# Random Magnetic Field and the Dirac Fermi Surface

Chao-Jung Lee  Department of Physics, California Institute of Technology,
Pasadena, CA 91125, USA Michael Mulligan Department of Physics and Astronomy,
University of California, Riverside, CA 92511, USA

###### Abstract

We study a single 2d Dirac fermion at finite density, subject to a quenched
random magnetic field. At low energies and sufficiently weak disorder, the
theory maps onto an infinite collection of 1d chiral fermions (associated to
each point on the Fermi surface) coupled by a random vector potential. This
low-energy theory exhibits an exactly solvable random fixed line, along which
we directly compute various disorder-averaged observables without the need for
the usual replica, supersymmetry, or Keldysh techniques. We find the
longitudinal dc conductivity in the collisionless
$\hbar\omega/k_{B}T\rightarrow\infty$ limit to be nonuniversal and to vary
continuously along the fixed line.

###### Contents

  1. 1 Introduction
  2. 2 Low-Energy Effective Theory
    1. 2.1 Low-Energy Limit
    2. 2.2 Electrical Current
    3. 2.3 Disorder Ensemble
    4. 2.4 Random Fixed Point and its Discrete Approximation
  3. 3 Observables along the Fixed Line
    1. 3.1 Diffusive Green’s Function and Density of States
    2. 3.2 Longitudinal Conductivity
  4. 4 Discussion and Summary
  5. A Disorder Average of Products of $U$ and $C$ Matrices

##  1 Introduction

In this paper, we introduce and analyze a new effective theory for studying
the influence of a quenched random magnetic field on a single Dirac fermion,
moving in two spatial dimensions (2d) and placed at finite density.
Applications of this theory include: graphene when restricted to a single
valley [1] (where the vector potential disorder describes the effects of
ripples in the graphene sheet [2, 3]) and the gapless surface states of a
time-reversal invariant 3d topological insulator [4]; an integer quantum Hall
plateau transition for spinless electrons in a periodic potential [5, 6]; and
Dirac composite fermion mean-field descriptions of the half-filled Landau
level and topological insulator surface state [7, 8, 9, 10], other even-
denominator metallic states [11, 12, 13], and the superconductor-insulator
transition [14]. In this last guise, the random vector potential arises from a
random scalar potential perturbation to the dual electron system.

In contrast to a 2d nonrelativistic fermion [15, 16], a single Dirac cone
enjoys a sort of topological protection against localization [17, 18, 19]: A
free Dirac fermion cannot be gapped without breaking on average either time-
reversal symmetry111Here we are explicitly referring to the symmetry of the
Lagrangian for a single Dirac fermion. This may be realized in a time-reversal
invariant way as the surface state of time-reversal invariant topological
insulator. In a purely 2d theory, the simplest model breaks time-reversal
microscopically [20, 21, 5]; the resulting theory preserves a nonlocal
particle-hole symmetry [7, 10], which we are here viewing as an effective
time-reversal symmetry. or charge conservation; the random vector potential is
unique (among random perturbations quadratic in the fermions) in that it
respects a particle-hole symmetry [6]. The Dirac theory therefore gives rise
to a delocalized, critical state for all values of the Fermi energy. The
problem here is to find an effective theory for this metal.

The conventional analytical approach to this problem, due to Pruisken and
others [22, 23, 24, 25, 26, 27] (see [28, 29] for reviews and [30, 31, 32] for
specific studies of a disordered Dirac fermion), coarse grains away the
elementary fermionic excitations in favor of a nonlinear sigma model with
topological term, in which renormalization group fixed points are
parameterized by the (disorder-averaged) Dirac fermion conductivity tensor.
Because the longitudinal resistivity $\sim 1/\sigma_{xx}$ plays the role of a
coupling constant in this model, it is a challenge to quantitatively extend
this description to the regime, $\sigma_{xx}\sim e^{2}/h$, relevant to
experiment (e.g., [33, 34]). Tsvelik [35] has conjectured a $PSL(2|2)_{8}$
Wess-Zumino-Novikov-Witten theory for the $\sigma_{xx}\sim e^{2}/h$ regime,
based on its consistency with various numerical studies of the Chalker-
Coddington model [36, 37] for the integer quantum Hall transition.
Unfortunately, there is as yet no consensus among the most recent numerical
works (e.g., [38, 39, 40, 41, 42]): For instance, there are statistically
significant deviations among the various predictions for the localization
length exponent (see Fig. 1 of [42]). These deviations are understood to be
the result of finite-size corrections to scaling [43, 44]; effects that are
exacerbated by a leading irrelevant perturbation that is close to marginality.
Motivated in part by the lack of numerical consensus, Zirnbauer [45, 46] has
recently proposed a fixed point description in terms of a CFT with only
marginal perturbations. Currently, this scenario appears to lack direct
numerical support [42, 47]. An alternative study of the Dirac theory using
(non-)Abelian bosonization by Ludwig et al. [6] makes use of an $SU(2)$
symmetry that is present only at zero density. The zero density theory
exhibits a fixed line (as a function of the disorder variance), along which
the dynamical critical exponent varies continuously while the longitudinal
conductivity is constant and of order $e^{2}/h$. Perturbation by a chemical
potential within this treatment leads to difficulty: The associated chemical
potential deformation breaks the $SU(2)$ symmetry and causes a flow to strong
coupling.222It may be surprising that the chemical potential perturbation—an
operator quadratic in the fermions—leads to strong coupling. The bosonization
used in [6] to treat the 2d disordered theory maps the chemical potential
operator to a nonlinear cosine term in the corresponding boson field with
imaginary coefficient.

Here we present a different approach to studying the finite density theory: We
first take the limit of the theory that focuses on the low-energy fluctuations
about the Dirac Fermi surface before incorporating the disorder. Taking
inspiration from [6], the idea is to try to study the effects of disorder on
the scale-invariant effective theory of (a system with) a Fermi surface [48,
49], rather than the “microscopic” Dirac theory that includes both particle
and antiparticle excitations. We show how this approach allows for an exact
solution of the effective theory—a random fixed line—provided the quenched
vector potential is suitably long ranged and weak, and for direct calculations
of the effects of disorder on various physical observables without the need
for replicas, supersymmetry, or Keldysh formalisms. In contrast to the zero
density theory [6], we find the longitudinal dc conductivity—calculated in the
collisionless $\hbar\omega/k_{B}T\rightarrow\infty$ limit [50]—to vary
continuously along this finite-density fixed line.

The remainder of the paper is organized as follows. In §2 we derive the
effective theory of a Dirac Fermi surface in a random magnetic field. This
theory takes the form of an infinite collection of 1d chiral fermions—one
fermion for each point on the Fermi surface—coupled by the quenched vector
potential disorder. The large emergent symmetry $U(N)$ with
$N\rightarrow\infty$ of the clean theory allows for an exact solution of the
disordered theory, for each disorder realization, in which the randomness is
removed entirely from the effective action by a gauge transformation.
Operators not invariant under the $U(N)$ symmetry, however, depend on the
disorder. In §3 we use the exact solution to directly compute various
disorder-averaged physical observables. We warm up by showing that the average
fermion Green’s function is short-ranged and that the system has a finite
density of states. We then turn to a calculation of the longitudinal
conductivity, finding the conductivity to vary continuously with the strength
of the disorder. In §4 we summarize and discuss possible directions of future
work. Appendix A contains relevant details for the calculations summarized in
the main text.

##  2 Low-Energy Effective Theory

In this section, we derive the low-energy effective theory of a 2d Dirac
fermion in a random magnetic field in terms of an infinite collection of 1d
chiral fermions, coupled via a random vector potential.

###  2.1 Low-Energy Limit

Our starting point is the theory of a two-component Dirac fermion $\Psi(t,{\bf
x})$ at finite density, coupled to a quenched random $U(1)$ vector potential
$A_{j}({\bf x})$:

$\displaystyle S=S_{0}+S_{1},$ (2.1)

where $S_{0}$ is the action of a free Dirac fermion in 2d,333This form of the
action corresponds to a relativistic one with $\gamma$ matrices:
$\big{(}\gamma^{0},\gamma^{1},\gamma^{2}\big{)}=\big{(}\sigma_{3},i\sigma_{2},-i\sigma_{1}\big{)}$.

$\displaystyle S_{0}=\int dtd^{2}{\bf x}\ \Psi^{\dagger}(t,{\bf
x})\big{(}i\sigma_{a}\partial_{a}+\mu\big{)}\Psi(t,{\bf x}),$ (2.2)

and $S_{1}$ is the coupling of the Dirac fermion to the vector potential
disorder,

$\displaystyle S_{1}=\int dtd^{2}{\bf x}\ \Psi^{\dagger}(t,{\bf
x})\sigma_{j}A_{j}({\bf x})\Psi(t,{\bf x}).$ (2.3)

Above, ${\bf x}=(x_{1},x_{2})$; $a\in\\{0,1,2\\}$; $\sigma_{0}$ is the
$2\times 2$ identity matrix and $\sigma_{j}$ for $j=1,2$ are standard Pauli
matrices; repeated indices are summed over unless otherwise specified. The
chemical potential $\mu$ is finite and nonzero, and we have set the velocity
of the Dirac fermion to unity. The vector potential $A_{j}({\bf x})$ is chosen
to be a zero-mean Gaussian random variable. We specify its disorder ensemble
in §2.3; for the present discussion, we require that $A_{j}({\bf x})$ is of
sufficiently long wavelength that its Fourier transform $A_{j}({\bf q})$444Our
Fourier transform convention: $A_{j}({\bf x})=\int{d^{2}{\bf
p}\over(2\pi)}\,e^{i{\bf p}\cdot{\bf x}}A_{j}({\bf p})$ and $\Psi(t,{\bf
x})=\int{d\omega d^{2}{\bf p}\over(2\pi)^{3}}\,e^{-i\omega t+i{\bf p}\cdot{\bf
x}}\Psi(\omega,{\bf p})$. is only nonzero for wave vectors $|{\bf q}|\ll
2|\mu|$. This condition on the vector potential restricts to small-angle
impurity scattering between fermionic excitations of nearby Fermi points.

Following [51], we derive the low-energy limit of this theory, in which we
first focus on the low-energy excitations near the Fermi surface, defined by
(2.2), and then incorporate scattering between Fermi points mediated by
$A_{j}$ in (2.3). This order of limits assumes that the fluctuations of
$A_{j}$ are sufficiently weak compared with $|\mu|$.

In Fourier space, the equation of motion following from (2.2) is

$\displaystyle\big{(}(\omega+\mu)\sigma_{0}-\,p_{j}\sigma_{j}\big{)}\,\Psi(\omega,{\bf
p})=0,$ (2.4)

where $\Psi(\omega,{\bf p})$ is the Fourier transform of $\Psi(t,{\bf x})$ and
${\bf p}=(p_{1},p_{2})$. The equation of motion (2.4) implies that the
particle/antiparticle excitations have the dispersion relation,
$\omega+\mu=\pm p$ with $p=|{\bf p}|\geq 0$. As such, (2.4) can be rewritten
as

$\displaystyle P^{(\mp)}({\bf p})\Psi(\omega,{\bf p})=0,$ (2.5)

where the projection matrices,

$\displaystyle P^{(\mp)}({\bf
p})\equiv\frac{1}{2}\Big{(}\sigma_{0}\mp\,\frac{p_{j}\sigma_{j}}{p}\Big{)}.$
(2.6)

Using the projection matrices, we may write (2.2) in Fourier space as

$\displaystyle S_{0}=\int{d\omega d^{2}{\bf p}\over(2\pi)^{3}}\
\Psi^{\dagger}(\omega,{\bf p})\Big{(}(\omega+\mu-p)P^{(+)}({\bf
p})+(\omega+\mu+p)P^{(-)}({\bf p})\Big{)}\Psi(\omega,{\bf p}).$ (2.7)

For a spherical Fermi surface, we parameterize ${\bf
p}=p\,\hat{r}_{\theta}\equiv p\,(\cos\theta,\sin\theta)$ with
$\theta\in[0,2\pi)$ labeling the points on the Fermi surface. The projection
matrices become

$\displaystyle P^{(\mp)}({\bf p})=\frac{1}{2}\begin{pmatrix}1&\mp
e^{-i\theta}\\\ \mp e^{+i\theta}&1\end{pmatrix}$ (2.8)

and we may expand $\Psi$ in terms of its particle ($R$) and antiparticle ($L$)
excitations as

$\displaystyle\Psi(t,{\bf x})$ $\displaystyle=\int{d\omega d^{2}{\bf
p}\over(2\pi)^{3}}\,e^{-i\omega t+i{\bf p}\cdot{\bf
x}}\Big{(}\frac{1}{\sqrt{2}}\begin{pmatrix}e^{-i\theta}\\\
1\end{pmatrix}R_{\theta}(\omega,p)+\frac{1}{\sqrt{2}}\begin{pmatrix}-e^{-i\theta}\\\
1\end{pmatrix}L_{\theta}(\omega,p)\Big{)},$ (2.9)

where

$\displaystyle P^{(+)}({\bf p})\Psi(\omega,{\bf p})$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}e^{-i\theta}\\\
1\end{pmatrix}R_{\theta}(\omega,p),$ (2.10) $\displaystyle\quad P^{(-)}({\bf
p})\Psi(\omega,{\bf p})$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}-e^{-i\theta}\\\
1\end{pmatrix}L_{\theta}(\omega,p).$ (2.11)

Inserting this expansion (2.9) into (2.7), the free Dirac fermion action in
Fourier space simplifies to

$\displaystyle S_{0}=\int{d\omega d^{2}{\bf
p}\over(2\pi)^{3}}\Big{(}R^{\ast}_{\theta}(\omega,p)\big{(}\omega+\mu-p\big{)}R_{\theta}(\omega,p)+L^{\ast}_{\theta}(\omega,p)\big{(}\omega+\mu+p\big{)}L_{\theta}(\omega,p)\Big{)}.$
(2.12)

For $\mu>0$, particles ($R$) with momentum $p\sim k_{F}\equiv|\mu|$ are light
and antiparticles ($L$) have energy $\omega\geq k_{F}$ ; for $\mu<0$, the
antiparticles are light and the particles are heavy. Therefore, depending on
the sign of $\mu$, the low-energy effective theory for excitations with $p\sim
k_{F}$ only retains the particles or antiparticles. Expanding about the Fermi
momentum $k_{F}$,

$\displaystyle p=k_{F}+p_{\perp},\quad|p_{\perp}|\ll k_{F},$ (2.13)

we have

$\displaystyle d^{2}{\bf p}=pdpd\theta\approx k_{F}dp_{\perp}d\theta,$ (2.14)

and, for $\mu>0$, $S_{0}$ becomes

$\displaystyle S_{0}=\int{d\omega dp_{\perp}d\theta\over(2\pi)^{3}}\
R^{\ast}_{\theta}(\omega,p_{\perp})\big{(}\omega-
p_{\perp}\big{)}R_{\theta}(\omega,p_{\perp}),$ (2.15)

where we replaced
$R_{\theta}(\omega,p)\rightarrow{1\over\sqrt{k}_{F}}R_{\theta}(\omega,p_{\perp})$.
For $\mu<0$, we substitute $R\rightarrow L$ and $\omega-
p_{\perp}\rightarrow\omega+p_{\perp}$. Note that the deviation $p_{\perp}$ of
the momentum about $k_{F}$ can be positive or negative. Depending on the sign
of $\mu$, we may interpret $S_{0}$ (2.15) as an infinite collection of 1d
chiral fermions $R_{\theta}(\omega,p_{\perp})$. We will only consider a single
species of fermions with $\mu>0$ so the low-energy theory will only involve
$R$ fermions.

The vector field $A_{j}({\bf x})$ couples the chiral fermions
$R_{\theta}(\omega,p_{\perp})$ to one another according to $S_{1}$ (2.3). To
derive this coupling, we first decompose the Fourier transform of the vector
potential $A_{j}({\bf q})$ in terms of its longitudinal $A_{L}({\bf q})$ and
transverse $A_{T}({\bf q})$ components:

$\displaystyle A_{j}({\bf q})=i{q_{j}\over q}A_{L}({\bf
q})+i\epsilon_{jk}{q_{k}\over q}A_{T}({\bf q}).$ (2.16)

The momentum transfer,

$\displaystyle{\bf q}={\bf p}-{\bf p}^{\prime},$ (2.17)

where ${\bf p}=(k_{F}+p_{\perp})\hat{r}_{\theta}$ and ${\bf
p}^{\prime}=(k_{F}+p^{\prime}_{\perp})\hat{r}_{\theta^{\prime}}$. Since $p\sim
p^{\prime}\approx k_{F}$, we take

$\displaystyle{q_{j}\over
q}\approx{\hat{r}_{\theta}-\hat{r}_{\theta^{\prime}}\over|\hat{r}_{\theta}-\hat{r}_{\theta^{\prime}}|}.$
(2.18)

Plugging the resulting decomposition (2.16) and the expansion (2.9) into
(2.3), we obtain the low-energy vector potential coupling for $\mu>0$:

$\displaystyle S_{1}=i\int{d\omega d^{2}{\bf p}d^{2}{\bf
p}^{\prime}\over(2\pi)^{5}}\
R^{\ast}_{\theta}(\omega,p)R_{\theta^{\prime}}(\omega,p^{\prime}){\rm
sign}(\theta-\theta^{\prime})e^{{i\over 2}(\theta-\theta^{\prime})}A_{T}({\bf
p}-{\bf p}^{\prime}),$ (2.19)

where ${\rm sign}(X)=1$ for $X\geq 0$ and ${\rm sign}(X)=-1$ for $X<0$. We
have dropped a particle-antiparticle coupling between $R$ and $L$ fermions
that is mediated by the longitudinal component $A_{L}$ of the vector
potential, since such a term has an energy cost $\sim 2k_{F}$. The
longitudinal component is therefore absent from this low-energy vector
potential coupling. For $\mu<0$, (2.19) acquires an overall minus sign and we
substitute $R\rightarrow L$.

In general, $A_{T}({\bf q})$ couples $R_{\theta}(\omega,p)$ and
$R_{\theta^{\prime}}(\omega,p^{\prime})$ for arbitrary $\theta$ and
$\theta^{\prime}$. By assuming that $A_{T}({\bf q})$ is only nonzero for
$|{\bf q}|\ll 2k_{F}$ (see §2.3 for further discussion), we may further
simplify the effective coupling (2.19). For $|{\bf q}|\ll 2k_{F}$, the
momentum transfer ${\bf q}={\bf p}-{\bf p}^{\prime}$ can be approximated as

$\displaystyle{\bf p}-{\bf
p}^{\prime}=\big{(}p_{\perp}-p^{\prime}_{\perp}\big{)}\hat{r}_{\theta^{\prime}}+k_{F}\big{(}\theta-\theta^{\prime}\big{)}{d\over
d\theta^{\prime}}\hat{r}_{\theta^{\prime}},$ (2.20)

where $\big{|}p_{\perp}-p^{\prime}_{\perp}\big{|}\ll 2k_{F}$ and
$\big{|}\theta-\theta^{\prime}\big{|}\ll 2$. Here,
$\hat{r}_{\theta^{\prime}}=(\cos\theta^{\prime},\sin\theta^{\prime})$ and
${d\over
d\theta^{\prime}}\hat{r}_{\theta^{\prime}}=(-\sin\theta^{\prime},\cos\theta^{\prime})$
are orthogonal unit vectors about the Fermi surface point $\theta^{\prime}$.
Defining

$\displaystyle
V_{\theta\theta^{\prime}}\big{(}p_{\perp}-p^{\prime}_{\perp}\big{)}\equiv
i\,k_{F}\,{\rm sign}(\theta-\theta^{\prime})\,e^{{i\over
2}(\theta-\theta^{\prime})}\,A_{T}({\bf p}-{\bf p}^{\prime}),$ (2.21)

and given (2.20), the vector potential coupling (2.19) becomes

$\displaystyle S_{1}=\int{d\omega dp_{\perp}d\theta
dp^{\prime}_{\perp}d\theta^{\prime}\over(2\pi)^{5}}\
R^{\ast}_{\theta}(\omega,p_{\perp})R_{\theta^{\prime}}(\omega,p^{\prime}_{\perp})V_{\theta\theta^{\prime}}\big{(}p_{\perp}-p_{\perp}^{\prime}\big{)}.$
(2.22)

(Recall the rescaling
$R_{\theta}(\omega,p)\rightarrow{1\over\sqrt{k}_{F}}R_{\theta}(\omega,p_{\perp})$.)
Note that the same $A_{T}({\bf q})$ can enter different components of
$V_{\theta\theta^{\prime}}\big{(}p_{\perp}-p^{\prime}_{\perp}\big{)}$ since a
given ${\bf q}$ can correspond to the momentum transfer between distinct pairs
of fermions.

We next perform the inverse Fourier transform555We have
$R_{\theta}(t,z)=\int{d\omega dp_{\perp}\over(2\pi)^{2}}\,e^{-i\omega
t+ip_{\perp}z}R_{\theta}(\omega,p_{\perp})$ and
$V_{\theta\theta^{\prime}}(z)=\int{dp_{\perp}\over
2\pi}\,e^{ip_{\perp}z}V_{\theta\theta^{\prime}}(p_{\perp})$.
$(\omega,p_{\perp})\rightarrow(t,z)$ on the fields appearing in the low-energy
forms of $S_{0}$ (2.15) and $S_{1}$ (2.22). For $\mu>0$, $S=S_{0}+S_{1}$
becomes

$\displaystyle S=\int{dtdzd\theta\over
2\pi}\,R^{\ast}_{\theta}(t,z)i\big{(}\partial_{t}+\partial_{z}\big{)}R_{\theta}(t,z)+\int{dtdzd\theta
d\theta^{\prime}\over(2\pi)^{2}}\,R^{\ast}_{\theta}(t,z)V_{\theta\theta^{\prime}}(z)R_{\theta^{\prime}}(t,z).$
(2.23)

The first term describes an infinite collection of 1d chiral fermions, labeled
by the Fermi point $\theta$. The second term is a (quenched) random matrix
interaction $V_{\theta\theta^{\prime}}(z)$ between fermions associated to the
Fermi points $\theta$ and $\theta^{\prime}$.

###  2.2 Electrical Current

The electrical current (density) is

$\displaystyle J_{j}(t,{\bf x})=\Psi^{\dagger}(t,{\bf x})\sigma_{j}\Psi(t,{\bf
x}).$ (2.24)

We are interested in the contribution to this current that arises from the
low-energy excitations near the Fermi surface when $\mu>0$. The simplest way
to obtain this low-energy current is to replace in (2.3) the quenched vector
potential $A_{j}({\bf x})$ by a slowly varying background vector field ${\cal
A}_{j}(t,{\bf x})$ and to take the low-energy limit described in the previous
section. (“Slowly varying” means $|q_{0}|,|{\bf q}|\ll k_{F}$.) This produces
the coupling (compare with (2.19)),

$\displaystyle S_{2}=\int{d\omega d^{2}{\bf p}d\omega^{\prime}d^{2}{\bf
p}^{\prime}\over(2\pi)^{6}}\
R^{\ast}_{\theta}(\omega,p)R_{\theta^{\prime}}(\omega^{\prime},p^{\prime})\Big{(}{e^{i\theta}+e^{-i\theta^{\prime}}\over
2}{\cal A}_{x}(q_{0},{\bf q})+{e^{i\theta}-e^{-i\theta^{\prime}}\over 2i}{\cal
A}_{y}(q_{0},{\bf q})\Big{)},$ (2.25)

where $q_{0}=\omega-\omega^{\prime}$ and ${\bf q}={\bf p}-{\bf p}^{\prime}$.
As before, $p=k_{F}+p_{\perp}$ and $p^{\prime}=k_{F}+p^{\prime}_{\perp}$. The
variational derivative $J_{j}(-q_{0},-{\bf q})=(2\pi)^{3}{\delta
S_{2}\over\delta{\cal A}_{j}(q_{0},{\bf q})}$ gives the Fourier transform of
the low-energy current:

$\displaystyle J_{x}(-q_{0},-{\bf q})$ $\displaystyle=\int{d\omega d^{2}{\bf
p}d^{2}{\bf p}^{\prime}\over(2\pi)^{3}}\delta({\bf q}-{\bf p}+{\bf
p}^{\prime})R^{\ast}_{\theta}(\omega,p)\Big{(}{e^{i\theta}+e^{-i\theta^{\prime}}\over
2}\Big{)}R_{\theta^{\prime}}(\omega-q_{0},p^{\prime}),$ (2.26) $\displaystyle
J_{y}(-q_{0},-{\bf q})$ $\displaystyle=\int{d\omega d^{2}{\bf p}d^{2}{\bf
p}^{\prime}\over(2\pi)^{3}}\delta({\bf q}-{\bf p}+{\bf
p}^{\prime})R^{\ast}_{\theta}(\omega,p)\Big{(}{e^{i\theta}-e^{-i\theta^{\prime}}\over
2i}\Big{)}R_{\theta^{\prime}}(\omega-q_{0},p^{\prime}).$ (2.27)

In computing the electrical conductivity, we will use a mixed Fourier space
representation of the ${\bf q}=0$ component of these currents. Fourier
transforming $J_{j}(-q_{0},{\bf q}=0)$ with respect to $q_{0}$,666We use our
previous convention for the Fourier transform of $R_{\theta}(t,z)$ and
$J_{j}(t,{\bf q})=\int{dq_{0}\over 2\pi}e^{-iq_{0}t}J_{j}(q_{0},{\bf q})$. we
have

$\displaystyle J_{x}(t)\equiv J_{x}(t,{\bf q}=0)=\int{dzd\theta\over
2\pi}\,R_{\theta}^{\ast}(t,z)\cos(\theta)R_{\theta}(t,z),$ (2.28)
$\displaystyle J_{y}(t)\equiv J_{y}(t,{\bf q}=0)=\int{dzd\theta\over
2\pi}\,R_{\theta}^{\ast}(t,z)\sin(\theta)R_{\theta}(t,z).$ (2.29)

$J_{j}(t)$ is the sum over the Fermi surface of the fermion density, weighted
by $\cos(\theta)$ or $\sin(\theta)$, according to the current component $j$.

###  2.3 Disorder Ensemble

We take the vector potential $A_{j}({\bf q})$ to be a zero-mean Gaussian
random variable with variance:

$\displaystyle\overline{A_{j}({\bf q})A_{k}({\bf
q}^{\prime})}=g\delta_{ij}f(|{\bf q}|)\delta({\bf q}+{\bf q}^{\prime}).$
(2.30)

Here, $g$ is a dimensionless constant that controls the overall scale of the
fluctuations of the vector potential; $g\,f(|{\bf q}|)$ is a unit-normalized
function with support $|{\bf q}|\ll k_{F}$. For definiteness, we choose
$f(|{\bf q}|)=\exp(-|{\bf q}|/M)$, where $M\ll k_{F}$ provides a smooth cutoff
on the momentum transfer ${\bf q}$ in a scattering process.

We would like to determine what (2.30) implies for the random matrix
interaction $V_{\theta\theta^{\prime}}(z)$. We will argue that
$V_{\theta\theta^{\prime}}(z)$ gives rise to a local interaction in the
effective 1d theory. To this end, we consider the disorder average:

$\displaystyle\overline{V_{\theta_{1}\theta_{2}}(z)V_{\theta_{3}\theta_{4}}(z^{\prime})}$
$\displaystyle=-k_{F}^{2}{\rm sign}(\theta_{1}-\theta_{2}){\rm
sign}(\theta_{3}-\theta_{4})e^{{i\over 2}(\theta_{1}-\theta_{2})}e^{{i\over
2}(\theta_{3}-\theta_{4})}$ (2.31)
$\displaystyle\times\int{dp_{\perp}dp^{\prime}_{\perp}\over(2\pi)^{2}}e^{ip_{\perp}z}e^{ip^{\prime}_{\perp}z^{\prime}}\overline{A_{T}({\bf
q})A_{T}({\bf q}^{\prime})},$ (2.32)

where (using our conventions from (2.20))

$\displaystyle{\bf q}$
$\displaystyle=p_{\perp}\hat{r}_{\theta_{2}}+k_{F}(\theta_{1}-\theta_{2}){d\over
d\theta_{2}}\hat{r}_{\theta_{2}},$ (2.33) $\displaystyle{\bf q}^{\prime}$
$\displaystyle=p^{\prime}_{\perp}\hat{r}_{\theta_{4}}+k_{F}(\theta_{3}-\theta_{4}){d\over
d\theta_{4}}\hat{r}_{\theta_{4}}.$ (2.34)

We decompose $A_{T}({\bf q})$ in terms of its Cartesian components by
replacing

$\displaystyle i\,{\rm sign}(\theta_{1}-\theta_{2})e^{{i\over
2}(\theta_{1}-\theta_{2})}A_{T}({\bf q})$
$\displaystyle={e^{i\theta_{1}}+e^{-i\theta_{2}}\over 2}A_{x}({\bf
q})+{e^{i\theta_{1}}-e^{-i\theta_{2}}\over 2i}A_{y}({\bf q}),$ (2.35)
$\displaystyle i\,{\rm sign}(\theta_{3}-\theta_{4})e^{{i\over
2}(\theta_{3}-\theta_{4})}A_{T}({\bf q}^{\prime})$
$\displaystyle={e^{i\theta_{3}}+e^{-i\theta_{4}}\over 2}A_{x}({\bf
q}^{\prime})+{e^{i\theta_{3}}-e^{-i\theta_{4}}\over 2i}A_{y}({\bf
q}^{\prime}),$ (2.36)

and then use (2.30) to find

$\displaystyle\overline{V_{\theta_{1}\theta_{2}}(z)V_{\theta_{3}\theta_{4}}(z^{\prime})}$
$\displaystyle={k_{F}^{2}\over
2}\int{dp_{\perp}dp^{\prime}_{\perp}\over(2\pi)^{2}}e^{ip_{\perp}z}e^{ip^{\prime}_{\perp}z^{\prime}}f(|{\bf
q}|)\delta({\bf q}+{\bf
q}^{\prime})\big{(}e^{i(\theta_{1}-\theta_{4})}+e^{-i(\theta_{2}-\theta_{3})}\big{)}.$
(2.37)

We will approximate (2.37) by

$\displaystyle\overline{V_{\theta_{1}\theta_{2}}(z)V_{\theta_{3}\theta_{4}}(z^{\prime})}=gk_{F}\delta(\theta_{1}-\theta_{4})\delta(\theta_{2}-\theta_{3})f(z-z^{\prime}),$
(2.38)

where

$\displaystyle f(z)=\int{dp_{\perp}\over 2\pi}\,e^{ip_{\perp}z}f(p_{\perp})$
(2.39)

and it is to be understood that $|\theta_{1}-\theta_{2}|\ll 1$ and
$|\theta_{3}-\theta_{4}|\ll 1$. The disorder (2.38) is local in $z$ since it
depends on the relative coordinate $z-z^{\prime}$. We expect (2.38) to be a
good approximation to (2.37) at small momentum transfers and sufficiently low
energies. More precisely, we require $M/k_{F}\ll 1$ and that the cutoff on
$p_{\perp}\in(-\Lambda,\Lambda)$ satisfies $\Lambda/M\ll 1$.

A detailed explanation for the approximation (2.38) goes as follows. We begin
by noting that for $M/k_{F}\ll 1$ and using $\delta({\bf q}+{\bf
q}^{\prime})$, $f({\bf q})=f({\bf q}^{\prime})$ fixes
$|\theta_{1}-\theta_{2}|\ll 1$ and $|\theta_{3}-\theta_{4}|\ll 1$. Next, we
expect the dominant contributions to (2.37) to arise when the terms in the sum
$\big{(}e^{i(\theta_{1}-\theta_{4})}+e^{-i(\theta_{2}-\theta_{3})}\big{)}$ are
in-phase. This sets $\theta_{1}=\theta_{4}+\theta_{3}-\theta_{2}+2\pi m$, for
an arbitrary integer $m$. Using the earlier two conditions, the in-phase
requirement implies $|\theta_{1}-\theta_{4}-\pi m|\ll 1$: This is modeled by
$\delta(\theta_{1}-\theta_{4}-\pi|m|)$. Substituting
$\theta_{1}=\theta_{4}+\pi m$ into $f({\bf q})$ and $\theta_{4}=\theta_{1}-\pi
m$ into $f({\bf q}^{\prime})$ fixes $|\theta_{4}-\theta_{2}+\pi m|\ll 1$ and
$|\theta_{3}-\theta_{1}+\pi m|\ll 1$. Since $\theta_{i}\in[0,2\pi)$, these two
conditions allow either $m=0$ or $|m|=1$. First consider $m=0$. Since
$\theta_{2}\approx\theta_{1}$ and $\theta_{4}\approx\theta_{1}$, ${\bf
q}\approx p_{\perp}\hat{r}_{\theta_{1}}+k_{F}(\theta_{1}-\theta_{2}){d\over
d\theta_{1}}\hat{r}_{\theta_{1}}$ and ${\bf q}^{\prime}\approx
p^{\prime}_{\perp}\hat{r}_{\theta_{1}}+k_{F}(\theta_{3}-\theta_{4}){d\over
d\theta_{1}}\hat{r}_{\theta_{1}}$. Because $\hat{r}_{\theta_{1}}$ and ${d\over
d\theta_{1}}\hat{r}_{\theta_{1}}$ are orthonormal, $\delta({\bf q}+{\bf
q}^{\prime})=\delta(p_{\perp}+p^{\prime}_{\perp})\delta\big{(}k_{F}(\theta_{1}-\theta_{2}+\theta_{3}-\theta_{4})\big{)}={1\over
k_{F}}\delta(p_{\perp}+p^{\prime}_{\perp})\delta(\theta_{3}-\theta_{2})$,
using $\delta(\theta_{1}-\theta_{4}-\pi|m|)$. For Gaussian $f(|{\bf q}|)$, the
dependence on $p_{\perp}$ and $\theta_{1}-\theta_{2}$ factorizes. Absorbing
into $g$ the variation of this Gaussian on $|\theta_{1}-\theta_{2}|\approx 0$,
we perform the integral over $p^{\prime}_{\perp}$ using
$\delta(p_{\perp}+p^{\prime}_{\perp})$ to arrive at (2.38). Next consider
$m=1$; the $m=-1$ case works similarly and will not be discussed. Following
the $m=0$ logic, the replacement $\theta_{4}=\theta_{1}-\pi$ introduces a
relative phase in the angular delta function, $\delta({\bf q}+{\bf
q}^{\prime})={1\over
k_{F}}\delta(p_{\perp}-p^{\prime}_{\perp})\delta\big{(}\theta_{1}-\theta_{2}-(\theta_{3}-\theta_{4})\big{)}={1\over
k_{F}}\delta(p_{\perp}-p^{\prime}_{\perp})\delta\big{(}2\theta_{1}-\theta_{2}-\theta_{3}-\pi\big{)}\approx{1\over
k_{F}}\delta(p_{\perp}-p^{\prime}_{\perp})\delta\big{(}\theta_{1}-\theta_{2}\big{)}$,
using $\theta_{3}\approx\theta_{1}-\pi$. The delta function has support when
$\theta_{1}=\theta_{2}$ (and similarly requires using the in-phase delta
function $\delta(\theta_{1}-\theta_{4}-\pi)$ and $f({\bf q}^{\prime})$ that
$\theta_{3}=\theta_{4}$). For $\Lambda/M\ll 1$, scattering along the Fermi
surface dominates and the relative contributions of $m\neq 0$ terms should be
suppressed. We therefore ignore the $|m|=1$ terms in the remainder.

###  2.4 Random Fixed Point and its Discrete Approximation

Under the renormalization group transformation777The scale transformation is
the following: $z\rightarrow\lambda z,t\rightarrow\lambda
t,\theta\rightarrow\theta,R_{\theta}(t,z)\rightarrow\lambda^{-1/2}R_{\theta}(t,z)$.
[48, 49] that leaves the $S_{0}$ part of (2.23) invariant, the leading flow
equation [52] for the disorder variance $g$ is

$\displaystyle{dg\over d\ell}=(3-2\Delta)g,$ (2.40)

where $\Delta=1$ is the scaling dimension of
$R_{\theta}^{\ast}(t,z)R_{\theta^{\prime}}(t,z)$ and $\ell$ is the
renormalization group length scale that increases as the energy is reduced. We
have substituted $f(z-z^{\prime})=\delta(z-z^{\prime})$ in deriving (2.40).
Randomness is therefore a relevant perturbation that drives the clean Dirac
theory towards strong disorder.

Because the randomness is ${\cal O}(1)$ relevant, perturbation theory about
the clean fixed point cannot access the strong disorder regime. Luckily, the
low-energy action (2.23) admits an exact solution for arbitrary $g$ (such that
the derivation in §2.1 holds), in which $V_{\theta\theta^{\prime}}(z)$ is
eliminated via the field redefinition [53, 54, 55]:

$\displaystyle\tilde{R}_{\theta}(t,z)=\int{d\theta^{\prime}\over
2\pi}U_{\theta\theta^{\prime}}(z)R_{\theta^{\prime}}(t,z),$ (2.41)

where the unitary matrix,

$\displaystyle U_{\theta\theta^{\prime}}(z)\equiv\Big{(}{\cal
T}_{z}e^{-i\int_{z_{0}}^{z}dz^{\prime}V(z^{\prime})}\Big{)}_{\theta\theta^{\prime}}.$
(2.42)

Here, ${\cal T}_{z}$ denotes path ordering along $z$ and $z_{0}$ is an
arbitrary base point. The resulting action simplifies to

$\displaystyle S=\int{dtdzd\theta\over
2\pi}\,\tilde{R}^{\ast}_{\theta}(t,z)i\big{(}\partial_{t}+\partial_{z}\big{)}\tilde{R}_{\theta}(t,z).$
(2.43)

The action (2.43) exactly describes the strong disorder regime of a Dirac
fermion at finite density, subject to a random vector potential of
sufficiently long wavelength. The random vector potential has been eliminated
from the effective action using the infinite-dimensional symmetry of the Fermi
surface [56, 57, 58].

The formal manipulations above are made concrete by discretizing the $\theta$
coordinate (say, by putting the system in a finite-size box). To this end, we
take the Fermi surface to consist of $N$ discrete points:
$\theta\rightarrow\theta_{I}=2\pi I/N$ with $I=1,\ldots,N$. The fields and
disorder are therefore replaced as

$\displaystyle R_{\theta}(t,z)$
$\displaystyle\rightarrow\sqrt{N}R_{\theta_{I}}(t,z)\equiv\sqrt{N}R_{I}(t,z)$
(2.44) $\displaystyle V_{\theta\theta^{\prime}}(z)$ $\displaystyle\rightarrow
V_{\theta_{I}\theta_{J}}(z)\equiv V_{IJ}(z).$ (2.45)

(The scaling of $R_{\theta_{I}}(t,z)$ by $\sqrt{N}$ is for notational
simplicity.) We are specifically interested in the limit $N\rightarrow\infty$.
Substituting in the discrete form of the angular integration
$\int{d\theta\over 2\pi}\rightarrow{1\over N}\sum_{I}$, we have the
(equivalent) discrete forms for the low-energy action,

$\displaystyle S$ $\displaystyle=\int
dtdz\sum_{I=1}^{N}\,R^{\ast}_{I}(t,z)i\big{(}\partial_{t}+\partial_{z}\big{)}R_{I}(t,z)+{1\over
N}\int dtdz\sum_{I,J=1}^{N}\,R^{\ast}_{I}(t,z)V_{IJ}(z)R_{J}(t,z)$ (2.46)
$\displaystyle=\int
dtdz\sum_{I=1}^{N}\,\tilde{R}^{\ast}_{I}(t,z)i\big{(}\partial_{t}+\partial_{z}\big{)}\tilde{R}_{I}(t,z),$
(2.47)

where the rotated fermions, in discrete form, are

$\displaystyle\tilde{R}_{I}(t,z)=\sum_{J=1}^{N}U_{IJ}(z)R_{J}(t,z),\quad
U_{IJ}(z)=\Big{(}{\cal T}_{z}e^{-{i\over
N}\int_{z_{0}}^{z}dz^{\prime}V(z^{\prime})}\Big{)}_{IJ}.$ (2.48)

The discrete action has an emergent $U(N)$ symmetry. In discrete form, the
low-energy currents (2.28) - (2.29) become

$\displaystyle J_{x}(t)$ $\displaystyle=\int
dz\sum_{I}R^{\ast}_{I}(t,z)\cos(\theta_{I})R_{I}(t,z),$ (2.49) $\displaystyle
J_{y}(t)$ $\displaystyle=\int
dz\sum_{I}R^{\ast}_{I}(t,z)\sin(\theta_{I})R_{I}(t,z).$ (2.50)

For discrete $\theta$, the disorder variance (2.38) becomes

$\displaystyle\overline{V_{IJ}(z)V_{KL}(z^{\prime})}=gk_{F}N\delta_{IL}\delta_{JK}f(z-z^{\prime}).$
(2.51)

The overall factor of $N$ arises from the discrete form of $\delta({\bf
q}+{\bf q}^{\prime})$ in (2.37) with $\theta_{I}=2\pi I/N$. It is the discrete
form of the action (2.47) that we will use in the next section.

##  3 Observables along the Fixed Line

In general, in the presence of quenched disorder $V$ with unit-normalized
distribution $P[V]$, the disorder-average of the correlation function of a
physical observable ${\cal O}$ is defined as

$\displaystyle\overline{\langle{\cal O}\rangle}\equiv\int
DVP[V]\,\langle\mathcal{O}\,\rangle_{V},$ (3.1)

where the correlation function $\langle\mathcal{O}\,\rangle_{V}$ in the
disorder realization $V$ is

$\displaystyle\langle\mathcal{O}\,\rangle_{V}\equiv\frac{\int
D\Phi\;\mathcal{O}\;e^{iS[\Phi,V]}}{\int D\Phi\;e^{iS[\Phi,V]}}.$ (3.2)

Here, we are momentarily denoting the dynamical fields of the theory by $\Phi$
and the action $S[\Phi,V]$ indicates the dependence upon both $\Phi$ and the
disorder $V$. In most theories, the presence of $V$ in the denominator of
$\langle\mathcal{O}\,\rangle_{V}$ renders the direct analytic integration over
all possible disorders in $\overline{\langle\mathcal{O}\rangle}$ difficult, if
not impossible. As such, various ingenious tricks—such as replica,
supersymmetric, and Keldysh formalisms—have been employed with various levels
of success. In this paper, we instead make use of the exact solution of the
low-energy effective theory presented in §2.4 to directly perform the disorder
average.

To see how this works, consider the correlation function of a local observable
${\cal O}(R)$, which is a function of the unrotated $R_{I}(t,z)$ fermion:

$\displaystyle\langle\mathcal{O}(R)\rangle_{V}=\frac{\int
DR^{\dagger}DR\;\mathcal{O}(R)\;e^{iS[R,V]}}{\int
DR^{\dagger}DR\;e^{iS[R,V]}}=\frac{\int
D\tilde{R}^{\dagger}D\tilde{R}\;\mathcal{O}(U^{\dagger}\tilde{R})\;e^{iS[\tilde{R}]}}{\int
D\tilde{R}^{\dagger}D\tilde{R}\;e^{iS[\tilde{R}]}}.$ (3.3)

In the first equality, $S[R,V]$ denotes the action (2.46); in the second
equality, $S[\tilde{R}]$ denotes (2.47) with $\tilde{R}$ the rotated fermion
(2.48) with rotation $U$ a function of $V$. After the rotation, the
denominator no longer depends on $V$ and, in this case, yields the usual
fermion determinant. The disorder $V$ now only appears in the expression for
the observable $\mathcal{O}(U^{\dagger}\tilde{R})$ and may, in principle, be
averaged over.888Note that any possible quantum anomalies [53] associated with
the unitary rotations (2.42) of the chiral fermion path integral measures,
being a function of the disorder $V$ only, mutually cancel between the
numerator and denominator. The key point is that the local observables we
consider separate into a sum of terms of the form,

$\displaystyle\mathcal{O}(U^{\dagger}\tilde{R})=\sum_{a,b}c_{ab}{\cal
A}_{a}(\tilde{R})\mathcal{B}_{b}(V),$ (3.4)

for some constants $c_{ab}$, such that the disorder-averaged correlation
function $\overline{\langle{\cal O}(R)\rangle}$ factorizes:

$\displaystyle\overline{\langle{\cal O}(R)\rangle}$
$\displaystyle=\sum_{a,b}c_{ab}\frac{\int
D\tilde{R}^{\dagger}D\tilde{R}\;\mathcal{A}_{a}(\tilde{R})\;e^{iS[\tilde{R}]}}{\int
D\tilde{R}^{\dagger}D\tilde{R}\;e^{iS[\tilde{R}]}}\cdot\int
dVP[V]\mathcal{B}_{b}(V)$ (3.5)
$\displaystyle\equiv\sum_{a,b}c_{ab}\langle{\cal
A}_{a}(\tilde{R})\rangle\cdot\overline{{\cal B}_{b}(V)}.$ (3.6)

We will show how this can be used to compute the fermion Green’s function and
the longitudinal conductivity at the random fixed point (2.47).

###  3.1 Diffusive Green’s Function and Density of States

We begin by calculating the disorder-averaged Green’s function and density of
states. We will find that the average Green’s function is short-ranged and
that the density of states is a positive constant. These calculations will
introduce the technique we will later use to calculate the conductivity.

Using the low-energy action (2.47), the fermion two-point function averaged
over the disorder is

$\displaystyle\overline{\langle R_{I}(t,z)R^{\dagger}_{I}(0,0)\rangle}$
$\displaystyle=\sum_{A,B}\langle\tilde{R}_{A}(t,z)\tilde{R}^{\dagger}_{B}(0,0)\rangle\cdot\overline{U^{\dagger}_{IA}(z)U_{BI}(0)}$
(3.7)
$\displaystyle=\sum_{A,B}\frac{i}{2\pi}\frac{\delta_{AB}}{(z-t)+i\alpha}\cdot\overline{U^{\dagger}_{IA}(z)U_{BI}(0)},$
(3.8)

where $U(z)$ is defined in (2.48), $\alpha$ is a short-distance cutoff, and
the index $I$ is not summed over. Following [59], we compute the disorder
average of $U$ matrices by decomposing the interval $[0,z]$ into $n$ steps
$z_{0}=0,z_{1}=\delta z,\ldots,z_{n}=n\delta z$ of length $\delta z=|z|/n$
such that

$\displaystyle U(z)=e^{-{i\over
N}\int_{z_{n-1}}^{z_{n}}dz^{\prime}V(z^{\prime})}\cdots e^{-{i\over
N}\int_{z_{1}}^{z_{2}}dz^{\prime}V(z^{\prime})}e^{-{i\over
N}\int_{z_{0}}^{z_{1}}dz^{\prime}V(z^{\prime})}.$ (3.9)

We have chosen the arbitrary reference point $z_{0}=0$. For sufficiently large
$n$ with fixed $n\delta z=|z|$, the argument of each exponential can be
approximated as

$\displaystyle{1\over
N}\int_{z_{j-1}}^{z_{j}}dz^{\prime}V(z^{\prime})\approx{1\over
N}V(z_{j})\delta z\equiv M_{j}.$ (3.10)

Using (2.51), $M_{j}$ is a zero-mean Gaussian random variable with variance,

$\displaystyle\overline{(M_{i})_{IJ}(M_{j})_{KL}}={gk_{F}\over
N}\delta_{IL}\delta_{JK}f(z_{i}-z_{j})\delta z^{2},$ (3.11)

where $f(z_{i}-z_{j})$ is given in (2.39). For discrete $z$, we take

$\displaystyle f(z_{i}-z_{j})\delta
z=f_{0}\delta_{|i-j|,0}+f_{1}\delta_{|i-j|,1}.$ (3.12)

The dimensionless coefficients $f_{0}$ and $f_{1}$ approximate a Gaussian
$f(z_{i}-z_{j})$ of finite width.

Using this, the disorder average of the product of $U$ matrices in (3.7)
becomes

$\displaystyle\sum_{A}\overline{U^{\dagger}_{IA}(z)U_{AI}(0)}$
$\displaystyle=\sum_{A}\overline{\Big{(}e^{i\,M_{1}}\cdots
e^{i\,M_{n}}\Big{)}_{IA}\times\bm{1}_{AI}}$ (3.16)
$\displaystyle=\overline{\Big{(}[1+iM_{1}-\frac{M_{1}^{2}}{2}+\ldots]\cdots[1+iM_{n}-\frac{M_{n}^{2}}{2}+\ldots]\Big{)}_{II}}$
$\displaystyle=\delta_{II}-gk_{F}[\frac{f_{0}}{2}\,n+f_{1}(n-1)]\delta
z\,\delta_{II}$
$\displaystyle=e^{-gk_{F}\big{(}\frac{f_{0}}{2}+f_{1}\big{)}|z|}\delta_{II}.$

In the third equality, we have dropped higher-order terms that vanish as
$n\rightarrow\infty$. Further details on such computations are given in
Appendix A. Defining

$\displaystyle g_{\rm eff}\equiv g\big{(}{f_{0}\over 2}+f_{1}\big{)},$ (3.17)

we obtain the disorder-averaged Green’s function,

$\displaystyle\overline{\langle
R_{I}(t,z)R^{\dagger}_{I}(0,0)\rangle}=\frac{i}{2\pi}\frac{e^{-g_{\rm
eff}k_{F}|z|}}{(z-t)+i\alpha}.$ (3.18)

This Green’s function has a spatial decay length $\lambda=1/g_{\rm eff}k_{F}$.

We now use the Green’s function (3.18) to check that the density of states is
finite. For this, we need the retarded Green’s function averaged over the
disorder:

$\displaystyle\overline{G_{II}^{R}}(t,z;0,0)=-i\Theta(t)\overline{\langle\\{R_{I}(t,z),R_{I}(0,0)\\}\rangle}=-{i\over\pi}{\alpha
e^{-g_{\rm eff}k_{F}|z|}\over(z-t)^{2}+\alpha^{2}},$ (3.19)

where $\Theta(t)$ is the step function. Fourier transforming
$\overline{G^{R}}(t,z;0,0)$ for $\alpha\rightarrow 0$, we obtain

$\displaystyle\overline{G_{II}^{R}}(\omega,p_{\perp})={1\over\omega-
p_{\perp}+ig_{\rm eff}k_{F}}.$ (3.20)

From (3.20), we obtain the density of states per unit volume,

$\displaystyle\rho(\omega)=-{1\over\pi}{\rm Im}\int{dp_{\perp}\over
2\pi}{1\over\omega-p_{\perp}+ig_{\rm eff}k_{F}}={1\over 2\pi}.$ (3.21)

###  3.2 Longitudinal Conductivity

We next turn to the disorder-averaged longitudinal conductivity
$\sigma_{xx}(\omega)$. The Kubo formula reads

$\displaystyle\sigma_{xx}(\omega)={1\over i\omega_{n}}{1\over L}{k_{F}\over
N}\int_{0}^{\beta}d\tau e^{i\omega_{n}\tau}\overline{\langle{\cal
T}_{\tau}J_{x}(\tau)J_{x}(0)\rangle}\,\Big{|}_{\omega_{n}\rightarrow\omega+i0^{+}},$
(3.22)

where $\tau=it$, $\omega_{n}=(2n+1)\pi\beta$ is a positive Matsubara frequency
at temperature $1/\beta$, and $L\cdot N/k_{F}$ is a spatial volume factor
equal to $\lim_{{\bf q}\rightarrow 0}\delta({\bf
q})=\lim_{p_{\perp}\rightarrow 0}\delta(p_{\perp})\cdot{1\over
2\pi}\delta(k_{F}\theta_{I})$ for some $I$.

Plugging in the expression for the current $J_{x}(\tau)$ (2.49) and using the
unitary (2.48), the current two-point function is

$\displaystyle\langle{\cal T}_{\tau}J_{x}(\tau)J_{x}(0)\rangle$
$\displaystyle=\int
dzdz^{\prime}\sum_{I,J,A,B,C,D}\langle\tilde{R}^{\dagger}_{A}(\tau,z)\tilde{R}_{B}(\tau,z)\tilde{R}^{\dagger}_{C}(0,z^{\prime})\tilde{R}_{D}(0,z^{\prime})\rangle$
(3.23)
$\displaystyle\cdot\overline{U_{AI}(z)\cos(\theta_{I})U^{\dagger}_{IB}(z)U_{CJ}(z^{\prime})\cos(\theta_{J})U^{\dagger}_{JD}(z^{\prime})}$
(3.24) $\displaystyle=\int dzdz^{\prime}\,\Big{(}{i\over
2\beta\sinh\big{(}{\pi(z-z^{\prime}+i(\tau-\tau^{\prime})\over\beta}\big{)}}\Big{)}^{2}\cdot{\rm
Tr}\,\overline{U(z)CU^{\dagger}(z)U(z^{\prime})CU^{\dagger}(z^{\prime})}.$
(3.25)

In the first equality, we introduced the diagonal matrix
$C_{II^{\prime}}=\cos\big{(}{2\pi I\over N}\big{)}\delta_{II^{\prime}}$; in
the second equality, we used (2.47) to compute the finite-temperature fermion
four-point function and expressed the product of $U$ and $C$ matrices using
standard matrix notation. The disorder average of the product of $U$ and $C$
matrices is computed as in the previous section. To this end, we partition the
interval $[0,z]$ into $n$ segments and the interval $[0,z^{\prime}]$ into $m$
segments, each of size $\delta z=z/n=z^{\prime}/m$, and decompose each $U$
matrix as in (3.9). We find

$\displaystyle{\rm Tr}\overline{\big{(}e^{-i\,M_{n}}\cdots
e^{-i\,M_{1}}\big{)}C\big{(}e^{i\,M_{1}}\cdots
e^{i\,M_{n}}\big{)}\big{(}e^{-i\,M_{m}}\cdots
e^{-i\,M_{1}}\big{)}\,C\big{(}e^{i\,M_{1}}\cdots e^{i\,M_{m}}\big{)}}$ (3.27)
$\displaystyle=\sum_{I}\cos^{2}\big{(}{2\pi I\over N}\big{)}e^{-2g_{\rm
eff}k_{F}|z-z^{\prime}|}$ (3.28) $\displaystyle={N\over 2}e^{-2g_{\rm
eff}k_{F}|z-z^{\prime}|},$ (3.29)

where $g_{\rm eff}$ is defined in (3.17). The details for the evaluation of
the disorder average are given in Appendix A. The sum over $I$ in the second
equality is performed for $N\rightarrow\infty$ using the continuum limit
${1\over N}\sum_{I}\rightarrow\int{d\theta\over 2\pi}$.

Inserting (3.27) into the current two-point function (3.23), the conductivity
(3.22) becomes

$\displaystyle\sigma_{xx}(\omega)={1\over i\omega_{n}}{k_{F}\over 2L}\int
dzdz^{\prime}d\tau\,e^{i\omega_{n}\tau}\Big{(}{i\over
2\beta\sinh^{2}\big{(}{\pi(z-z^{\prime}+i(\tau-\tau^{\prime})\over\beta}\big{)}}\Big{)}^{2}e^{-2g_{\rm
eff}k_{F}|z-z^{\prime}|}\Big{|}_{\omega_{n}\rightarrow\omega+i0^{+}}.$ (3.30)

Shifting $z\rightarrow z+z^{\prime}$ and $\tau\rightarrow\tau+\tau^{\prime}$,
the integral over $z^{\prime}$ produces a factor of $L$; we next calculate the
integral over $\tau$ in the zero temperature limit $\beta\rightarrow\infty$.
The relevant term is

$\displaystyle\int_{0}^{\infty}d\tau e^{i\omega_{n}\tau}\Big{(}{i\over
2\pi}{1\over\big{(}z+i\tau\big{)}}\Big{)}^{2}=-{1\over
4\pi}|\omega_{n}|\,e^{-\omega_{n}z}\Theta\big{(}\omega_{n}z\big{)}.$ (3.31)

Performing the remaining integral over $z$, we obtain the conductivity

$\displaystyle\sigma_{xx}(\omega)={1\over 8}\cdot{1\over g_{\rm
eff}-i{\omega\over 2vk_{F}}}\quad\text{(units of $e^{2}/h$)},$ (3.32)

where we have restored the fermion velocity $v$, previously set to one. This
is the main result of this paper. The dc longitudinal conductivity varies as
$1/g_{\rm eff}$ along the random fixed line. An identical calculation with
$\cos\big{(}{2\pi I\over N}\big{)}\rightarrow\sin\big{(}{2\pi I\over
N}\big{)}$ produces $\sigma_{yy}(\omega)=\sigma_{xx}(\omega)$. The Hall
conductivity $\sigma_{xy}(\omega)$ vanishes because there is no time-reversal
symmetry breaking on average. This follows from an explicit computation
similar to the above, in which $\sum_{I}\cos^{2}\big{(}{2\pi I\over
N}\big{)}\rightarrow\sum_{I}\cos\big{(}{2\pi I\over N}\big{)}\sin\big{(}{2\pi
I\over N}\big{)}=0$.

We may crudely estimate the regime of validity of (3.32) as follows. Recall
from (2.30) that $g_{\rm eff}\sim g$ characterizes the scale of the
fluctuations of the random vector potential ${\bf A}({\bf q})$, which in turn
determines the random coupling
$|V_{\theta\theta^{\prime}}(p_{\perp}-p^{\prime}_{\perp})|\sim k_{F}|{\bf
A}({\bf q})|\sim k_{F}\sqrt{g_{\rm eff}}$ in (2.22). General effective field
theory considerations require $k_{F}\sqrt{g_{\rm eff}}\leq\Lambda$, where
$\Lambda\ll k_{F}$ is the cutoff on momenta transverse to the Fermi surface.
Inserting this inequality into (3.32) at $\omega=0$, we find

$\displaystyle\sigma_{xx}(\omega=0)\geq{1\over
8}\cdot{k_{F}^{2}\over\Lambda^{2}}.$ (3.33)

$\Lambda\ll k_{F}$ (rather than $\Lambda\leq k_{F}$) ensures the scattering is
primarily tangential to the Fermi surface, instead of perpendicular to it (see
§2.3). A study of the effects of the various leading corrections to the
effective theory (2.23) could potentially clarify the bound (3.33).

##  4 Discussion and Summary

We have studied the effects of a quenched random, transverse magnetic field on
a 2d Dirac fermion placed at finite density. For weak disorder of sufficiently
long wavelength, we showed how the effective theory reduces to an infinite
collection of chiral fermions coupled by the vector potential. This
simplification allows for an exact treatment of the effects of the disorder.
We found a line of fixed points along which the longitudinal dc conductivity
(3.32) varies continuously with the disorder variance.

The dc conductivity was calculated in the collisionless
$\hbar\omega/k_{B}T\rightarrow\infty$ limit. It is important to extend our
study to the opposite order of limits $\hbar\omega/k_{B}T\rightarrow 0$, the
so-called incoherent regime [50], relevant to experiment. (See [60] for a
study of distinct dc limits of the ac conductivity of a clean Dirac fermion at
zero density.) This question is pertinent to the expected universality of the
conductivity at a quantum phase transition [61]. Numerical studies999We are
grateful to Prashant Kumar for discussions about this. [62, 63, 64] of the
integer quantum Hall transition appear to be roughly consistent with
experiment (e.g., [65, 66]), giving a value for the dc longitudinal
conductivity $\sigma_{xx}\sim(.54-.60)\,e^{2}/h$.

We focused exclusively on the point where the Dirac fermion is massless. In
the clean limit at finite density, a metal intervenes between integer quantum
Hall states with $\sigma_{xy}=\pm{1\over 2}{e^{2}\over h}$ as the mass $m$ is
tuned between $\pm\mu$, where $\mu$ is the chemical potential. Based on
numerics (e.g., [41]), the metallic region is absent in the presence of
disorder and a direct integer quantum Hall transition should obtain. It would
be interesting to redo our analysis with a finite mass $m$ to try to find the
localization length exponent for this transition.

The disorder we studied was of sufficiently long wavelength that it mediated
elastic scattering between nearby Fermi points only. The opposite regime, in
which all Fermi points are coupled by the disorder, might be interesting to
consider. The action of the theory at energy $\omega=0$ takes the form:

$\displaystyle S_{\omega=0}=\int
dz\sum_{I,J=1}^{N}\,R^{\ast}_{I}(z)\Big{(}i\partial_{z}\delta_{IJ}+J_{IJ}\Big{)}R_{J}(z),$
(4.1)

with random $J_{IJ}$ coupling all Fermi points $I,J$, subject to a given
ensemble. Interpreting $z$ as “time,” this action is reminiscent of the
quadratic Sachdev-Ye-Kitaev model [67, 68] with complex fermions.

The similarity of the effective theory (2.46) of the random Dirac Fermi
surface to the theory of $N$ chiral free fermions in 1d suggests a possible
route towards a non-Fermi liquid generalization, in which disorder may be
studied simultaneously. For example, we may consider two independent Dirac
fermions—with chemical potentials that are of equal magnitude and opposite
sign—in the presence of quenched vector potential disorder. The effective
action is $S_{R}+S_{L}$, where

$\displaystyle S_{\rm R}$ $\displaystyle=\int
dtdz\sum_{I,J=1}^{N}\,R^{\ast}_{I}(t,z)\Big{(}i\big{(}\partial_{t}+\partial_{z}\big{)}\delta_{IJ}+{1\over
N}V_{IJ}(z)\Big{)}R_{J}(t,z),$ (4.2) $\displaystyle S_{\rm L}$
$\displaystyle=\int
dtdz\sum_{I,J=1}^{N}\,(L^{\ast}_{I}(t,z)\Big{(}i\big{(}\partial_{t}-\partial_{z}\big{)}\delta_{IJ}-{1\over
N}V_{IJ}(z)\Big{)}L_{J}(t,z),$ (4.3)

and $R_{I}$ ($L_{J}$) is the low-energy excitation about the Fermi surface
defined by positive (negative) chemical potential. Couplings between right
($R^{\ast}_{I}R_{I}$) and left ($L^{\ast}_{J}L_{J}$) densities leads to
Luttinger liquid-like behavior. So long as the couplings preserve a diagonal
subgroup of the $U(N)\times U(N)$ symmetry of the Fermi surface, the
interactions and disorder can be studied simultaneously. The implications and
possible microscopic origin of such a theory are unclear.

## Acknowledgments

We thank Prashant Kumar and Sri Raghu for useful conversations. C.-J. Lee was
partially supported by the scholarship of the Taiwan Ministry of Education.
This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of Basic Energy Sciences under Award No. DE-
SC0020007.

##  Appendix A Disorder Average of Products of $U$ and $C$ Matrices

In this appendix, we detail the evaluation of the disorder average of products
of $U$ and $C$ matrices, focusing on the product that appears in (3.23):

$\displaystyle{\rm
Tr}\,\overline{U(z)CU^{\dagger}(z)U(z^{\prime})CU^{\dagger}(z^{\prime})}\equiv\overline{U_{AI}(z)C_{II^{\prime}}U^{\dagger}_{I^{\prime}B}(z)U_{BJ}(z^{\prime})C_{JJ^{\prime}}U^{\dagger}_{J^{\prime}A}(z^{\prime})},$
(A.1)

where $U_{IJ}(z)$ is defined in (2.48), $C_{II^{\prime}}=\cos\big{(}{2\pi
I\over N}\big{)}\delta_{II^{\prime}}$, and the sums over
$A,I,I^{\prime},B,J,J^{\prime}=1,\ldots,N$ are understood. The computation of
(3.16) is similar and will not be discussed.

To calculate (A.1), we discretize the $z$ direction into segments
$[z_{k-1},z_{k}]$ of length $\delta z>0$, where $z_{k}=k\delta z$ for all
integer $k$. We take $z=z_{n}$, $z^{\prime}=z_{m}$, and consider the limit
$n,m\rightarrow\infty$ with $z_{n}$ and $z_{m}$ fixed. $U(z)$ is decomposed as

$\displaystyle U(z)=e^{-{i\over
N}\int_{z_{n-1}}^{z_{n}}dz^{\prime}\,V(z^{\prime})}\cdots e^{-{i\over
N}\int_{z_{1}}^{z_{2}}dz^{\prime}\,V(z^{\prime})}e^{-{i\over
N}\int_{z_{0}}^{z_{1}}dz^{\prime}\,V(z^{\prime})}$ (A.2)

and similarly for $U(z^{\prime})$. Since $\delta z$ is infinitesimal, we
approximate

$\displaystyle{1\over
N}\int_{z_{j-1}}^{z_{j}}dz^{\prime}\,V(z^{\prime})\approx{1\over
N}V(z_{j})\delta z\equiv M_{j}.$ (A.3)

The decomposition of $U$ becomes

$\displaystyle U(z)=e^{-i\,M_{n}}e^{-iM_{n-1}}\cdots e^{-i\,M_{1}}.$ (A.4)

From (2.51), $M_{j}$ is a zero-mean Gaussian random variable with variance,

$\displaystyle\overline{(M_{i})_{IJ}(M_{j})_{KL}}={gk_{F}\over
N}\delta_{IL}\delta_{JK}f(z_{i}-z_{j})\delta z^{2},$ (A.5)

where $f(z_{i}-z_{j})$ is defined by

$\displaystyle f(z_{i}-z_{j})\delta
z=f_{0}\delta_{|i-j|,0}+f_{1}\delta_{|i-j|,1}+f_{2}\delta_{|i-j|,2}+...+f_{k}\delta_{|i-j|,k}.$
(A.6)

We refer to $f_{0}$ as the on-site correlation coefficient, $f_{1}$ as the
$1^{st}$ neighbor correlation coefficient, $f_{2}$ as the $2^{nd}$ neighbor
correlation coefficient, etc. Using (A.6), we write the disorder average (A.1)
as

$\displaystyle{\rm
Tr}\,\overline{U(z)CU^{\dagger}(z)U(z^{\prime})CU^{\dagger}(z^{\prime})}=\text{Tr}[C\,C]+W_{0}+W_{1}+\ldots+W_{k},$
(A.7)

where $W_{j}$ denotes the contribution from $f_{j}$. The first term
$\text{Tr}[C\,C]$ is the constant term without any Wick contraction due to the
disorder averaging.

###  $1^{st}$-neighbor correlation: $W_{1}$

We begin with the $1^{st}$ neighbor contribution $W_{1}$. We substitute (A.4)
into (A.7) and expand the exponentials to obtain

$\displaystyle W_{1}$ $\displaystyle=\overline{\Big{(}e^{-i\,M_{n}}\cdots
e^{-i\,M_{1}}\Big{)}_{AI}C_{II^{\prime}}\Big{(}e^{i\,M_{1}}\cdots
e^{i\,M_{n}}\Big{)}_{I^{\prime}B}}$ (A.8)
$\displaystyle\times\overline{\Big{(}e^{-i\,M_{m}}\cdots
e^{-i\,M_{1}}\Big{)}_{BJ}\,C_{JJ^{\prime}}\Big{(}e^{i\,M_{1}}\cdots
e^{i\,M_{m}}\Big{)}_{J^{\prime}A}}\Big{|}_{f_{1}}$ (A.9)
$\displaystyle\approx\overline{\Big{(}(1-iM_{n})\cdots(1-iM_{1})\Big{)}_{AI}C_{II^{\prime}}\Big{(}(1+iM_{1})\cdots(1+iM_{n})\Big{)}_{I^{\prime}B}}$
(A.10)
$\displaystyle\times\overline{\Big{(}(1-iM_{m})\cdots(1-iM_{1})\Big{)}_{BJ}C_{JJ^{\prime}}\Big{(}(1+iM_{1})\cdots(1+iM_{m})\Big{)}_{J^{\prime}A}}\Big{|}_{f_{1}},$
(A.11)

where $\Big{|}_{f_{1}}$ indicates that we only consider the contributions due
to the $1^{st}$ neighbor correlation $f_{1}$ in (A.6). Note that the disorder
average in the first equality is being performed over the entire product, not
separately over each term in the product. Assuming $n>m$, there are four
possible contractions involving $M_{n}$ that we isolate by writing $W_{1}$ as

$\displaystyle\overline{\Big{(}(-iM_{n})(-iM_{n-1})\Big{)}_{AI}}C_{II^{\prime}}\delta_{I^{\prime}B}\;\delta_{BJ}C_{JJ^{\prime}}\delta_{J^{\prime}A}+\overline{(-iM_{n})_{AI}(iM_{n-1})_{I^{\prime}B}}\;C_{II^{\prime}}\delta_{BJ}C_{JJ^{\prime}}\delta_{J^{\prime}A}$
(A.12)
$\displaystyle+\overline{(-iM_{n-1})_{AI}(iM_{n})_{I^{\prime}B}}C_{II^{\prime}}\;\delta_{BJ}C_{JJ^{\prime}}\delta_{J^{\prime}A}+\delta_{AI}C_{II^{\prime}}\overline{\Big{(}(iM_{n-1})(iM_{n})\Big{)}_{I^{\prime}B}}\;\delta_{BJ}C_{JJ^{\prime}}\delta_{J^{\prime}A}$
(A.13) $\displaystyle+\overline{\Big{(}e^{-i\,M_{n-1}}\cdots
e^{-i\,M_{1}}\Big{)}_{AI}C_{II^{\prime}}\Big{(}e^{i\,M_{1}}\cdots
e^{i\,M_{n-1}}\Big{)}_{I^{\prime}B}\Big{(}e^{-i\,M_{m}}\cdots
e^{-i\,M_{1}}\Big{)}_{BJ}\,C_{JJ^{\prime}}\Big{(}e^{i\,M_{1}}\cdots
e^{i\,M_{m}}\Big{)}_{J^{\prime}A}}\Big{|}_{f_{1}}$ (A.14)
$\displaystyle=2g\,f_{1}\,k_{F}\,\delta
z\,\Big{(}-\delta_{AI}\delta_{I^{\prime}B}+\frac{1}{N}\delta_{II^{\prime}}\delta_{AB}\Big{)}\delta_{BJ}\hat{C}_{II^{\prime}}\hat{C}_{JJ^{\prime}}\delta_{J^{\prime}A}$
(A.15) $\displaystyle+\overline{\Big{(}e^{-i\,M_{n-1}}\cdots
e^{-i\,M_{1}}\Big{)}_{AI}C_{II^{\prime}}\Big{(}e^{i\,M_{1}}\cdots
e^{i\,M_{n-1}}\Big{)}_{I^{\prime}B}\Big{(}e^{-i\,M_{m}}\cdots
e^{-i\,M_{1}}\Big{)}_{BJ}\,C_{JJ^{\prime}}\Big{(}e^{i\,M_{1}}\cdots
e^{i\,M_{m}}\Big{)}_{J^{\prime}A}}\Big{|}_{f_{1}}.$ (A.16)

The subleading term $\propto\frac{1}{N}$, which comes from “crossing-
contractions,” vanishes because
$C_{II^{\prime}}\delta_{II^{\prime}}=\text{Tr}[C]=0$.

We continue as above sequentially contracting $1^{st}$ neighbor pairs
involving $M_{n-1},M_{n-2},\ldots,M_{m+2}$ to arrive at

$\displaystyle
W_{1}=-2g\,f_{1}\,k_{F}\text{Tr}[CC]\Big{(}n-(m+2)+1\Big{)}\;\delta
z+\Big{(}\text{Contractions of $M_{m+1},\ldots,M_{1}$}\Big{)}$ (A.18)

There are eight contractions that involve $M_{m+1}$ and $M_{m}$ in the second
term in (A.18):

$\displaystyle\Big{[}\overline{\Big{(}(-iM_{m+1})(-iM_{m})\Big{)}_{AI}}C_{II^{\prime}}\delta_{I^{\prime}B}\delta_{BJ}C_{JJ^{\prime}}\delta_{J^{\prime}A}+\overline{(-iM_{m+1})_{AI}(iM_{m})_{I^{\prime}B}}C_{II^{\prime}}\delta_{BJ}C_{JJ^{\prime}}\delta_{J^{\prime}A}$
(A.19)
$\displaystyle+\overline{(-iM_{m+1})_{AI}(-iM_{m})_{BJ}}C_{II^{\prime}}\delta_{I^{\prime}B}C_{JJ^{\prime}}\delta_{J^{\prime}A}+\overline{(-iM_{m+1})_{AI}(+iM_{m})_{J^{\prime}A}}C_{II^{\prime}}\delta_{I^{\prime}B}\delta_{BJ}C_{JJ^{\prime}}\Big{]}$
(A.20)
$\displaystyle+\Big{[}\overline{(M_{m})_{AI}(M_{m+1})_{I^{\prime}B}}C_{II^{\prime}}\delta_{BJ}C_{JJ^{\prime}}\delta_{J^{\prime}A}-\overline{(M_{m}M_{m+1})_{I^{\prime}B}}\delta_{AI}C_{II^{\prime}}\delta_{BJ}C_{JJ^{\prime}}\delta_{J^{\prime}A}$
$\displaystyle+\delta_{AI}C_{II^{\prime}}\overline{(M_{m+1})_{I^{\prime}B}(M_{m})_{BJ}}C_{JJ^{\prime}}\delta_{J^{\prime}A}-\delta_{AI}C_{II^{\prime}}\overline{(M_{m+1})_{I^{\prime}B}(M_{m})_{J^{\prime}A}}\delta_{BJ}C_{JJ^{\prime}}\Big{]}.$
(A.21)

In the first $[\ldots]$, the first term cancels with the fourth and the second
term cancels with the third; the same pairs mutually cancel in the second
$[\ldots]$. Next we consider the contractions involving $M_{m}$ and $M_{m-1}$,
and we again obtain zero. These cancellations continue through to contractions
involving $M_{1}$. This makes sense since we expect the result to only depend
on the difference $(n-m)$. Summarizing, we have

$\displaystyle W_{1}=-\text{Tr}[CC]\Big{(}2g\,f_{1}\,k_{F}\,[n-(m+2)+1]\delta
z\Big{)}.$ (A.22)

###  $2^{nd}$-neighbor correlation and beyond: $W_{2},\ldots,W_{k}$

Following the logic used to compute the $1^{st}$ neighbor correlations, we
find

$\displaystyle W_{2}$ $\displaystyle=\overline{\Big{(}e^{-i\,M_{n}}\cdots
e^{-i\,M_{1}}\Big{)}_{AI}C_{II^{\prime}}\Big{(}e^{i\,M_{1}}\cdots
e^{i\,M_{n}}\Big{)}_{I^{\prime}B}}$ (A.23)
$\displaystyle\times\overline{\Big{(}e^{-i\,M_{m}}\cdots
e^{-i\,M_{1}}\Big{)}_{BJ}\,C_{JJ^{\prime}}\Big{(}e^{i\,M_{1}}\cdots
e^{i\,M_{m}}\Big{)}_{J^{\prime}A}}\Big{|}_{f_{1}}$ (A.24)
$\displaystyle\approx\overline{\Big{(}(1-iM_{n})\cdots(1-iM_{1})\Big{)}_{AI}C_{II^{\prime}}\Big{(}(1+iM_{1})\cdots(1+iM_{n})\Big{)}_{I^{\prime}B}}$
(A.25)
$\displaystyle\times\overline{\Big{(}(1-iM_{m})\cdots(1-iM_{1})\Big{)}_{BJ}C_{JJ^{\prime}}\Big{(}(1+iM_{1})\cdots(1+iM_{m})\Big{)}_{J^{\prime}A}}\Big{|}_{f_{2}}$
(A.26) $\displaystyle=-\text{Tr}[CC]\Big{(}2g\,f_{2}\,k_{F}\,[n-(m+3)+1]\delta
z\Big{)}.$ (A.27)

Generalizing to $k^{th}$ neighbor correlations, we have

$\displaystyle
W_{k}=-\text{Tr}[CC]\Big{(}2g\,f_{k}\,k_{F}\,[n-(m+k+1)+1]\delta z\Big{)}.$
(A.28)

### On-site Correlation: $W_{0}$

Unlike the off-site correlations discussed above, we have to expand the
exponentials to quadratic order to obtain the contribution from on-site
correlations:

$\displaystyle W_{0}$ $\displaystyle=\overline{\Big{(}e^{-i\,M_{n}}\cdots
e^{-i\,M_{1}}\Big{)}_{AI}C_{II^{\prime}}\Big{(}e^{i\,M_{1}}\cdots
e^{i\,M_{n}}\Big{)}_{I^{\prime}B}}$ (A.29)
$\displaystyle\times\overline{\Big{(}e^{-i\,M_{m}}\cdots
e^{-i\,M_{1}}\Big{)}_{BJ}\,C_{JJ^{\prime}}\Big{(}e^{i\,M_{1}}\cdots
e^{i\,M_{m}}\Big{)}_{J^{\prime}A}}\Big{|}_{f_{0}}$ (A.30)
$\displaystyle\approx\overline{\Big{(}(1-iM_{n}-\frac{M_{n}^{2}}{2})\cdots(1-iM_{1}-\frac{M_{1}^{2}}{2})\Big{)}_{AI}\hat{C}_{II^{\prime}}\Big{(}(1+iM_{1}-\frac{M_{1}^{2}}{2})\cdots(1+iM_{n}-\frac{M_{n}^{2}}{2})\Big{)}_{I^{\prime}B}}$
(A.31)
$\displaystyle\times\overline{\Big{(}(1+iM_{m}-\frac{M_{m}^{2}}{2})\cdots(1+iM_{1}-\frac{M_{1}^{2}}{2})\Big{)}_{BJ}\hat{C}_{JJ^{\prime}}\Big{(}(1+iM_{1}-\frac{M_{1}^{2}}{2})\cdots(1+iM_{m}-\frac{M_{m}^{2}}{2})\Big{)}_{J^{\prime}A}}\Big{|}_{f_{0}}$
(A.32) $\displaystyle=gf_{0}k_{F}\delta
z\Big{(}-\delta_{AI}\delta_{I^{\prime}B}+\frac{2}{N}\delta_{II^{\prime}}\delta_{AB}\Big{)}\delta_{BJ}\hat{C}_{II^{\prime}}\hat{C}_{JJ^{\prime}}\delta_{J^{\prime}A}+\Big{(}\text{Contractions
of $M_{n-1},\ldots,M_{1}$}\Big{)}.$ (A.33)

The crossing term $\propto{1\over N}$ again vanishes. Continuing in this way
we arrive at

$\displaystyle W_{0}=-\text{Tr}[CC]\Big{(}g\,f_{0}\,k_{F}\,[n-(m+1)+1]\delta
z\Big{)}+\Big{(}\text{Contractions of $M_{m},\ldots,M_{1}$}\Big{)}.$ (A.35)

Similar to the cancellations that were discussed in the context of the
$1^{st}$ neighbor correlations, the remaining contractions of
$M_{m},\ldots,M_{1}$ equal zero. Thus, we have

$\displaystyle W_{0}=-\text{Tr}[CC]\Big{(}g\,f_{0}\,k_{F}\,[n-(m+1)+1]\delta
z\Big{)}.$ (A.36)

### Final Result

Collecting all correlations $f_{0},f_{1},\ldots,f_{k}$ and replacing
$\lim_{n\rightarrow\infty}n\delta z=z$ and $\lim_{m\rightarrow\infty}m\delta
z=z^{\prime}$, we have

$\displaystyle{\rm
Tr}\,\overline{U(z)CU^{\dagger}(z)U(z^{\prime})CU^{\dagger}(z^{\prime})}$
$\displaystyle=\text{Tr}[CC]\Big{(}1-2gk_{F}\delta z\big{(}{f_{0}\over
2}(n-m)+\sum_{j=1}^{k}f_{j}[n-m-k]\big{)}\Big{)}$ (A.37)
$\displaystyle=\text{Tr}[CC]e^{-2gk_{F}|z-z^{\prime}|\big{(}{f_{0}\over
2}+f_{1}+\ldots f_{k}\big{)}}.$ (A.38)

Note that each term proportional to $k$ in the sum vanishes at large $n$ and
$m$. This concludes our calculation of the disorder average in (3.27). In the
main text, $f_{0}$ and $f_{1}$ are nonzero, and $f_{k\geq 2}=0$.

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