# Theory of a Strange Metal in a Quantum Superconductor to Metal Transition

Jaewon Kim  Department of Physics, University of California, Berkeley, CA
94720, USA  Erez Berg  Department of Condensed Matter Physics, Weizmann
Institute of Science, Rehovot 76100, Israel  Ehud Altman  Department of
Physics, University of California, Berkeley, CA 94720, USA

###### Abstract

Recent experiments with nanopatterned thin films revealed an unusual quantum
superconductor to metal phase transition (QSMT) with a linear in temperature
resistivity. In contrast, most known examples of such transitions and standard
theoretical considerations predict a temperature-independent sheet resistance
of order $R_{Q}=\frac{\hbar}{e^{2}}$. We propose an effective theory of a
disordered superconductor which features a QSMT with robust $T$ linear
resistivity at the critical point. The crucial ingredient in our model is
spatial disorder in the pairing interaction. Such random pairing mirrors the
emergent phase disorder seen in a recent mean field study of a microscopic
$d$-wave superconductor subject to potential disorder. We also make the
prediction that in such systems the diamagnetic susceptibility diverges as
$\log\frac{\Lambda}{T}$, which starkly differs from the power law divergence
in standard XY transitions.

##  I Introduction

More than three decades since their discovery, the cuprates continue to pose a
significant challenge to theory. In addition to high $T_{c}$
superconductivity, the strong electronic correlations give rise to
unconventional metallic phases, including the pseudogap state in the under-
doped regime Warren  _et al._ (1989); Alloul  _et al._ (1989); Doiron-Leyraud
_et al._ (2007), linear in $T$ resistivity near optimal doping Gurvitch and
Fiory (1987); Takagi  _et al._ (1992); Taillefer (2010) and a range of
competing ordered states.

One way to achieve theoretical control over the complex behavior of the
cuprates is to tune the system to a quantum phase transition at which the
superconductivity is lost. There, one should be able to focus on a single
critical fixed-point rather than an array of competing phases. But even this
attempted simplification leads to intriguing surprises. In particular it was
found that different ways of driving the superconductor to normal state
transition result in distinct critical behaviors. A transition tuned by
introducing zinc impurities, substituting for Cu atoms Fukuzumi  _et al._
(1996), follows the expected scaling of a two dimensional $XY$ model. In
particular, the critical sheet resistivity approaches a constant of order
resistivity quantum $R_{Q}\simeq\hbar/4e^{2}$, in the low temperature limit.
This scaling is of course not special to the cuprates and is seen in many
other quantum superconductor to metal transitions Hebard and Paalanen (1990);
Paalanen  _et al._ (1992); Seidler  _et al._ (1992); van der Zant  _et al._
(1992); Wang  _et al._ (1991). On the other hand recent experiments with YBCO
films where superconductivity is suppressed by nano-patterning with an array
of holes of varying sizes show linear in $T$ resistivity with very small
residual resistivity at the critical point Yang  _et al._ (2022). Previous
theoretical work on disordered $d$-wave superconductors could capture a small
residual critical resistivity $R_{0}\ll R_{Q}$, but not the linear temperature
dependenceHerbut (2000); Podolsky  _et al._ (2007).

In this paper we present a model of a disordered $d$-wave superconductor that
leads to $T$ linear resistivity at the critical point over a broad temperature
range. As in the earlier theories the effective model we study includes a
fluctuating $d$-wave order parameter coupled to gapless fermions occupying a
Fermi sea. The key new element is that we take the dominant disorder driving
the transition to be a random pairing interaction, or more precisely disorder
in the strength and sign of the coupling between the fermions and the
fluctuating $d$-wave pairing field (see Fig. 1).

Including this kind of disorder is motivated by recent mean field studies of a
d-wave superconductor, showing that short range correlated potential disorder
gives rise to emergent longer scale disorder in the sign of the pairing field
Li  _et al._ (2021a). To account for the longer scale of this disorder we take
it to be correlated over a length $\xi>2\pi/k_{F}$.

Figure 1: (Left) An illustration of our model. Our model is a disordered
$d$-wave superconductor with spatially random sign changes of the order
parameter. (Right) Schematic phase diagram of our model. The strange metal
behavior occur at the critical point of the quantum superconductor to metal
transition.

We solve the effective model in a controlled large $N$ limit, where the
numbers of fermion and boson (order parameter) species are taken to infinity
together. The main advantage of this approach, developed in Refs. Kim  _et
al._ (2021); Aldape  _et al._ (2022); Esterlis  _et al._ (2021); Patel  _et
al._ (2022) to describe critical points in itinerant fermion systems, is that
it preserves the mutual feedback between the critical fluctuations (order
parameter fields) and the fermions. This allows us to capture a strong
coupling critical point, exhibiting linear in $T$ resistivity within the large
$N$ saddle point.

In addition to the $T$ linear resistivity our model also shows linear
magnetoresistance at the critical point with a universal relaxation rate
$1/\tau\sim\mu_{B}B$. Such behavior has been observed in a number of
correlated materials Giraldo-Gallo  _et al._ (2018); Cooper  _et al._ (2009);
Hayes  _et al._ (2021); Ataei  _et al._ (2022); Jaoui  _et al._ (2022) and in
the patterned cuprate films discussed above. Within our model, however, the
linear magnetoresistance is found only when the cyclotron frequency
$\omega_{cf}$ is in the rather restricted range $\omega_{cf}\in
k_{B}T[(k_{F}\xi)^{-4},(k_{F}\xi)^{-3}]$, which may be too small to explain
the experiments.

The rest of the paper is organized as follows. In section II we introduce the
model and give an overview of the main results. In section III we detail the
solution for the single particle Greens functions in the large $N$ limit. This
allows us to derive the conductivity in section IV and the magnetic
susceptibility in section V. We conclude with a discussion in section VI.

##  II Model and overview

In this section we present and motivate the effective model we employ to
understand the superconductor to metal quantum phase transition in patterned
cuprate films and summarize the main results from its analysis. The model
consists of a field theory for a complex field describing $d$-wave pair
fluctuations, coupled to a Fermi surface through a random pairing interaction
in 2+1 dimensions. The action is given by,

$\displaystyle\mathcal{S}=S_{b}+S_{f}+S_{int}\,,\textrm{ where }$
$\displaystyle
S_{b}=\int_{\tau}\sum_{x}\Delta_{b}^{0}|\Psi_{x}|^{2}+\frac{U}{2}|\Psi_{x}|^{4}+\sum_{\braket{xx^{\prime}}}t_{b}\Psi_{x}^{\dagger}\Psi_{x^{\prime}}$
(1a) $\displaystyle
S_{f}=\int_{\tau,k}c_{k}^{\dagger}\left(\partial_{\tau}+\epsilon_{k}-\mu\right)c_{k}$
(1b) $\displaystyle
S_{int}=\int_{\tau}\sum_{x}Jg_{x}\Psi_{x}\left(c^{T}_{x}\sigma^{y}c_{x+a\hat{x}}-c^{T}_{x}\sigma^{y}c_{x+a\hat{y}}\right)+c.c.$
(1c)

Here, $\Psi_{x}$ is a complex boson field at position $x$ describing the
Cooper pair fluctuations, and $c_{x}^{T}=(c_{x,\uparrow},c_{x,\downarrow})$,
the fermion field at position $x$. For simplicity, in (1b) we consider a
quadratic dispersion $\epsilon_{k}=k^{2}/2m_{f}$. $g_{x}$ is a quenched
disorder field with zero mean and correlation length $\xi$ that satisfies

$\overline{g_{x}g_{x^{\prime}}}\simeq e^{-|x-x^{\prime}|^{2}/\xi^{2}}$

The Ginsburg-Landau action $S_{b}$, in itself, is a standard description of a
superconductor to insulator quantum phase transition. The disordered coupling
of the bosonic field to a Fermi surface is designed to stabilize a metallic
normal state in place of the insulator. Note that a uniform coupling to the
fermions cannot achieve this. Even in the gapped phase of $S_{b}$, integrating
out $\Psi$ in presence of such uniform coupling generates attractive
interactions between the fermions leading to a BCS instability. Hence the
transition is eliminated and the system is superconducting throughout. The
random coupling $g_{x}$ provides an IR cutoff to the BCS instability at the
length scale $\xi$. As a result, superconductivity onsets only beyond a finite
coupling strength $J$. The quantum phase transition can also be tuned at
constant $J$ by varying the disorder correlation length $\xi$.

It is worth noting that the conventional disorder present in physical systems
at short scales couples directly to the fermion density rather than to the
pairing field. The pairing disorder in our model is a phenomenological
description of inhomogeneity that can emerge at longer scales due to the
interplay of the short-range disorder and d-wave pairing. Self consistent mean
field calculations have shown that this interplay gives rise to geometrically
frustrated phase disorder at longer scales, evidenced by emergence of isolated
$\pi$ vortices Li  _et al._ (2021a). Instead of this geometric frustration,
the random sign of the coupling in the model (1) induces internal frustration
between the boson and fermion species.

To facilitate a controlled solution we consider an extension of the model to a
large number $N$ of fermion and boson species. Accordingly, the Yukawa
coupling $g$ between the species is promoted to a random tensor. Similar
techniques have been developed recently to deal with strongly coupled quantum
critical points, first in Dirac systems Kim  _et al._ (2021) and later in
quantum critical points with a metallic Fermi surface Aldape  _et al._ (2022);
Esterlis  _et al._ (2021); Patel  _et al._ (2022). Keeping the ratio of
fermionic and bosonic species constant in the large $N$ limit ensures that the
mutual feedback between them is preserved. In particular, the bosons are
Landau damped due to coupling to the fermions, generating a self-energy that
scales as $\alpha|\omega|$ at the QCP. At the same time the fermions become a
marginal Fermi liquid, due to coupling to the critical bosons, exhibiting a
self-energy that scales as $\Sigma(k,\omega)\sim\gamma_{k}\omega\log\omega$.

The fermion self-energy implies a scattering rate that grows linearly with
temperature as $1/\tau_{k}\approx\gamma_{k}T$. The angle dependence of
$\gamma_{k}$ stems from the $d$-wave nature of the coupling and for
sufficiently smooth disorder (i.e. $k_{F}\xi\gg 1)$ leads to strongly
suppressed scattering at the nodes $1/\tau_{\textrm{node}}\sim
T/(k_{F}\xi)^{3}$. As a result, the nodes become a significant bottleneck in
momentum relaxation, leading to the inverse transport time
$\frac{1}{\tau_{\textrm{tr}}}\sim(k_{F}\xi)^{-4}T$. Thus, the quasiparticle
scattering rate is linear in $T$, but sub-Planckian for smooth disorder.

In section IV we calculate the conductivity using the quantum Boltzmann
equation. The conductivity is dominated by the quasiparticle (fermion) current
and is given by $\sigma\approx(k_{F}\xi)^{4}e^{2}\epsilon_{F}/T$; The critical
pairing fluctuations, on the other hand, contribute a sub-leading logarithmic
temperature dependence $\sigma_{b}\sim\log(\Lambda/T)$. Hence the system
exhibits strange metal behavior with
$\rho\simeq(k_{F}\xi)^{-4}(T/e^{2}\epsilon_{F})$.

In section IV.1 we investigate the effect of adding weak conventional
potential disorder coupled to the local fermion density at short scales. This
scattering generates a small constant decay rate $1/\tau_{0}$ that ultimately
leads to a residual resistivity $\rho_{0}\sim(e^{2}\epsilon_{F}\tau_{0})^{-1}$
at zero temperature. Scattering from potential disorder also eliminates the
nodal bottleneck for quasi-particle relaxation, which in turn alters the slope
of the $T$ linear resistivity below a crossover temperature scale.

In section IV.2 we study the magnetoresistance at the critical point. In
particular, we find a linear in $B$ magnetoresistance for the range of
magnetic fields such that the fermion cyclotron frequency $\omega_{cf}$ is
within the range $(k_{F}\xi)^{-4}T\ll\omega_{cf}\ll(k_{F}\xi)^{-3}T$.

In section V we study the diamagnetic susceptibility $\chi(T)$ in the critical
regime. We predict a logarithmic divergence $\chi\sim\log(\Lambda/T)$ which is
contrasted with the power law divergence expected in a conventional XY quantum
critical point in $2+1D$. This much slower divergence is a result of strong
Landau damping of the order parameter fluctuations and offers a sharp testable
prediction of our theory.

##  III Large N Model and Solution

In this section we describe the calculation of the fermion and boson (pairing
field) Green’s functions in the special large $N$ limit described above. To
this end we extend the model to $N$ flavors of (spinful) fermions
$c_{i},i=1,...,N$ and the same number N of boson flavors $\Psi_{n},n=1,...,N$
111In principle, the number of bosons and fermions can be different as long as
they are taken to infinity together (i.e. with a constant ratio), but this
detail does not greatly affect the physics of our model.. Accordingly, the
disorder in the pairing $g$ is also promoted to a random rank 3 tensor
$g_{ij}^{n}$. In this large $N$ limit, we may solve a closed set of saddle
point (Schwinger-Dyson) equations to obtain the exact boson and fermion
correlation functions and their corresponding self-energies. Since we keep the
ratio of the number of fermion to boson flvaors constant, the correlation
functions and the self-energies that we compute from the saddle point
equations contain the mutual feedback between the bosons and fermions.

Upon generalizing (1) to $N$ flavors of fermions and bosons, our action is
given by,

$\begin{split}\mathcal{S}_{N}(c^{\dagger},c,\Psi)&=\int_{k}\sum_{i=1}^{N}{c_{ik}^{\dagger}}(\partial_{\tau}+\epsilon_{k}-\mu)c_{ik}+\sum_{n=1}^{N}\left(\Delta_{b}^{0}+\frac{k^{2}}{2m_{b}}\right)\Psi_{nk}^{\dagger}\Psi_{nk}+\int_{x}\frac{U}{2N}\left(\sum_{n=1}^{N}\Psi_{nx}^{\dagger}\Psi_{nx}\right)^{2}\\\
&+\int_{q,q^{\prime},k}\sum_{ij,n}^{N}J_{k}g_{ij,q}^{n}\Psi_{q^{\prime}}^{n}\left({c}^{T}_{i\frac{q+q^{\prime}}{2}+k}\sigma_{y}c_{j\frac{q+q^{\prime}}{2}-k}\right)_{x}+c.c.\,,\textrm{
where }J_{k}=J(\cos k_{x}a-\cos k_{y}a)\end{split}$ (2)

Here, we have written the pairing interaction in momentum space. As such, the
pairing interaction has a momentum dependence $J_{k}$, which vanishes at the
nodes of $k_{x}=\pm k_{y}$. In addition, the disorder in momentum space
$g_{q}$ is of zero mean and its variance satisfies,

$\begin{split}\overline{g_{ij,q}^{n}g_{j^{\prime}i^{\prime},q^{\prime}}^{n^{\prime}}}&=\frac{\pi\xi^{2}}{N^{2}}\delta_{nn^{\prime}}\delta_{ii^{\prime}}\delta_{jj^{\prime}}\delta(q+q^{\prime})e^{-\xi^{2}q^{2}}\,,\\\
\end{split}$ (3)

Accordingly, the disorder in the pairing $g$ can scatter momentum up to
$\mathcal{O}(\pi/\xi)$.

We now perform disorder averaging and take the saddle point, which gives the
following Schwinger-Dyson equations (Also illustrated pictorially in Fig.2).

Figure 2: Graphic representation of the large $N$ saddle point equations.
Solid (wavy) lines denote fermions (bosons, resp.). Dashed lines denote
disorder contraction. a) Fermion self-energy diagram. b) Boson self-energy
diagrams.

$\begin{split}&G(k,\omega)=\frac{1}{\mathbf{i}\omega+\Sigma(k,\omega)-(\epsilon_{k}-\mu)}\,,\
\
\Sigma(k,t)=\int_{|q^{\prime}|\lesssim\frac{\pi}{\xi}}\int_{q}2J^{2}_{k-\frac{q+q^{\prime}}{2}}\xi^{2}F(q,t)G(-k+q+q^{\prime},-t)\,,\\\
&F(q,\omega)=\frac{1}{\Delta_{b}^{0}-\Pi_{f}(q,\omega)+q^{2}/2m_{b}}\,,\ \
\Pi=\Pi_{b}+\Pi_{f}\,,\textrm{ where }\\\
&\Pi_{f}(q,t)=\int_{|q^{\prime}|\lesssim\frac{\pi}{\xi}}\int_{k}2J^{2}_{k-\frac{q+q^{\prime}}{2}}\xi^{2}G(k,t)G(-k+q+q^{\prime},t)\,,\
\ \Pi_{b}(q,\omega)=-U\int_{k,\omega}F(k,\omega)\,.\end{split}$ (4)

Here, we denote with $G=\frac{1}{N}\sum_{i}\braket{c_{i}c^{\dagger}_{i}}$, the
fermion Green’s functions, and
$F=\frac{1}{N}\sum_{n}\braket{\Psi_{n}\Psi^{\dagger}_{n}}$, the boson Green’s
functions. $\Sigma$ and $\Pi$ denote the fermion and boson self-energies,
respectively. We work from the metallic side of the transition, so that
superconducting correlation functions such as $\braket{\psi\psi}$ are zero.

The fermion self-energy $\Sigma$ is simply a bubble diagram of a fermion and
boson propagator. On the other hand, the boson self-energy $\Pi$ is a sum of
two parts, $\Pi_{b}$ and $\Pi_{f}$. The boson contribution $\Pi_{b}$ is due to
the quartic boson interaction and is a tadpole diagram. Conversely, the
fermion contribution $\Pi_{f}$ is due to the random pairing interaction and is
a bubble diagram of two fermion propagators.

It is important to note that due to (LABEL:eq:ddisorderp), the disorder lines
can only carry momentum $q^{\prime}\lesssim\frac{\pi}{\xi}$. This fact is
reflected in the integral over $q^{\prime}$ in (4) for $\Sigma$ and $\Pi_{f}$.

We begin solving (4) by first determining the boson self-energy $\Pi$. In
particular, let us first focus on its frequency and momentum dependence. This
comes solely from $\Pi_{f}$, the self-energy due to the random pairing
interaction; since $\Pi_{b}$ is a tadpole diagram, it is a constant
independent of frequency and momentum.

For small $q$ and $\omega$, $\Pi_{f}(q,\omega)$ is given as (See Appendix.B
for details),

$\begin{split}&\Pi_{f}(q,\omega)=\Pi_{f}(0)-\frac{q^{2}}{2M}-\alpha|\omega|+\mathbf{i}\eta\omega\,,\textrm{
where }\\\ &\frac{1}{M}=\frac{J^{2}\xi^{2}k_{F}}{8\pi^{2}v_{F}}\,,\
\alpha=\frac{J^{2}\xi k_{F}}{4\pi v_{F}^{2}}\,,\
\eta\simeq\frac{J^{2}\log\frac{\Lambda_{f}}{v_{F}/\xi}}{v_{F}^{2}}\,.\end{split}$
(5)

(5) has two important consequences. First, the boson propagator which was
originally independent of frequency now obtains a frequency dependence; it is
Landau damped ($\alpha$) and particle-hole asymmetric ($\eta$). In particular,
the Landau damping is very strong compared to the particle-hole asymmetry with
$\frac{\alpha}{\eta}\simeq\frac{k_{F}\xi}{\log k_{F}\xi}\gg 1$.

Second, the momentum dependence of $\Pi$ means that the boson mass $m_{b}$ is
renormalized to $M_{b}=(m_{b}^{-1}+M^{-1})^{-1}$. Together with the particle-
hole asymmetry, the effective boson mass $\tilde{M}_{b}$ is given by $\eta
M_{b}\simeq\frac{\log k_{F}\xi}{(k_{F}\xi)^{2}}m_{f}$. For $k_{F}\xi\gg 1$,
$\tilde{M}_{b}$ is extremely light; this means that the boson low energy
degrees of freedom are essentially restricted to small momenta of order
$\mathcal{O}(1/\xi)$.

We now turn to $\Pi(0)$, the boson self-energy at zero frequency and momentum.
It renormalizes the boson gap $\Delta_{b}$ as,

$\displaystyle\Delta_{b}=\Delta_{b}^{0}-\Pi_{b}(0)-\Pi_{f}(0)\,,\textrm{ where
}$ (6a) $\displaystyle\Pi_{b}(0)=-U\int_{k,\omega}F(k,\omega)\,,$ (6b)
$\displaystyle\Pi_{f}(0)=\int_{q^{\prime}\lesssim\pi/\xi}\int_{k,\omega}2J_{k-\frac{q^{\prime}}{2}}^{2}G_{k+q^{\prime},\omega}G_{-k,-\omega}$
$\displaystyle\quad\quad\ \ \simeq\frac{J^{2}k_{F}}{v_{F}}\log k_{F}\xi$ (6c)

Here, both $\Pi_{b}$ and $\Pi_{f}$ participate in the boson gap
renormalization. The boson contribution $\Pi_{b}(0)$ (6b) increases the gap.
On the other hand, the fermion contribution $\Pi_{f}(0)$ (6c) decreases the
gap and drives the bosons toward superconductivity.

Crucially, $\Pi_{f}$ is IR convergent. This is in contrast to an IR divergence
seen in standard BCS theories, where the ground state inevitably becomes
superconducting due to the divergence. Within our model, the disorder in the
pairing $g$ is providing an IR cutoff lengthscale $\xi$ that eliminates this
BCS instability. Therefore, we have a quantum phase transition between the
superconducting phase ($\Delta_{b}<0$) and the metallic phase ($\Delta_{b}>0$)
as we tune the interaction strength $J$ or the disorder correlation length
$\xi$. In the rest of this paper we focus on the critical point, defined by
$\Delta_{b}=0$.

Putting the above results together gives us the boson propagator at the QCP,

$\begin{split}&F(k,\omega)=\frac{1}{k^{2}/2M_{b}+\alpha|\omega|-\mathbf{i}\eta\omega}\,.\end{split}$
(7)

We can now use the boson propagator to determine the fermion self-energy
$\Sigma$, which is given by the convolution of the boson and fermion
propagators as shown in (Fig.2a). At the QCP, this convolution is dominated by
the critical bosons (7), leading to a marginal Fermi liquid Varma (2016) self-
energy, albeit with a momentum dependent prefactor $\gamma_{k}$ (See
Appendix.B):

$\begin{split}&\Sigma(k,\omega)\sim\mathbf{i}\gamma_{k}\omega\log\frac{\Lambda}{|\omega|}\,,\\\
&\textrm{ where }\gamma_{k}\simeq\frac{2}{\pi}\left(\frac{2\pi}{k_{F}\xi}+\cos
4\theta_{k}\sin\frac{2\pi}{k_{F}\xi}\right)\,.\end{split}$ (8)

$\gamma_{k}$ generally scales as $\frac{\cos^{2}2\theta_{k}}{k_{F}\xi}$, due
to the $d$-wave nature of the interaction that has a momentum dependence of
$J_{k}\sim J\cos 2\theta_{k}$.

Importantly, even at the nodes of $J_{k}$ at
$\theta_{k}=\pm\frac{\pi}{4},\pm\frac{3\pi}{4}$, $\gamma_{k}$ is nonzero, but
rather takes the small value $\gamma_{k}\sim(k_{F}\xi)^{-3}\ll 1$. The nonzero
scatterring rate at the nodes is due to the momentum carried by the pairing
disorder $g$ that allows for scattering within angles $(k_{F}\xi)^{-1}$ of the
nodes to contribute to the nodal self-energy.

Having derived the critical propagators, we now generalize the results to non
vanishing temperatures in the critical regime. Crucially, the boson propagator
develops a thermal gap at non-vanishing temperatures. The main contribution to
this gap is from the four-boson interactions as previously discussed in
Podolsky  _et al._ (2007). This contribution scales as
$\Delta_{b}\approx\Pi_{b}(0)\sim\frac{UM_{b}T}{\log\Lambda/T}$ 222Although the
fermions also contribute to the thermal boson gap, this contribution scales as
$T^{2}/\Lambda_{f}$ and may be ignored. Therefore, the boson propagator at
temperature $T$ has the form,

$\begin{split}F(k,\omega_{n})=\frac{1}{\Delta_{b}+\frac{k^{2}}{2M_{b}}+\alpha|\omega_{n}|-\mathbf{i}\eta\omega_{n}}\,.\end{split}$
(9)

where $\omega_{n}=2\pi nT$ are the Matsubara frequencies. Analytical
continuation of (9) to real frequencies results in the following retarded
boson correlation function:

$F_{R}(k,\omega)=\frac{1}{\Delta_{b}+\frac{k^{2}}{2M_{b}}-\mathbf{i}\alpha\omega-\eta\omega}\,.$
(10)

We now turn to the fermions and compute their retarded self-energy
$\Sigma_{R}$; from $\Sigma_{R}$ we can extract the quasi-particle relaxation
rate $1/\tau$.

We first determine $\Re\\{\Sigma_{R}\\}$ and the corresponding fermion quasi-
particle weight $Z$. Due to the thermal boson gap, the Euclidean fermion self-
energy deviates slightly from the marginal Fermi liquid behavior of (8). At
low frequencies $|\omega|<\frac{\Delta_{b}}{\alpha}$, it is linear in
$\omega$, scaling as
$\Sigma(k,\omega)\sim\mathbf{i}\gamma_{k}\log\left(\Lambda/\Delta_{b}\right)\omega$
(See Appendix.B). Analytically continuing to real time, we find that the real
part of $\Sigma_{R}$ is given as
$\Sigma_{R}(k,\omega)\sim\gamma_{k}\log(\Lambda/\Delta_{b})\omega\,.$ The
corresponding quasi-particle weight $Z$ is,

$Z=\frac{1}{1+\gamma_{k}\log\frac{\Lambda}{\Delta_{b}}}\,.$ (11)

$Z$ renormalizes the quasi-particle effective mass to $m_{f}/Z$. However,
recall that the maximum value of $\gamma_{k}$ at most scales as
$(k_{F}\xi)^{-1}$. This means that for moderate values of $k_{F}\xi$, the
quasi-particle weight is essentially 1 at physically relevant temperatures.
Thus, we may safely take the fermion mass to be its bare mass $m_{f}$.

We now turn to $\mathrm{Im}\\{\Sigma_{R}(k,\omega)\\}$. By carrying out the
convolution of the boson and fermion correlation functions in the Keldysh-
framework (See Appendix.B), we obtain

$\begin{split}&\mathrm{Im}\\{\Sigma_{R}(k,\omega)\\}\simeq-\gamma_{k}T\Big{(}\lambda+\pi\log\cosh\frac{\beta\omega}{2}\Big{)}\,.\end{split}$
(12)

Here, $\lambda\sim\log\frac{\alpha T}{\Delta_{b}}=\log\log\frac{\Lambda}{T}$.
Henceforth we shall treat it as an $O(1)$ constant.

Putting (11) and (12) together, we find that $\tau_{k}^{-1}$, the scattering
rate for a quasi-particle at momentum $k$ has the form,

$\tau_{k}^{-1}=Z\mathrm{Im}\\{\Sigma_{R}\\}\simeq\gamma_{k}T\,.$ (13)

The scattering rate is most suppressed at the nodes, where it scales as
$\tau_{\textrm{node}}^{-1}\sim(k_{F}\xi)^{-3}T$; elsewhere, it is much faster,
scaling as $\tau_{k}^{-1}\sim\frac{\cos^{2}2\theta_{k}}{k_{F}\xi}T$.

It is important to note that the scattering rate of (13) is different from the
momentum relaxation rate. After each collision, the momentum distribution of
an excitation broadens by order $1/\xi$. For complete momentum relaxation, the
distribution has to spread over the entire Fermi surface. This means that an
excitation needs to undergo a number of collisions of the order of
$(k_{F}\xi)^{2}$ to relax its momentum. During this process, the nodes become
a significant bottleneck due to the suppressed scattering rate in their
vicinity. In the next section we use the quantum Boltzmann equation to account
for the angular dependence of the scattering. As we will show, the resulting
overall momentum relaxation rate for an excitation scales as
$1/\tau_{re}\sim(k_{F}\xi)^{-4}T$ and is notably sub-Planckian for $\xi\gg 1$.

##  IV Transport

In this section, we employ the quantum Boltzmann equation to compute the
transport properties. In the DC limit it is given by Mahan (2000)

$\displaystyle
A_{F}(q,\Omega)^{2}\mathrm{Im}\\{\Pi_{R}(q,\Omega)\\}\frac{2eq\cdot
E}{M_{b}}{b_{0}}^{\prime}(\Omega)=\Pi^{>}F^{<}-\Pi^{<}F^{>}\,,$ (14a)
$\displaystyle
A_{G}(k,\omega)^{2}\mathrm{Im}\\{\Sigma_{R}(k,\omega)\\}\frac{ek\cdot
E}{m_{f}}{f_{0}}^{\prime}(\omega)=\Sigma^{>}G^{<}-\Sigma^{<}G^{>}\,.$ (14b)

Here $f_{0},b_{0}=\frac{1}{e^{\beta\omega}\pm 1}$ are the Fermi-Dirac / Bose-
Einstein distribution functions, and $A_{G,F}$ are the fermion/boson spectral
functions. This equation takes the Green’s functions computed in the previous
section as input. It then describes the evolution of the generalized
distribution functions in both momentum and frequency space in response to the
external field, thereby allowing for calculation of transport even in the
absence of sharp quasiparticles Kim  _et al._ (1995); Nave and Lee (2007);
Senthil  _et al._ (2004); Maslov  _et al._ (2011). The equilibrium
distributions of the fermions and bosons are given respectively by

$n_{f}^{0}(k,\omega)=A_{G}(k,\omega)f_{0}(\omega),\
n_{b}^{0}(q,\Omega)=A_{F}(q,\Omega)b_{0}(\Omega)\,.$

Inserting the Green’s functions computed in the previous section we obtain the
quantum Boltzmann equation for the deviations $\delta f(k,\omega),\delta
b(q,\Omega)$ from the equilibrium distributions (See Appendix.C for details),

$\displaystyle
A_{F}(q,\Omega)^{2}\frac{2eq\cdot\vec{E}}{M_{b}}{b_{0}}^{\prime}(\Omega)=\delta
b(q,\Omega)\,,$ (15a)
$\displaystyle\frac{ev_{F}\hat{k}\cdot\vec{E}f_{0}^{\prime}(\omega)}{16\pi}=\int_{|\hat{k}+\hat{k^{\prime}}|<\frac{\pi}{k_{F}\xi},\
\omega^{\prime}}\cos^{2}2\theta_{\hat{k}-\hat{k^{\prime}}}B(\omega+\omega^{\prime})\big{(}1-f_{0}(\omega)\big{)}\big{(}1-f_{0}(\omega^{\prime})\big{)}\\{g_{f}(\hat{k},\omega)+g_{f}(\hat{k^{\prime}},\omega^{\prime})\\}\,,$
(15b) $\displaystyle\textrm{ where }\delta f(\hat{k},\omega)=\int_{\Delta
k}\delta f\big{(}(k_{F}+\Delta
k)\hat{k},\omega\big{)}=f_{0}(\omega)\big{(}1-f_{0}(\omega)\big{)}g_{f}(\hat{k},\omega)\,,\
\ B(\omega)=b_{0}(\omega)\tan^{-1}\frac{\alpha\omega}{\Delta_{b}}\,.$

Here we have set $\hbar,k_{B}=1$. The left hand sides of (15) denote the
evolution of the boson (fermion) distribution function in response to the
external field. The right hand sides of the equations are the collision
integrals that account for the relaxation processes due to the fermion-boson
scattering processes. Note that in the fermion quantum Boltzmann equation
(15b), we have used the fact that the fermion spectral function is still
sharply peaked and integrated out the momentum direction perpendicular to the
Fermi surface for simplicity.

The strategy to compute the conductivity from (15) is then the following: We
first solve (15) for $\delta f$ and $\delta b$, from which we calculate the
current of each species $J_{f,b}$. The conductivity then naturally follows
from the relation $\sigma=J/E$.

Let us start by computing $\sigma_{b}$, the boson contribution to the
conductivity. The boson current is given by,
$J_{b}=\int_{q,\Omega}\frac{2eq}{M_{b}}\delta b(q,\Omega)$. Dividing by the
electric field, we find,

$\sigma_{b}^{xx}=\frac{e^{2}\beta}{4M_{b}^{2}}\int_{q,\Omega}\left(qA_{F}(q,\Omega)\
\mathrm{csch}\frac{\beta\Omega}{2}\right)^{2}\,.\\\ $ (16)

Evaluating the boson conductivity (16) by inserting (9), we find,

$\sigma_{b}^{xx}\simeq\frac{\alpha e^{2}}{M_{b}U}\log\frac{\Lambda}{T}\,.$
(17)

Hence the Cooper pair fluctuations contribute a conductivity that scales
logarithmically with temperature.

We now turn to the fermion conductivity, $\sigma_{f}$. We demonstrate through
a scaling argument that it is inversely proportional to the temperature. The
key point is the following: the solution to (15b) for an electric field
$\vec{E}=E\hat{x}$ can be written as $g(\hat{k},\omega)=ev_{F}\beta^{2}E\
\tilde{g}(\hat{k},\tilde{\omega}=\beta\omega)$, where
$\tilde{g}(\hat{k},\tilde{\omega})$ is a dimensionless function independent of
temperature (up to logarithmic corrections) that satisfies,

$\begin{split}\frac{\cos\theta_{\hat{k}}\tilde{f}_{0}(\tilde{\omega})}{16\pi}&=\int_{\hat{k^{\prime}},\tilde{\omega}^{\prime}}^{|\hat{k}+\hat{k^{\prime}}|<\frac{\pi}{k_{F}\xi}}\cos^{2}2\theta_{\hat{k}-\hat{k^{\prime}}}\tilde{B}(\tilde{\omega}+\tilde{\omega}^{\prime})\big{(}1-\tilde{f}_{0}(\tilde{\omega}^{\prime})\big{)}\big{\\{}\tilde{g}(\hat{k},\tilde{\omega})+\tilde{g}(\hat{k^{\prime}},\tilde{\omega}^{\prime})\big{\\}}\,,\\\
&\textrm{where
}\tilde{B}(\tilde{\omega})=\frac{\tan^{-1}\frac{\alpha\tilde{\omega}}{\tilde{\Delta}_{b}}}{e^{\tilde{\omega}}-1},\
\ \tilde{f}_{0}(\tilde{\omega})=\frac{1}{1+e^{\tilde{\omega}}}\,.\end{split}$
(18)

To see that $\tilde{g}$ is independent of temperature, notice that (18) is
dependent only on dimensionless variables such as $\frac{1}{k_{F}\xi}$ and
$\tilde{\Delta}_{b}=\frac{\Delta_{b}}{T}\sim\frac{UM_{b}}{\log\Lambda/T}$.

Since $J_{f}=2\int_{\hat{k},\omega}ek_{F}\hat{k}\delta f(\hat{k},\omega)$, we
find that the longitudinal and Hall fermion conductivity are each given by,

$\begin{split}\sigma_{f}^{xx}&=\frac{2e^{2}\epsilon_{F}}{T}\int_{\hat{k},\tilde{\omega}}\cos\theta_{k}\tilde{f}_{0}(\tilde{\omega})\big{(}1-\tilde{f}_{0}(\tilde{\omega})\big{)}\tilde{g}(\hat{k},\tilde{\omega})\,,\\\
\sigma_{f}^{xy}&=\frac{2e^{2}\epsilon_{F}}{T}\int_{\hat{k},\tilde{\omega}}\sin\theta_{k}\tilde{f}_{0}(\tilde{\omega})\big{(}1-\tilde{f}_{0}(\tilde{\omega})\big{)}\tilde{g}(\hat{k},\tilde{\omega})\,.\end{split}$
(19)

Note that the integrals of the right hand sides of (19) is independent of
temperature: It is therefore clear that the fermion conductivities are
inversely proportional to $T$.

To determine the proportionality factor in (19), we need to solve (18). This
can be done numerically, and the result is displayed in Fig. 3. We find
333$\sigma_{f}^{xy}$ is identically zero without a magnetic field present.,

$\sigma_{f}^{xx}\simeq 0.180\ (k_{F}\xi)^{4}\frac{e^{2}\epsilon_{F}}{T}$ (20)

Figure 3: $\sigma_{f}T/e^{2}\epsilon_{F}$ vs $k_{F}\xi$ plotted in log-log
scale for $\tilde{\Delta}_{b}=1$, obtained from a numerical solution of the
quantum Boltzmann equation, (15b).

This scaling behavior can be understood through a simple semiclassical
argument. We focus our attention on the bottleneck of the momentum relaxation
process at the nodes. In particular, let us take an excitation at an angle
$\theta_{k}=\frac{\pi}{4}+\phi$ without loss of generality. Recall that the
scattering rate at angle $\theta_{k}$ is $\tau_{k}$, and that after each
collision the momentum changes at most by $\frac{1}{\xi}$. This means that the
rate at which its variance in momentum increases is given as,
$\partial_{t}{\braket{p^{2}}}_{k}\sim\xi^{-2}\tau_{k}^{-1}$. As a result, the
rate at which the angular variance increase is given by,
$\partial_{t}\braket{\theta^{2}}_{k}\sim(k_{F}\xi)^{-2}\tau_{k}^{-1}$. From
this, we estimate the momentum relaxation time $\tau_{\textrm{tr}}$ as,

$\tau_{\textrm{tr}}\simeq\int_{\theta_{k}}(k_{F}\xi)^{2}\tau_{k}\simeq\int_{\phi}\frac{(k_{F}\xi)^{5}/T}{1+(k_{F}\xi)^{2}\phi^{2}}\sim(k_{F}\xi)^{4}/T\,.$

Finally, using a Drude approximation of
$\sigma_{f}\simeq\frac{ne^{2}\tau_{\textrm{tr}}}{m}$, we arrive at,

$\sigma_{f}\sim(k_{F}\xi)^{4}\frac{e^{2}\epsilon_{F}}{T}$

in agreement with the numerical solution of the quantum Boltzmann equations
(20). It follows that at low temperatures $\sigma_{f}\gg\sigma_{b}$. Hence the
charge transport is dominated by the fermion contribution leading to $T$
linear resistivity

$\rho_{tot}\simeq\frac{5.56}{(k_{F}\xi)^{4}}\frac{T}{e^{2}\epsilon_{F}}\,.$
(21)

###  IV.1 Effects of Potential Disorder

So far we have only considered the disorder in the boson-fermion pairing
interaction. However it is natural to expect at least a small amount of
regular single particle disorder coupling to the fermion density. Indeed, we
argued in section II that the long scale pairing disorder can be produced by
short scale potential disorder. In this section we address the effect of the
residual short scale potential disorder. Specifically we consider the
following random potential coupling to the fermion density:

$H_{rp}=V_{x}c^{\dagger}_{x}c_{x}\,,\textrm{ where }\overline{V_{x}}=0,\
\overline{V_{x}V_{x^{\prime}}}\sim W\delta_{xx^{\prime}}\,.$ (22)

Scattering off this random potential results in the following standard
contribution to the fermion self-energy,

$\Sigma_{rp}(\omega)=\frac{\mathbf{i}\mathrm{sgn}(\omega)}{2\tau_{0}}\,,\textrm{
where }\frac{1}{\tau_{0}}=m_{f}W\,.$ (23)

Because the short range disorder scatters at all angles, unlike the
interaction disorder, the above self energy is directly related to the
momentum relaxation rate.

The self energy (23) adds a contribution to the collision integral in the
right hand side of the of the quantum Boltzmann equation (18) of
$\frac{\tilde{f_{0}}(\tilde{\omega})}{16\pi\tau_{0}T}\tilde{g}(\hat{k},\tilde{\omega})$
(See Appendix.C,(50)). Consequently $\tilde{g}$, and therefore the conductance
depend on $\tau_{0}T$.

In particular, in the limit $T\to 0$ the potential disorder leads to a
saturation of the resistivity to a non vanishing constant

$\rho_{0}=\frac{2\pi}{e^{2}\epsilon_{F}\tau_{0}}\,.$ (24)

Note that for weak potential disorder $\tau_{0}$ is large and $\rho_{0}$ can
be several orders of magnitude smaller than the resistance quantum $R_{Q}$.

Comparing the two contributions to the collision integral suggests a crossover
at a temperature scale $T\gg T_{0}=\frac{(k_{F}\xi)^{4}}{\tau_{0}}$ between
the low temperature limit affected by the potential disorder to the higher
temperature regime dominated by the interaction disorder. As expected, the
resistivity for $T\gg T_{0}$ approaches the result 21. On the other hand the
leading temperature dependence for $T\ll T_{0}$ can be obtained by solving the
quantum Boltzmann equation perturbatively around the zero temperature
solution. This gives

$\rho_{low}-\rho_{0}\simeq 16.8(k_{F}\xi)^{-3}\frac{T}{e^{2}\epsilon_{F}}\,.$
(25)

That is, we find linear in $T$ resistivity also in the low temperature limit,
but with a larger slope compared to the regime $T>T_{0}$. The reason for the
stronger temperature dependence is that the scattering from the potential
disorder eliminates the nodal bottleneck for relaxation. As a result,
scattering off of Cooper pairs contribute a faster relaxation rate of
$(k_{F}\xi)^{-2}\tau_{\textrm{antinode}}^{-1}\sim(k_{F}\xi)^{-3}T$ that is
reflected in (25). The resistivity obtained from numerical solution of the
quantum Boltzmann equation, shown in Fig.4, corroborates the analytic results.

Figure 4: Log-log plot of $\frac{\rho(T)-\rho_{0}}{T}$ versus temperature in
the presence of potential disorder for various disorder correlation lengths.
The temperature axis has been rescaled by the electron mean free time
$\tau_{0}$. At high temperatures $\tau_{0}T\gg(k_{F}\xi)^{4}$, the resistivity
scales linearly in $T$ as given by (21). At low temperatures resistivity
saturates to a constant $\rho_{0}$ (24). Additionally, resistivity increases
linearly with $T$ at low temperatures, albeit with a different slope than at
high $T$. The inset graph shows how this slope scales with the disorder
correlation length. It scales as,
$\frac{\Delta\rho_{low}}{T}\sim\frac{(k_{F}\xi)^{-3}}{\epsilon_{F}}$.

###  IV.2 Magnetotransport

We have seen above that the fermions dominate the transport. We consider their
contribution to the magnetoresistance first. We then confirm that the boson
contribution is negligible.

The quantum Boltzmann equation for the fermion distribution is modified by a
perpendicular magnetic field, bringing the left hand side of(18) into the
following form

$\frac{\cos\theta_{k}\tilde{f}_{0}(\tilde{\omega})}{16\pi}-\frac{\omega_{cf}/T}{16\pi}\tilde{f}_{0}(\tilde{\omega})\partial_{\theta_{k}}g(\hat{k},\tilde{\omega})\,,$
(26)

where $\omega_{cf}=eB/m_{f}$ is the fermion cyclotron frequency. We see that
the magnetic field and the temperature appear in this scaled equation only
through the dimensionless ratio $b=\omega_{cf}/T$ and therefore the solution
$\tilde{g}$ must also depend only on $b$. The conductivity, determined from
$\tilde{g}$ through the relation (19) must then take the scaling form
$\sigma=e^{2}\epsilon_{F}/T\,\phi(\omega_{cf}/T)$. Similarly for the
magnetoresistance
$\rho(T,B)-\rho(T,0)=(T/e^{2}\epsilon_{F})\psi(\omega_{cf}/T)$.

Fig.5 shows the behavior of the magnetoresistance by plotting the scaling
function $\psi(b)$ obtained from numerical solution of the quantum Boltzmann
equation for different values of the scaled disorder correlation length
$k_{f}\xi$. We find that the magnetoresistance is linear in $B$ within the
range of magnetic fields such that $T(k_{F}\xi)^{-4}\ll\omega_{cf}\ll
T(k_{F}\xi)^{-3}$. Note that the slope of the linear $B$ dependence is
independent of the disorder (i.e. of $\xi$) and is given by
$\tilde{\mu}_{B}/(e^{2}\epsilon_{F})$, where
$\tilde{\mu}_{B}=\frac{e}{m_{f}}$.

Figure 5: Log-log plot of $B/T$ vs $\frac{\rho(T,B)-\rho(T,0)}{T}$ for various
disorder correlation lengths. For magnetic fields such that
$(k_{F}\xi)^{-5}\ll b=\frac{\tilde{\mu}_{B}B}{T}\ll(k_{F}\xi)^{-3}$,
longitudinal resistivity is linear in $B$ with slope
$\tilde{\mu}_{B}=\frac{e}{m_{f}}$ (Dashed line). For weak fields of
$b\ll(k_{F}\xi)^{-5}$, resistivity behaves as (20). Inset shows that for
strong fields $b\gg(k_{F}\xi)^{-3}$, resistivity saturates to scaling as
$\rho^{*}\sim(k_{F}\xi)^{-3}\frac{T}{e^{2}\epsilon_{F}}$.

The linear in field magnetoresistance arises from the strongly anisotropic
quasi-particle scattering rate (13) as we now explain. Recall that in absence
of a magnetic field the nodes at
$\theta_{k}=\pm\frac{\pi}{4},\pm\frac{3\pi}{4}$ act bottlenecks for momentum
relaxation. Once a magnetic field is turned on, quasiparticles begin to rotate
around the Fermi surface at a rate $\omega_{cf}$. This can overcome the nodal
bottleneck if the rotation is faster than the rate at which momentum relaxes
at the nodes, i.e. $\omega_{cf}\gg(k_{F}\xi)^{-4}T$. In this case the time
scale for momentum relaxation rate can be replaced by
$\omega_{cf}=\tilde{\mu}_{B}B$, leading to linear magneto resistance. However
if $\omega_{cf}$ exceeds the rate of momentum relaxation away from the nodes,
i.e. $\omega_{cf}\gg T(k_{F}\xi)^{-3}$, then this anti-nodal scattering
becomes the bottleneck for momentum relaxation and the resistivity saturates
to $\rho^{*}\simeq(k_{F}\xi)^{-3}T/(e^{2}\epsilon_{F})$.

We can now explain why the contribution of the bosons to the magnetoresistance
is negligible compared with that of the fermions. In the regime with linear
magnetoresistance the fermion conductivity is reduced from
$e^{2}(\epsilon_{F}/T)(k_{F}\xi)^{4}$ to
$e^{2}(\epsilon_{F}/T)(k_{F}\xi)^{3}$. This change is much larger than the
entire boson contribution to the conductivity, which is, up to logarithmic
corrections, just of order of the resistance quantum. As a consequence, any
changes in the boson conductivity is washed out by the change in fermion
conductivity. We leave a more detailed derivation of the boson
magnetotransport to Appendix.C.

##  V Diamagnetic Susceptibility

The diamagnetic susceptibility can be calculated from the transverse current-
current correlator at $\omega=0$

$\chi=\lim_{q\to 0}\partial^{2}_{q}\left\langle
J_{x}(q\hat{y},0)J_{x}(-q\hat{y},0)\right\rangle.$ (27)

The scaling of $\chi$ with temperature at the critical point provides a
further test of our theory of the QSMT.

The diamagnetic response is a sum of the fermionic and bosonic contributions
given by

$\displaystyle\chi_{f}$
$\displaystyle=-\frac{e^{2}T}{(2\pi)^{2}}\partial_{q}^{2}\sum_{\omega_{n}}\int_{k}\frac{k_{x}^{2}}{m_{f}^{2}}\left[G_{k+\frac{q\hat{y}}{2}}(\omega_{n})G_{k-\frac{q\hat{y}}{2}}(\omega_{n})\right]\,,$
(28a) $\displaystyle\chi_{b}$
$\displaystyle=-\frac{4e^{2}T}{(2\pi)^{2}}\partial_{q}^{2}\sum_{\omega_{n}}\int_{k}\frac{k_{x}^{2}}{M_{b}^{2}}\left[F_{k+\frac{q\hat{y}}{2}}(\omega_{n})F_{k-\frac{q\hat{y}}{2}}(\omega_{n})\right]\,,$
(28b)

in the limit of $q\rightarrow 0$.

The fermions contribution gives essentially the standard, temperature
independent Landau diamagnetic response (See Appendix.D),

$\chi_{f}\simeq\frac{e^{2}}{12\pi m_{f}}\,.$ (29)

The bosons on the other hand give rise to a singular contribution at the
critical point. Using the boson propagator (9) in (28b) and integrating over
the momentum gives

$\chi_{b}=\frac{e^{2}}{6\pi\beta}\sum_{\omega_{n}}\frac{1}{\Delta_{b}+\alpha|\omega_{n}|}\,.$

The contribution to this sum fron the zeroth Matsubara frequency diverges
logarithmically with temperature because $\Delta_{b}\sim
UM_{b}T/\log\frac{\Lambda}{T}$). The sum over the non-zero frequencies also
gives a logarithmic divergence. So, together we obtain

$\chi_{b}\simeq\left(\frac{e^{2}}{UM_{b}}+\frac{e^{2}}{12\pi^{2}\alpha}\right)\log\frac{\Lambda}{T}$
(30)

We conclude that the total diamagnetic response diverges logarithmically with
temperature at the critical point. Note that whereas the charge transport was
dominated by the fermionic quasiparticles, the critical diamagnetic response
is governed by the critical bosonic (Cooper pair) fluctuations.

The logarithmically divergent susceptibility in our model should be compared
with the diamagnetic response in a standard XY transition Doniach (1981).
Within a Gaussian theory the result is $\chi\sim 1/T$ for the quantum critical
point in two spatial dimensions. This would be modified to a non-trivial
power-law in the true (Willson-Fisher) fixed point in 2+1 dimensdions,
markedly different from the logarithmic temperature dependece we obtain.

##  VI Summary and Discussion

We presented a theory for an unusual quantum superconductor to metal
transition (QSMT) observed in nano-patterned YBCO films Yang  _et al._ (2022).
The film in this experiment exhibits $T$-linear resistivity with very small
residual resistivity at the QSMT tuned by the size of nano-patterned holes.
This stands in contrast to the Fermi liquid phenomenology characterizing the
superconductor to metal transition obtained in cuprates by doping Zinc
impurities Fukuzumi  _et al._ (1996) or at the overdoped edge of the
superconducting dome Cooper  _et al._ (2009). The essential ingredient in our
model, leading to the destruction of quasi-particles and the $T$-linear
resistivity, is a disordered and frustrated coupling between the quasipaticles
and the critical bosonic fluctuations (i.e. $d$-wave Cooper pairs). Results of
mean field calculations Li  _et al._ (2021a) provide indirect evidence that
such effective interactions can emerge at intermediate scales due to the
interplay of d-wave pairing and a disorder potential. Our results show that
$T$ linear resistivity can be obtained over a broad range if the emergent
disordered interaction is stronger than the usual disordered potential
coupling to the fermion density. It is an open question how this situation
emerges from the microscopics, but it is possible that patterning the cuprate
film with holes, as done in Ref. Yang  _et al._ (2022), arranges the required
conditions. A direct microscopic derivation of the effective interactions
remain a challenge for future work.

Within the effective model we studied, the dominant contribution to the
conductivity, scaling as $1/T$, comes from fermionic quasi-particles decaying
into Landau damped critical Cooper pairs. The critical bosons themselves
contribute only a sub-leading logarithmic divergence
$\sigma_{b}\sim\log\Lambda/T$. In contrast, the authors of the experimental
paper Yang  _et al._ (2022) argue that the transport is dominated by the
critical bosons. To support this interpretation they show oscillations of the
magnetoresistance with a period set by the superconducting flux quantum $h/2e$
through a unit cell of the patterned superlattice. We note, however, that such
oscillations can also arise from the the sub-leading contribution of the
bosons in our model. As we show in Appendix A, the temperature dependence of
the magnetoresistance oscillation shown in Ref. Yang  _et al._ (2022), is
consistent with the $\log(\Lambda/T)$ behavior of the bosonic contribution to
the resistivity in our model.

It is worth noting that similar phenomenology has been observed in nano-
patterned films of the iron based superconductor FeSe Li  _et al._ (2021b).
While here we focused on $d$-wave pairing, our model can be easily adapted to
describe a QSMT involving other pairing symmetries. In particular, interplay
of a random potential and the sign structure of the $s_{\pm}$ pairing wave-
function proposed for this superconductor can lead to a disordered sign-
changing pairing interaction through the same mechanism as in $d$-wave
superconductors. The transport calculations would be almost identical except
for minor differences due to the absence of nodes in the coupling of Fermi
surface quasi-particles to the critical fluctuations. Because the nodal
bottleneck for momentum relaxation is removed, we expect
$\rho\sim(k_{F}\xi)^{-3}\epsilon_{F}/T$ in the nodeless case versus
$\rho\sim(k_{F}\xi)^{-4}\epsilon_{F}/T$ for a nodal pairing interaction.

Finally, having discussed the critical point with T linear resistivity, it is
worth noting that the samples which we regarded as being on the
superconducting side of it in Ref. Yang  _et al._ (2022) are not truly
superconducting. At the lowest measured temperatures they behave as failed
superconductors (or anomalous metals) with very low resistivity Kapitulnik
_et al._ (2019)). Developing an effective description of this phase is an
interesting problem for future work.

###### Acknowledgements.

We would like to thank Yimu Bao, Xiangyu Cao, Shubhayu Chatterjee, Debanjan
Chowdhury, Hart Goldman, Tobias Holder, Ruihua Fan, Chunxiao Liu, Aavishkar
Patel, Xuepeng Wang, and Zack Weinstein for helpful discussions. E.B.
acknowledges support from NSF-BSF award DMR-2000987 and from the European
Research Council (ERC) under grant HQMAT (grant agreement No. 817799). Part of
this work was done in the Aspen Center for Physics, supported by National
Science Foundation grant PHY-2210452.

## References

  * Warren  _et al._ (1989) Jr. Warren, W. W., R. E. Walstedt, G. F. Brennert, R. J. Cava, and R. Tycko, “Cu spin dynamics and superconducting precursor effects in planes above Tc in YBa2Cu3O6.7,” Phys. Rev. Lett. 62, 1193–1196 (1989).
  * Alloul  _et al._ (1989) H. Alloul, T. Ohno, and P. Mendels, “89Y NMR evidence for a fermi-liquid behavior in YBa2Cu3O6+x,” Phys. Rev. Lett. 63, 1700–1703 (1989).
  * Doiron-Leyraud  _et al._ (2007) Nicolas Doiron-Leyraud, Cyril Proust, David Leboeuf, Julien Levallois, Jean-Baptiste Bonnemaison, Ruixing Liang, D. A. Bonn, W. N. Hardy, and Louis Taillefer, “Quantum oscillations and the Fermi surface in an underdoped high-Tc superconductor,” Nature (London) 447, 565–568 (2007), arXiv:0801.1281 [cond-mat.supr-con] .
  * Gurvitch and Fiory (1987) M. Gurvitch and A. T. Fiory, “Resistivity of ${\mathrm{la}}_{1.825}$${\mathrm{sr}}_{0.175}$${\mathrm{cuo}}_{4}$ and ${\mathrm{yba}}_{2}$${\mathrm{cu}}_{3}$${\mathrm{o}}_{7}$ to 1100 k: Absence of saturation and its implications,” Phys. Rev. Lett. 59, 1337–1340 (1987).
  * Takagi  _et al._ (1992) H. Takagi, B. Batlogg, H. L. Kao, J. Kwo, R. J. Cava, J. J. Krajewski, and W. F. Peck, “Systematic evolution of temperature-dependent resistivity in ${\mathrm{la}}_{2\mathrm{-}\mathit{x}}$${\mathrm{sr}}_{\mathit{x}}$${\mathrm{cuo}}_{4}$,” Phys. Rev. Lett. 69, 2975–2978 (1992).
  * Taillefer (2010) Louis Taillefer, “Scattering and Pairing in Cuprate Superconductors,” Annual Review of Condensed Matter Physics 1, 51–70 (2010), arXiv:1003.2972 [cond-mat.supr-con] .
  * Fukuzumi  _et al._ (1996) Y. Fukuzumi, K. Mizuhashi, K. Takenaka, and S. Uchida, “Universal superconductor-insulator transition and ${T}_{c}$ depression in zn-substituted high- ${T}_{c}$ cuprates in the underdoped regime,” Phys. Rev. Lett. 76, 684–687 (1996).
  * Hebard and Paalanen (1990) A. F. Hebard and M. A. Paalanen, “Magnetic-field-tuned superconductor-insulator transition in two-dimensional films,” Phys. Rev. Lett. 65, 927–930 (1990).
  * Paalanen  _et al._ (1992) M. A. Paalanen, A. F. Hebard, and R. R. Ruel, “Low-temperature insulating phases of uniformly disordered two-dimensional superconductors,” Phys. Rev. Lett. 69, 1604–1607 (1992).
  * Seidler  _et al._ (1992) G. T. Seidler, T. F. Rosenbaum, and B. W. Veal, “Two-dimensional superconductor-insulator transition in bulk single-crystal ${\mathrm{yba}}_{2}$${\mathrm{cu}}_{3}$${\mathrm{o}}_{6.38}$,” Phys. Rev. B 45, 10162–10164 (1992).
  * van der Zant  _et al._ (1992) H. S. J. van der Zant, F. C. Fritschy, W. J. Elion, L. J. Geerligs, and J. E. Mooij, “Field-induced superconductor-to-insulator transitions in josephson-junction arrays,” Phys. Rev. Lett. 69, 2971–2974 (1992).
  * Wang  _et al._ (1991) T. Wang, K. M. Beauchamp, D. D. Berkley, B. R. Johnson, J.-X. Liu, J. Zhang, and A. M. Goldman, “Onset of high-temperature superconductivity in the two-dimensional limit,” Phys. Rev. B 43, 8623–8626 (1991).
  * Yang  _et al._ (2022) Chao Yang, Haiwen Liu, Yi Liu, Jiandong Wang, Dong Qiu, Sishuang Wang, Yang Wang, Qianmei He, Xiuli Li, Peng Li, Yue Tang, Jian Wang, X. C. Xie, James M. Valles, Jie Xiong, and Yanrong Li, “Signatures of a strange metal in a bosonic system,” Nature (London) 601, 205–210 (2022), arXiv:2105.02654 [cond-mat.supr-con] .
  * Herbut (2000) Igor F. Herbut, “Zero-temperature $\mathit{d}$-wave superconducting phase transition,” Phys. Rev. Lett. 85, 1532–1535 (2000).
  * Podolsky  _et al._ (2007) Daniel Podolsky, Ashvin Vishwanath, Joel Moore, and Subir Sachdev, “Thermoelectric transport near pair breaking quantum phase transition out of $d$-wave superconductivity,” Phys. Rev. B 75, 014520 (2007).
  * Li  _et al._ (2021a) Zi-Xiang Li, Steven A. Kivelson, and Dung-Hai Lee, “Superconductor-to-metal transition in overdoped cuprates,” npj Quantum Materials 6, 36 (2021a), arXiv:2010.06091 [cond-mat.supr-con] .
  * Kim  _et al._ (2021) Jaewon Kim, Ehud Altman, and Xiangyu Cao, “Dirac Fast Scramblers,” Phys. Rev. B 103, 081113 (2021), arXiv:2010.10545 [cond-mat.str-el] .
  * Aldape  _et al._ (2022) Erik E. Aldape, Tessa Cookmeyer, Aavishkar A. Patel, and Ehud Altman, “Solvable theory of a strange metal at the breakdown of a heavy Fermi liquid,” Phys. Rev. B 105, 235111 (2022), arXiv:2012.00763 [cond-mat.str-el] .
  * Esterlis  _et al._ (2021) Ilya Esterlis, Haoyu Guo, Aavishkar A Patel, and Subir Sachdev, “Large-n theory of critical fermi surfaces,” Physical Review B 103, 235129 (2021).
  * Patel  _et al._ (2022) Aavishkar A. Patel, Haoyu Guo, Ilya Esterlis, and Subir Sachdev, “Universal, low temperature, $T$-linear resistivity in two-dimensional quantum-critical metals from spatially random interactions,” (2022), arXiv:2203.04990 [cond-mat.str-el] .
  * Giraldo-Gallo  _et al._ (2018) P. Giraldo-Gallo, J. A. Galvis, Z. Stegen, K. A. Modic, F. F. Balakirev, J. B. Betts, X. Lian, C. Moir, S. C. Riggs, J. Wu, A. T. Bollinger, X. He, I. Bozović, B. J. Ramshaw, R. D. McDonald, G. S. Boebinger, and A. Shekhter, “Scale-invariant magnetoresistance in a cuprate superconductor,” Science 361, 479–481 (2018), arXiv:1705.05806 [cond-mat.str-el] .
  * Cooper  _et al._ (2009) R. A. Cooper, Y. Wang, B. Vignolle, O. J. Lipscombe, S. M. Hayden, Y. Tanabe, T. Adachi, Y. Koike, M. Nohara, H. Takagi, Cyril Proust, and N. E. Hussey, “Anomalous Criticality in the Electrical Resistivity of La2-xSrxCuO4,” Science 323, 603 (2009).
  * Hayes  _et al._ (2021) Ian M. Hayes, Nikola Maksimovic, Gilbert N. Lopez, Mun K. Chan, B. J. Ramshaw, Ross D. McDonald, and James G. Analytis, “Superconductivity and quantum criticality linked by the Hall effect in a strange metal,” Nature Physics 17, 58–62 (2021), arXiv:1912.06130 [cond-mat.str-el] .
  * Ataei  _et al._ (2022) Amirreza Ataei, A. Gourgout, G. Grissonnanche, L. Chen, J. Baglo, M. E. Boulanger, F. Laliberté, S. Badoux, N. Doiron-Leyraud, V. Oliviero, S. Benhabib, D. Vignolles, J. S. Zhou, S. Ono, H. Takagi, C. Proust, and Louis Taillefer, “Electrons with Planckian scattering obey standard orbital motion in a magnetic field,” Nature Physics 18, 1420–1424 (2022), arXiv:2203.05035 [cond-mat.str-el] .
  * Jaoui  _et al._ (2022) Alexandre Jaoui, Ipsita Das, Giorgio Di Battista, Jaime Díez-Mérida, Xiaobo Lu, Kenji Watanabe, Takashi Taniguchi, Hiroaki Ishizuka, Leonid Levitov, and Dmitri K. Efetov, “Quantum critical behaviour in magic-angle twisted bilayer graphene,” Nature Physics 18, 633–638 (2022), arXiv:2108.07753 [cond-mat.str-el] .
  * Note (1) In principle, the number of bosons and fermions can be different as long as they are taken to infinity together (i.e. with a constant ratio), but this detail does not greatly affect the physics of our model.
  * Varma (2016) Chandra M. Varma, “Quantum-critical fluctuations in 2D metals: strange metals and superconductivity in antiferromagnets and in cuprates,” Reports on Progress in Physics 79, 082501 (2016), arXiv:1601.03403 [cond-mat.str-el] .
  * Note (2) Although the fermions also contribute to the thermal boson gap, this contribution scales as $T^{2}/\Lambda_{f}$ and may be ignored.
  * Mahan (2000) G. D. Mahan,  _Many Particle Physics, Third Edition_ (Plenum, New York, 2000).
  * Kim  _et al._ (1995) Yong Baek Kim, Patrick A. Lee, and Xiao-Gang Wen, “Quantum Boltzmann equation of composite fermions interacting with a gauge field,” Phys. Rev. B 52, 17275–17292 (1995), arXiv:cond-mat/9504063 [cond-mat] .
  * Nave and Lee (2007) Cody P. Nave and Patrick A. Lee, “Transport properties of a spinon Fermi surface coupled to a U(1) gauge field,” Phys. Rev. B 76, 235124 (2007), arXiv:0708.1850 [cond-mat.str-el] .
  * Senthil  _et al._ (2004) T. Senthil, Matthias Vojta, and Subir Sachdev, “Weak magnetism and non-Fermi liquids near heavy-fermion critical points,” Phys. Rev. B 69, 035111 (2004), arXiv:cond-mat/0305193 [cond-mat.str-el] .
  * Maslov  _et al._ (2011) Dmitrii L. Maslov, Vladimir I. Yudson, and Andrey V. Chubukov, “Resistivity of a Non-Galilean-Invariant Fermi Liquid near Pomeranchuk Quantum Criticality,” Phys. Rev. Lett. 106, 106403 (2011), arXiv:1012.0069 [cond-mat.str-el] .
  * Note (3) $\sigma_{f}^{xy}$ is identically zero without a magnetic field present.
  * Doniach (1981) S. Doniach, “Quantum fluctuations in two-dimensional superconductors,” Phys. Rev. B 24, 5063–5070 (1981).
  * Li  _et al._ (2021b) Yanan Li, Yi Liu, Haiwen Liu, Shichao Qi, Haoran Ji, Wenfeng Dong, Yi Sun, Wenhao Zhang, Chengcheng Ji, Zihan Cui, Nitin Samarth, Lili Wang, X. C. Xie, Qi-Kun Xue, and Jian Wang, “Bosonic metal states in crystalline iron-based superconductors at the two-dimensional limit,” arXiv e-prints , arXiv:2111.15488 (2021b), arXiv:2111.15488 [cond-mat.supr-con] .
  * Kapitulnik  _et al._ (2019) Aharon Kapitulnik, Steven A. Kivelson, and Boris Spivak, “Colloquium: Anomalous metals: Failed superconductors,” Rev. Mod. Phys. 91, 011002 (2019).
  * Little and Parks (1962) W. A. Little and R. D. Parks, “Observation of quantum periodicity in the transition temperature of a superconducting cylinder,” Phys. Rev. Lett. 9, 9–12 (1962).

##  Appendix A Little-Parks Effect

The authors of the experimental paper on nanopatterned YBCO films Ref. Yang
_et al._ (2022) interpreted $T$ linear metallic conductivity as arising from
critical bosonic Cooper pairs. This interpretation was based on the
oscillation of the conductivity as a function of magnetic field with a period
corresponding to a bosonic flux quantum $h/2e$ similar to the Little-Parks
effect Little and Parks (1962).

Here we argue that these oscillations are consistent with our theory wherein
the linear in T resistivity comes from the fermions and the bosons add a sub-
leading contribution. Indirect evidence for this interpretation is given in
Fig. 6; it shows that the oscillation amplitude $A_{\sigma}$ shows a
logarithmic temperature dependence, matching the predicted scaling of the sub-
leading bosonic conductivity.

Figure 6: $A_{\sigma}$ versus $T$ ($\log$ scale) at the critical point for
sample $f2$.

##  Appendix B Derivation of the Schwinger-Dyson Equations and Self Energies

In this appendix, we derive the large $N$ Schwinger-Dyson equations and
provide the details behind the various self-energy calculations. In our model,
the interaction between the fermions
$c_{i,k}^{T}=(c_{i\uparrow,k},c_{i\downarrow,k})$ and bosons $\Psi_{n,q}$ is
given by,

$H_{int}=\int\frac{d^{2}k\,d^{2}q\,d^{2}q^{\prime}}{(2\pi)^{6}}\sum_{ij,n}^{N}J_{k}g_{ij,q}^{n}\Psi_{n,q^{\prime}}\left({c}^{T}_{i\frac{q+q^{\prime}}{2}+k}\sigma_{y}c_{j\frac{q+q^{\prime}}{2}-k}\right)_{x}+c.c.$

Averaging over the random disorder $g_{ij,q}^{n}$ results in the following
contribution to the action

$S_{int}=\frac{\pi\xi^{2}}{N^{2}}\int_{k,q,q^{\prime},\tau,\tau^{\prime}}\sum_{ij,n}\Big{(}{c}^{T}_{i\frac{q+q^{\prime}}{2}+k}\sigma_{y}c_{j\frac{q+q^{\prime}}{2}-k}\Big{)}_{\tau}\Big{(}{c}^{\dagger}_{j\frac{q+q^{\prime}}{2}-k}\sigma_{y}{c}^{*}_{i\frac{q+q^{\prime}}{2}+k}\Big{)}_{\tau^{\prime}}J_{k}^{2}e^{-\xi^{2}{q^{\prime}}^{2}}\Psi^{\dagger}_{n}(q,\tau)\Psi_{n}(q,\tau^{\prime})\,.$
(31)

We now introduce the fermion and boson Green’s functions,

$G(x,\tau,x^{\prime},\tau^{\prime})=\frac{1}{2N}\sum_{i,\sigma}\braket{c_{i\sigma}(x,\tau)c^{\dagger}_{i\sigma}(x^{\prime},\tau^{\prime})}\,,\quad
F(x,\tau,x^{\prime},\tau^{\prime})=\frac{1}{N}\sum_{n}\braket{\Psi_{n}(x,\tau)\Psi^{\dagger}_{n}(x^{\prime},\tau^{\prime})}\,,$

which we impose through the integral expression for the delta function:

$\begin{split}\delta\Big{(}\sum_{i\sigma}c_{i\sigma}(x,\tau)c^{\dagger}_{i\sigma}(x^{\prime},\tau^{\prime})-2NG_{x-x^{\prime},\tau-\tau^{\prime}}\Big{)}=&\int
D\Sigma\exp\Big{(}\Sigma_{x^{\prime}-x,\tau^{\prime}-\tau}\big{(}\sum_{i\sigma}c_{i\sigma}(x,\tau)c^{\dagger}_{i\sigma}(x^{\prime},\tau^{\prime})-2NG_{x-x^{\prime},\tau-\tau^{\prime}}\big{)}\Big{)}\\\
\delta\Big{(}\sum_{n}\Psi_{n}(\tau,x)\Psi^{\dagger}_{n}(\tau^{\prime},x^{\prime})-NF_{\tau-\tau^{\prime},x-x^{\prime}}\Big{)}=&\int
D\Pi\exp\Big{(}\Pi_{\tau^{\prime}-\tau,x^{\prime}-x}\big{(}\sum_{n}\Psi_{n}(\tau,x)\Psi_{n}^{\dagger}(\tau^{\prime},x^{\prime})-NF_{\tau-\tau^{\prime},x-x^{\prime}}\big{)}\Big{)}\end{split}$
(32)

With the introduction of the Green’s functions and their corresponding self
energies, the effective action takes the form

$\begin{split}\frac{\beta
S(G,\Sigma,F,\Pi)}{N}&=\sum_{\omega_{n}}\int_{k}-2\log\big{(}-\mathbf{i}\omega_{n}-\Sigma_{k,\omega_{n}}+v_{F}(k-k_{F})\big{)}+\log(\Delta_{b}+k^{2}/2m_{b}-\Pi_{k,\omega_{n}})\\\
&\ \
+2\Sigma_{k,\omega_{n}}G_{k,\omega_{n}}+\Pi_{k,\omega_{n}}F_{k,\omega_{n}}-\int_{\tau,\tau^{\prime},k,q,q^{\prime}}2\pi\xi^{2}J_{k}^{2}e^{-\xi^{2}q^{2}}\
G_{k+\frac{q+q^{\prime}}{2},\tau-\tau^{\prime}}G_{-k+\frac{q+q^{\prime}}{2},\tau-\tau^{\prime}}F_{q,\tau^{\prime}-\tau}\end{split}$
(33)

The saddle point of this action gives the Schwinger-Dyson equations (4).

We now calculate the self energies in (4). We begin with the boson self-energy
$\Pi$ (5), which is given by the following integral

$\begin{split}\Pi(q,\Omega)&=\int\frac{kdkd\theta_{k}}{(2\pi^{2})}\int_{\mathcal{B}\left(\frac{\pi}{\xi}\right)}\frac{d^{2}q^{\prime}}{(2\pi)^{2}}\int\frac{d\omega}{2\pi}2\xi^{2}J_{k}^{2}\
G(k+(q+q^{\prime})/2,\omega+\Omega/2)G(-k+(q+q^{\prime})/2,-\omega+\Omega/2)\\\
&=\int\frac{kdkd\theta_{k}}{(2\pi^{2})}\int_{\mathcal{B}\left(\frac{\pi}{\xi}\right)}\frac{d^{2}q^{\prime}}{(2\pi)^{2}}\int\frac{d\omega}{2\pi}2\xi^{2}J_{k}^{2}\
\frac{G_{-k+(q+q^{\prime})/2,-\omega+\Omega/2}-G_{k+(q+q^{\prime})/2,\omega+\Omega/2}}{2\mathbf{i}\omega+\Sigma_{k+(q+q^{\prime})/2,\omega+\Omega/2}-\Sigma_{-k+(q+q^{\prime})/2,-\omega+\Omega/2}+v_{F}(q+q^{\prime})\cos\theta_{qk}}\,.\end{split}$
(34)

In the second line, we assumed a circular Fermi surface for simplicity. We
also approximate $J_{k}\sim J\cos 2\theta_{k}$.

To see the general structure of the solution we first calculate this using the
free Green’s functions of the fermions (i.e. neglecting the fermion self
energy correction). At this point we also linearize the dispersion about the
Fermi energy and neglect the Fermi surface curvature, thereby imposing an
effective particle hole symmetry. Later we discuss the particle hole symmetry
breaking terms. With the above approximations the boson self energy is given
by the integral,

$\begin{split}\Pi(q,\Omega)&=\frac{2J^{2}\xi^{2}k_{F}}{(2\pi)^{5}v_{F}}\int
d\theta_{k}\cos^{2}2\theta_{k}\int_{\mathcal{B}\left(\frac{\pi}{\xi}\right)}d^{2}q^{\prime}\int
d\omega\frac{\mathbf{i}\pi\big{(}\mathrm{sgn}(\omega+\Omega/2)+\mathrm{sgn}(\omega-\Omega/2)\big{)}}{2\mathbf{i}\omega+v_{F}(q+q^{\prime})\cos\theta_{q+q^{\prime},k}}\\\
&=\frac{2\pi^{2}J^{2}\xi^{2}k_{F}}{(2\pi)^{5}v_{F}}\int
d\omega\int_{\mathcal{B}\left(\frac{\pi}{\xi}\right)}d^{2}q^{\prime}\frac{\mathrm{sgn}(\omega)\big{(}\mathrm{sgn}(\omega+\Omega/2)+\mathrm{sgn}(\omega-\Omega/2)\big{)}}{\Big{(}4\omega^{2}+v_{F}^{2}(q+q^{\prime})^{2}\Big{)}^{1/2}}\end{split}$
(35)

Now we can expand this for small frequency $\Omega$ and momentum $q$. Setting
$\Omega=0$, the momentum dependence of $\Pi$ is given by

$\begin{split}\Pi(q,0)&=\frac{4\pi^{2}J^{2}\xi^{2}k_{F}}{(2\pi)^{5}v_{F}}\int
d\omega\int_{\mathcal{B}\left(\frac{\pi}{\xi}\right)}d^{2}q^{\prime}(4\omega^{2}+v_{F}^{2}(q+q^{\prime})^{2})^{-1/2}\simeq\Pi(0,0)+\frac{q^{2}}{2M}\\\
\frac{1}{M}&\simeq\nabla_{q}^{2}\Pi|_{q=0,\omega=0}=\frac{4\pi^{2}J^{2}\xi^{2}v_{F}k_{F}}{(2\pi)^{5}}\int_{\omega}\int_{\mathcal{B}\left(\frac{\pi}{\xi}\right)}\frac{-8\omega^{2}+v_{F}^{2}q^{\prime
2}}{(4\omega^{2}+v_{F}^{2}q^{\prime
2})^{5/2}}\simeq-\frac{J^{2}\xi^{2}k_{F}}{8\pi^{2}v_{F}}\end{split}$ (36)

This result defines the mass renormalization of the boson from the bare mass
$m_{b}$ to $M_{b}=\Big{(}{M^{-1}+m_{b}^{-1}}\Big{)}^{-1}$.

The frequency dependence of the self energy is found by setting $q=0$

$\begin{split}\Pi(0)-\Pi(0,\Omega)&=\frac{4\pi^{2}J^{2}k_{F}}{(2\pi)^{5}v_{F}}\int_{-\Omega/2}^{\Omega/2}d\omega\int^{\pi/\xi}_{0}2\pi
qdq\Big{(}4\omega^{2}+v_{F}^{2}q^{2}\Big{)}^{-1/2}\\\
&\simeq\frac{J^{2}k_{F}\xi|\Omega|}{4\pi v_{F}^{2}}\,.\end{split}$ (37)

This is reminiscent of the familiar Landau damped form of the self energy
$\Delta\Pi(q,\Omega)\propto|\Omega|/q$ with the disorder scale $\xi$ playing
the role of $1/q$.

The correction to $\Pi$ from breaking of particle hole symmetry is given by

$\begin{split}\Delta\Pi(\Omega)&=\frac{2\pi
J^{2}\xi^{2}}{(2\pi)^{5}}\int\Delta kd\Delta k\int qdqd\theta_{q}\int
d\omega\frac{G(k+q/2,\omega+\Omega/2)-G(-k+q/2,-\omega+\Omega/2)}{2\mathbf{i}\omega+v_{F}q\cos\theta_{qk}}\\\
&-\frac{2\pi J^{2}\xi^{2}k_{F}}{(2\pi)^{5}}\int d\Delta k\int
qdqd\theta_{q}\int
d\omega\frac{G(k+q/2,\omega+\Omega/2)-G(-k+q/2,-\omega+\Omega/2)}{(2\mathbf{i}\omega+v_{F}q\cos\theta_{qk})^{2}}\frac{q\Delta
k\cos\theta_{qk}}{m}.\end{split}$ (38)

Here $\Delta k=|k|-k_{F}$ denotes the distance from the Fermi surface. The
first term above accounts for the non vanishing curvature of the Fermi surface
(i.e. the correction to approximating $kdk$ as $k_{F}dk$ in (35)). The second
term stems from the non-linear correction to the dispersion near the Fermi
surface. Computing the integral we find,

$\begin{split}\Delta\Pi(\Omega)&\simeq\mathbf{i}\left(\frac{J^{2}}{v_{F}^{2}}\log\frac{\Lambda}{v_{F}/\xi}\right)\Omega\end{split}$
(39)

which gives the value of $\eta$ appearing in (5).

We now turn to the fermions and calculate their self-energy at zero
temperature. This will then allow us to show that the above form of $\Pi$ is
self consistent and not modified except by a possible prefactor. The fermion
self energy at momentum $k$ on the fermi surface is given by

$\begin{split}\Sigma(k,\Omega)&=2J^{2}\xi^{2}\int\frac{d^{2}qd^{2}q^{\prime}d\omega}{(2\pi)^{5}}\cos^{2}2\theta_{k-\frac{q^{\prime}}{2}}F(q,\omega+\Omega)G(-k+q^{\prime},\omega)\\\
&=\frac{J^{2}\xi^{2}M_{b}}{\pi}\int\frac{dq^{\prime}_{\perp}dq^{\prime}_{\parallel}d\omega}{(2\pi)^{3}}\cos^{2}2\theta_{k-\frac{q^{\prime}}{2}}\log\frac{\Lambda}{\alpha|\omega+\Omega|}\frac{1}{-\mathbf{i}\omega-\Sigma_{-k+q^{\prime},\omega}-v_{F}q^{\prime}_{\perp}}\\\
&=\frac{\mathbf{i}J^{2}\xi^{2}M_{b}}{2\pi
v_{F}}\int\frac{dq^{\prime}_{\parallel}d\omega}{(2\pi)^{2}}\cos^{2}2\theta_{k-\frac{q^{\prime}}{2}}\log\frac{\Lambda}{\alpha|\omega+\Omega|}\
\mathrm{sgn}(\omega)\\\
&=\mathbf{i}\gamma_{k}\Omega\log\frac{\Lambda}{\Omega}\,,\end{split}$ (40)

where,

$\begin{split}\gamma_{k}&=\frac{J^{2}\xi^{2}M_{b}}{4\pi^{3}v_{F}}\int_{-\pi/\xi}^{\pi/\xi}dq^{\prime}_{\parallel}\cos^{2}(2\theta_{k}-q^{\prime}_{\parallel}/k_{F})\\\
&=\frac{J^{2}\xi^{2}M_{b}}{4\pi^{3}v_{F}}\left\\{\frac{\pi}{\xi}+\frac{k_{F}\cos
4\theta_{k}\sin(2\pi/k_{F}\xi)}{2}\right\\}\\\
&\simeq\frac{2}{\pi}\left\\{\frac{2\pi}{k_{F}\xi}+\cos
4\theta_{k}\sin(2\pi/k_{F}\xi)\right\\}\end{split}$

We see that the fermion self-energy scales with frequency as
$\Omega\log\Lambda/\Omega$. Importantly, it has a prefactor $\gamma_{k}$ which
is at most $\mathcal{O}(1/k_{F}\xi)$. This means that the fermion self-energy
$\Sigma$ is parametrically smaller than $\omega$ in the limit of large
$k_{F}\xi$. Ergo, including the fermion self-energy in (35) results in only a
small correction that can be neglected for large values of $k_{F}\xi$.

We now determine the self-energies at finite temperatures. The effect of the
temperature on the boson self energy is to generate a thermal gap $\Delta_{b}$
(i.e. a frequency and momentum independent contribution to the self energy)
Podolsky  _et al._ (2007). This thermal gap in turn leads to a modification of
the fermion self-energy (40) to,

$\begin{split}\Sigma(k,\Omega)&=\mathbf{i}\gamma_{k}\int\frac{\omega}{2\pi}\log\frac{\Lambda}{\Delta_{b}+\alpha|\omega+\Omega|}\mathrm{sgn}(\omega)\\\
&\simeq\mathbf{i}\gamma_{k}\log\frac{\Lambda}{\Delta_{b}}\Omega\,,\textrm{ for
}\Omega\ll\Delta_{b}.\end{split}$ (41)

Analytically continuing (41) to real time, we find the real part of the
retarded fermion self energy
$\Re\\{\Sigma_{R}\\}\simeq\gamma_{k}\log\frac{\Lambda}{\Delta_{b}}\Omega\,,$.
This subsequently results in a renormalization of the quasi-particle weight to
(11) of the main text.

We now calculate the imaginary part of the retarded fermion self-energy, which
(together with the quasi-particle weight $Z$) is related to the quasi-particle
decay rate. Using the standard Keldysh formalism, the retarded fermion self-
energy is given as,

$\begin{split}\Sigma_{R}(k,t)&=J^{2}\xi^{2}\int\frac{d^{2}qd^{2}q^{\prime}}{(2\pi)}\cos^{2}2\theta_{k-\frac{q+q^{\prime}}{2}}\Big{(}F_{R}(q,t)G_{K}(-k+q^{\prime},-t)-F_{K}(q,t)G_{R}(-k+q^{\prime},-t)\Big{)}\\\
&\textrm{ where
}F_{R}(t)=-\mathbf{i}\theta(t)\big{(}F_{>}(t)-F_{<}(t)\big{)}\,,\ \
F_{K}(t)=F_{>}(t)+F_{<}(t)\,,\\\ &\quad
G_{R}(t)=-\mathbf{i}\theta(t)\big{(}G_{>}(t)+G_{<}(t)\big{)}\,,\ \
G_{K}(t)=G_{>}(t)-G_{<}(t)\,.\end{split}$

where the greater and lesser Green’s functions are defined as,

$G_{>}(t)=\braket{\psi(t)\psi^{\dagger}(0)},\
G_{<}(t)=\braket{\psi^{\dagger}(0)\psi(t)}\,,\
F_{>}(t)=\braket{\Psi(t)\Psi^{\dagger}(0)},\
F_{<}(t)=\braket{\Psi^{\dagger}(0)\Psi(t)}\,.$

Using the fluctuation-dissipation theorem, the Keldysh Green’s functions can
be found from the spectral functions of each species:

$F_{K}(\omega)=A_{F}(\omega)\coth\frac{\beta\omega}{2}\,,\quad
G_{K}(\omega)=A_{G}(\omega)\tanh\frac{\beta\omega}{2}\,.$

With this, we find that the imaginary part of the self-energy, i.e., the
quasi-particle decay rate, can be written as:

$\mathrm{Im}\\{\Sigma_{R}(k,\Omega)\\}=\frac{J^{2}\xi^{2}}{4\pi}\int\frac{d\omega}{2\pi}\int\frac{d^{2}q}{(2\pi)^{2}}\int\frac{d^{2}q^{\prime}}{(2\pi)^{2}}\cos^{2}2\theta_{k-q^{\prime}/2}A_{F}(q,\omega)A_{G}(-k+q^{\prime},\omega-\Omega)\left(\coth\frac{\beta\omega}{2}-\tanh\frac{\beta(\omega-\Omega)}{2}\right)$
(42)

The momentum integrals over the boson momentum $q$ and the momentum carried by
the disorder $q^{\prime}$ are independent and can be performed separately.
Integrating over the $q$ dependent terms of (42) gives,

$\begin{split}&\int^{\pi/\xi}_{0}\frac{d^{2}q}{(2\pi)^{2}}A_{F}(q,\omega)=-2\int^{\pi/\xi}_{0}\frac{qdq}{2\pi}\mathrm{Im}\left\\{\big{(}\Delta_{b}+q^{2}/2M_{b}-\eta\omega-\mathbf{i}\alpha\omega\big{)}^{-1}\right\\}\\\
&\quad\quad=-\frac{2M_{b}}{\pi}\left(\pi\Theta(\eta\omega-\Delta_{b})+\tan^{-1}\frac{\alpha\omega}{\Delta_{b}-\eta\omega}\right)\\\
&\quad\quad\simeq-\frac{2M_{b}}{\pi}\left(\tan^{-1}\frac{\alpha\omega}{\Delta_{b}}\right)\end{split}$

Here in the third line we have made use of the fact that $\eta$ is an order of
magnitude smaller than $\alpha$.

On the other hand, integrating over the $q^{\prime}$ dependent terms of (42)
gives,

$\begin{split}&\int^{\pi/\xi}_{0}\frac{d^{2}q^{\prime}}{(2\pi^{2})}\cos^{2}2\theta_{k-q^{\prime}/2}A_{G}(-k+q^{\prime}/2,\omega)=2\int\frac{dq^{\prime}_{\parallel}}{2\pi}\frac{dq^{\prime}_{\bot}}{2\pi}\cos^{2}(2\theta_{k}-q^{\prime}_{\parallel}/k_{F})\mathrm{Im}\left\\{\big{(}\omega-\Sigma_{R}(k,\omega)-v_{F}\Delta
k+v_{F}q_{\bot}\big{)}^{-1}\right\\}\\\ &\quad\quad=\frac{1}{2\pi
v_{F}}\int_{-\pi/\xi}^{\pi/\xi}dq_{\parallel}\cos^{2}(2\theta_{k}-q_{\parallel}/k_{F})\end{split}$

Here, we have made use of the assumption that at low temperatures, our
effective theory is limited to momentum $k$ is close to the fermi surface, so
that $|k-k_{F}|\ll\frac{\pi}{\xi}$. $q^{\prime}_{\bot}$
($q^{\prime}_{\parallel}$) denotes the momentum perpendicular (parallel) to
the fermi surface, respectively.

Therefore, combining these two results, we find that the imaginary part of the
self-energy is given as,

$\begin{split}&\mathrm{Im}\\{\Sigma_{R}(k,\omega)\\}=-\gamma_{k}\int\frac{d\omega}{2\pi}\tan^{-1}\frac{\alpha\omega}{\Delta_{b}}\left(\coth\frac{\beta\omega}{2}+\tanh\frac{\beta(\Omega-\omega)}{2}\right)\\\
&\simeq-\gamma_{k}T\log\frac{\alpha
T}{\Delta_{b}}-\gamma_{k}Tf(\Omega,\Delta_{b})\,,\textrm{ where
}f(\Omega,\Delta_{b})=\int\frac{d\omega}{2\pi}\left(\tan^{-1}\frac{\alpha\omega}{\Delta_{b}}-\tan^{-1}\frac{\alpha(\omega+\Omega)}{\Delta_{b}}\right)\tanh\frac{\beta\omega}{2}\end{split}$
(43)

Let us make a few remarks about this dimensionless function $f$. By
definition, $f(0,\Delta_{b})=0$. Also, at low frequencies,

$f(\Omega,\Delta_{b})\simeq\frac{\pi+\pi\tan^{2}\Delta_{b}/2\alpha
T}{8}(\beta\Omega)^{2}$

On the other hand, for large frequencies $\Omega\gg\Delta_{b}$,
$f(\Omega,\Delta_{b})\simeq\pi\log\cosh\frac{\beta\Omega}{2}$.

##  Appendix C Quantum Boltzmann Equation

The quantum Boltzmann equations model the evolution of a generalized
probability distribution and allows for the calculation of transport even in
the absence of sharply defined quasiparticles, such as in our model. The
quantum Boltzmann equation in the DC limit reads Mahan (2000),

$\displaystyle
A_{F}(q,\Omega)^{2}\mathrm{Im}\\{\Pi_{R}(q,\Omega)\\}\frac{2eq\cdot
E}{M_{b}}{b_{0}}^{\prime}(\Omega)=\Pi^{>}F^{<}-\Pi^{<}F^{>}\,,$ (44a)
$\displaystyle
A_{G}(k,\omega)^{2}\mathrm{Im}\\{\Sigma_{R}(k,\omega)\\}\frac{ek\cdot
E}{m_{f}}{f_{0}}^{\prime}(\omega)=\Sigma^{>}G^{<}-\Sigma^{<}G^{>}\,.$ (44b)

Here, $f_{0},b_{0}=\frac{1}{e^{\beta\omega}\pm 1}$ denote the Fermi-
Dirac/Bose-Einstein distribution functions, and $A_{G,F}$, the fermion/boson
spectral functions. The left hand side denotes the evolution of the boson and
fermion distribution function in response to the external field, and the right
hand side, the collision integral that account for the relaxation processes
due to the fermion - boson scattering.

In the presence of an external field, the greater and lesser Green’s functions
shift from their equilibrium value and we have,

$G^{<,>}(k,\omega)=\frac{A_{G}(k,\omega)}{e^{\pm\beta\omega}+1}+\delta
f(k,\omega)\,,\quad
F^{<,>}(q,\Omega)=\frac{A_{F}(q,\Omega)}{e^{\pm\beta\Omega}-1}+\delta
b(q,\Omega)\,,$ (45)

where $\delta f$ and $\delta b$ denote stand for the deviation from the
fermion and boson equilibrium distribution. As explained in the main text, the
fermion and boson currents are then determined from $\delta f$ and $\delta b$,
from which we subsequently find the charge conductance.

In this appendix, we derive the simplified quantum Boltzmann equations (15)
which was used in the main text to determine $\delta f$ and $\delta b$. We
start with the derivation of the boson quantum Boltzmann equation (15a) from
(44a). To this end, we require the greater and lesser boson self-energies
$\Pi^{>,<}$ which appear in the collision integral. These are given as,

$\Pi^{>,<}(q,\Omega)=\int_{|k+k^{\prime}-q|<\frac{\pi}{\xi}}\frac{d^{2}kd^{2}k^{\prime}d\omega}{(2\pi)^{5}}2J^{2}\xi^{2}\cos^{2}2\theta_{\hat{k}-\hat{k^{\prime}}}G^{>,<}(k,\omega+\Omega)G^{>,<}(k^{\prime},-\omega)\,.$

At equilibrium,
$G^{>,<}(k,\omega)=\frac{A_{G}(k,\omega)}{e^{\pm\beta\omega}+1}$ and we find,

$\begin{split}\Pi^{>}(q,\Omega)&\simeq\frac{J^{2}k_{F}\xi}{2\pi
v_{F}^{2}}\frac{\Omega}{1-e^{-\beta\Omega}}\,,\ \
\Pi^{<}(q,\Omega)\simeq\frac{J^{2}k_{F}\xi}{2\pi
v_{F}^{2}}\frac{\Omega}{e^{\beta\Omega}-1}\end{split}$ (46)

Applying (45) and (46) together on the collision integral, and dividing either
side by $\alpha\Omega$ brings us to (15a) of the main text.

We now derive the fermion quantum Boltzmann equation (15b) from (44b).
Foremost, the non-equilibrium fermion self-energy is given by,

$\begin{split}\Sigma^{>}(k,\omega)&=\int\frac{d^{2}qd^{2}q^{\prime}d\omega^{\prime}}{(2\pi)^{5}}2J^{2}\xi^{2}\cos^{2}2\theta_{k-\frac{q+q^{\prime}}{2}}F^{>}(q,\omega+\omega^{\prime})G^{<}(-k+q+q^{\prime},\omega^{\prime})\\\
&=\int\frac{d^{2}qd^{2}q^{\prime}d\omega^{\prime}}{(2\pi)^{5}}2J^{2}\xi^{2}\cos^{2}2\theta_{k-\frac{q+q^{\prime}}{2}}\big{(}b_{0}(\omega+\omega^{\prime})+1\big{)}A_{F}(q,\omega+\omega^{\prime})G^{<}(-k+q+q^{\prime},\omega^{\prime})\\\
&\simeq\int\frac{d^{2}q^{\prime}d\omega^{\prime}}{(2\pi)^{3}}\frac{2J^{2}M_{b}\xi^{2}}{\pi}\cos^{2}2\theta_{k-\frac{q+q^{\prime}}{2}}\big{(}b_{0}(\omega+\omega^{\prime})+1\big{)}\tan^{-1}\frac{\alpha(\omega+\omega^{\prime})}{\Delta_{b}}G^{<}(-k+q+q^{\prime},\omega^{\prime})\,.\end{split}$
(47)

In the second line, we have made the assumption that the bosons are in thermal
equilibrium. This amounts to neglecting “drag” effects, by which the boson
fluid is driven out of equilibrium when the fermions carry a current. Because
of the disordered nature of the interaction, fermion-boson scattering does not
conserve momentum, and hence drag effects are not likely to qualitatively
change our results.

We now perform a change of variables from $k=(k_{F}+\Delta k)\hat{k}$ to
$\hat{k}$ and $\Delta k=|k|-k_{F}$. The self-energy is mostly independent of
$\Delta k$ allowing us to write
$\Sigma_{R}(k,\omega)=\Sigma_{R}(\hat{k},\omega)$. We now define
$f(\hat{k},\omega)$ as the generalized distribution function for fermions
pointing in the $\hat{k}$ direction with frequency $\Omega$. They are given
as,

$f(\hat{k},\omega)=v_{F}\int_{\Delta k}G^{<}\big{(}(k_{F}+\Delta
k)\hat{k},\omega\big{)}\,,\ \ 1-f(\hat{k},\omega)=v_{F}\int_{\Delta
k}G^{>}\big{(}(k_{F}+\Delta k)\hat{k},\omega\big{)}$

Upon this change of variables (and integrating out $\Delta k$), (44b) becomes,

$\begin{split}e\hat{k}\cdot
Ef_{0}^{\prime}(\omega)&=\frac{16\pi}{v_{F}}\int_{|\hat{k}+\hat{k^{\prime}}|<\frac{\pi}{k_{F}\xi}}\frac{d\hat{k^{\prime}}d\omega^{\prime}}{(2\pi)^{2}}\cos^{2}2\theta_{\hat{k}-\hat{k^{\prime}}}\tan^{-1}\frac{\alpha(\omega+\omega^{\prime})}{\Delta_{b}}\\\
&\quad\Big{[}\big{(}b_{0}(\omega+\omega^{\prime})+1\big{)}f(\hat{k},\omega)f(\hat{k^{\prime}},\omega^{\prime})-b_{0}(\omega+\omega^{\prime})\big{(}1-f(\hat{k},\omega)\big{)}\big{(}1-f(\hat{k^{\prime}},\omega^{\prime})\big{)}\Big{]}\\\
&=\frac{16\pi}{v_{F}}\int_{|\hat{k}+\hat{k^{\prime}}|<\frac{\pi}{k_{F}\xi}}\frac{d\hat{k^{\prime}}d\omega^{\prime}}{(2\pi)^{2}}\cos^{2}2\theta_{\hat{k}-\hat{k^{\prime}}}B(\omega+\omega^{\prime})(1-f_{0}(\omega))(1-f_{0}(\omega^{\prime}))\big{\\{}g(\hat{k},\omega)+g(\hat{k^{\prime}},\omega^{\prime})\big{\\}}\\\
&\textrm{ where }B(\nu)=\tan^{-1}\frac{\alpha\nu}{\Delta_{b}}\
b_{0}(\nu)\end{split}$ (48)

where $f(\hat{k},\omega)=f_{0}(\omega)+\delta f(\hat{k},\omega)$, and $\delta
f(\hat{k},\omega)=f_{0}(\omega)(1-f_{0}(\omega))g(\hat{k},\omega)$. This
ultimately brings us to (15b) of the main text.

###  C.1 Potential Disorder

We now consider the effects of the following potential disorder acting on the
fermions.

$H_{rp}=V_{x}c^{\dagger}_{x}c_{x}\,,\textrm{ where }\overline{V_{x}}=0,\
\overline{V_{x}V_{x^{\prime}}}\sim W\delta_{xx^{\prime}}\,.$

This random potential results in an additional self-energy
$\Sigma^{>,<}_{rp}(\omega)=W\int_{k}G^{>,<}(k,\omega)$ on the fermions. In
turn, this extra self energy results in an additional term in the collision
integral of the fermion quantum Boltzmann equation and we have,

$\begin{split}e\hat{k}\cdot
Ef_{0}^{\prime}(\omega)&=\frac{16\pi}{v_{F}}\int_{|\hat{k}+\hat{k^{\prime}}|<\frac{\pi}{k_{F}\xi}}\frac{d\hat{k^{\prime}}d\omega^{\prime}}{(2\pi)^{2}}\cos^{2}2\theta_{\hat{k}-\hat{k^{\prime}}}B(\omega+\omega^{\prime})\big{(}1-f_{0}(\omega)\big{)}\big{(}1-f_{0}(\omega^{\prime})\big{)}\big{\\{}g(\hat{k},\omega)+g(\hat{k^{\prime}},\omega^{\prime})\big{\\}}\\\
&\quad+\frac{1}{\tau_{0}v_{F}}f_{0}(\omega)\big{(}1-f_{0}(\omega)\big{)}g(\hat{k},\omega)\end{split}$
(49)

This means that (18), the equation for $\tilde{g}$ is modified as,

$\begin{split}\frac{\cos\theta_{\hat{k}}\tilde{f}_{0}(\tilde{\omega})}{16\pi}&=\int_{|\hat{k}+\hat{k^{\prime}}|<\frac{\pi}{k_{F}\xi}}\frac{d\hat{k^{\prime}}d\tilde{\omega}^{\prime}}{(2\pi)^{2}}\cos^{2}2\theta_{\hat{k}-\hat{k^{\prime}}}\tilde{B}(\tilde{\omega}+\tilde{\omega}^{\prime})\big{(}1-\tilde{f}_{0}(\tilde{\omega}^{\prime})\big{)}\big{\\{}\tilde{g}(\hat{k},\tilde{\omega})+\tilde{g}(\hat{k^{\prime}},\tilde{\omega}^{\prime})\big{\\}}+\frac{\tilde{f_{0}}(\tilde{\omega})}{16\pi\tau_{0}T}\tilde{g}(\hat{k},\tilde{\omega})\,.\end{split}$
(50)

Solving (50) for $\tilde{g}$, and applying it to (19) we can obtain the
longitudinal conductivity as a function of $\tau T$ and $\xi$.

###  C.2 Magnetotransport

We now turn to magnetotransport within our model. We will derive the quantum
Boltzmann equation used in the main text (26) to describe transport under a
magnetic field. Furthermore, we will demonstrate that the boson
magnetotransport is negligble in comparison to that of the fermions.

The QBE (15) is modified in the following manner in the presence of a
perpendicular magnetic field $B$:

$\displaystyle
A_{F}(q,\Omega)^{2}\mathrm{Im}\\{\Pi_{R}(q,\Omega)\\}\frac{2eq\cdot
E}{M_{b}}{b_{0}}^{\prime}(\Omega)+\frac{2eq\times
B\hat{z}}{M_{b}}\cdot\nabla_{q}F^{<}=\Pi^{>}F^{<}-\Pi^{<}F^{>}\,,$ (51a)
$\displaystyle
A_{G}(k,\omega)^{2}\mathrm{Im}\\{\Sigma_{R}(k,\omega)\\}\frac{ek\cdot
E}{m_{f}}{f_{0}}^{\prime}(\omega)+\frac{ek\times
B\hat{z}}{m_{f}}\cdot\nabla_{k}G^{<}=\Sigma^{>}G^{<}-\Sigma^{<}G^{>}\,.$ (51b)

We first derive the fermion quantum Boltzmann equation of the main text (26).
As in the derivation of (48), we integrate over $\Delta k$: We find that (51b)
then becomes,

$\begin{split}&e\hat{k}\cdot
Ef_{0}^{\prime}(\omega)-\frac{\omega_{cf}}{v_{F}}f_{0}(\omega)\big{(}1-f_{0}(\omega)\big{)}\partial_{\theta_{k}}g(\hat{k},\omega)\\\
&\qquad=\frac{16\pi}{v_{F}}\int_{|\hat{k}+\hat{k^{\prime}}|<\frac{\pi}{k_{F}\xi}}\frac{d\hat{k^{\prime}}d\tilde{\omega}^{\prime}}{(2\pi)^{2}}\cos^{2}2\theta_{\hat{k}-\hat{k^{\prime}}}B(\omega+\omega^{\prime})\big{(}1-f_{0}(\omega)\big{)}\big{(}1-f_{0}(\omega^{\prime})\big{)}\big{\\{}g(\hat{k},\omega)+g(\hat{k^{\prime}},\omega^{\prime})\big{\\}}\,,\end{split}$
(52)

where $\omega_{cf}=\frac{eB}{m_{f}}$ denotes the fermion cyclotron frequency.
Rescaling frequency with temperature, we arrive at (26) of the main text and
find that the magnetic field influences the fermion conductance as a function
of $b=\frac{\omega_{cf}}{T}$.

We now turn to boson magnetotransport. There, the rotation symmetry
facilitates a simple solution to (51a). Let us switch to polar coordinates in
momentum space. We find that (51a) is given as,

$\alpha\Omega
A_{F}(|q|,\Omega)^{2}\frac{2e|q|E\cos\theta_{q}}{M_{b}}{b_{0}}^{\prime}(\Omega)+\omega_{cb}\nabla_{\theta_{q}}\delta
b(|q|,\theta_{q},\Omega)=\alpha\Omega\delta b(|q|,\theta_{q},\Omega)\,.$

Here, $\omega_{cb}=\frac{2eB}{M_{b}}$ denotes the boson cyclotron frequency,
and we have set the electric field in the $\hat{x}$ direction without loss of
generality. Performing a Fourier transform in the angular coordinate
$\theta_{q}$, we find that (51a) simplifies to,

$A_{F}(|q|,\Omega)^{2}\frac{2e|q|E}{M_{b}}b_{0}^{\prime}(\Omega)\frac{1}{2}\\{\delta_{l,1}+\delta_{l,-1}\\}=\left(1-\frac{\mathbf{i}l\omega_{cb}}{\alpha\Omega}\right)\delta
b_{l}(|q|,\Omega)$ (53)

where $\delta
b_{l}(|q|,\Omega)=\int_{\theta_{q}}e^{-\mathbf{i}l\theta_{q}}\delta
b(|q|,\theta_{q},\Omega)$ denotes the nonequilibrium boson distribution with
angular momentum $l$.

From (53), we find that $\delta b_{l}$ is zero other than for $l=\pm 1$, which
are given by,

$\delta b_{\pm
1}(q,\Omega)=\frac{eqEb_{0}^{\prime}(\Omega)}{M_{b}}\frac{A_{F}(q,\Omega)^{2}}{1\mp\frac{\mathbf{i}\omega_{cb}}{\alpha\Omega}}$

The longitudinal and Hall currents can be calculated from $\delta b_{1,-1}$
through the relation,

$\begin{split}&J_{x}=\int\frac{d^{2}qd\Omega}{(2\pi)^{3}}\frac{2eq\cos\theta}{M_{b}}\delta
b(q,\Omega)=\int\frac{d|q|d\Omega}{(2\pi)^{2}}\frac{eq^{2}}{2\pi
M_{b}}\big{(}\delta b_{1}(q,\Omega)+\delta b_{-1}(q,\Omega)\big{)}\,,\\\
&J_{y}=\int\frac{d|q|d\Omega}{(2\pi)^{2}}\frac{\mathbf{i}eq^{2}}{2\pi
M_{b}}\big{(}\delta b_{1}(q,\Omega)-\delta
b_{-1}(q,\Omega)\big{)}\,.\end{split}$

Putting this altogether, we find that the boson magnetotransport can be
computed through the following integral equations:

$\begin{split}\sigma_{b}^{xx}&=\int\frac{d|q|d\Omega}{(2\pi)^{2}}\frac{e^{2}q^{3}b_{0}^{\prime}(\Omega)A_{F}(q,\Omega)^{2}}{2\pi
M_{b}^{2}}\frac{\alpha^{2}\Omega^{2}}{\alpha^{2}\Omega^{2}+\omega_{cb}^{2}}\,,\\\
&=\frac{e^{2}}{4\pi^{3}}\int_{-\infty}^{\infty}d\tilde{\Omega}\int_{0}^{\infty}d\tilde{x}\frac{\tilde{x}\tilde{\Omega}^{2}\mathrm{csch}^{2}\frac{\tilde{\Omega}}{2}}{\big{(}\tilde{\Delta}_{b}/\alpha+\tilde{x}-\frac{\eta}{\alpha}\tilde{\Omega}\big{)}^{2}+\tilde{\Omega}^{2}}\frac{\tilde{\Omega}^{2}}{\tilde{\Omega}^{2}+\tilde{b}^{2}}\\\
\sigma_{b}^{xy}&=\int\frac{d|q|d\Omega}{(2\pi)^{2}}\frac{e^{2}q^{3}b_{0}^{\prime}(\Omega)A_{F}(q,\Omega)^{2}}{2\pi
M_{b}^{2}}\frac{\alpha\Omega\omega_{cb}}{\alpha^{2}\Omega^{2}+\omega_{cb}^{2}}\\\
&=\frac{e^{2}}{4\pi^{3}}\int_{-\infty}^{\infty}d\tilde{\Omega}\int_{0}^{\infty}d\tilde{x}\frac{\tilde{x}\tilde{\Omega}^{2}\mathrm{csch}^{2}\frac{\tilde{\Omega}}{2}}{\big{(}\tilde{\Delta}_{b}/\alpha+\tilde{x}-\frac{\eta}{\alpha}\tilde{\Omega}\big{)}^{2}+\tilde{\Omega}^{2}}\frac{\tilde{\Omega}\tilde{b}}{\tilde{\Omega}^{2}+\tilde{b}^{2}}\end{split}$
(54)

We see from (54) that the boson conductance is a function of
$\tilde{\Delta}_{b}/\alpha$, $\eta/\alpha$, and
$\tilde{b}=\frac{\omega_{cb}}{\alpha T}$. Here, $\tilde{\Delta}_{b}/\alpha$
and $\eta/\alpha\sim(k_{F}\xi)^{-1}$ are constants independent of temperature.
Therefore, the dimensionless ratio
$\tilde{b}\sim(k_{F}\xi)\frac{\omega_{cf}}{T}$ determines the boson
magnetotransport.

Figure 7: $\sigma_{b}^{xx},\ \sigma_{b}^{xy}$ as a function of
$b=\omega_{cf}/T$, plotted in log-log scale for $\tilde{\Delta}_{b}=1$,
$\alpha=10$, $\eta=1$ which roughly corresponds to $k_{F}\xi\sim 10$. Inset of
the $\sigma_{b}^{xx}$ plot shows
$|\Delta\sigma_{b}^{xx}|=\sigma_{b}^{xx}(0,T)-\sigma_{b}^{xx}(B,T)$ as a
function of $b$ in log-log scale. Dashed lines in both the left inset and
right figures are of slope 1.

In Fig.7 we display the magnetic field dependence of $\sigma_{b}^{xx,xy}$
obtained by numerically integrating (54). For direct comparison with fermion
magnetotransport, we plot the x axis as a function of
$b=\frac{\omega_{cf}}{T}$. We see that the change in the longitudinal boson
conductivity $|\Delta\sigma_{b}^{xx}|$ scales linearly with
$\frac{\omega_{cf}}{T}$. At low temperatures, this change is minuscule in
comparison to that of the fermions. In the linear magnetoresistance regime of
$b\in[(k_{F}\xi)^{-4},(k_{F}\xi)^{-3}]$, recall that
$\sigma_{f}^{xx}\sim\epsilon_{F}/\omega_{cf}=\epsilon_{F}/Tb^{-1}$. For
typical temperatures $\epsilon_{F}/T$ is a large number, and so is $b^{-1}$ in
the LMR regime. Therefore, the change in the boson magnetoresistance that
scales linearly with $b$ is completely negligible.

##  Appendix D Diamagnetic Susceptibility

In order to compute the diamagnetic susceptibility, we take the current-
current correlator $\braket{J\cdot J}$ and set $\Omega=0$ and take the limit
of $q\rightarrow 0$. The diamagnetic susceptibility is given as,

$\chi=-\lim_{q\rightarrow 0}\partial_{q}^{2}\braket{J_{x}\cdot
J_{x}}(q\hat{y},\omega=0)$

Let us for simplicity assume a quadratic dispersion for the fermions,
$\epsilon_{k}=\frac{k^{2}}{2m_{f}}$. Then we find,

$\begin{split}\chi_{f}&=\frac{e^{2}}{(2\pi)^{2}\beta
m_{f}^{2}}\partial_{q}^{2}\sum_{\omega_{n}}\int\frac{d^{2}k}{(2\pi)^{2}}k^{2}\cos^{2}{\theta}\
G_{k+q\hat{y}/2}(\omega_{n})G_{k-q\hat{y}/2}(\omega_{n})\\\
&=-\frac{e^{2}}{(2\pi)^{2}\beta
m_{f}^{2}}\sum_{\omega_{n}}\int\frac{d^{2}k}{(2\pi)^{2}}\frac{\frac{k^{2}}{2m_{f}}\cos^{2}{\theta}\Big{(}\frac{k^{2}}{2m_{f}}(\cos^{2}\theta-\sin^{2}\theta)-\mu+\mathbf{i}\omega_{n}+\Sigma(k,\omega_{n})\Big{)}}{\Big{(}\frac{k^{2}}{2m_{f}}-\mu+\mathbf{i}\omega_{n}+\Sigma(k,\omega_{n})\Big{)}^{4}}\\\
&=-\frac{e^{2}}{8\pi\beta
m_{f}}\sum_{\omega_{n}}\int\frac{d\epsilon_{k}}{2\pi}\frac{\epsilon_{k}^{2}+2\epsilon_{k}\big{(}-\mathbf{i}\omega-\mu-\Sigma(k,\omega)\big{)}}{\big{(}-\mathbf{i}\omega_{n}-\Sigma(k,\omega_{n})-(\epsilon_{k}-\mu)\big{)}^{4}}\\\
&=-\frac{e^{2}}{24\pi
m_{f}\beta}\sum_{\omega_{n}}\frac{1}{\mathbf{i}\omega_{n}+\Sigma-\mu}\\\
&\simeq\frac{e^{2}}{24\pi m_{f}}\end{split}$ (55)

In the third line we used the fact that the self-energy $\Sigma\in
T[(k_{F}\xi)^{-3},(k_{F}\xi)^{-1}]$ and is small. Consequently, we find that
the low temperature diamagnetic response of fermions $\chi_{f}$ is similar to
the Landau diamagnetism of free fermions. Similar results have been seen in
Aldape  _et al._ (2022).

We now calculate the diamagnetic response due to Cooper pairs. Evaluating the
boson current-current correlator, we find,

$\begin{split}\chi_{b}&=-\frac{4e^{2}}{(2\pi)^{2}\beta
M_{b}^{2}}\partial_{q}^{2}\sum_{\omega_{n}}\int\frac{d^{2}k}{(2\pi)^{2}}k^{2}\cos^{2}{\theta}\
F_{k+q\hat{y}/2}(\omega_{n})F_{k-q\hat{y}/2}(\omega_{n})\\\
&=\frac{e^{2}}{6\pi\beta
M_{b}}\sum_{\omega_{n}}\frac{1}{\Delta_{b}+\alpha|\omega_{n}|}\\\ \end{split}$
(56)

which brings us to $\chi_{b}$ in (30).