacing smaller than 0.09 fm to constrain the
possible violations, which are discussed in detail in Cossu:2015kfa . Given
the subspace of paired eigenmodes $\phi_{1}$ and $\phi_{2}$ for the hermitian
$H=\gamma_{5}D$ ($H\phi_{1,2}=\pm\lambda\phi_{1,2}$, $\lambda\in\mathbb{R}$)
the corresponding eigenmodes of $D$ are $\psi_{+,-}=(\phi_{1}\pm
i\phi_{2})/\sqrt{2}$ so that the local norm and the local chirality are just
the sum of the corresponding $\phi_{1,2}$ matrix elements. These are the modes
studied in this work.

$\beta$ | $m$ | $N_{s}^{3}\times N_{t}\,(\times L_{s})$ |  $T$ (MeV)
---|---|---|---
4.18 | 0.009 | $16^{3}\times 32\,(\times 12)$ | 0
4.10* | 0.01 | $16^{3}\times 8\,(\times 12)$ | 216(1)
4.10* | 0.01 | $32^{3}\times 8\,(\times 12)$ | 216(1)
4.18 | 0.01 | $16^{3}\times 8\,(\times 12)$ | 257(1)
4.18 | 0.01 | $32^{3}\times 8\,(\times 12)$ | 257(1)
4.30 | 0.01 | $32^{3}\times 8\,(\times 12)$ | 330(3)
4.30 | 0.005 | $32^{3}\times 8\,(\times 12)$ | 330(3)
4.18 | 0.01 | $32^{3}\times 12\,(\times 16)$ | 171(1)
4.23 | 0.01 | $32^{3}\times 12\,(\times 16)$ | 191(1)
4.23 | 0.005 | $32^{3}\times 12\,(\times 16)$ | 191(1)
4.23 | 0.0025 | $32^{3}\times 12\,(\times 16)$ | 191(1)
4.24 | 0.01 | $32^{3}\times 12\,(\times 16)$ | 195(1)
4.24 | 0.0025 | $32^{3}\times 12\,(\times 16)$ | 195(1)
4.30 | 0.01 | $32^{3}\times 12\,(\times 16)$ | 220(2)
4.30 | 0.005 | $32^{3}\times 12\,(\times 16)$ | 220(2)
Table 1: Lattice ensembles used in this study. Bare quark mass $m$ and the
lattice size $N_{s}^{3}\times N_{t}\,(\times L_{s})$ are given in lattice
units. The ensembles $\beta=4.10$ with $N_{t}=8$, shown by (*) have higher
violation of chirality than the one required for Section 3.3 (see discussion
in the text). They are only used in Section 2.2 where the effect of violations
on the localisation is irrelevant.

###  2.2 Eigenmode localisation

A basic quantity to understand the localisation properties is the
participation ratio (PR) that is defined as

$\displaystyle
PR_{n}=\Bigl{(}V\int|\psi_{n}(x)|^{4}d^{4}x\Bigr{)}^{-1},\qquad\int|\psi_{n}(x)|^{2}d^{4}x=1$
(1)

where the $\psi_{n}$ is the $n$-th mode of the Dirac operator, normalised to
1. $PR_{n}$ is constant ($=1$) in the purely metallic phase with a completely
delocalised eigenfunction, $|\psi_{n}(x)|^{2}=1/V$. It is, on the other hand,
proportional to $1/V$ if the mode is completely localised ($\delta$-function
on some $\bar{x}$), which implies an insulator phase. With this definition the
PR represents the fraction of the total volume occupied by the support of the
eigenmode $\psi_{n}$. Therefore the scaling with the volume of the eigenmodes
will distinguish the localised and delocalised states. At the Anderson
transition $\lambda_{c}$ the eigenmodes manifest a multifractal behaviour that
has been observed also in lattice QCD Ujfalusi:2015nha . This regime is
characterised by multiple scaling exponents depending on the scale.

The scaling of low modes has been already studied in detail by Kovacs et al.
using staggered fermions PhysRevLett.105.192001 ; PhysRevD.86.114515 . We
report here an example of our results as a warm-up and confirmation that the
same phenomena observed in pure gauge ensembles and with different fermion
discretisations PhysRevLett.105.192001 ; PhysRevD.86.114515 ; Giordano:2013taa
; Pittler:2014qea  are found also with chiral fermions on dynamical
configurations. In Figure 1 we illustrate the scaling of the participation
ratio for a couple of ensembles at $\beta=4.10,4.18$ and $m=0.01$ where two
different volumes $16^{3}\times 8$ and $32^{3}\times 8$ are available. It is
evident in the high-mode region that the PR approaches 1 and it is almost
independent of the volume, a characteristic of delocalised modes. The scaling
in the low-mode region is clearer in the inset, where the PR is multiplied by
the total volume. The size of the lowest modes in this scale is independent of
the volume implying a characteristic length: they are localised modes.

Figure 1: Scaling of the participation ratio for the ensembles $\beta=4.10$,
$m=0.01$ (red circles) and $\beta=4.18$, $m=0.01$ (black squares), in log-
scale. Filled and empty symbols correspond to the $16^{3}\times 8$ and
$32^{3}\times 8$ volumes. The corresponding temperatures are $T=216,257$ MeV
and the bare quark mass is $17$ and $20$ MeV respectively. In the inset the
participation ratio is multiplied by the volume. Figure 2: Dependence of the
participation ratio on the boundary conditions for the ensemble $32^{3}\times
12$, $\beta=4.30$, $m=0.01$. The corresponding temperature is $T=220$ MeV and
the bare quark mass is 26 MeV.

The result for the dependence on the boundary conditions of the PR is reported
in Figure 2, for one ensemble. The localisation length of the lowest modes for
the periodic boundary condition on the same configurations is quite different,
occupying a larger fraction of the volume. It is an indication of a possible
delocalisation but this conclusion cannot be inferred without a proper scaling
analysis. In the following we will discuss other observables that complete the
picture, showing delocalisation of the lowest modes when the boundary
conditions are changed.

Figure 3: Dependence on the temperature of the physical size of the lowest 10
eigenmodes, averaged over the ensemble. The ensembles are ordered by
increasing temperature. Volume effects visible in the higher modes for the
$16^{3}\times 8$ ensemble (yellow box), that are limited by the total volume
being $\sim 2.7$ fm4. Compare to the correspoding $32^{3}\times 8$ ensemble
(crosses). Figure 4: Temperature dependence of the lowest non-zero eigenmode
localisation length $l_{n}=v_{n}^{1/4}$ times the temperature $T$. The grey
band indicates the estimated transition temperature, 175 MeV. The localisation
length is governed by the temperature scale in a wide range of temperatures.

The temperature dependence of the size of the localised modes is presented in
Figure 3. Here we plot the physical size $v_{n}=PR_{n}\cdot V$ (in fm4) of the
first 10 non-zero eigenmodes against their eigenvalue, after averaging over
the ensemble.

The localisation length $l_{n}=v_{n}^{1/4}$ does not show an appreciable
dependence on the mass of the valence quarks. The dimensionless quantity
$l_{n}T$ is approximately constant $\sim 1.3$ for the lowest mode, see Figure
4. This results indicates that the typical localisation length is governed by
the temperature scale (see also PhysRevD.86.114515  for an analogous result).

###  2.3 Level spacing distribution

The unfolded level spacing distribution (ULSD) is another trait that
distinguishes the insulator and metallic phases in the Anderson model,
together with localisation. The ULSD is the probability distribution $P(s)$ of

$\displaystyle
s_{i}=\frac{\lambda_{i+1}-\lambda_{i}}{\langle\lambda_{i+1}-\lambda_{i}\rangle},$
(2)

the distance of two consecutive eigenvalues in the unfolded spectrum of a
matrix RMTHandBook . The spectral density is normalised to a constant, 1, to
disentangle non-universal properties, a common procedure in this analyses. The
ULSD can be analytically calculated for the three canonical RMT ensembles:
Gaussian orthogonal ensemble (GOE), unitary (GUE), and symplectic (GSE). The
eigenmodes in these ensembles are strongly correlated, with repulsion of
nearest modes and suppression of distant modes. Completely decorrelated
eigenvalues are scattered according to the Poisson distribution. The insulator
phase of the Anderson model, with strong disorder and localisation, has
Poisson distributed eigenvalues while the weak disorder and metallic phase
show delocalised modes following RMT. An interesting insight on this
difference is given by the Bohigas-Giannoni-Smidth conjecture PhysRevLett.52.1
. It states that quantum systems with chaotic classical counterparts have a
nearest-neighbour spacing distribution given by RMT whereas systems whose
classical counterparts are integrable obey a Poisson distribution (thus
regular classical dynamics). Therefore a specific form of $P(s)$ is often
taken as a criterion for the presence or absence of quantum chaos.

By universality arguments the low temperature chirally broken phase in QCD is
expected to follow the predictions of one of the RMT ensembles. For an $SU(3)$
chiral operator in the fundamental representation, the chiral unitary ensemble
(chGUE) describes the low temperature level spacing.333GUE and chGUE are
equivalent for the local fluctuations in the bulk. They differ for the
$\epsilon$-regime predictions Verbaarschot:2000dy . This has been extensively
tested, for examples see the review Verbaarschot:2000dy  and our data on the
right panel of Figure 5. The $P(s)$ is in good accordance with the GUE
distribution in three different regions of the spectrum, from the near-zero to
the high eigenvalue region. This is quite different from the distribution of
the local fluctuations at high temperature, as we are going to show.

In the insulator phase the typical feature of the Anderson model is presence
of the uncorrelated (Poisson) low modes. This is the kind of behaviour
observed in the QCD spectrum for $T>T_{c}$ (see PhysRevLett.105.192001 ;
PhysRevD.86.114515 ; Giordano:2013taa ; Giordano:2014qna  for previous
detailed works). On the left panel of Figure 5 we report one example from our
ensembles, where $T\sim 1.2~{}T_{c}$. Three distributions are computed in
different sectors of the spectrum, separated by the eigenvalues
$\lambda=[0.01;0.04],[0.04;0.075],[0.075;0.1]$. The near-zero sector follows
the Poisson distribution (localised modes), while the bulk part is in
accordance with RMT. In the middle an intermediate distribution is realised.
Such kind of interpolating distributions were studied in Nishigaki:1999zz .
These results confirm the idea that the system shows similarities with an
Anderson model. This hypothesis has been quantitatively proved in
Giordano:2013taa  by a scaling analysis using staggered fermions. They showed
that the critical exponent for the correlation length $\nu$, is in agreement
with the one of the 3d Anderson model.

By changing the boundary condition on the same ensemble, Figure 6, the ULSD
changes radically and it is in accordance with RMT, similar to the low
temperature case. This is expected from our discussion on the PR. In our
perspective it poses the question of how the eigenmode localisation depends on
the boundary conditions. The background gauge field is unchanged, the lattice
physical dimensions are unchanged, in particular we still have a compact
dimension, but the modes are now delocalised. In the following we investigate
this difference. Notice also that the spectral density has a gap in the anti-
periodic case and the gap disappears in the periodic case with non zero
density at the origin. This is a clear signal of a chirally broken phase. A
monopole instanton picture that addresses this behaviour is discussed in
Section 4.

Figure 5: Distribution $P(s)$ of the level spacings for the unfolded Dirac
spectrum in several sectors. The left and right panels are for the high
temperature $T=220$ MeV and low temperature case $T=171$ MeV respectively.
Continuous lines are the analytic predictions for totally uncorrelated
eigenvalues (Poisson, red-dashed curve) and random matrix theory from the
unitary ensemble (GUE, blue curve), with no free parameters. Figure 6:
Distribution $P(s)$ of the level spacings for the unfolded spectrum at $T=220$
MeV and bare quark mass of 26 MeV with periodic boundary condition. Continuous
lines are as in Figure 5.

##  3 Numerical analysis of the background gauge field

We now discuss the numerical results on the background gauge field supporting
the eigenmodes, with the intent of identifying the source of localisation. We
address the interpretation later in Section 4. We are looking for a gauge
invariant characterization of the background gauge field.

###  3.1 Correlation with the local Polyakov line

We begin with the investigation of the correlation of the eigenmodes with the
trace of the Polyakov line, or holonomy, in a gauge theory with $N$ colors

$\displaystyle P(x)=\frac{1}{N}{\rm Tr\,}L(x)=\frac{1}{N}{\rm
Tr\,}\exp\Bigl{(}i\oint A_{4}{\rm d}\tau\Bigr{)}.$ (3)

$\tau$ being the compact temporal dimension. This is a gauge invariant
quantity related to confinement. Similar studies on this observable were
conducted by other groups on pure gauge configurations or using effective
models Bruckmann:2011cc ; Giordano:2015vla .

The Polyakov line is measured after suppressing UV fluctuations with the
Wilson flow (other smoothing methods like cooling are equivalent
Bonati:2014tqa ). In the high temperature phase the spatial average of the
Polyakov line $P(x)$ gives a finite value $\rightarrow 1$, but we are
interested in its local structure. We then correlate, for several temperatures
and masses, the local norm of the non-zero eigenmodes $|\psi(x)|^{2}$, with
the real part of the Polyakov line trace. The results are reported in Figure
7. Every single line represents one eigenmode on one configuration. We average
the norm of the mode in small intervals of the Polyakov line axis to clear up
the presentation. The colour scale represents the position in the spectrum of
the eigenmode, see the colour bar. The green horizontal line stands for a
normalised flat mode, everywhere constant. A mode with no correlation with the
Polyakov line would follow this line.

The plots show that all the modes just below the phase transition (upper left
panel, $T=171$ MeV) have a mild dependence of $|\psi(x)|^{2}$ on the Polyakov
line, notice the log-scale. This is in marked contrast with the highest
temperature case (lower left panel, $T=220$ MeV). At this temperature the
lowest eigenmodes are localised, i.e. the norm is higher, where the Polyakov
line is mostly negative. The highest modes, delocalised, show again no
particular dependence on ${\rm Re}P(x)$. The norm of the low modes on the
regions where ${\rm Re}P(x)\rightarrow-1/3$ increases with the temperature,
see the sequence of the first five panels. The modes are more and more
localised in these areas. These are the regions where two of the eigenvalues
of the Polyakov loop are close to $-1$. In the majority of the volume at this
temperature, on the other hand, the Polyakov loop eigenmodes are all similar
and approach 1.

We also consider the mass dependence of the correlation but we cannot observe
any strong dependence in a range of masses spanning one order of magnitude,
although they are not shown here.

The last panel shows that by changing the boundary condition the correlations
are washed out and all the modes are flattened. This is another signal of the
delocalisation associated to the periodic boundary condition.

These results indicate that all the localised modes are centred in the regions
where the holonomy is far from its average for anti-periodic boundary
condition. They repel the region where the Polyakov line is 1, which at high
temperature is the most common value. The change of boundary condition to
periodic cancels this localisation, even if the “wrong” $P$ islands are still
present in the background. The modes are now delocalised and do not show any
preference on the Polyakov line value, a situation similar to the low
temperature case. The boundary condition affect the way the eigenmodes can
diffuse in the volume. These observations will be useful in the discussion of
Section 4.

Figure 7: The local squared norm of the eigenmode in correlation with the
local Polyakov line, real part. Temperature increases from left to right, top
to bottom. The last panel is for eigenmodes with periodic boundary conditions.
The colours represent the eigenvalue, as indicated by the colour bar attached
to each plot. The green horizontal line represents a vector with a constant
norm. Log scale for the norm axis is used. The vertical red line indicates the
location of ${\rm Re}P=-1/3$.

###  3.2 Correlation with action and topology density

Two other relevant gauge invariant quantities that characterise the background
configuration are the action density $s(x)=F_{\mu\nu}F_{\mu\nu}(x)$ and the
topological charge density $q(x)=F_{\mu\nu}\tilde{F}_{\mu\nu}(x)$, without the
canonical constant $8/32\pi^{2}$ in front. For a self-dual field
$\tilde{F}=F$, the ratio $|q|/s=1$. As with the Polyakov line, these variables
are measured after smearing with the Wilson flow. The topological charge has
been renormalised multiplicatively using a global fit on the whole ensemble
DelDebbio:2002xa .

Before analysing the ensemble averages, it is instructive to look at one
single configuration. In Figure 8 we present a typical situation for the
eigenmode localisation in relation to $s(x)$ and $q(x)$. Here we pick one
configuration at high temperature ($T\sim 1.2~{}T_{c}$) and without zero
modes. The first panel is a scatter plot of $q(x)$ versus $s(x)$ where every
point corresponds to one lattice site. It shows two branches that demonstrate
that the regions where the action is larger coincide with the regions where
the topological density $q(x)$ increases in absolute value. The branches
follow the two continuous lines that are the sector secants, $s=|q|$. On the
branches the action and the topological charge are the same on average: they
represent almost self-dual fluctuations of the gauge field. This is a common
result on every configuration we analysed. Notice that several self dual
fluctuations can be present in the configuration, though the plot does not
capture this information. Figure 9 depicts the actual three-dimensional
distribution of action and topological charge in a z-slice where the action
has the highest peak. Several peaks are visible besides the largest one. The
action tends to cluster in a tube in the temporal direction.

The following panels in Figure 8 add to the scatter plot the information of
the eigenmode norm $|\psi_{n}(x)|^{2}$ on site $x$. The colour and the size of
the dots represent the local norm of the eigenmode. Note that the top end of
the colour range changes in the panels. The mode number increases from left to
right, top to bottom. The lowest modes are localised on the branches, i.e. on
the regions where the action and the topological charge are higher.

Figure 8: Correlations between the action density $s(x)$ and topological
charge density $q(x)$ (first panel) and the eigenmode local norm, following
panels. Configuration number 3500 for the $32^{3}\times 12$, $\beta=4.30$,
$m=0.01$ ensemble. This configuration does not contain zero modes.

To systematically study the correlation with the action density of the
eigenmodes we computed the following weighted average:

$\displaystyle\bar{s}_{n}=\frac{\int\rho_{n}(x)^{2}s(x)d^{4}x}{\int\rho_{n}(x)^{2}d^{4}x},\qquad\rho_{n}(x)=|\psi_{n}(x)|^{2}$
(4)

that enhances the relevance of localised peaks. The data for some ensembles
are reported in Figure 10, where we plot the ratio $\bar{s}_{n}/s$ ($s$ is the
volume average of $s(x)$). To simplify the plots we averaged the results for
modes in bins $[\lambda,\lambda+\delta]$. The lowest modes at all temperatures
are localised with the peaks in regions with higher action. The average action
ratio increases with the temperature. We observe again the effect of the
boundary condition on the localisation. The average action as seen by the
lowest modes with periodic boundary condition is suppressed (empty symbol),
almost similar to the low temperature case.

A systematic view of the dominance of the regions where $|q|/s\sim 1$ is shown
in Figure 11. For every eigenmode we averaged the quantity $|q|/s$ in a norm
bin. The three panels represent different regions of the spectrum, from the
localised to the delocalised region.

Figure 9: A three-dimensional section ($z=9$) of the action, left, and the
topological charge, right, for the configuration n. 3500 of the ensemble
$32^{3}\times 12$, $\beta=4.30$, $m=0.01$, same as Figure 8. The action is
clustered in tube-like regions along the temporal direction, T-axis, with
strong correlation among the time-slices.

The picture that arises is that lowest modes are localised where the Polyakov
loop is negative and precisely in the region where only two of its eigenvalues
are equal and negative $\sim 1$, as discussed in Section 3.1. The same regions
are characterised by an action several times higher than the average and an
almost self dual gauge field. These are some gauge invariant properties of the
underlying fluctuations supporting the low modes at finite temperature. On the
same configurations the lowest modes of the operator with periodic boundary
condition show a much milder dependence, if none, on these features. The
localisation mechanism depends on the boundary conditions.

Figure 10: Ratio of the eigenmode weighted average of the action $\bar{s}_{n}$
and the average $s$, in correlation with the eigenvalue. Lowest modes favour
higher action regions. Temperatures ranging from $[0.9,1.9]~{}T_{c}$. Figure
11: Dependence of the $|q|/s$ ratio on the norm for several eigenvalue
sectors. Ensemble $32^{3}\times 12$, $\beta=4.30$, $m=0.01$, $T=220$ MeV.

###  3.3 Left and right projections

In this section we present the study on the correlation between the
localisation and the eigenmodes’ chiral properties. The intent is to find
signals of a connection between the Anderson localisation mechanism and the
chiral phase transition (via the Banks-Casher relation).

The non-zero eigenmodes of the non-hermitian Dirac operator $D$ have zero
total integrated chirality:

$\displaystyle\chi_{n}=\int\psi_{n}(x)^{\dagger}\gamma_{5}\psi_{n}(x)d^{4}x=0$
(5)

thus any positive fluctuation of the local chirality
$\psi_{n}(x)^{\dagger}\gamma_{5}\psi_{n}(x)$ is balanced by a negative one. We
identified the positive and negative fluctuations by projecting the eigenmodes
onto their left- and right-handed components. The idea behind this separation
is to study the chiral properties of the localised modes and the interaction
between the left and right projections.

For localised modes the typically observed spatial structure is a single large
fluctuation of the norm density $\rho(x)$ that contains two internal
fluctuations of the chirality. To clarify this point, the ideal situation is
depicted in Figure 12 as a sum and difference of two gaussians at different
separations. Notice that this ideal situation is never perfectly realised,
since several peaks can be present in the eigenmode landscape. Nevertheless it
is an helpful simplified model that drives this analysis.

Figure 12: Simplified model description of a localised mode. Two gaussian
distributions of the same weight and width represent the left and right
components. The norm and the chirality are respectively the sum and the
difference of the two gaussians. Left and right panels illustrate two cases
depending on the separations.

The interaction between the left and the right projections can be parametrised
by the spatial overlap among the two distributions. Defining $\rho_{L,R}$ as
the square norm of the left and right projection of every eigenmode of $D$,

$\displaystyle\psi_{L,R}(x)=\frac{1\pm\gamma_{5}}{2}\psi(x),\qquad\rho_{L,R}=|\psi_{L,R}(x)|^{2},$
(6)

we construct the relative overlap as follows

$\displaystyle
O_{n}=\frac{\int\rho_{n,L}(x)\rho_{n,R}(x)d^{4}x}{\frac{1}{2}\Bigl{[}\int(\rho_{n,L}(x)^{2}+\rho_{n,R}(x)^{2})d^{4}x\Bigr{]}},\qquad
O_{n}\in[0,1],$ (7)

that represents the integrated probability of finding both projections at a
given site normalized by their average size, their participation ratio. This
is zero for a zero mode by definition.

When two infinitely distant zero modes, with opposite topological charge, are
brought closer the interaction of their wave functions will raise their
eigenvalues from zero. It is a simple system described by a $2\times 2$ matrix
with only off-diagonal matrix elements, the modes interaction. We parametrised
the mode interaction with the $O_{n}$. In this simplified picture the left and
right modes increase their eigenvalue with the overlap.

The numerical results for the high temperature ensembles are reported in
Figure 13. We plot the dependence of $O_{n}$ on the eigenvalue in the physical
scale, after averaging in eigenvalue bins. The most relevant feature is the
monotonic increase of the overlap, with respect to the eigenvalue. There is a
mild dependence on the temperature in a wide range $[0.9,1.9]T_{d}$ and a
negligible dependence on the quark mass.

In the second panel, we show the same quantity for ensembles at zero
temperature, and for the operator with periodic boundary conditions on finite
temperature configurations. Non-zero overlap is observed ever for near-zero
modes, and this is not surprising due to the delocalised nature of the
eigenmodes for these ensembles. However the monotonic behaviour persists.

Figure 13: Overlap of the left and right projection for several ensembles, eq.
(7). Left panel shows the result for ensembles with localised modes. Right
panel for ensembles with only delocalised modes.

To accumulate further knowledge on the relation between the localisation and
the left-right mode separation we considered the dependence of the overlap
$O_{n}$ on the participation ratio. The $PR_{n}$ has been rescaled by the
volume $V$ to get the effective volume occupied by an eigenmode. In order to
compare different ensembles we computed $V\cdot PR_{n}$ in physical units. The
plot in Figure 14 shows the results for ensembles where the lowest modes are
localised. Only in these ensembles the rescaled PR has a physical meaning as
the size of the localised mode (i.e. independent of the total volume, see
Figure 3).

Figure 14: Dependence of the overlap $O_{n}$ of the left and right components
on the physical size of the localised modes. The averages of the first 10 non-
zero modes are plotted.

For every ensemble we plotted the averages of $V\cdot PR_{n}$ and the overlap
$O_{n}$ for the first 10 non-zero eigenmodes. The ensembles are ordered by
increasing temperature. There is an evident correlation between the size of
the eigenmode and the overlap of its left and right components. As the
temperature increases, the typical size shrinks while the overlap increases.
This observation, in combination with the monotonic relation between the
overlap and the eigenvalue, pushes the eigenvalues up in the spectrum. The
conclusion of this section is that the increasing localisation of the low
modes is responsible for the diminishing spectral density around the origin in
the chirally symmetric phase.

##  4 Interpretation in terms of topological objects

We now turn to an interpretation of the results presented in the previous
sections. Anticipating in short our conclusions, the background configurations
supporting the near-zero modes have the characteristics of monopole-instanton
(sometime called dyon in the literature) pairs. In order to keep the paper
self-contained, we first summarise some basic properties of the topological
objects called calorons, and their constituents, the monopole-instantons. For
a detailed discussion on their construction and their relation to confinement
we recommend some reviews, articles Diakonov:2009jq ; Bruckmann:2003yq ;
Shuryak:2012aa ; Poppitz:2012nz  and the references therein.

At zero temperature the relevant gauge field topological fluctuations are the
instantons. Instantons are saddle-point configurations that satisfy the
classical equation of motion $D_{\mu}F_{\mu\nu}=0$ Belavin:1975fg . They also
support zero modes of the Dirac operator Atiyah:1963zz ; Atiyah:1968mp . It is
straightforward to extend such solutions in the case of trivial holonomy by
making the fields periodic Harrington:1978ve ; Harrington:1978ua . However, at
finite temperature these are no longer solutions and they were extended by
Kraan, van Baal and Lee, Lu Kraan:1998pm ; Kraan:1998kp ; Kraan:1998sn ;
Lee:1998vu ; Lee:1998bb ; Lee:1997vp  to accommodate a background with a non-
trivial holonomy. Such saddle-point solutions are called calorons, KvBLL
calorons from the authors. KvBLL proved that the caloron solution has monopole
constituents with fractional topological charge Kraan:1998sn . These
constituent monopoles are BPS monopoles and are static, time-independent self-
dual solutions of the equations of motion. They possess a non-zero
chromomagnetic charge and correspond to non-trivial eigenvalues of the
holonomy.

A caloron with unit topological charge in $SU(N)$ is a magnetically neutral
combination of $N$ monopoles. There are $N-1$ BPS monopoles, which we call
$M$-type in this section, each carrying unitary magnetic charge in one of the
Abelian Cartan subgroups, associated with the simple roots of the $SU(N)$
group (forming the $A_{N-1}$ Lie algebra):

$A_{N-1}=\\{\alpha_{i}|\alpha_{i}=e_{i}-e_{i+1}\\},\qquad i\in\\{1\dots
N-1\\}$ (8)

in which $e_{i}$ is the unit vector in $\mathbb{R}^{N}$ along the direction
$i$. The $N$-th monopole has magnetic charge in each of these subgroups and it
is associated with the affine root

$\alpha_{N}=e_{N}-e_{1}=-\sum_{i}^{N-1}\alpha_{i}.$ (9)

The latter is the so called Kaluza-Klein (KK) monopole Lee:1997vp , which we
call $L$-type, and has a twisted gauge field along the compact dimension, so
it is not time-independent. To fix the notation let us write the expressions
for the asymptotic holonomy phases, $\mu_{m}\in[-\frac{1}{2},\frac{1}{2}]$

$\displaystyle L(\infty)=\exp\Bigl{[}2\pi i\,{\rm
diag~{}}(\mu_{1},\mu_{2},\dots,\mu_{N})\Bigr{]},$
$\displaystyle\mu_{1}\leq\mu_{2}\leq\dots\leq\mu_{N}\leq\mu_{1}+1\equiv\mu_{N+1},\qquad\sum_{m=1}^{N}\mu_{m}=0.$
(10)

after a suitable gauge transformation to order the eigenvalues. A summary of
the properties of the monopole solutions is given in Table 2, where $\nu_{m}$
is the difference between two consecutive eigenvalues of the holonomy,
$\mu_{m}$ and $\mu_{m+1}$,

$\displaystyle\nu_{m}\equiv\mu_{m+1}-\mu_{m},\qquad\sum_{m=1}^{N}\nu_{m}=1,$
(11)

the last relation coming from the determinant constraint. The naming
convention for the monopoles adheres to Diakonov:2009jq . The value of the
gauge field at the centre of a $m$ monopole has been derived for all simple
Lie groups by Weinberg:1979zt ; Davies:2000nw  and for $SU(N)$ is:

$\displaystyle L(0)=\exp\Bigl{[}2\pi i\,{\rm
diag~{}}\Bigl{(}\mu_{1},\mu_{2},\dots,\mu_{m-1},\underbrace{\frac{\mu_{m}+\mu_{m+1}}{2}}_{m},\underbrace{\frac{\mu_{m}+\mu_{m+1}}{2}}_{m+1},\dots\Bigr{)}\Bigr{]},$
(12)

which is cyclic for the $N$-th KK monopole, i.e. $m=N$ and $m+1=N+1\equiv 1$.
At least two of the eigenvalues must coincide in the centre of a monopole.

| $M_{m}$ | $\bar{M}_{m}$ | $L$ | $\bar{L}$
---|---|---|---|---
E-charge | + | + | $-(N-1)$ | $-(N-1)$
B-charge | + | $-$ | $-(N-1)$ | $+(N-1)$
action S, $\frac{8\pi^{2}}{g^{2}}$ | $\nu_{m}$ | $\nu_{m}$ | $1-\sum\nu_{m}$ | $1-\sum\nu_{m}$
top. charge Q | $\nu_{m}$ | $-\nu_{m}$ | $1-\sum\nu_{m}$ | $\sum\nu_{m}-1$
Table 2: List of characteristics of the self-dual solutions for $SU(N)$
groups. Here the index $m\in[1\dots N-1]$, distinguishes the BPS ($M$-type)
monopoles. $\nu_{m}$ phase is in unit of $2\pi$. The bar denotes the anti-
monopoles solutions. $L$-type monopoles are the Kaluza-Klein monopoles. A
$Q=1$ caloron is composed by the sum of the different $N-1$ $M$-type monopoles
and the $L$-type. It is neutral and has the canonical action
$\frac{8\pi^{2}}{g^{2}}$. $L$ monopoles have B-charge $\pm 1$ in each one of
the $N-1$ Cartan subgroups.

Below the phase transition the average Polyakov line is zero and thus the
expected values for $\nu_{m}$ are all equal to $1/N$ on average, the maximal
distance between all the eigenvalues. In this symmetric case the action and
the topological charge are the same for all types of monopoles. Above the
phase transition the Polyakov line tends to a configuration where all the
eigenvalues are identical to one of the elements of the centre of $SU(N)$.
This results in $M$-type monopoles to be light, $\nu_{m}\ll 1$, and $L$-type
monopoles to be heavy, $\nu_{N}\sim 1$, see Table 2. The monopoles are not
particles, therefore do not have a mass. We keep using the wording light/heavy
with the caveat that it only refers to their carried action.

The topological charge fluctuation will be mostly localised around the heavy
monopoles. A caloron is composed by the sum of the different $N-1$ $M$-type
monopoles and one of the $L$-type. It is neutral, has the canonical action
$\frac{8\pi^{2}}{g^{2}}$ and carries topological charge 1.

For the sake of clarity let us write the explicit form of the above equations
for the case of an $SU(3)$ gauge theory by referring to Figure 15. The
holonomy at spatial infinity at very large temperature approaches

$\displaystyle L(r\rightarrow\infty)_{T\rightarrow\infty}=\exp\Bigl{[}2\pi
i\,{\rm diag~{}}(0,0,0)\Bigr{]}$ (13)

where $\mu_{1}=\mu_{2}=\mu_{3}=0$, and $r$ is the distance from the monopole
core. The corresponding differences are $\nu_{1}=\nu_{2}=0$ that give the 2
$M$-type monopoles, and $\nu_{3}=1$ for the heavy $L$ monopole. The action is
zero for the two light monopoles and it is concentrated on the heavy one. Also
the topological charge is carried only by the $L$ monopole. In this regime the
holonomy at the centre of the monopoles, according to 12 is given by

$\displaystyle L(0)_{M}=\exp\Bigl{[}2\pi i\,{\rm
diag~{}}(0,0,0)\Bigr{]},\qquad L(0)_{L}=\exp\Bigl{[}2\pi i\,{\rm
diag~{}}\Bigl{(}\frac{1}{2},\frac{1}{2},0\Bigr{)}\Bigr{]}$ (14)

modulo gauge rotation of the eigenvalues. They give $P=(1,0)$ and $P=(-1/3,0)$
respectively for the complex Polyakov line. This is of course an idealised
situation, it is nevertheless a useful guide to understand the numerical data.

Figure 15: Example of holonomy eigenvalue $\mu_{m}$ (black dots) in the unit
circle for $T\rightarrow\infty$. The dashed lines show the difference
$\nu_{m}$ that are related to the action and the topological charge. The grey
square and diamond indicate the fermionic phase for the respective boundary
conditions. In this specific example the zero mode for APBC will be sitting in
the $L$ monopole and on the $M_{2}$ monopole for PBC. The value of the
holonomy at the centre of a specific monopole is read by taking the middle
point of the corresponding arc (doubly degenerate) and the remaining
eigenvalue (red lines in the figure showing the case of the $L$ monopole).

The relation between the calorons and the zero modes of the Dirac operator has
been explored analytically in GarciaPerez:1999ux ; Chernodub:1999wg ;
Nye:2000eg ; Poppitz:2008hr , where the exact expression for $SU(2)$ and
$SU(N>2)$ fermionic zero-modes was derived for a caloron with $Q=1$ and non-
trivial holonomy (see also GarciaPerez:2009mg  for a discussion on the modes
of the Dirac operator in the adjoint representation). A remarkable result of
these papers is the dependence of the eigenmode location from the boundary
conditions of the fermionic field.

The situation is best illustrated in the case of $SU(2)$ where only two
species form the caloron GarciaPerez:1999ux ; Poppitz:2008hr . Let’s start
with a background configuration for a caloron made of the combination $ML$.
The zero mode is localised on one of the constituents and distinguishes
between the periodic and anti-periodic boundary conditions. The general result
states that for the boundary condition $\psi(t+T)=\exp(2\pi i\phi)\psi(t)$ the
eigenmode is localised on the monopole $m$ that satisfies
$\phi\in[\mu_{m},\mu_{m+1}]$. At high temperature periodic zero modes are
localised on the light $M$-constituents while anti-periodic zero-modes
localise around the heavier $L$ monopoles, see Figure 15. This is true for
$SU(N)$ groups too, with some exceptions Chernodub:1999wg  that are irrelevant
in the deconfined phase. Figure 15 exemplifies the case of $N_{c}=3$ and
Figure 16 depicts the typical distribution of the norm of the eigenmodes in
the Polyakov loop plane for the low-mode and the high-mode regions in the high
temperature phase.

Figure 16: Density plot for the norm of the eigenmodes in the Polyakov loop
plane. Average of the mode by mode results of Figure 7. Ensemble $32^{3}\times
12$, $\beta=4.30$, $m=0.01$, $T=220$ MeV. (left) Lower eigenmodes with
$\lambda<0.03$. (right) higher eigenmodes with $\lambda>0.1$. Analogous
results by filtering using the eigenmodes in Bornyakov:2014esa ;
Bornyakov:2015xao .

The results of this paper suggest that these properties of the zero-modes for
the classical Yang-Mills (YM) background do apply also for low-lying non-zero
modes. The underlying assumption is that topological fluctuations are stable
under deformations. The numerical evidences in Section 3, on the properties of
the background configurations supporting the low-lying modes with anti-
periodic boundary conditions, indicate that these modes are localised around
the heavy KK monopole fluctuations of the gauge field. We measured a set of
gauge invariant observables characterising the gauge fluctuations. The values
of the Polyakov line at the centre of the eigenmodes, the averaged action, the
self-duality of the gauge field and the boundary condition dependence are all
in accordance with the characteristic of the monopole-instantons. We also
cross-checked the dependence on the boundary condition by confirming that the
number of zero modes is unchanged by the boundary condition.

These monopole-instantons are not necessarily part of a caloron: in the case
they support non-zero modes, the results from Section 3 support the picture
that they are coupled in topologically neutral pairs, $L\bar{L}$ pairs. Other
works in the literature have presented evidence of monopole-instanton (dyonic)
structures near the phase transition by using the low modes as UV filter for
the topological charge, see Bornyakov:2014esa ; Bornyakov:2015xao  and
references therein. Recently a work with chiral fermions investigated the
topological features of the high temperature phase Dick:2015twa .

Here we showed using chiral fermions that, mode by mode, the disorder
underlying the Anderson localisation is composed by $L$-type monopoles which
have increasing action with the temperature, and this is reflected by the
increasing localisation of the near-zero modes. At finite temperature these
heavy $L$-type monopoles excitations are expected to be suppressed while most
of the lattice points support lighter fluctuations. This is confirmed
qualitatively by our results, but we did not estimate the $L$ monopole
density. The situation changes below the phase transition where there is no
difference in the monopole actions on average, and the $L$-type monopoles are
now easier to appear as fluctuations. If the low-modes support mechanism is
the same on both sides, it is not difficult to qualitatively explain the
delocalisation below $T_{d}$, and above $T_{d}$ by changing the boundary
conditions, in connection with the expected relevant monopole abundance.

In this paper we do not address the task of estimating the abundance of the
monopole structures and its dependence on the temperature and mass of the sea
quarks. This will be the subject of a future study. The statistical mechanics
of monopole instanton ensembles has been studied in Faccioli:2013ja .

The relations of other pairs of monopoles ($L\bar{M}$, $M\bar{L}$, $M\bar{M}$)
to the lowest eigenmodes is disfavoured by the numerical evidence (with APBC).
The support of the near-zero modes strongly favours large action for both
positive and negative topology fluctuations. It also favours a region where
the Polyakov loop has only two coinciding eigenvalues (${\rm
Re}P\rightarrow-1/3$). All the other couples except $L\bar{L}$ do not satisfy
these requirements. This does not exclude their presence in the gauge
configurations but it rules out their relation to the Anderson localisation
mechanism. By changing the boundary conditions the situation is reversed and
the eigenmodes are allowed to hop between $M$-type monopoles. Results still
seem to exclude that mixed pairs are correlated to the low energy spectrum.

An interesting result is that we could not observe a strong dependence of the
results on the fermion masses. If there is any dependence, it is below the
sensitivity of our probes. A mass dependence of the interaction among monopole
pairs was predicted in some models Shuryak:2012aa .444For discussion of the
dyon-antidyon liquid model see Liu:2015ufa ; Liu:2015jsa .

We have observed in Section 3.3 that the increasing localisation with the
temperature, coming from the heavy monopoles, is associated with an increasing
overlap of the lowest modes. In the monopole picture two infinitely distant
$L$-type monopoles will support two zero modes, when surrounded by $M$-type
monopoles. Any overlap of their wave function will result in an increased
eigenvalue. This picture explains the measured monotonic dependence between
the eigenvalue $\lambda_{n}$ and the overlap $O_{n}$. Together with the
localisation-overlap relation, this can be the mechanism that suppresses the
spectral density at high temperature, eventually leading to the chirally
symmetric phase in the thermodynamical limit. This is a conjecture that needs
further investigation. If proven true can be the leading indication of a
deeper connection between the deconfinement phase transition and the chiral
phase transition, via localisation and Banks-Casher relation. In principle two
distinct mechanism are at work here that can drive the chiral restoration. The
first is the abundance of $L$-monopoles that is suppressed with the
temperature. The second one is the formation of $L\bar{L}$ molecules that can
lead to a spectral gap, Figure 13. The current numerical results are not
sufficient to differentiate among the two and clarify their relative
importance.

Below the phase transition the action of the monopoles is similar and thus the
relative abundance of $M$ and $L$ monopoles would be the same. The consequence
is an independence of the results on the boundary conditions. The opposite
case happens at finite temperature, as demonstrated by the data in Section 3.
Models have been discussed in Shuryak:2012aa  that reached similar
conclusions. The dependence of the chiral condensate on the boundary
conditions in the deconfined phase has been reported in Bilgici:2008qy ;
Bilgici:2009tx . In this conjecture the change in density of the $L$
monopoles, strictly intertwined with the Polyakov line, is the trigger of the
chiral phase transition and can explain our results of Section 3.3 as well as
Bilgici:2008qy ; Bilgici:2009tx , hence $T_{c}\geq T_{d}$. Notice that the
Anderson localisation and the presence of monopoles Bornyakov:2014esa  has
been observed also in pure gauge setups Giordano:2015vla  where all fermion
interactions are switched off. This seems to suggest that the fermion
interactions could be a second order effect in the mechanism of localisation.

##  5 Conclusions and future works

We have studied the finite temperature phase of two-flavor QCD using chiral
fermions around the phase transition, $T\in[0.9,1.9]T_{c}$. At fixed finite
temperature $T$, the Dirac spectrum undergoes a quantum phase transition that
is in accordance with the prediction of the Anderson model for localisation
Giordano:2013taa ; Ujfalusi:2015nha . The near-zero Dirac eigenmodes at high
temperature are localised up to a critical eigenvalue $\lambda_{c}(T)$. The
Anderson model for the metal-insulator transition describes a localisation
mechanism of wave functions due to the presence of disorder in the crystal. We
studied the properties of the gauge configurations that can cause disorder and
localise the lowest Dirac eigenmodes.

We presented several numerical evidences that the background gauge
configurations supporting each near-zero eigenmode of the Dirac operator in
the fundamental representation are self-dual and carry higher action than the
volume average. They are strongly related to fluctuations of the Polyakov line
where two of the eigenvalues of the holonomy are identical (equal to $-1$ at
high temperature). The data show localisation of the lowest eigenmodes in such
regions of negative Polyakov line, large action and self-dual gauge fields.
Changing the boundary conditions of the Dirac operator on the same finite
temperature ensembles washes up the localised states and all modes become
delocalised.

Furthermore, we studied the chiral properties and in particular the overlap of
the left and right chiral projections of the non-zero eigenmodes. This overlap
is a monotonically increasing function of the eigenvalue with mild dependence
on the temperature. By varying the temperature the overlap shows also inverse
correlation with the participation ratio, meaning that the more localised
modes show larger overlap of their left and right components.

We proposed an interpretation of the numerical results in terms of monopole-
instanton fluctuations of the background gauge field. Monopole-instantons are
self-dual solutions of the Yang-Mills equations of motion in a non-trivial
holonomy background. In the deconfined phase there is a class of monopoles
that support larger action (in our terminology “heavier”). The properties of
the observed gauge background are in accordance with the properties of heavy
monopole-antimonopole molecules. These fluctuations trap the lowest eigenmodes
and trigger localisation.

In the monopole model the support of the eigenmodes is expected to change with
the boundary conditions. At high temperature, heavier monopoles are favoured
by the physical case of anti-periodic boundary conditions. The eigenmode
support moves to the lighter monopoles with periodic boundary conditions. We
observed a dependence on the boundary conditions of the eigenmodes consistent
with this expectation.

Based on the current results, we conjectured a relation of the deconfinement
phase transition, driven by the Polyakov line, and the chiral phase
transition. The key connection element is the heavy monopole action that is
linked to the localisation of the low-modes, and the relation of the overlap
between the left and right components of the eigenmode and its eigenvalue.
Further study is needed to strengthen the connection and the next step would
be a proper estimate of the density of the several monopole species and
monopole-pairs and the dependence of these characteristics on the temperature
and mass of fermions. The connection between the localisation mechanism and
the chiral phase transition has been considered in GarciaGarcia:2006gr  and in
the model based approach Pittler:2014qea ; Giordano:2016cjs .

We are currently investigating a non-perturbative derivation of an _effective
potential_ that drives the localisation of the fermionic modes. This would be
a particularly useful tool to understand Anderson localisation mechanism in
QCD, especially the appearance of a mobility edge at finite temperature. The
results will appear in future works.

An interesting extension of this study would be the research on the adjoint
fermions case. The two phase transition are separated in the $SU(3)$ case
Karsch:1998qj . The different localisation properties of the adjoint
eigenmodes GarciaPerez:2009mg  can account for this difference.

SU(3) with adjoint fermions in $R^{3}\times S^{1}$ Cossu:2009sq ;
Misumi:2014raa ; Cossu:2013ora  is another theory that has a peculiar phase
structure where the average Polyakov line can get non-trivial values. In the
completely broken $U(1)\times U(1)$ phase where new magnetically charged and
topologically neutral molecules of BPS and anti-KK monopoles, called bions are
expected to be relevant for confinement Unsal:2007jx ; Misumi:2014raa . It
would be interesting to test the background configurations for bions using the
methods illustrated in this paper.

###### Acknowledgements.

We thank the members of the JLQCD collaboration, especially H. Fukaya and A.
Tomiya, for many discussions and for sharing the work of generating and
analyzing the finite temperature lattice ensembles. GC would like to thank M.
Giordano, M. Ünsal and T. Kanazawa for several very useful discussions during
the work on this subject and for helping to improve the original manuscript.
Numerical simulations are performed on IBM System Blue Gene Solution at High
Energy Accelerator Research Organization (KEK) under a support for its Large
Scale Simulation Program (No. 14/15-10). This work is supported in part by the
Grant-in-Aid of the Ministry of Education (No. 26247043, 15K05065) and by MEXT
SPIRE and JICFuS.

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