# Localization by particle-hole symmetry breaking:
a loop expansion

K. Ziegler  Institut für Physik, Universität Augsburg
D-86135 Augsburg, Germany

###### Abstract

Localization by a broken particle-hole symmetry in a random system of non-
interacting quantum particles is studied on a $d$–dimensional lattice. Our
approach is based on a chiral symmetry argument and the corresponding
invariant measure, where the latter is described by a Grassmann functional
integral. Within a loop expansion we find for small loops diffusion in the
case of particle-hole symmetry. Breaking the particle-hole symmetry results in
the creation of random dimers, which suppress diffusion and lead to
localization on the scale $\sqrt{D/|\mu|}$, where $D$ is the effective
diffusion coefficient at particle-hole symmetry and $\mu$ is the parameter
related to particle-hole symmetry breaking.

##  I Introduction

There is strong evidence that disordered systems with particle-hole (PH)
symmetry can avoid Anderson localization in any spatial dimension. Such a
behavior was observed for one-dimensional systems at the band center of a one-
dimensional tight-binding model with random hopping some time ago [1, 2, 3].
There is also numerical evidence for extended states at the band center in
disordered two-dimensional lattice models [4, 5]. A renewed interest in this
problem appeared with the discovery of two-dimensional Dirac-like materials,
such as graphene [6, 7, 8]. Graphene is a semimetal with a very robust
conductivity at the PH-symmetric Dirac point. On the other hand, after
breaking the PH symmetry by doping, its conductivity changes substantially; it
is either enhanced for weak disorder or reduced for strong disorder [9, 10,
11, 12]. Despite of a substantial effort, there was no conclusive confirmation
of localization away from the PH-symmetric Dirac point, neither from the
theory side [13, 14] nor from the experiment [9, 10, 11, 12].

While in the PH-symmetric case diffusion was identified as the dominant
behavior of non-interacting Dirac particles in a random environment, the
breaking of the PH symmetry was accompanied with the creation of random dimers
[15]. A preliminary work, based on a perturbative renormalization group
analysis, did not reveal localization, though [16]. In the following we will
analyze the competition of diffusing quantum particles and randomly
distributed dimers on a $d$–dimensional lattice without employing a
perturbation theory.

Starting from a PH-symmetric random Hamiltonian $H$, a symmetry-breaking term
is defined by a uniform shift $\mu$ of the PH-symmetric Hamiltonian. The
resulting Hamiltonian is still invariant under a chiral transformation. This
invariance is relevant for the analysis of spontaneous chiral-symmetry
breaking and the corresponding long-scale behavior of the model, such that we
reduce our description to the corresponding invariant measure (IM). Then the
IM, written in terms of a Grassmann functional integral [17], is represented
by a loop expansion, which consists of graphs with 4-vertices. Only the
smallest loops are taken into account, which is known as the nonlinear sigma
model approximation. This is used as the starting point for the analysis of
the spatial correlations. As mentioned above, there is diffusion in the
presence of PH symmetry with a diffusion coefficient $D$, which depends on the
scattering rate $\eta$ and the average Hamiltonian $\langle H\rangle$. After
breaking the PH symmetry with $\mu\neq 0$, we create small loops with two
sites on the lattice which represent repulsive lattice dimers. They act as
obstacles for the diffusion and suppress the latter on large scales, which
leads to an exponential decay of the particle correlation. Since the density
of the dimers is proportional to $|\mu|$, this localization effect increases
with this parameter. An estimation of the decay length gives $\sqrt{D/|\mu|}$.

##  II Model: Invariant measure

We briefly recapitulate the main ideas which were developed in the previous
work on the loop expansion and the IM for non-interacting particles in a
random environment [18, 19]. For this purpose a random Hamiltonian matrix $H$
on a lattice $\Lambda$ is considered. It is assumed that this Hamiltonian has
an internal spinor structure; its matrix elements are of the form $H_{{\bf
r}j,{\bf r}^{\prime}j^{\prime}}$, where ${\bf r},{\bf r}^{\prime}\in\Lambda$
and $j,j^{\prime}=1,2,...,N$ is a spinor index, where the latter can also be a
band index of a multiband Hamiltonian. We further assume that there is an
$N\times N$ unitary matrix $U$ that (i) acts only on the spinor or band index
and (ii) for which the Hamiltonian matrix obeys the relation

$UH^{T}U^{\dagger}=-H\ .$ (1)

This relation implies a particle-hole transformation in the following sense:
Since $H$ is Hermitean, we have $H^{T}=H^{*}$ and the relation (1) implies for
the eigenstate $\Psi_{E}$ of $H$ with the real eigenvalue $E$

$H^{*}U^{\dagger}\Psi_{E}=-U^{\dagger}H\Psi_{E}=-EU^{\dagger}\Psi_{E}\ .$

Complex conjugation of this equation yields

$H(U^{\dagger}\Psi_{E})^{*}=-E(U^{\dagger}\Psi_{E})^{*}\ ,$

such that $\Psi_{-E}=(U^{\dagger}\Psi_{E})^{*}$ is eigenstate of $H$ with
eigenvalue $-E$. Thus, there is a PH symmetry for $E=0$. The PH symmetry of
$H$ is broken by $\mu$ for $H_{\pm}:=H\pm\mu\sigma_{0}$, where $\sigma_{0}$ is
the $N\times N$ unit matrix. In the following we will analyze the effect of a
shift of $E=0$ by $\mu$, using the nonlinear sigma model approach.

Following Refs. [16, 15], we extend $H$ to the random Hamiltonian matrix

${\hat{H}}=\pmatrix{H_{+}&0&0&0\cr 0&H_{-}&0&0\cr 0&0&H_{+}^{T}&0\cr
0&0&0&H_{-}^{T}\cr}\equiv\pmatrix{{\bar{H}}&0\cr 0&{\bar{H}}^{T}\cr}\ .$ (2)

At the PH symmetric point ($\mu=0$) it satisfies the relation
${\hat{H}}{\hat{S}}_{0}=-{\hat{S}}_{0}{\hat{H}}$ for

${\hat{S}}_{0}=\pmatrix{0&0&\varphi_{3}U&\varphi_{1}U\cr
0&0&\varphi_{2}U&\varphi_{4}U\cr\varphi^{\prime}_{3}U^{\dagger}&\varphi^{\prime}_{2}U^{\dagger}&0&0\cr\varphi^{\prime}_{1}U^{\dagger}&\varphi^{\prime}_{4}U^{\dagger}&0&0\cr}$
(3)

with some general parameters $\varphi_{j}$. For a broken PH symmetry ($\mu\neq
0$) it satisfies ${\hat{H}}{\hat{S}}=-{\hat{S}}{\hat{H}}$ for

${\hat{S}}=\pmatrix{0&0&0&\varphi_{1}U\cr 0&0&\varphi_{2}U&0\cr
0&\varphi^{\prime}_{2}U^{\dagger}&0&0\cr\varphi^{\prime}_{1}U^{\dagger}&0&0&0\cr}\
.$ (4)

The PH-symmetric case was studied previously, such that we can focus
subsequently on the broken PH symmetry. Then the relation
${\hat{H}}{\hat{S}}=-{\hat{S}}{\hat{H}}$ implies for ${\hat{H}}$ the chiral
symmetry $e^{\hat{S}}{\hat{H}}e^{\hat{S}}={\hat{H}}$ of the extended
Hamiltonian defined in Eq. (2). The chiral symmetry reveals some interesting
properties, which will be discussed next.

First, we construct the IM that is associated with the chiral symmetry. For
the matrix ${\hat{S}}$ and the graded determinant (cf. Eq. (30)) we get

$detg(e^{2{\hat{S}}})=\exp[Trg(\log e^{2{\hat{S}}})]=\exp[2Trg({\hat{S}})]=1\
,$

since the graded trace vanishes: $Trg({\hat{S}})=0$. Together with the chiral
symmetry of ${\hat{H}}$ this implies immediately

$detg({\hat{H}}+i\eta)=detg({\hat{H}}+i\eta)detg(e^{2{\hat{S}}})=detg(e^{\hat{S}}{\hat{H}}e^{\hat{S}}+i\eta
e^{2{\hat{S}}})=detg({\hat{H}}+i\eta e^{2{\hat{S}}})\ ,$ (5)

where $detg({\hat{H}}+i\eta)=1$, according to the definition of the graded
determinant. Thus, $detg({\hat{H}}+i\eta e^{2{\hat{S}}})$ is invariant under
the chiral transformation.

From relation (5) we can construct the IM through substituting the general
parameters $\varphi_{j}$ by a spatial Grassmann field $\varphi_{{\bf r}j}$. As
explained in App. A, this leads to the lattice version of the IM with

$J_{j}=\det({\bf
1}+\varphi^{\prime}_{j}\varphi_{j}-\varphi^{\prime}_{j}h\varphi_{j}h^{\dagger})^{-1}=\det({\bf
1}+\varphi_{j}\varphi^{\prime}_{j}-h\varphi_{j}h^{\dagger}\varphi^{\prime}_{j})\
\ \ (j=1,2),$ (6)

where the relation is derived in (34), with

$h={\bf 1}+2i\eta{\bar{G}}_{0}\ ,\ \ \
{\bar{G}}_{0}=({\bar{H}}_{0}+i\epsilon-i\eta)^{-1}\ .$ (7)

$J_{j}$ is an invariant for a global chiral transformation, provided that $h$
is unitary. This is the case for $\epsilon=0$: $hh^{\dagger}={\bf
1}+O(\epsilon)$ [19]. It is important to realize that the random Hamiltonian
${\hat{H}}$ has been replaced by its average
${\bar{H}}_{0}:=\langle{\bar{H}}\rangle$ in the IM. In Refs. [18, 19] this IM
was associated with the correlator $K_{{\bf R}{\bf R}^{\prime}}$ between the
lattice sites ${\bf R}$ and ${\bf R}^{\prime}$ through the relation

$K_{{\bf R}{\bf R}^{\prime}}=\frac{1}{\cal N}\int_{\Lambda}J_{j}\varphi_{{\bf
R}j}\varphi_{{\bf R}^{\prime}j}\ ,\ \ \ {\cal
N}=\int_{\Lambda}J_{1}=\int_{\Lambda}J_{2}$ (8)

of a two-component Grassmann field $\varphi_{{\bf r}j}$ (${\bf r}\in\Lambda$).
$\int_{\Lambda}$ is the functional integral with respect to the Grassmann
field on the lattice $\Lambda$. This correlator indicates localization on the
localization length $\xi$ when it decays exponentially on the scale $\xi$. It
is identical for both components of the Grassmann field $j=1,2$, such that we
can drop this index subsequently. Another important feature of the Green’s
function $h$ is that the relation (1) implies for the PH-symmetric case
$\mu=0$ the relation

$Uh^{\dagger}U^{\dagger}=h\ ,$ (9)

which does not hold for $\mu\neq 0$.

The Grassmann field can be expressed by its Fourier components as

$\varphi_{\bf r}=\sum_{\bf k}e^{-i{\bf k}\cdot{\bf r}}{\tilde{\varphi}}_{\bf
k}={\tilde{\varphi}}_{0}+\sum_{{\bf k}\neq 0}e^{-i{\bf k}\cdot{\bf
r}}{\tilde{\varphi}}_{\bf k}\equiv{\tilde{\varphi}}_{0}+\tau_{{\bf r}}\ ,$
(10)

where the zero mode ${\tilde{\varphi}}_{0}$ does not depend on ${\bf r}$. Then
the normalization becomes

${\cal N}=\int_{\Lambda}\det\\{{\bf
1}+[{\tilde{\varphi}}_{0}+\tau-h({\tilde{\varphi}}_{0}+\tau)h^{\dagger}]({\tilde{\varphi}}_{0}^{\prime}+\tau^{\prime})\\}$

and with $hh^{\dagger}={\bf 1}-\epsilon\bar{\Gamma}+O(\epsilon^{2})$ we get

${\tilde{\varphi}}_{0}-h{\tilde{\varphi}}_{0}h^{\dagger}={\tilde{\varphi}}_{0}({\bf
1}-hh^{\dagger})=\epsilon\bar{\Gamma}{\tilde{\varphi}}_{0}+O(\epsilon^{2})\ ,$

since the space-independent ${\tilde{\varphi}}_{0}$ commutes with $h$.
$\epsilon$ can be absorbed into the Grassmann integration by rescaling
${\tilde{\varphi}}_{0}\to{\bar{\tau}}/\epsilon$. This implies for the
normalization

${\cal N}=\epsilon\int_{\Lambda}\det[{\bf
1}+(\bar{\Gamma}{\bar{\tau}}+\tau-h\tau
h^{\dagger})({\tilde{\varphi}}_{0}^{\prime}+\tau^{\prime})]\ ,$ (11)

where we have neglected terms of order $\epsilon$ inside the integrand. For
the unnormalized correlator this rescaling argument for
${\tilde{\varphi}}_{0}$ provides a constant term
${\tilde{\varphi}}_{0}{\tilde{\varphi}}^{\prime}_{0}$ from the external
Grassmann variable $\varphi_{\bf R}\varphi^{\prime}_{{\bf R}^{\prime}}$:

$\int_{\Lambda}J\varphi_{{\bf R}}\varphi_{{\bf
R}^{\prime}}=\int_{\Lambda}J({\tilde{\varphi}}_{0}+\tau_{{\bf
R}})({\tilde{\varphi}}^{\prime}_{0}+\tau^{\prime}_{{\bf
R}^{\prime}})=\epsilon\int_{\Lambda}J{\bar{\tau}}({\tilde{\varphi}}^{\prime}_{0}+\tau^{\prime}_{{\bf
R}^{\prime}})+\int_{\Lambda}J\tau_{{\bf
R}}({\tilde{\varphi}}^{\prime}_{0}+\tau^{\prime}_{{\bf R}^{\prime}})\ ,$ (12)

where only the first term on the right-hand side vanishes with $\epsilon\to
0$. This implies for the normalized correlator that the second term diverges
for $\epsilon\to 0$, which is a consequence of the broken translational
invariance due to the factor $\tau_{\bf R}$.

##  III Loop expansion

The IM of Eq. (6) can be rewritten as

$J=\exp\\{Tr([WW^{\dagger}]_{d}\varphi\varphi^{\prime})\\}\det({\bf 1}-\varphi
X^{\dagger}\varphi^{\prime}-X\varphi\varphi^{\prime}-W\varphi
W^{\dagger}\varphi^{\prime})$ (13)

when we define $h=[h]_{d}+W$, $X=W[h]_{d}^{\dagger}$, where $[h]_{d}$ and
$[WW^{\dagger}]_{d}$ are the spatial diagonal parts of $h$ and $WW^{\dagger}$,
respectively. Here we implicitly assume the limit $\epsilon\to 0$. Using the
determinant identity $\det(A)=\exp[Tr(\log A)]$ this yields for the IM after
expanding the logarithm

$\log J=Tr([WW^{\dagger}]_{d}\varphi\varphi^{\prime})-\sum_{l\geq
1}\frac{1}{l}Tr[(\varphi
X^{\dagger}\varphi^{\prime}+X\varphi\varphi^{\prime}+W\varphi
W^{\dagger}\varphi^{\prime})^{l}]\ .$ (14)

This sum terminates on a finite lattice $\Lambda$ for $l=|\Lambda|$ due to the
Grassmann field. The trace term with the power $l$ represent a sum of loops of
length $l$ on the lattice, such that the sum can be considered as a loop
expansion of the IM [18]. Integration with respect to the Grassmann field
yields graphs with 4-vertices from the term $W\varphi
W^{\dagger}\varphi^{\prime}$ because the non-zero Grassmann integral requires
at each site ${\bf r}$ the product $\varphi_{\bf r}\varphi^{\prime}_{\bf r}$.
The 4-vertex graphs reflect the equivalence of the expression (14) with the
random phase representation of the IM [19].

###  III.1 Nonlinear sigma model

As a special case of the loop expansion (14) only the smallest loops are
considered, namely only loops that contain at most two hopping matrices,
either $X$ or $W$. This approximation is known as the nonlinear sigma model
[20] and becomes in the present case

$\log J_{NLSM}=$ $Tr([WW^{\dagger}]_{d}\varphi\varphi^{\prime})-Tr(W\varphi
W^{\dagger}\varphi^{\prime})-Tr(\varphi
X^{\dagger}\varphi^{\prime}+X\varphi\varphi^{\prime})-Tr[(\varphi
X^{\dagger}\varphi^{\prime}+X\varphi\varphi^{\prime})(\varphi
X^{\dagger}\varphi^{\prime}+X\varphi\varphi^{\prime})]\ ,$

where the third term vanishes due to the trace of an off-diagonal matrix $X$,
such that the IM reduces to

$\log J_{NLSM}=Tr([WW^{\dagger}]_{d}\varphi\varphi^{\prime})-Tr(W\varphi
W^{\dagger}\varphi^{\prime})+Tr(X^{\dagger}\varphi\varphi^{\prime}X^{\dagger}\varphi\varphi^{\prime}-X\varphi\varphi^{\prime}X\varphi\varphi^{\prime})\
.$ (15)

The first two terms represent a diffusion propagator

$Tr_{N}([WW^{\dagger}]_{{\bf r}{\bf r}})\delta_{{\bf r}{\bf
r}^{\prime}}-Tr_{N}(W_{{\bf r}{\bf r}^{\prime}}W^{\dagger}_{{\bf
r}^{\prime}{\bf r}})=\sum_{{\bf r}^{\prime\prime}}Tr_{N}(W_{{\bf r}{\bf
r}^{\prime\prime}}W^{\dagger}_{{\bf r}^{\prime\prime}{\bf r}})\delta_{{\bf
r}{\bf r}^{\prime}}-Tr_{N}(W_{{\bf r}{\bf r}^{\prime}}W^{\dagger}_{{\bf
r}^{\prime}{\bf r}})\ ,$ (16)

where $Tr_{N}$ is the trace with respect to the spinor index. The quartic term
reads

$Tr(X^{\dagger}\varphi\varphi^{\prime}X^{\dagger}\varphi\varphi^{\prime}-X\varphi\varphi^{\prime}X\varphi\varphi^{\prime})=\sum_{{\bf
r},{\bf r}^{\prime}}\beta_{{\bf r}{\bf r}^{\prime}}\varphi_{{\bf
r}}\varphi^{\prime}_{{\bf r}}\varphi_{{\bf r}^{\prime}}\varphi^{\prime}_{{\bf
r}^{\prime}}\ .$

$\beta$ is an imaginary matrix due to

$\beta_{{\bf r}{\bf r}^{\prime}}:=Tr_{N}(X^{\dagger}_{{\bf r}{\bf
r}^{\prime}}X^{\dagger}_{{\bf r}^{\prime}{\bf r}}-X_{{\bf r}{\bf
r}^{\prime}}X_{{\bf r}^{\prime}{\bf r}})\ .$ (17)

Relation (9) implies that $\beta$ vanishes in the presence of the PH symmetry
(i.e., for $\mu=0$). This was also observed in Refs. [16, 15].

Next, we represent the functional integral with a quadratic form in the
Grassmann field, using a coupling of the latter to a real Gaussian field
$Q_{\bf r}$. This can be achieved by exploiting the relation

$m_{\bf r}{\bar{\beta}}_{{\bf r}{\bf r}^{\prime}}m_{{\bf r}^{\prime}}-(Q_{\bf
r}+V_{{\bf r}{\bf r}^{\prime\prime}}m_{{\bf r}^{\prime\prime}})V^{-1}_{{\bf
r}{\bf r}^{\prime}}(Q_{{\bf r}^{\prime}}+V_{{\bf r}^{\prime}{\bf
r}^{\prime\prime\prime}}m_{{\bf r}^{\prime\prime\prime}})=-Q_{\bf
r}V^{-1}_{{\bf r}{\bf r}^{\prime}}Q_{{\bf r}^{\prime}}-2m_{\bf r}Q_{\bf r}$
(18)

with the sum convention for paired indices and with the correlation matrix
$V=i\alpha{\bf 1}+{\bar{\beta}}$ and ${\bar{\beta}}=\beta/i|\mu|$. We note
that $m_{\bf r}V_{{\bf r}{\bf r}^{\prime}}m_{{\bf r}^{\prime}}=m_{\bf
r}{\bar{\beta}}_{{\bf r}{\bf r}^{\prime}}m_{{\bf r}^{\prime}}$ for Grassmann
variables $m_{\bf r}$. Then we introduce the Gaussian integral

$\langle...\rangle_{Q}:=\frac{(i|\mu|/\pi)^{|\Lambda|/2}}{\sqrt{\det
V}}\int_{Q}\exp(-i|\mu|\sum_{{\bf r},{\bf r}^{\prime}}Q_{\bf r}V^{-1}_{{\bf
r}{\bf r}^{\prime}}Q_{{\bf r}^{\prime}})...\prod_{{\bf r}\in\Lambda}dQ_{\bf
r}\ ,$ (19)

where we have set the free positive parameter $\alpha$ such that $V$ is non-
singular and its eigenvalues have positive real parts. The Grassmann
integration can be performed because the argument of the exponential function
is a quadratic form of the Grassmann field:

$K_{{\bf R}{\bf R}^{\prime}}=\frac{1}{{\cal
N}}\Big{\langle}\int_{\Lambda}\exp\left\\{{\tilde{\gamma}}_{0}\sum_{\bf
r}\varphi_{{\bf r}}\varphi^{\prime}_{{\bf r}}-\sum_{{\bf r},{\bf
r}^{\prime}}\gamma_{{\bf r}{\bf r}^{\prime}}\varphi_{{\bf
r}}\varphi^{\prime}_{{\bf r}^{\prime}}-2i|\mu|\sum_{{\bf r}\in\Lambda}Q_{\bf
r}\varphi_{{\bf r}}\varphi^{\prime}_{{\bf r}}\right\\}\varphi_{\bf
R}\varphi^{\prime}_{{\bf R}^{\prime}}\Big{\rangle}_{Q}$ (20) $=\frac{1}{{\cal
N}}\langle adj_{{\bf R}{\bf
R}^{\prime}}({\tilde{\gamma}}_{0}-\gamma-2i|\mu|Q)\rangle_{Q}\ .$

with $\gamma_{{\bf r}{\bf r}^{\prime}}=Tr_{N}(W_{{\bf r}{\bf
r}^{\prime}}W^{\dagger}_{{\bf r}^{\prime}{\bf r}})$ and
${\tilde{\gamma}}_{0}=\sum_{{\bf r}^{\prime}}\gamma_{{\bf r}{\bf
r}^{\prime}}$. The adjugate matrix can be expressed by the determinant as

$adj_{{\bf R}{\bf
R}^{\prime}}({\tilde{\gamma}}_{0}-\gamma-2i|\mu|Q)=\det({\tilde{\gamma}}_{0}-\gamma-2i|\mu|Q)({\tilde{\gamma}}_{0}-\gamma-2i|\mu|Q)^{-1}_{{\bf
R}{\bf R}^{\prime}}\ ,$ (21)

provided that ${\tilde{\gamma}}_{0}-\gamma+2i|\mu|Q$ is not singular. The
latter can always be arranged by a deformation of the $Q_{\bf r}$ path of
integration in the complex plane. The deformation moves the poles of the
inverse matrix away from the real axis, which results in an exponentially
decay with respect to $|{\bf R}-{\bf R}^{\prime}|$.

###  III.2 Estimation of the localization length

To analyze the spatial behavior of the inverse matrix we use the plane wave
eigenvector $\Phi_{\bf k}=(\exp[i{\bf k}\cdot{\bf r}])$ of the translational
invariant matrix $\gamma$ with eigenvalue ${\tilde{\gamma}}_{\bf k}$:

$({\tilde{\gamma}}_{0}-\gamma-2i|\mu|Q)\Phi_{\bf
k}=({\tilde{\gamma}}_{0}-{\tilde{\gamma}}_{\bf k}-2i|\mu|Q)\Phi_{\bf k}$ (22)

with a diagonal matrix on the right-hand side. Since we have

$\Phi_{\bf
k}=({\tilde{\gamma}}_{0}-\gamma-2i|\mu|Q)^{-1}({\tilde{\gamma}}_{0}-\gamma-2i|\mu|Q)\Phi_{\bf
k}=({\tilde{\gamma}}_{0}-\gamma-2i|\mu|Q)^{-1}({\tilde{\gamma}}_{0}-{\tilde{\gamma}}_{\bf
k}-2i|\mu|Q)\Phi_{\bf k}\ ,$ (23)

we get for $\Phi^{\prime}_{\bf k}:=({\tilde{\gamma}}_{0}-{\tilde{\gamma}}_{\bf
k}-2i|\mu|Q)\Phi_{\bf k}$ the equation

$({\tilde{\gamma}}_{0}-\gamma-2i|\mu|Q)^{-1}\Phi^{\prime}_{\bf k}=\Phi_{\bf
k}=({\tilde{\gamma}}_{0}-{\tilde{\gamma}}_{\bf
k}-2i|\mu|Q)^{-1}\Phi^{\prime}_{\bf k}\ .$ (24)

Then we consider the basis $\\{\Phi^{\prime}_{\bf k}\\}$ and define the vector
$\phi^{\prime}_{{\bf R}}:=\int_{\bf k}\phi_{{\bf k},{\bf R}}\Phi^{\prime}_{\bf
k}$. This gives with Eq. (24)

$({\tilde{\gamma}}_{0}-\gamma-2i|\mu|Q)^{-1}\phi^{\prime}_{{\bf R}}=\int_{\bf
k}\phi_{{\bf k},{\bf R}}({\tilde{\gamma}}_{0}-{\tilde{\gamma}}_{\bf
k}-2i|\mu|Q)^{-1}\Phi^{\prime}_{\bf k}=\int_{\bf k}\phi_{{\bf k},{\bf
R}}\Phi_{\bf k}\ .$ (25)

Using the special local vector $\phi_{{\bf R}^{\prime}}=(\delta_{{\bf
R}^{\prime}{\bf r}})$, we get

$\phi_{{\bf
R}^{\prime}}\cdot({\tilde{\gamma}}_{0}-\gamma-2i|\mu|Q)^{-1}\phi^{\prime}_{{\bf
R}}=\int_{\bf k}\phi_{{\bf k},{\bf R}}(\phi_{{\bf
R}^{\prime}}\cdot({\tilde{\gamma}}_{0}-{\tilde{\gamma}}_{\bf
k}-2i|\mu|Q)^{-1}\Phi^{\prime}_{\bf k})=\int_{\bf k}\phi_{{\bf k},{\bf
R}}(\phi_{{\bf R}^{\prime}}\cdot\Phi_{\bf k})=\int_{\bf k}\phi_{{\bf k},{\bf
R}}e^{i{\bf k}\cdot{\bf R}^{\prime}}\ .$ (26)

Finally, we define the expansion coefficients as $\phi_{{\bf k},{\bf
R}}=e^{-i{\bf k}\cdot{\bf R}}/({\tilde{\gamma}}_{0}-{\tilde{\gamma}}_{\bf
k}-2i|\mu|Q_{{\bf R}})$ and obtain

$\phi_{{\bf
R}^{\prime}}\cdot({\tilde{\gamma}}_{0}-\gamma-2i|\mu|Q)^{-1}\phi^{\prime}_{{\bf
R}}=\int_{\bf k}\frac{e^{i{\bf k}\cdot({\bf R}^{\prime}-{\bf
R})}}{{\tilde{\gamma}}_{0}-{\tilde{\gamma}}_{\bf k}-2i|\mu|Q_{{\bf R}}}\ ,\ \
\phi^{\prime}_{{\bf R}}=\int_{\bf k}\frac{e^{-i{\bf k}\cdot{\bf
R}}}{{\tilde{\gamma}}_{0}-{\tilde{\gamma}}_{\bf k}-2i|\mu|Q_{{\bf
R}}}({\tilde{\gamma}}_{0}-{\tilde{\gamma}}_{\bf k}-2i|\mu|Q)\Phi_{\bf k}\ .$
(27)

The decay of this expression with respect to $|{\bf R}-{\bf R}^{\prime}|$
characterizes the localization even before averaging, i.e., for any
realization of $Q_{\bf R}$. To calculate the decay we choose the contour
$\Gamma$ of the $Q_{\bf R}$ integration as $Q_{\bf R}=e^{i{\rm
sgn}({\bar{Q}})\zeta}{\bar{Q}}+i\eta$ with real $-\infty<{\bar{Q}}<\infty$,
$\eta>0$ and $0<\zeta<\pi/4$, as visualized in Fig. 1. Moreover, we assume
that ${\tilde{\gamma}}_{0}-{\tilde{\gamma}}_{\bf k}=Dk^{2}$, which is the
inverse diffusion propagator with diffusion coefficient $D$. Then the $k_{1}$
integration can be carried out for the decay along the 1–direction of the
lattice as

$\int_{\bf k}\frac{e^{i{\bf k}\cdot({\bf R}^{\prime}-{\bf
R})}}{{\tilde{\gamma}}_{0}-{\tilde{\gamma}}_{\bf k}-2i|\mu|Q_{\bf
R}}=\int_{{\bf
k}_{\perp}}\int_{-\infty}^{\infty}\frac{e^{ik_{1}(R_{1}^{\prime}-R_{1})}}{Dk_{1}^{2}+Dk_{\perp}^{2}-2i|\mu|Q_{\bf
R}}dk_{1}=\pi\int_{{\bf
k}_{\perp}}\frac{e^{-|R_{1}^{\prime}-R_{1}|\sqrt{k_{\perp}^{2}-2i|\mu|(e^{i{\rm
sgn}({\bar{Q}})\zeta}{\bar{Q}}+i\eta)/D}}}{D\sqrt{k_{\perp}^{2}-2i|\mu|(e^{i{\rm
sgn}({\bar{Q}})\zeta}{\bar{Q}}+i\eta)/D}}\ ,$ (28)

where ${\bf k}_{\perp}$ is the ${\bf k}$ vector perpendicular to the
1–direction. From the decay of the exponential function in this expression we
extract a localization length as

$\xi=\frac{1}{Re\left(\sqrt{k_{\perp}^{2}-2i|\mu|(e^{i{\rm
sgn}({\bar{Q}})\zeta}{\bar{Q}}+i\eta)/D}\right)}\leq
c(\eta,|{\bar{Q}}|,\zeta)\sqrt{D/|\mu|}$ (29)

where the coefficient $c(\eta,|{\bar{Q}}|,\zeta)$ of the upper bound is of
order 1.

Figure 1: Contour $\Gamma$ of the $Q_{\bf R}$ integration in the complex
plane. The angle with respect to the abscissa is $\pm\zeta$ ($0<\zeta<\pi/4$).

##  IV Discussion and conclusion

The effect of PH-symmetry breaking is characterized by the appearance of small
repulsive dimers in a lattice system that is diffusive at the PH-symmetric
point. This means that diffusion, a classical process, is disturbed by a
random distribution of small obstacles. These obstacles are complex (i.e.,
they have a phase factor) due to the imaginary matrix $\beta$ in Eq. (17).
This effect can be seen as a quantum effect, since for classical diffusion in
a random environment there are only real obstacles. The dimers are represented
by a Gaussian field with a complex correlation matrix in the Grassmann
functional integral, as given in Eqs. (19), (20). This reduces the original
problem of the IM in Eqs. (6) and (8) to the rather elementary case of
diffusion in the presence of random obstacles. The effect of the latter has
been estimated and gives a localization length, whose upper bound is
$\sqrt{D/|\mu|}$ with the diffusion coefficient $D$ and the PH-symmetry
breaking parameter $\mu$. This surprisingly elementary result, which depends
only on the ratio of the two model parameters, reflects the competition
between diffusion and PH-symmetry breaking. The diffusion coefficient, which
is determined through the expression
${\tilde{\gamma}}_{0}-{\tilde{\gamma}}_{\bf k}=Dk^{2}$, depends on the average
Hamiltonian $H_{0}$, the scattering rate $\eta$ and $\mu$ according to the
expressions in Eq. (7). On the other hand, the localization effect due to PH-
symmetry breaking does not agree with the conventional picture of a mobility
edge somewhere in the band of the random Hamiltonian and the related second
order phase transition for dimensionality $d>2$ [21, 22]. Moreover, our result
is different from the self-consistent approach to Anderson localization by
Vollhardt and Wölfle [23, 24], who found that the diffusion coefficient $D$
vanishes at the transition to Anderson localization but leaves a pole of the
effective propagator on the real axis instead of moving it away into the
complex plane. These differences might be related to the fact that we have
considered a special class of Hamiltonians, based on the property (1) and the
chiral symmetry. Moreover, the type of PH-symmetry breaking in Eq. (2) is
special. With our results we cannot rule out that there are other types of PH-
symmetry breaking which lead to the conventional Anderson transition.

The type of Hamiltonian obeying (1) is known in the form of the Dirac
Hamiltonian, which is realized for low-energy quasi particles in graphene. As
mentioned in the Introduction, this material has been the subject of intense
experimental as well as theoretical research for a number of years. Our result
of a finite localization length in the case of a broken PH symmetry might be
useful for the characterization of transport in doped graphene. On the other
hand, finite systems sizes prevent us to distinguish extended states from
localized states whose localization length is larger than the system size.
This could be important for experiments with graphene flakes and for numerical
simulations of the localization effect away from the Dirac point, especially
for a small PH-symmetry breaking parameter $\mu$. In those cases the
localization effect should be observable for localization lengths smaller than
the system size.

Acknowledgment:

This research was supported by a grant of the Julian Schwinger Foundation for
Physics Research.

##  Appendix A The invariant measure

Since ${\hat{S}}^{k}=0$ for $k>2$ we can write for the matrix of the IM

${\hat{H}}_{0}+i\epsilon+i\eta
e^{2{\hat{S}}}={\hat{H}}_{0}+i\epsilon+i\eta(1+2{\hat{S}}+2{\hat{S}}^{2})={\hat{H}}_{0}+i\epsilon-i\eta+2i\eta({\bf
1}-{\hat{S}})^{-1}=({\bf 1}-{\hat{S}})^{-1}\hat{\cal G}^{-1}({\bf 1}-\hat{\cal
G}{\hat{S}}){\hat{G}}_{0}^{-1}$

with

${\hat{G}}_{0}^{-1}={\hat{H}}_{0}+i\epsilon-i\eta\ ,\ \ \hat{\cal
G}=\pmatrix{h&0\cr 0&h^{T}\cr}\ .$

The definition of graded trace is

$Trg\pmatrix{A&B\cr C&D\cr}=TrA-TrD$

and of the graded determinant is

$detg\pmatrix{A&B\cr C&D\cr}=\frac{\det A}{\det D}\det({\bf
1}-BD^{-1}CA^{-1})\ ,$ (30)

where the latter implies

$detg(\hat{\cal G})=detg({\hat{G}}_{0})=1\ .$

This gives for the IM

$J=detg({\bf 1}-{\hat{S}})^{-1}detg({\bf 1}-\hat{\cal G}{\hat{S}}))\ .$

Moreover, using

${\hat{S}}=\pmatrix{0&{\bar{S}}\cr{\bar{S}}^{\prime}&0\cr},\
{\bar{S}}=\pmatrix{0&\varphi_{1}U\cr\varphi_{2}U&0\cr}=\pmatrix{\varphi_{1}U&0\cr
0&\varphi_{2}U\cr}\sigma_{1},\
{\bar{S}}^{\prime}=\pmatrix{0&\varphi^{\prime}_{2}U^{\dagger}\cr\varphi^{\prime}_{1}U^{\dagger}&0\cr}=\sigma_{1}\pmatrix{\varphi^{\prime}_{1}U^{\dagger}&0\cr
0&\varphi^{\prime}_{2}U^{\dagger}\cr},$

we can express the IM via (30) in terms of determinants as

$J=\det({\bf 1}+\bar{S}\bar{S}^{\prime}-h\bar{S}h^{T}\bar{S}^{\prime})\ .$
(31)

With $H_{0}:=\langle H\rangle$ and

$h=\pmatrix{(H_{0}+\mu+i\epsilon-i\eta)(H_{0}+\mu+i\epsilon+i\eta)^{-1}&0\cr
0&(H_{0}-\mu+i\epsilon-i\eta)(H_{0}-\mu+i\epsilon+i\eta)^{-1}\cr}$

we have $hh^{\dagger}={\bf 1}+O(\epsilon)$ and

$\sigma_{1}Uh^{T}U^{\dagger}\sigma_{1}=\pmatrix{(H_{0}+\mu-i\epsilon+i\eta)(H_{0}+\mu-i\epsilon-i\eta)^{-1}&0\cr
0&(H_{0}-\mu-i\epsilon+i\eta)(H_{0}-\mu-i\epsilon-i\eta)^{-1}\cr}=h^{\dagger}\
.$

This implies for Eq. (31)

$J=\det({\bf
1}+\varphi_{1}\varphi^{\prime}_{1}-h\varphi_{1}h^{\dagger}\varphi^{\prime}_{1})\det({\bf
1}+\varphi_{2}\varphi^{\prime}_{2}-h\varphi_{2}h^{\dagger}\varphi^{\prime}_{2})\
.$ (32)

Finally, we apply the determinant identity to write

$\det({\bf
1}+\varphi_{j}\varphi^{\prime}_{j}-h\varphi_{j}h^{\dagger}\varphi^{\prime}_{j})=\exp\\{-\sum_{l\geq
1}\frac{1}{l}Tr[(-\varphi_{j}\varphi^{\prime}_{j}+h\varphi_{j}h^{\dagger}\varphi^{\prime}_{j})^{l}]\\}\
.$ (33)

The properties of the Grassmann variables imply

$\sum_{l\geq
1}\frac{1}{l}Tr[(-\varphi_{j}\varphi^{\prime}_{j}+h\varphi_{j}h^{\dagger}\varphi^{\prime}_{j})^{l}]=Tr(\varphi^{\prime}_{j}\varphi_{j})-\sum_{l\geq
1}\frac{1}{l}Tr[(\varphi^{\prime}_{j}h\varphi_{j}h^{\dagger})^{l}]=-\sum_{l\geq
1}\frac{1}{l}Tr[(-\varphi^{\prime}_{j}\varphi_{j}+\varphi^{\prime}_{j}h\varphi_{j}h^{\dagger})^{l}]$

such that

$\det({\bf
1}+\varphi_{j}\varphi^{\prime}_{j}-h\varphi_{j}h^{\dagger}\varphi^{\prime}_{j})=\det({\bf
1}+\varphi^{\prime}_{j}\varphi_{j}-\varphi^{\prime}_{j}h\varphi_{j}h^{\dagger})^{-1}\
.$ (34)

This is the relation in Eq. (6).

## References

  * Dyson [1953] F. J. Dyson, The dynamics of a disordered linear chain, Phys. Rev. 92, 1331 (1953).
  * Theodorou and Cohen [1976] G. Theodorou and M. H. Cohen, Extended states in a one-demensional system with off-diagonal disorder, Phys. Rev. B 13, 4597 (1976).
  * Eggarter and Riedinger [1978] T. P. Eggarter and R. Riedinger, Singular behavior of tight-binding chains with off-diagonal disorder, Phys. Rev. B 18, 569 (1978).
  * Soukoulis  _et al._ [1982] C. M. Soukoulis, I. Webman, G. S. Grest, and E. N. Economou, Study of electronic states with off-diagonal disorder in two dimensions, Phys. Rev. B 26, 1838 (1982).
  * Inui  _et al._ [1994] M. Inui, S. A. Trugman, and E. Abrahams, Unusual properties of midband states in systems with off-diagonal disorder, Phys. Rev. B 49, 3190 (1994).
  * Novoselov  _et al._ [2005] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Two-dimensional gas of massless dirac fermions in graphene, Nature 438, 197 (2005).
  * Castro Neto  _et al._ [2009] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81, 109 (2009).
  * Abergel  _et al._ [2010] D. Abergel, V. Apalkov, J. Berashevich, K. Ziegler, and T. Chakraborty, Properties of graphene: a theoretical perspective, Advances in Physics 59, 261 (2010), https://doi.org/10.1080/00018732.2010.487978 .
  * Chen  _et al._ [2008] J.-H. Chen, C. Jang, S. Adam, M. S. Fuhrer, E. D. Williams, and M. Ishigami, Charged-impurity scattering in graphene, Nature Physics 4, 377 (2008).
  * Elias  _et al._ [2009] D. C. Elias, R. R. Nair, T. M. G. Mohiuddin, S. V. Morozov, P. Blake, M. P. Halsall, A. C. Ferrari, D. W. Boukhvalov, M. I. Katsnelson, A. K. Geim, and K. S. Novoselov, Control of graphene’s properties by reversible hydrogenation: Evidence for graphane, Science 323, 610 (2009), https://www.science.org/doi/pdf/10.1126/science.1167130 .
  * Chen  _et al._ [2009] J.-H. Chen, W. G. Cullen, C. Jang, M. S. Fuhrer, and E. D. Williams, Defect scattering in graphene, Phys. Rev. Lett. 102, 236805 (2009).
  * Bostwick  _et al._ [2009] A. Bostwick, J. L. McChesney, K. V. Emtsev, T. Seyller, K. Horn, S. D. Kevan, and E. Rotenberg, Quasiparticle transformation during a metal-insulator transition in graphene, Phys. Rev. Lett. 103, 056404 (2009).
  * Das Sarma  _et al._ [2011] S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Electronic transport in two-dimensional graphene, Rev. Mod. Phys. 83, 407 (2011).
  * Peres  _et al._ [2006] N. M. R. Peres, F. Guinea, and A. H. Castro Neto, Electronic properties of disordered two-dimensional carbon, Phys. Rev. B 73, 125411 (2006).
  * Ziegler, Klaus [2013] Ziegler, Klaus, Dynamical symmetry breaking in a 2d electron gas with a spectral node, Eur. Phys. J. B 86, 391 (2013).
  * Sinner and Ziegler [2012] A. Sinner and K. Ziegler, Renormalized transport properties of randomly gapped two-dimensional dirac fermions, Phys. Rev. B 86, 155450 (2012).
  * Negele and Orland [2019] J. Negele and H. Orland,  _Quantum Many-particle Systems_, Advanced Book Classics (Taylor & Francis Group, 2019).
  * Ziegler [2009] K. Ziegler, Diffusion in the random gap model of monolayer and bilayer graphene, Phys. Rev. B 79, 195424 (2009).
  * Ziegler [2015] K. Ziegler, Quantum transport with strong scattering: beyond the nonlinear sigma model, Journal of Physics A: Mathematical and Theoretical 48, 055102 (2015).
  * Gell-Mann and Lévy [1960] M. Gell-Mann and M. Lévy, The axial vector current in beta decay, Il Nuovo Cimento 16, 705 (1960).
  * Abrahams  _et al._ [1979] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979).
  * Wegner [1979] F. Wegner, The mobility edge problem: Continuous symmetry and a conjecture, Zeitschrift für Physik B Condensed Matter 35, 207 (1979).
  * Vollhardt and Wölfle [1980] D. Vollhardt and P. Wölfle, Diagrammatic, self-consistent treatment of the anderson localization problem in $d\leq 2$ dimensions, Phys. Rev. B 22, 4666 (1980).
  * Wölfle and Vollhardt [2010] P. Wölfle and D. Vollhardt, Self-consistent theory of anderson localization: General formalism and applications, International Journal of Modern Physics B 24, 1526–1554 (2010).