# Vortex Fermi Liquid and Strongly Correlated Quantum Bad Metal

Nayan Myerson-Jain  Department of Physics, University of California, Santa
Barbara, CA 93106  Chao-Ming Jian  Department of Physics, Cornell University,
Ithaca, New York 14853, USA  Cenke Xu  Department of Physics, University of
California, Santa Barbara, CA 93106

###### Abstract

The semiclassical description of two-dimensional ($2d$) metals based on the
quasiparticle picture suggests that there is a universal threshold of the
resistivity: the resistivity of a $2d$ metal is bounded by the so called Mott-
Ioffe-Regal (MIR) limit, which is at the order of $h/e^{2}$. If a system
remains metallic while its resistivity is beyond the MIR limit, it is referred
to as a “bad metal”, which challenges our theoretical understanding as the
very notion of quasiparticles is invalidated. The description of the system
becomes even more challenging when there is also strong correlation between
the electrons. Partly motivated by the recent experiment on transition metal
dichalcogenides moiré heterostructure, we seek for understanding of strongly
correlated bad metals whose resistivity far exceeds the MIR limit. For some
strongly correlated bad metals, though a microscopic description based on
electron quasiparticles fails, a tractable dual description based on the
“vortex of charge” is still possible. We construct a concrete example of such
strongly correlated bad metals where vortices are fermions with a Fermi
surface, and we demonstrate that its resistivity can be exceptionally large at
zero temperature. And when extra charge $\delta n_{e}$ is doped into the
system away from half-filling, a small Drude weight proportional to $(\delta
n_{e})^{2}$ will emerge in the optical conductivity .

##  I Introduction

The most rudimentary description of a metal relies on the notion of
quasiparticles, $i.e.$ an electron near the Fermi surface can be well
approximated as a wave packet between two consecutive elastic scatterings with
impurities. This picture requires that $l_{\mathrm{m}}k_{F}>1$, where
$l_{\mathrm{m}}$ is the mean free path of the electrons from scattering with
the impurities Emery and Kivelson (1995). When $l_{\mathrm{m}}k_{F}\sim 1$,
the resistivity of a two dimensional system is of the order of $h/e^{2}$,
which is also known as the Mott-Ioffe-Regal (MIR) limit. The common wisdom is
that, for noninteracting electrons, when the resistivity of a $2d$ metal
exceeds the MIR limit, not only would the rudimentary description of the
system fail, the system would actually become an insulator due to the Anderson
localization. The potential metal-insulator transition (MIT) of a
noninteracting $2d$ electron system Evers and Mirlin (2008) (within certain
symmetry class such as the symplectic) should happen when the resistivity is
of order $h/e^{2}$.

In strongly interacting electron systems, the universal threshold of
resistivity $h/e^{2}$ still appears to hold. For electrons at half-filling (on
average one electron per site) on a lattice, the competition between the
interaction and kinetic energy can lead to an interaction-driven MIT between a
metal and a Mott insulator phase. When the insulator is a particular type of
spin liquid phase, this MIT can be understood through a parton construction
Lee and Lee (2005); Senthil (2008), and the total resistivity follows the
Ioffe-Larkin rule Ioffe and Larkin (1989) $\rho=\rho_{b}+\rho_{f}$, where
$\rho_{b}$ and $\rho_{f}$ are resistivity from the bosonic and fermionic
partons respectively. $\rho_{f}$ is a smooth function across the MIT, and at
low temperature $\rho_{f}$ mostly arises from disorder, which is expected to
be small if we assume weak disorder. Hence at the MIT, the critical
resistivity is mostly dominated by the bosonic parton $\rho_{b}$. The critical
resistivity $\rho_{b}$ is expected to be of order $h/e^{2}$ (though in the DC
limit $\rho_{b}$ may acquire an extra factor of $7\sim 8$, based on analytical
evaluation in certain theoretical limit Witczak-Krempa et al. (2012)) 111Here
we also note that a weak disorder, or Umklapp process is needed to remove the
logarithmic divergence of conductivity due to thermal fluctuation, which is
predicted by hydrodynamics Delacrétaz (2020); Kovtun (2012)..

Hence in most noninteracting as well as strongly interacting systems that we
have understood, the resistivity of a $2d$ metallic system should be roughly
bounded by the MIR limit. Hence if a $2d$ system remains metallic while its
resistivity far exceeds the MIR limit, it challenges our theoretical
understanding. These exotic metals are referred to as “bad metals” Emery and
Kivelson (1995). The recent experiment on transition metal dichalcogenides
(TMD) revealed the existence of a novel interaction-driven MIT Li et al.
(2021), where the universal MIR limit is violated: the DC critical resistivity
at the MIT exceeds the MIR limit by nearly two orders of magnitude. The system
is supposedly modelled by an extended Hubbard model of spin-1/2 electrons on a
triangular moiré lattice Wu et al. (2018); Tang et al. (2019), but the
experimental finding is qualitatively beyond the previous theory of MIT. A few
recent theoretical proposals Xu et al. (2022); Kim et al. (2022) were made in
order to understand this exotic MIT. The experiment mentioned above only
revealed a critical point whose resistivity is clearly beyond the MIR limit.
Given the current experimental finding and the strongly interacting nature of
the system, it is natural to ask, can there also be a stable bad metal phase
of strongly correlated electrons, whose properties can be evaluated in a
tractable way?

In this work we discuss the construction of a quantum bad metal state with
longitudinal transport only; the electrical resistivity $\rho_{e}$ of the
state can far exceed the MIR limit even with weak disorder, at zero and low
temperature. It is worth noting that the phenomenology of the state we
construct is different from the original example of “bad metal” (hole doped
cuprates) discussed in Ref. Emery and Kivelson, 1995, where the resistivity
increases with temperature monotonically and exceeds the MIR limit at high
temperature; while the resistivity of our “quantum bad metal” remains finite
and large at zero temperature, and clearly violates the MIR bound. Our
construction is formulated through the dual degrees of freedom of “charge
vortex”. The particle-vortex duality has a long history Peskin (1978);
Dasgupta and Halperin (1981); Fisher and Lee (1989). This duality was
originally discussed for bosons, but recent developments have generalized the
duality to fermion-vortex duality Son (2015); Wang and Senthil (2015a);
Metlitski and Vishwanath (2016); Mross et al. (2016); Kachru et al. (2017), as
well as Chern-Simons matter theory to free Dirac or Majorana fermion duality
Aharony (2016); Seiberg et al. (2016); Hsin and Seiberg (2016); Aharony et al.
(2017); Metlitski et al. (2017); Chen et al. (2018); Son et al. (2019); Jian
et al. (2019). And since the particle-vortex duality is still a “strong-weak”
duality, when the charges are strongly correlated which invalidates a
perturbative description based on quasiparticles, the vortices are weakly
interacting through the dual gauge field, which facilitates a rudimentary
description.

Hence one way to construct a quantum bad metal for charges is to drive the
vortices into a good metal. The vortices can naturally form a good metal as
long as (1) the vortex is a fermion, and (2) the fermionic vortices form a
Fermi surface with a finite density of states. In the next section we will
discuss how exactly a vortex becomes a fermion in our construction, and how to
derive the charge responses of the system from the physics of vortices. We
would like to clarify that we are not the first to investigate correlated
electrons as vortex liquid. Besides the more well-known interpretation of
composite fermions as “vortex liquid” in the context of half-filled Landau
level (and similarly for charged bosons at filling 1) Halperin et al. (1993);
Read (1998); Son (2015); Wang and Senthil (2016a, b); a metallic phase with
anomalously large conductivity that emerges in amorphous thin film also
motivated discussions of exotic physics of superconductor vortices Galitski et
al. (2005); Wu and Phillips (2006); Kapitulnik et al. (2019). We will compare
our construction with the previous works.

##  II construction of the quantum bad metal

###  II.1 General considerations

Before we detail our construction, some general considerations can already be
made.

(1) As was pointed out in previous literatures, at least for charged bosons,
the product of the conductivity of the charges and the conductivity of the
vortices is a constant Fisher et al. (1990); Gazit (2015), $i.e.$
$\sigma_{e}\sim 1/\sigma_{v}$. If this relation still (at least approximately)
holds in our construction, it implies that if the vortex conductivity
$\sigma_{v}$ follows the standard behavior of a good metal at finite
temperature, then the resistivity $\rho_{e}(T)$ of charge should decrease with
$T$, at least below certain characteristic energy scale.

(2) A charge vortex can generally be viewed as a point defect with circulating
vorticity of charge current. A charge vortex must become an anti-vortex under
spatial reflection $\mathcal{P}$. This is because the electric current
circulation will reverse its orientation under reflection. If the vortices
form a Fermi surface, in general it would break $\mathcal{P}$, as a Fermi
surface usually is not invariant under the particle-hole transformation. The
same observation can be made for time-reversal $\mathcal{T}$: since charge
density is invariant under $\mathcal{T}$, time-reversal would reverse the
direction of electric current circulation. We will discuss later how to
preserve $\mathcal{P}$ and $\mathcal{T}$ in our construction, by enforcing
certain particle-hole symmetry of the fermionic vortices.

(3) As was pointed out in Ref. Wang and Senthil, 2016a, b, the Wiedemann-Franz
law $\kappa\sim T\sigma$ should generally be strongly violated in a vortex
liquid, as the vortices carry entropy, but no charge. In the state we
construct this is still true, the modified Wiedmann-Franz law should be
$\kappa\sim LT\sigma_{v}\sim LT\rho_{e}$. The Lorenz number $L$ is about
$\kappa/(T\sigma_{e})\sim\rho_{e}^{2}$, which can be exceedingly larger than
an ordinary metal. Here we remind the readers that our state has longitudinal
transport only, and it can have both large resistivity and thermal
conductivity.

(4) For a strongly interacting electron system, the relaxation of the electric
current is pretty much independent from the relaxation of a single particle.
Hence the physics of a strongly interacting electron liquid may be only
captured by some hydrodynamical description without microscopic particles
Delacrétaz (2020); Kovtun (2012); Hartnoll (2014); Hartnoll and Mackenzie
(2021); Lucas and Fong (2018); Hartnoll et al. (2016), as hydrodynamics is
defined at a much larger length scale. But since the particle-vortex duality
is a strong-weak duality, the interaction between electron density becomes

$\displaystyle\sum_{i,j}V_{i,j}n_{i}n_{j}\sim\int d^{2}x\
\frac{1}{g}(\vec{\nabla}\times\vec{a})^{2}$ (1)

in the dual picture, where $g$ can be viewed as the gauge coupling of the
gauge charges (vortices), and also the charge compressibility $\kappa_{e}$.
The stronger the charge interaction is, the weaker is the bare gauge coupling
of the vortices. The common “patch theory” for analyzing the RG flow of a
Fermi surface coupled with a U(1) gauge field predicts that the gauge coupling
would flow to a strongly coupled fixed point eventually Nayak and Wilczek
(1994a, b); Lee (2009); Metlitski and Sachdev (2010); Mross et al. (2010). But
this patch theory breaks down when there is disorder, as disorder would mix
different patches in the momentum space. But at least when the bare gauge
coupling $g$ is weak enough (which corresponds to a strong charge density-
density interaction), there should be a sufficient window for the gauge
coupling to be viewed as a perturbation, and the momentum of the vortices can
be transferred to the photons, and then relax through disorder before “feeding
back” to the vortices. Hence in this sense we can view the dual vortex system
as an approximate vortex Fermi liquid.

Figure 1: The resistivity $\rho(T)$ as a function of temperature at low
temperature, computed using the composition formula Eq. 17. We have chosen
$\sigma_{b}=\exp(-\Delta_{b}/T)$ where $\Delta_{b}=1$, and $\sigma_{f}\sim
200$. $\rho$ is measured in unit of $h/e^{2}$ in this plot. We can also give
$\sigma_{f}$ a Fermi-liquid like temperature dependence, the plot remains
qualitatively unchanged.

###  II.2 Quantum Bad metal at half-filling on a lattice

The system we begin with is a strongly interacting electron system with half-
filling (one electron per site on average) on a lattice, later we will discuss
what happens when the system is doped away from half-filling. We start with
the standard “$\mathrm{SU}(2)$ slave rotor” theory for the electron operator
Affleck et al. (1988); Dagotto et al. (1988); Wen and Lee (1996); Lee et al.
(1998):

$\displaystyle c_{\uparrow}$ $\displaystyle=$ $\displaystyle
f_{\uparrow}z_{1}-f^{\dagger}_{\downarrow}z^{\dagger}_{2},$ (2) $\displaystyle
c_{\downarrow}$ $\displaystyle=$ $\displaystyle
f_{\downarrow}z_{1}+f^{\dagger}_{\uparrow}z^{\dagger}_{2}.$ (4)

Here $(f_{\uparrow},f_{\downarrow})$ is a fermionic spinon doublet (fermionic
partons) that carries spin-1/2, and $(z_{1},z_{2})$ are slave rotors (bosonic
partons) carrying the electric charge. This formalism can maximally host a
SU(2) charge transformation and a SU(2) gauge transformation, and both
transformations can be made explicit by rewriting Eq. 4 in a matrix form (see
$e.g.$ Ref. Hermele, 2007; Ran et al., 2008 and references therein) 222An even
more complete parton construction can accommodate an SO(4) gauge symmetry Xu
and Sachdev (2010). But for our purpose it suffices to assume all SU(2)
transformations, including the spin symmetry are broken down to U(1). In fact,
on a lattice with frustration, both the charge SU(2) and the gauge SU(2)
transformations are broken down to U(1) by the most natural mean field states
of $z_{\alpha}$ and $f_{\alpha}$. The assignment of the electric charge
symmetry $\mathrm{U}(1)_{e}$, gauge symmetry $\mathrm{U}(1)_{g}$ and spin
symmetry $\mathrm{U}(1)_{s}$ on the partons is

$\displaystyle\mathrm{U}(1)_{e}$ $\displaystyle:$ $\displaystyle
z_{1}\rightarrow e^{\mathrm{i}\theta_{e}}z_{1},\ \ \ z_{2}\rightarrow
e^{-\mathrm{i}\theta_{e}}z_{2},\ \ \ f_{\alpha}\rightarrow f_{\alpha};$ (5)
$\displaystyle\mathrm{U}(1)_{g}$ $\displaystyle:$ $\displaystyle
z_{1}\rightarrow e^{\mathrm{i}\theta_{g}}z_{1},\ \ \ z_{2}\rightarrow
e^{\mathrm{i}\theta_{g}}z_{2},\ \ \ f_{\alpha}\rightarrow
e^{-\mathrm{i}\theta_{g}}f_{\alpha},$ (7) $\displaystyle\mathrm{U}(1)_{s}$
$\displaystyle:$ $\displaystyle z_{\alpha}\rightarrow z_{\alpha},\ \ \
f_{\uparrow}\rightarrow e^{\mathrm{i}\theta_{s}}f_{\uparrow},\ \ \
f_{\downarrow}\rightarrow e^{-\mathrm{i}\theta_{s}}f_{\downarrow}.$ (9)

The system being at half-filling implies that the total rotor number of
$z_{1}$ and $z_{2}$ are equal: $\sum_{i}n_{1,i}=\sum_{i}n_{2,i}$. There is a
dynamical $\mathrm{U}(1)_{g}$ gauge field $a_{\mu}$ that couples to both
$z_{\alpha}$ and $f_{\alpha}$. The U(1) gauge constraint demands that on every
site $i$,
$\sum_{\alpha}n_{\alpha,i}+1=\sum_{\alpha}f^{\dagger}_{\alpha,i}f_{\alpha,i}$.
A detailed discussion of the physical meaning of the partons introduced can be
found in Ref. Ran et al., 2008.

This slave rotor parton construction allows us to construct many states of the
strongly interacting electron system which are difficult to visualize using
free or weakly interacting electrons. For example, if the bosonic partons
$z_{\alpha}$ are in a trivial bosonic Mott insulator state with
$n_{1,i}=n_{2,i}=0$ (meaning
$\sum_{\alpha}f^{\dagger}_{\alpha,i}f_{\alpha,i}=1$ on each site), the system
becomes a Mott insulator of electrons with a charge gap, and the spin physics
of the Mott insulator depends on the state of $f_{\alpha}$. Various spin
liquid states can be designed and classified depending on the mean field band
structure of $f_{\alpha}$ Wen (2002).

The many-body state of electrons is determined by the states of the partons.
In the last decade the study of symmetry protected topological states (SPT)
significantly broadened our understanding of the states of matter Chen et al.
(2013, 2012), which also allows us to construct even more novel states of
electrons using the partons. We first use $z_{\alpha}$ to define two other
composite bosonic fields: $\phi_{e}=z^{\dagger}_{1}z_{2}$,
$\phi_{g}=z_{1}z_{2}$. $\phi_{e}$ and $\phi_{g}$ carry charge $(2,0)$ and
$(0,2)$ under $\left(\mathrm{U}(1)_{e},\mathrm{U}(1)_{g}\right)$. Then we
drive the composite fields $\phi_{e}$ and $\phi_{g}$ into a bosonic SPT state
with $\mathrm{U}(1)_{e}$ and $\mathrm{U}(1)_{g}$ symmetries, which is the bSPT
state for two flavors of bosons constructed in Ref. Levin and Senthil, 2004.
The physics of this bSPT state is analogous to the quantum spin Hall
insulator: the vortex of $\phi_{e}$ carries charge of $\phi_{g}$, and vice
versa. If we follow the Chern-Simons description of the bSPT Lu and Vishwanath
(2012), this state is

$\displaystyle\mathcal{L}_{\mathrm{bSPT}}=\frac{\mathrm{i}K^{IJ}}{2\pi}\tilde{a}_{I}\wedge
d\tilde{a}_{J}+\frac{\mathrm{i}2}{2\pi}\tilde{a}_{1}\wedge
da+\frac{\mathrm{i}2}{2\pi}\tilde{a}_{2}\wedge dA^{e},$ (10)

where $K^{IJ}$ takes the same form as the Pauli matrix $\sigma^{x}$. Here
$\ast d\tilde{a}_{1}$ and $\ast d\tilde{a}_{2}$ are the dual of the currents
of $\phi_{g}$ and $\phi_{e}$ respectively. This bSPT state of the rotors also
has a particle-hole symmetry of the bosonic rotors, and in this bSPT state the
expectation value of the total rotor number of both $z_{1}$ and $z_{2}$ is
zero. Please note that the total rotor number being zero does not imply a
trivial vacuum state, as the rotor number (just like a spin $S^{z}$ operator)
can take both positive and negative values.

A bSPT state is gapped, and also nondegenerate, hence it is safe to integrate
out the bosonic degree of freedom, and obtain the response to the gauge
fields. After integrating out $\tilde{a}_{1,2}$ from Eq. 10, we obtain:

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{F}(f_{\alpha},a_{\mu})+\frac{4\mathrm{i}}{2\pi}a\wedge
dA^{e}+\frac{1}{g}(\vec{\nabla}\times\vec{a})^{2}+\cdots.$ (11)

One can also introduce the external gauge field for the spin symmetry
$\mathrm{U}(1)_{s}$, but it won’t have a nontrivial response from the bSPT
state.

The mutual Chern-Simons term in the last term of Eq. 11 fundamentally changes
the physics of the system in the following way:

The electric charge current, which is defined as
$J^{e}=\delta\mathcal{L}/\delta A_{e}$, is identified as

$\displaystyle J^{e}=e\frac{4}{2\pi}\ast da,$ (12)

meaning the flux of $a_{\mu}$ now carries electric charge $4e$. Hence the bSPT
state so constructed turns the gauge field $a_{\mu}$ into the dual of the
charge current in the sense of the particle-vortex duality Peskin (1978);
Dasgupta and Halperin (1981); Fisher and Lee (1989), and turns the gauge
charge of $a_{\mu}$ into the charge vortex. The fermionic parton $f_{\alpha}$,
which carries gauge charge $1$ under $a_{\mu}$, now automatically becomes the
vortex of the electric charge, as when a charge (now the flux of $a_{\mu}$)
circulates the gauge charged $f_{\alpha}$, it would accumulate a Berry’s
phase. It is worth noting that, for more general bSPT states of $z_{\alpha}$
with only mutual $\mathrm{U}(1)_{e}\times\mathrm{U}(1)_{g}$ Chern-Simons
response, the electric charge carried by the flux of $a_{\mu}$ has to be an
integer multiple of $4e$. Hence, the bSPT described above is the minimal non-
trivial choice.

Now we take the long wavelength limit, and integrate out both $f_{\alpha}$ and
$a_{\mu}$; we also choose the temporal gauge with $a_{\tau}=0$. The response
Lagrangian in terms of $A^{e}$ is

$\displaystyle\mathcal{L}_{\mathrm{res}}=\sum_{\omega}\frac{1}{2}\frac{4}{\pi^{2}}\frac{\omega^{2}}{\Pi_{f}(\omega)}|\vec{A}^{e,t}(\omega)|^{2}.$
(13)

$\vec{A}^{e,t}$ is the transverse component of $\vec{A}^{e}$.
$\Pi_{f}(\omega)$ is the polarization of $f_{\alpha}$, and it should be
proportional to $\mathrm{i}\omega\sigma_{f}(\omega)$ after analytic
continuation Lee and Nagaosa (1992), where $\sigma_{f}(\omega)$ is the
conductivity of the fermionic parton ($i.e.$ also the vortex) $f_{\alpha}$.
This implies that the electrical resistivity of the system should be

$\displaystyle\rho_{e}(\omega)=\frac{\pi^{2}}{4}\sigma_{f}(\omega)=\frac{\pi^{2}}{4}\left(\frac{\sigma_{0}}{1-\mathrm{i}\omega\tau_{v}}\right).$
(14)

We note that here $\rho_{e}$ is measured with unit $\hbar/e^{2}$; $\sigma_{f}$
is computed in the convention that $f_{\alpha}$ carries charge 1. Here we can
evaluate the conductivity of the good metal of $f_{\alpha}$ using the
rudimentary Drude formula. $\sigma_{0}$ is the conductivity of the
$f_{\alpha}$ in the DC limit. It was shown recently that even when a Fermi
surface is coupled to a dynamical gauge field, the response to the gauge field
is still exactly the same as what is computed by the Drude theory (at least
when there is no disorder) Shi et al. (2022). We also exploit the fact that,
when the electron density has a strong interaction, the bare gauge coupling
becomes weak, and the photon-vortex interaction remains perturbative at least
within a large window of scale. An analysis of the fermions interacting with
gauge field in a disordered environment can be found in Ref. Kumar et al.,
2022.

In Eq.14, $\sigma_{0}$ can be rather large, namely the vortices form a good
metal, when there is a finite Fermi surface of $f_{\alpha}$ and the disorder
is weak. In this case the electrical resistivity of the system can be far
beyond the MIR limit, $i.e.$ the system is a very bad metal.

Now we investigate the spatial reflection symmetry $\mathcal{P}$ of our
system. And let us use $\mathcal{P}_{y}:y\rightarrow-y$ as an example. We
assume that $c_{\alpha}$ changes up to a sign under $\mathcal{P}_{y}$, then
this leads to the transformation of $f_{\alpha}$, $z_{\alpha}$:

$\displaystyle\mathcal{P}_{y}:~{}~{}f_{\alpha}\rightarrow(\sigma^{x})_{\alpha\beta}f_{\alpha}^{\dagger},\
\ \ z_{\alpha}\rightarrow(\sigma^{x})_{\alpha\beta}z^{\dagger}_{\beta}.$ (15)

The $\mathrm{U}(1)_{e}$ and $\mathrm{U}(1)_{g}$ charges are even and odd under
$\mathcal{P}_{y}$ respectively. This means that $\tilde{a}_{1,2}$ transform
oppositely under $\mathcal{P}$, and the bSPT state preserves $\mathcal{P}$
based on Eq. 10. In order to ensure the reflection symmetry, we also need the
band structure of $f_{\alpha}$ to satisfy
$\varepsilon_{\uparrow}(k_{x},k_{y})=-\varepsilon_{\downarrow}(k_{x},-k_{y})$.

Our construction also preserves a (special) time-reversal symmetry
$\mathcal{T}$ defined as following: the electron operator is still invariant
under $\mathcal{T}$ (up to an extra sign). We can choose the following
transformations of $z_{\alpha}$ and $f_{\alpha}$ to ensure the desired
transformation of the electrons:

$\displaystyle\mathcal{T}:~{}~{}z_{\alpha}\rightarrow(\sigma^{x})_{\alpha\beta}z^{\dagger}_{\beta},\
\ \ f_{\alpha}\rightarrow(\sigma^{x})_{\alpha\beta}f^{\dagger}_{\beta}.$ (16)

As we can see, the $\mathrm{U}(1)_{e}$ and $\mathrm{U}(1)_{g}$ charges are
again even and odd under $\mathcal{T}$ respectively. This means that
$\tilde{a}_{1,2}$ transform oppositely under $\mathcal{T}$, and the bSPT state
preserves $\mathcal{T}$ based on Eq. 10. In order to ensure the time-reversal
symmetry, we also need the band structure of $f_{\alpha}$ to satisfy
$\varepsilon_{\uparrow}(\vec{k})=-\varepsilon_{\downarrow}(-\vec{k})$. More
precisely, here the time-reversal is a product between the particle-hole
transformation and a spatial-inversion 333Without the spatial-inversion
transformation, the time-reversal transformation alone would demand
$\varepsilon_{\uparrow}(\vec{k})=-\varepsilon_{\downarrow}(\vec{k})$, which
would make the Fermi pockets of the two spinon bands overlap with each other,
and lead to a potential exciton instablity due to nesting. . A more realistic
time-reversal symmetry for electrons with $\mathcal{T}^{2}=-1$ can be defined
and preserved if we introduce another orbital flavor to the electrons.

So far we have ignored the conductivity of the bosonic partons, which is valid
when the energy scale is much smaller than the gap of the bosons $\Delta_{b}$.
With finite frequency $\omega$, the bosons will also make two new nonzero
contributions to the longitudinal response of $A^{e}$. The first of which is a
simple addition to the response Lagrangian of the boson polarization $i.e.$
$\mathcal{L}^{b}_{\mathrm{res}}\sim\frac{1}{2}\Pi_{b}(\omega)|\vec{A}^{e,t}(\omega)|^{2}$,
for which $\Pi_{b}(\omega)\rightarrow\mathrm{i}\omega\sigma_{b}(\omega)$ after
analytic continuation. This only modifies the charge conductivity by shifting
it a value of $\sigma_{b}(\omega)$. The second, more interesting contribution,
is that a longitudinal term will be generated for the internal gauge field
$a_{\mu}$ as well. In principle $\Pi_{b}(\omega)$ can be different for
$A^{e}_{\mu}$ and $a_{\mu}$, but without loss of generality we assume that
they are the same. Since the internal gauge field and electromagnetic gauge
field transform differently under $\mathcal{P}$ and $\mathcal{T}$ and so do
the $\phi_{g}$ and $\phi_{e}$ current, there cannot be mixed terms between $a$
and $A^{e}$ that would lead to mutual longitudinal response in the effective
Lagrangian. Then eventually the electrical conductivity of the system follows
the following composition rule

$\displaystyle\sigma_{e}=\sigma_{b}+\frac{4}{\pi^{2}}\left(\frac{1}{\sigma_{f}+\sigma_{b}}\right).$
(17)

In this equation, $\sigma_{e}$ is measured in the unit of $e^{2}/\hbar$;
$\sigma_{b}$ and $\sigma_{f}$ are computed in the convention of charge $1$ and
$\hbar=1$. This composition is very different from the Ioffe-Larkin rule Ioffe
and Larkin (1989), and we expect this composition rule to be valid for
temperature far smaller than the gap of the bosonic parton, $i.e.$ when the
thermally activated bosonic partons are very dilute. In Fig. 1 we plot
$\rho_{e}(T)=1/\sigma_{e}(T)$ measured in unit of $h/e^{2}$ using the
composition rule Eq. 17.

Another quantity of interest is the charge compressibility. The
compressibility can be computed from the charge density-density correlation
function which can be attained by reading off the $A_{\tau}(q)A_{\tau}(-q)$
term in $\mathcal{L}_{\text{res}}$ at vanishing frequency after integrating
out all the matter and the internal gauge fields. As we discussed before, one
contribution to the compressibility is proportional to the gauge coupling $g$,
which is evident after integrating out $\vec{a}$ from Eq. 11. Eventually the
compressibility should involve the bare charge density-density interaction,
after being renormalized by integrating out the fermions $f_{\alpha}$. In the
limit $\omega\rightarrow 0$ and large gap of bosonic rotors, the total
compressibility is given by

$\displaystyle\kappa_{e}(\mathbf{q})=\frac{1}{(\kappa_{0})^{-1}+\frac{\pi^{2}}{4}\frac{\Pi_{f}(q)}{|\mathbf{q}|^{2}}},$
(18)

where $\kappa_{0}$ is the “bare” compressibility of the system when the gauge
field $a_{\mu}$ is not coupled to any matter fields, hence $\kappa_{0}$ is
proportional to the gauge coupling $g$: $\kappa_{0}=g(4/\pi^{2})$. If we
choose a simple quadratic dispersion or a circular Fermi surface for the
vortices, then the result for the fermionic vortex polarization at zero
frequency is well-known to be Nave et al. (2007); Lee and Nagaosa (1992)
$\Pi_{f}(\mathbf{q})\sim|\mathbf{q}|^{2}/(12\pi m_{v})$ where $m_{v}$ is the
effective mass of the fermionic vortices. Like the charge conductivity, this
gives us a composition rule for the charge compressibility in terms of linear
response functions for the different species of partons. For the more general
case of a finite boson gap, this composition rule involves both the boson
compressibility and boson polarization $\Pi_{b}\sim\chi_{b}|\mathbf{q}|^{2}$:

$\displaystyle\kappa_{e}(\mathbf{q})=\kappa_{b}(\mathbf{q})+\frac{1}{(\kappa_{0})^{-1}+(\chi_{b}+\chi_{f})\pi^{2}/4}.$
(19)

Here, $\chi_{b}$ is the magnetic susceptibility of the bosons. We expect this
composition rule to hold at small finite temperature much below the boson gap.

###  II.3 Physics at weak doping

Figure 2: The optical conductivity $\mathrm{Re}[\sigma(\omega)]$ near
$\omega=0$ at half-filling, and with small doping $\delta n_{e}$.

We would also like to consider the effects of weak charge doping $\delta
n_{e}$ away from one electron per site. As the charge density is now bound
with the internal gauge flux through the mutual CS term in Eq. 11, weakly
doping the system corresponds to adding a background $\mathrm{U}(1)_{g}$ gauge
field with small average magnetic flux. Since the vortices (spinon
$f_{\alpha}$) form a good metal, their conductivity may still be evaluated
through the semiclassical Drude theory. Using the semiclassical equation of
motion with magnetic field, it is straight forward to compute the Drude
conductivity of $f_{\alpha}$:

$\displaystyle\sigma_{f}(\omega)=\bigg{(}\frac{\frac{1}{\tau_{v}}-i\omega}{\omega_{c}^{2}+(\frac{1}{\tau_{v}}-i\omega)^{2}}\bigg{)}\frac{1}{\tau_{v}}\sigma_{0}.$
(20)

$\omega_{c}$ is the cyclotron frequency of the vortices
$\omega_{c}=\bar{b}/m_{v}$, and $\bar{b}\sim\delta n_{e}$ is the average flux
seen by the fermionic vortices. Note that the conductivity
$\sigma_{f}(\omega)$ of $f_{\alpha}$ only contains a longitudinal component
given by the expression above. The Hall conductivities from $f_{\uparrow}$ and
$f_{\downarrow}$ cancel each other due to the time-reversal symmetry
$\mathcal{T}$ which involves a particle-hole transformation of the spinons
$f_{\alpha}$. From the relation between the charge conductivity and vortex
conductivity in Eq. 17, if the boson gap is taken to infinity, we can extract
the main interesting piece of the charge conductivity, which is given by

$\displaystyle\sigma_{e}(\omega)\sim\frac{4}{\pi^{2}\sigma_{f}(\omega)}=\frac{4\tau_{v}}{\pi^{2}\sigma_{0}}\bigg{[}\frac{\omega_{c}^{2}}{\frac{1}{\tau_{v}}-i\omega}+\bigg{(}\frac{1}{\tau_{v}}-i\omega\bigg{)}\bigg{]}.$
(21)

At zero doping, the optical conductivity does not have a Drude weight; but the
presence of added charge has introduced a new Drude peak to the optical
conductivity with Drude weight:

$\displaystyle
D\sim\frac{2\omega_{c}^{2}\tau_{v}}{\pi^{2}\sigma_{0}}\sim(\delta n_{e})^{2},$
(22)

and the Drude weight is proportional the square of the doped charge density,
contrary to the ordinary Drude theory where the Drude weight is linear with
the charge density.

The DC resistivity now takes the form of

$\displaystyle\rho_{e}=1/\sigma_{e}(0)=\frac{\pi^{2}\sigma_{0}}{4}\frac{1}{1+\tau_{v}^{2}\omega_{c}^{2}}.$
(23)

The Lorenz number, defined as $L=\kappa/(T\sigma_{e})$ becomes

$\displaystyle
L=\frac{\kappa}{T\sigma_{e}}\sim\frac{L_{0}\sigma_{f}}{\sigma_{e}}\sim\frac{\pi^{2}}{4}\frac{L_{0}\sigma_{0}^{2}}{(1+\tau_{v}^{2}\omega_{c}^{2})^{2}}.$
(24)

Here $\kappa$ represents the thermal conductivity. Both the resistivity, and
Lorenz number decrease with the doped charge density. Combing with the
emergence of the Drude weight under doping, it suggests that doping would
eventually drive the the system more like a normal metal. Here we have ignored
the thermal conductivity arising from the gauge bosons, which also transports
heat without charge, hence it also contributes to the violation of the
Wiedemann-Franz law.

###  II.4 Nearby Phases

(1) Spin liquid Mott insulator, and metal

The quantum bad metal state constructed above can be driven to a Mott
insulator which is also a $\mathrm{U}(1)$ spin liquid with a Fermi surface of
spinon $f_{\alpha}$, by driven $(z_{1},z_{2})$ into a trivial bosonic Mott
insulator, with zero rotor number of $z_{1}$ and $z_{2}$. In the Mott
insulator, $f_{\alpha}$ still has Fermi pockets, but the gauge flux of
$a_{\mu}$ no longer carries any nontrivial quantum number. This is one of the
most studied spin liquid states in the literature Lee and Lee (2005);
Motrunich (2005); Ran et al. (2007), with potential applications to a variety
of materials.

The quantum phase transition between a bSPT state and a trivial Mott insulator
of the boson is described by the $N=2$ QED Grover and Vishwanath (2013); Lu
and Vishwanath (2012), and this theory is part of a web of duality involving
also the easy-plane deconfined quantum critical point Xu and You (2015); Mross
et al. (2016); Hsin and Seiberg (2016); Potter et al. (2017); Wang et al.
(2017); Senthil et al. (2019). The original theory of the bSPT-MI transition
is now coupled to an extra dynamical gauge field $a_{\mu}$. When there is no
disorder, the dynamics of $a_{\mu}$ is overdamped by the Fermi surface of
$f_{\alpha}$, then we do not expect the gauge field $a_{\mu}$ to change the
infrared fate of the transition. The presence of disorder may complicate the
nature of the bSPT-MI transition.

The quantum bad metal phase can also be driven into an ordinary metal phase by
condensing either $z_{1}$ or $z_{2}$. The $\mathrm{U}(1)_{g}$ gauge field will
be gapped out by the Higgs mechanism, and the spinon operator $f_{\alpha}$
becomes the electron operator due to the condensate of, $e.g.$ $z_{1}$,
similar to the previous theory of interaction-driven MIT Lee and Lee (2005);
Senthil (2008).

(2) Charge-$4e$ superconductor

Starting with the quantum bad metal, we can also drive the spinon $f_{\alpha}$
into a trivial insulator without special topological response, likely through
a Lifshitz transition where the Fermi pockets shrink to zero. Then the action
of $a_{\mu}$ is just the ordinary Maxwell term, which describes a photon
phase. The monopole which creates and annihilates the gauge flux is prohibited
here as the gauge flux carries charge-$4e$ as we discussed before, and the
electric charge is a conserved quantity. The photon phase of the gauge field
$a_{\mu}$ is also dual to the condensate of its flux, $i.e.$ a condensate of
charge$-4e$, or in other words a charge$-4e$ superconductor.

(3) $Z_{2}$ spin-charge topological order

We can also consider a situation where the fermionic vortices $f_{\alpha}$
form a “superconductor”, $i.e.$ the Cooper pair of $f_{\alpha}$ condenses.
This condensation will gap out $a_{\mu}$ through a Higgs mechanism, and break
$a_{\mu}$ down to a $Z_{2}$ gauge field, which supposedly forms a $Z_{2}$
topological order. Like all the $Z_{2}$ topological orders, here there are
three types of anyons with mutual semionic statistics. One type of anyon is
$f_{\alpha}$, another is the half-flux of $a_{\mu}$. Since the flux of
$a_{\mu}$ carries charge-$4e$, the half-flux of $a_{\mu}$ carries charge-$2e$.

###  II.5 Other Constructions

One can also construct a similar quantum bad metal phase starting with a
charge-$2e$ spin-singlet superconductor. Let us assume there are two flavors
of bosons, $b_{1}$ and $b_{2}$, which carry charge $\pm 2e$ respectively. We
take the following parton treatment for $b_{\alpha}$:

$\displaystyle b_{1}=\psi_{1}f,\ \ \ b_{2}=\psi_{2}f,$ (25)

where all the partons are complex fermions. Apparently there is also a gauge
$U(1)_{g}$ shared by the partons. $\psi_{1,2}$ carries electric charge $\pm
2$, and gauge charge $+1$ of a dynamical internal gauge field; $f$ carries
gauge charge $-1$ of the internal gauge field. We now consider the following
state: $f$ again forms a band structure which is a good metal; $\psi_{\alpha}$
forms a quantum “psudospin” Hall insulator, in the sense that the flavor index
of $\psi_{\alpha}$ is viewed as a pseudospin index. After integrating out
$\psi_{\alpha}$, a mutual Chern-Simons term between the external EM field
$A^{e}$ and the internal gauge field $a$ is generated, with the same form as
Eq. 11. We assume that there is no other conserved charges other than the
charge $\mathrm{U}(1)$. The charge response of this construction can be
evaluated following the steps of the previous section.

The bSPT state is one of the states that the bosonic partons $(z_{1},z_{2})$
can form that make the charge vortex a fermion. There are other options which
achieve the similar effect, if we allow topological degeneracy. For example,
$z_{1}$ and $z_{2}$ can each form a bosonic fractional quantum Hall state with
Hall conductivity $\pm 1/(2k)$ where $k$ is an integer. Although there is a
bit subtlety of integrating out a topological order, suppose we can do this,
the response mutual CS theory in Eq. 11 would have level $2/k$. The rest of
the discussion follows directly.

We would like to compare our state with other “vortex liquids” discussed in
previous literature, for example the well-known Dirac vortex liquid in the
context of half-filled Landau level Son (2015); Wang and Senthil (2016b). The
electrical conductivity tensor of the system reads
$\sigma_{ij}=\delta_{ij}\sigma^{\ast}+\frac{e^{2}}{2h}\epsilon_{ij}$, where
$\sigma^{\ast}\sim 1/\sigma_{v}$, and $\sigma_{v}$ is the conductivity of the
Dirac composite fermions (the vortices). Although the longitudinal
conductivity of the system can be small when $\sigma_{v}$ is large, the
longitudinal resistivity would still be small due to the nonzero Hall
conductivity. Hence in the simplest experimental set-up where the transport is
measured along the $\hat{x}$ direction while the $\hat{y}$ direction of the
sample has an open boundary, the measured longitudinal resistivity along the
$x$ direction would be small.

We can also put $c_{\uparrow}$ in the Dirac vortex liquid of the half-filled
Landau level; and $c_{\downarrow}$ in the time-reversal conjugate state of
$c_{\uparrow}$. In this case in order to correctly extract the longitudinal
electrical resistivity, we need to introduce both the external electromagnetic
field, and a “spin gauge field” $A^{s}$:

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{D}(\psi_{1},a_{1})+\frac{\mathrm{i}}{4\pi}a_{1}\wedge
A_{1}+\frac{\mathrm{i}}{8\pi}A_{1}\wedge dA_{1},$ (26) $\displaystyle+$
$\displaystyle\mathcal{L}_{D}(\psi_{2},a_{2})-\frac{\mathrm{i}}{4\pi}a_{2}\wedge
A_{2}-\frac{\mathrm{i}}{8\pi}A_{2}\wedge dA_{2}.$ (28)

$\psi_{1,2}$ are the composite Dirac fermions, and their Lagrangian
$\mathcal{L}_{D}$ should in general have a nonzero chemical potential. We have
introduced two external gauge fields $A_{1}=A^{e}+A^{s}$, and
$A_{2}=A^{e}-A^{s}$. The system does not have a net charge Hall response, but
there is a spin-Hall effect, $i.e.$ there is a mutual CS term between the
electromagnetic field and the spin gauge field. The existence of the spin Hall
response will again lead to a small longitudinal electrical resistivity when
$\psi_{1,2}$ are good metals. By contrast, in our construction presented in
the previous section, there is only longitudinal transport, hence a large
vortex conductivity would ensure a large longitudinal electrical resistivity.

Another vortex liquid with fermionic vortex was discussed in Ref. Galitski et
al., 2005, aiming to understand the observed metallic state with an
anomalously large conductivity between the superconductor and insulator in
amorphous thin films. There the vortex is turned into a fermion through manual
flux attachment.

Vortices can also play an important role in quantum magnets. Exotic quantum
spin liquid states were also constructed through fermionic spin vortices in
previous literature Alicea et al. (2005a, b, 2006); Hermele (2009); Wang and
Senthil (2015b). These works generally use two approaches to generate
fermionic vortices: one can either introduce fermionic partons for the vortex
operator, or turn a vortex into a fermion through flux attachment when the
vortex sees a background magnetic field (dual of fractional spin density). In
our work, instead of granting existing vortices fermionic statistics, an
interpretation of the fermionic partons as charge vortices naturally emerges
due to the topological physics of the bosonic sector.

##  III Summary

We present a construction of a strongly interacting quantum bad metal phase,
$i.e.$ at zero and low temperature the resistivity is finite but exceedingly
larger than the MIT limit $h/e^{2}$, by making the charge vortices a good
metallic phase with a vortex Fermi surface. In this construction the charge
vortex is naturally a fermion by driving the charge degree of freedom into a
bosonic symmetry protected topological state. The quantum bad metal so
constructed has the following features: (1) its resistivity can be exceedingly
larger than the MIR limit; (2) a small Drude weight proportional to $(\delta
n_{e})^{2}$ emerges under weak charge doping away from half-filling (one
electron per unit cell); (3) like previously discussed vortex liquids, our
construction should also have strong violation of the Wiedemann-Franz law. We
also demonstrated that this quantum bad metal phase is next to a charge-$4e$
superconductor, a $Z_{2}$ spin-charge topological order, the Mott insulator
phase which is also a well-studied spin liquid, and a normal metal phase.

The exoticness of the state we constructed is of “quantum nature”, as strictly
speaking a bSPT state that our construction strongly relies on is only sharply
defined at zero temperature. At high temperature all the partons will be
confined, and our state no longer enjoys a tractable description in terms of
the dual weakly interacting vortices. Hence our state is different from the
original example of bad metal discussed in Ref. Emery and Kivelson, 1995,
$i.e.$ the cuprates materials with hole doping, where the resistivity of the
system in each $2d$ layer reaches the threshold of bad metal at finite
temperature.

The authors thank Matthew Fisher and Steve Kivelson for very helpful
discussions. We also thank Umang Mehta and Xiao-Chuan Wu for participating in
the early stage of the work. This work is supported by the NSF Grant No.
DMR-1920434, and the Simons Investigator program. C.-M. J. is supported by a
faculty startup grant at Cornell University

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