# Universal scaling behavior of resistivity under two-dimensional
superconducting phase fluctuations

Zongsheng Zhou  Beijing National Laboratory for Condensed Matter Physics and
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China  Kang
Wang  Beijing National Laboratory for Condensed Matter Physics and Institute
of Physics, Chinese Academy of Sciences, Beijing 100190, China  School of
Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049,
China  Hai-Jun Liao  Beijing National Laboratory for Condensed Matter Physics
and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China  Zi-
Xiang Li<EMAIL_ADDRESS>Beijing National Laboratory for Condensed
Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China  School of Physical Sciences, University of Chinese Academy of
Sciences, Beijing 100049, China  Tao Xiang<EMAIL_ADDRESS>Beijing National
Laboratory for Condensed Matter Physics and Institute of Physics, Chinese
Academy of Sciences, Beijing 100190, China  School of Physical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, China  Beijing
Academy of Quantum Information Sciences, Beijing, 100190, China

###### Abstract

In superconductors with relatively low superfluid density, such as cuprate
high-$T_{c}$ superconductors, the phase fluctuations of the superconducting
order parameter are remarkable, presumably playing a nonnegligible role in
shaping many distinctive physical properties. This work systematically
investigates the electrical transport properties arising from thermal
superconducting phase fluctuations in two-dimensional superconductors.
Employing the Monte Carlo procedure, we access the numerically exact
properties of a microscopic model of superconductivity, in which the classical
XY model governs the thermal phase fluctuations of the superconducting order
parameter. For both $s$-wave and $d_{x^{2}-y^{2}}$-wave pairings, the
electrical resistivity exhibits a universal scaling behavior in the
temperature range above $T_{c}$. Our numerical results demonstrate that the
scaling behavior of the quasiparticle lifetime is associated with the
correlation length of the superconducting order parameter, yielding the
universal scaling behavior of electrical resistivity determined by the
Berezinskii-Kosterlitz-Thouless critical scaling of the correlation length.
Furthermore, we discuss the dependence of the electrical resistivity
coefficient on the pairing amplitude and the possible implication on recent
transport experiments.

##  I Introduction

Superconductivity (SC) is a macroscopic quantum phenomenon stemming from the
condensation of paired electrons, which is characterized by a complex pairing
order parameter with both amplitude and phase [1, 2]. For BCS mean-field
theory, it is widely assumed that the phase fluctuations of the order
parameter are negligible for determining important physical properties such as
transition temperature $T_{c}$, which generically holds for SC in many
conventional metals [3]. However, phase fluctuations in conventional
superconductors could be enlarged by disorder [4, 5, 6]. Additionally, for
unconventional superconductors, including cuprates [7, 8, 9, 10], organic
superconductors [11, 12], and twisted bilayer graphene[13, 14], the superfluid
density is low owing to their low charge carrier density, giving rise to the
strong phase fluctuations that presumably accounts for many intriguing
physical phenomena [15]. In the past decades, phase fluctuations in
unconventional superconductors have been extensively investigated from both
theoretical [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32] and experimental [33, 34, 8, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]
perspectives. In the presence of strong phase fluctuations, the phase
coherence temperature is lower than the temperature at which the Copper
pairing forms, leading to the opening of the spectral gap above
superconducting $T_{c}$ [16, 15, 17, 18, 21, 22, 23, 30]. It is also
theoretically proposed that phase fluctuations could trigger novel phenomena
such as the Fermi arc [22, 23, 24, 25, 45] observed in underdoped cuprates
[46, 47, 48], and the possibility of charge-4e SC [49, 50, 51, 52, 53, 54, 55,
56] as a vestigial order.

Nevertheless, the transport properties of unconventional superconductors,
including electrical resistivity arising from superconducting phase
fluctuations in the temperature regime above $T_{c}$, are rarely studied.
Recently, armed with the development of thin-film technology, strange-metal
electrical transport behaviors, which were unveiled in underdoped and
optimally doped cuprates soon after the discovery of high-$T_{c}$
superconductors [57, 58, 59], have been observed in a wide doping range of
electron-doped cuprates $\rm La_{2-x}Ce_{x}CuO_{4}$ [60, 61] and iron-pnictide
superconductors $\rm FeSe$ [62], along with ${\rm Ca}_{10}\left({\rm
Pt}_{4}{\rm As}_{8}\right)\left(\left({\rm Fe}_{0.97}{\rm
Pt}_{0.03}\right)_{2}{\rm As}_{2}\right)_{5}$ [63]. More remarkably, the
coefficient of linear-$T$ resistivity exhibits a quadratic dependence on the
superconducting transition temperature $T_{c}$ [61, 62, 63], which implies the
possible relation between anomalous metallic transport and the SC or other
ingredients associated with superconducting pairing mechanisms. Hence,
exploring the electrical resistivity driven by the superconducting phase
fluctuations above $T_{c}$ is desirable, which potentially sheds new insights
into understanding the possible underlying relation between strange metal
dissipation and SC observed in unconventional superconductors.

In this work, we systematically investigate the properties of electrical
transports of a microscopic BdG Hamiltonian involving superconducting pairing
and thermal phase fluctuations above the superconducting critical temperature
$T_{c}$. The amplitude of the superconducting order parameter is fixed as a
spatially independent value. Distinct from the BCS mean-field Hamiltonian, the
model under consideration incorporates the spatial fluctuations of phase. The
phase fluctuations are governed by a classical XY model, characterizing the
thermal fluctuations of the order parameter’s phase with varying temperatures.
Employing large-scale Monte Carlo simulation, we study the transport
properties of the model at finite temperatures. The numerical results reveal
that the electrical resistivity induced by the thermal phase fluctuations in
the low-temperature regime above $T_{c}$ obeys the scaling behavior
$\rho\left(T\right)\propto A\mathrm{exp}({-b}/{\sqrt{T/T_{c}-1}})$, which is
consistent with the critical scaling relation of the correlation length for
the Berezinskii-Kosterlitz-Thouless (BKT) transition [64]. We further evaluate
the quasiparticle lifetime of electrons versus temperature. Interestingly, we
achieve the same scaling behavior as that of electrical resistivity. Hence,
our numerical results establish that thermal phase fluctuations yield the
anomalous temperature dependence of electrical resistivity in two-dimensional
superconductors, which is determined by the BKT critical scaling of the
correlation length.

This paper is organized as follows: Section II introduces the model
Hamiltonian, which leverages the classical two-dimensional XY model to
simulate the superconducting phase fluctuations, along with the approach for
determining resistance values through finite lattice scaling analysis. Section
III presents the main results. Firstly, we elucidate the temperature
dependence of DC resistivity, revealing universal BKT behavior, and show
numerical evidence for a quadratic scaling relationship between the
coefficient of temperature-dependent electrical resistivity and the
superconducting pairing amplitude. Secondly, we focus on determining the
superconducting transition temperature $T_{c}$ from the superfluid density.
The paper concludes with a summary of the key findings in Section IV.

##  II Model

In superconductors characterized by pronounced superconducting fluctuations,
such as cuprates, the superconducting transition temperature $T_{c}$ is often
significantly lower than the transition temperature $T_{\rm MF}$ predicted by
the mean-field theory. Within the temperature regime above $T_{c}$ yet
substantially below $T_{\rm MF}$, superconducting coherence is compromised due
to phase fluctuations, whose effects can be effectively modeled by the BdG
Hamiltonian:

$\mathcal{H}_{0}\\!=\\!\sum_{\bm{r},\bm{r}^{\prime}}\begin{pmatrix}c_{\bm{r}\uparrow}^{\dagger},c_{\bm{r}\downarrow}\end{pmatrix}\\!\begin{pmatrix}H_{0}(\bm{r},\bm{r^{\prime}})&\Delta(\bm{r},\bm{r}^{\prime})\\\
\Delta^{*}(\bm{r},\bm{r}^{\prime})&H_{0}(\bm{r},\bm{r}^{\prime})\end{pmatrix}\\!\begin{pmatrix}c_{\bm{r}^{\prime}\uparrow}\\\
c_{\bm{r}^{\prime}\downarrow}^{\dagger}\end{pmatrix},$ (1)

where $c_{\bm{r}\sigma}^{\dagger}$ is the creation operator of an electron at
site $\bm{r}$ with spin $\sigma$ and $H_{0}$ is the Hamiltonian of non-
interacting electrons defined on a square lattice,

$H_{0}(\bm{r},\bm{r}^{\prime})=-t\delta_{\bm{r}^{\prime},\bm{r}\pm\bm{e}}-\mu\delta_{\bm{r},\bm{r}^{\prime}},$
(2)

with $t$ the hopping integral between two nearest-neighboring sites,
$\bm{e}=(\bm{e}_{x},\bm{e}_{y})$ the two unit lattice vectors, and $\mu$ the
chemical potential. Variable $\theta$ is the phase of the superconducting gap
parameter $\Delta(\bm{r},\bm{r}^{\prime})$

$\Delta(\bm{r},\bm{r}^{\prime})=\left|\Delta(\bm{r},\bm{r}^{\prime})\right|e^{i\theta(\bar{\bm{r}})},$
(3)

where $\bar{\bm{r}}=(\bm{r}+\bm{r}^{\prime})/2$. In the temperature regime
$T_{c}<T\ll T_{\rm MF}$, we assume the gap amplitude to be a constant,
$\left|\Delta(\bm{r},\bm{r}^{\prime})\right|=\Delta$, independent of
$(\bm{r},\bm{r}^{\prime})$ and $T$.

In this work, two kinds of pairing states are considered. One is the $s$-wave
pairing state, and the other is the $d_{x^{2}-y^{2}}$-wave pairing state. The
$s$-wave pairing state only involves the on-site pairing, whose gap parameter
is defined by

$\Delta(\bm{r},\bm{r}^{\prime})=\Delta
e^{i\theta\left(\bm{r}\right)}\delta_{\bm{r},\bm{r}^{\prime}}.$ (4)

The $d_{x^{2}-y^{2}}$-wave pairing state, on the other hand, arises from the
pairing of two electrons on two nearest-neighboring sites $\bm{r}$ and
$\bm{r}^{\prime}=\bm{r}\pm\bm{e}$ and the phase parameter is defined on the
corresponding link of the square lattice.

$\Delta(\bm{r},\bm{r}^{\prime})=\Delta
e^{i\theta(\bar{\bm{r}})}\left(\delta_{{\bm{r}}^{\prime},{\bm{r}}\pm{\bm{e}}_{x}}-\delta_{{\bm{r}}^{\prime},{\bm{r}}\pm{\bm{e}}_{y}}\right).$
(5)

In the temperature range considered here, we assume that the fluctuation of
$\theta$ is slow in comparison with the quasiparticle excitations so that the
dynamics of $\theta$ are purely governed by the classical XY model, described
by the Hamiltonian

$\mathcal{H}_{\theta}=-\sum_{\left\langle\bar{\bm{r}},\bar{\bm{r}}^{\prime}\right\rangle}J_{\theta}\mathrm{cos}\left[\theta\left(\bar{\bm{r}}\right)-\theta\left(\bar{\bm{r}}^{\prime}\right)\right],$
(6)

where $\langle\cdots\rangle$ denotes summations over the nearest-neighboring
sites for the $s$-wave pairing state or bond centers for the
$d_{x^{2}-y^{2}}$-wave pairing state. The employment of the classical Monte
Carlo method allows for the generation of a series of configurations of
$\theta$ using this model at a given temperature. To mitigate self-
correlation, a single configuration is selected from every 20,000
configurations produced by the Monte Carlo simulation. By inserting each of
these phase configurations into Eq. (1), it becomes possible to determine the
eigenvalues $\varepsilon_{a}$ and their corresponding eigenvectors $\psi_{a}$
of ${\cal H}_{0}$ by exact diagonalization.

In the numerical simulations, we set the hopping integral $t=1$ as the unit of
energy, the coupling constant $J_{\theta}=0.1$, the gap amplitude
$\Delta=0.3$, and the electron filling factor $n_{e}=0.83$ (the filling factor
is 2 if a band is fully filled) if not explicitly stated. We take these values
in our numerical simulations such that there is a relatively wide temperature
window above the superconducting transition but lower than the energy scale of
pairing amplitude. Furthermore, periodic boundary conditions are assumed.

##  III Electromagnetic response functions

In the linear response theory, the electromagnetic response function is
determined by the current-current correlation function

$\Lambda_{\alpha\beta}(\bm{q},\omega)=\int_{0}^{\infty}dte^{i\omega
t}\left[j^{\alpha}(\bm{q},t),j^{\beta}(-\bm{q},0)\right],$ (7)

where $j^{\alpha}(\bm{q},t)$ is the Fourier component of the current operator
along the $\alpha$-direction

$j^{\alpha}(\bm{r})=it\sum_{\sigma}\left(c_{\bm{r},\sigma}^{\dagger}c_{\bm{r}+\bm{e}_{\alpha},\sigma}-c_{\bm{r}+\bm{e}_{\alpha},\sigma}^{\dagger}c_{\bm{r},\sigma}\right).$
(8)

Using the eigenpairs of ${\cal H}_{0}$, the current-current correlation can be
expressed as

$\Lambda^{\alpha\beta}(\bm{q},\omega)=\frac{1}{N}\sum_{a\neq
b}M_{ab}(\bm{q})M_{ba}(-\bm{q})\frac{f(\varepsilon_{a})-f(\varepsilon_{b})}{\omega-\varepsilon_{a}+\varepsilon_{b}+i\eta},$
(9)

where $N=L\times L$ is the lattice size, $f\left(\varepsilon\right)$ is the
Fermi distribution function, $\eta$ is a positive number that tends to zero in
the thermodynamic limit $N\rightarrow\infty$, and

$\displaystyle M_{ab}\left(\bm{q}\right)$ $\displaystyle=$
$\displaystyle\sum_{\bm{r}\sigma}\left[\psi_{a}^{*}(\bm{r},\sigma)\psi_{b}(\bm{r}+\bm{e}_{\alpha},\sigma)\right.$
(10)
$\displaystyle-\left.\psi_{a}^{*}(\bm{r}+\bm{e}_{\alpha},\sigma)\psi_{b}(\bm{r},\sigma)\right]e^{i\bm{q}\cdot\bm{r}}.$

###  III.1 Resistivity

Figure 1: Frequency dependence of the inverse of $\mathrm{Re}\sigma(\omega)$
obtained on five different $L\times L$ square lattices at $T=0.1$ for the two
phase fluctuating states with (a) $s$-wave and (b) $d_{x^{2}-y^{2}}$-wave
pairing states.

The resistivity is the reciprocal of the real part of optical conductivity
$\sigma\left(\omega\right)$ in the $\omega\rightarrow 0$ limit. The optical
conductivity, on the other hand, is determined by the imaginary part of the
current-current correlation function in the long-wavelength limit

$\displaystyle\mathrm{Re}\sigma\left(\omega\right)=\frac{\mathrm{Im}\Lambda^{xx}\left(\bm{q}=\bm{0},\omega\right)}{\omega}.$
(11)

Using the expression (9), it is straightforward to show that

$\mathrm{Re}\sigma\left(\omega\right)=\sum_{a\neq
b}W_{ab}\delta(\omega-\varepsilon_{a}+\varepsilon_{b})$ (12)

in the limit $\eta\rightarrow 0$. In this equation,

$W_{ab}=-\frac{\pi}{N}M_{ab}(\bm{0})M_{ba}(\bm{0})\frac{f(\varepsilon_{a})-f(\varepsilon_{b})}{\varepsilon_{a}-\varepsilon_{b}}.$
(13)

Hence, only the terms satisfying $\omega=\varepsilon_{a}-\varepsilon_{b}$
contribute to the conductivity.

In a finite lattice system, the eigenvalues of ${\cal H}_{0}$ are discrete.
Therefore, $\sigma(\omega)$ determined using Eq. (12) is also discrete. To
extrapolate the value of the conductivity to the thermodynamic limit, we first
calculate the total weight of the conductivity for the frequency from 0 to
$\omega$

$S_{\mathrm{tot}}(\omega)=\sum_{\begin{subarray}{c}a\not=b\\\
{\footnotesize{0<\varepsilon_{a}-\varepsilon_{b}<\omega}}\end{subarray}}W_{ab}.$
(14)

We then determine the conductivity from the numerical derivative of
$S_{\mathrm{tot}}$

$\sigma(\omega)=\frac{S_{\mathrm{tot}}(\omega+d\omega)-S_{\mathrm{tot}}(\omega-d\omega)}{2d\omega},$
(15)

where $d\omega$ is a small parameter that should be taken such that at least
one eigenvalue difference $\varepsilon_{a}-\varepsilon_{b}$ falls between the
interval ($\omega-d\omega,\omega+d\omega$). Furthermore, we need to do an
average of $\sigma(\omega)$ over different phase configurations generated from
the Monte-Carlo sampling of the XY model at a given temperature. The
resistivity is finally obtained by extrapolating the inverse of the
conductivity $\sigma(\omega)$ to the limit $\omega\rightarrow 0$.

Fig. 1 is an illustrative example, demonstrating the variation in optical
conductivity with frequency $\omega$ at $T=0.1$ according to the scheme above.
Notably, the outcomes across various system sizes converge onto a single curve
for both $s$\- and $d_{x^{2}-y^{2}}$-wave pairing states. This convergence
suggests that the scheme is highly effective in reducing the impact of finite-
size effects, thereby enabling accurate extrapolation of resistivity.

Figure 2: Temperature dependence of the DC resistivity for the $s$-wave (left
panel) and $d_{x^{2}-y^{2}}$-wave (right panel) pairing states in three
different square lattice systems. The numerical results are well described by
the formula $\rho\left(T\right)=A\mathrm{exp}({-b}/{\sqrt{T/T_{c}-1}})$,
depicted by the solid curves. From the fitting, the superconducting transition
temperature $T_{c}$ is found to be $T_{c}=0.089(2)$ in both cases.

Resistivity is determined by extrapolating $1/\mathrm{Re}\sigma(\omega)$ to
the $\omega=0$ limit. Fig. 2 depicts how resistivity varies with temperature
for both $s$\- and $d_{x^{2}-y^{2}}$-wave pairing states. This analysis is
based on an average of over 72 independent phase configurations for each
system size at a specific temperature. The findings reveal that resistivity
stabilizes as system size increases, indicating negligible finite-size
effects. Although there are quantitative differences in their absolute values,
a universal temperature dependence emerges for both pairing states.
Specifically, resistivity escalates with temperature, displaying a downward
curvature that contrasts with the $T^{2}$ behavior characteristic of Fermi
liquids.

To delve deeper into the temperature-dependent behavior of resistivity, we
conduct a scaling analysis on the numerical data. In the system under study,
resistivity results from electron scattering by the fluctuating phase field.
Given that the correlation length of the fluctuating pairing phase constitutes
the sole length scale in the XY model, we propose that the electron scattering
length $l_{s}$, which is a product of the scattering time $\tau_{s}$ and the
Fermi velocity, is proportional to the correlation length $\xi$ of fluctuating
phase, i.e., $l_{s}\propto\xi$. As the temperature approaches the BKT critical
temperature $T_{c}$, the correlation length $\xi$ is expected to diverge,
following the theory outlined by Kosterlitz in 1974 [64]

$\xi(T)=a\,\mathrm{exp}\left({\frac{b}{\sqrt{T/T_{c}-1}}}\right),$ (16)

where $b\simeq 1.5$ and $a$ is a non-universal coefficient. If the Fermi
velocity remains constant within the considered temperature range, it is
anticipated that the resistivity obeys the scaling law

$\rho(T)\sim\xi^{-1}\propto
A\mathrm{exp}\left(-\frac{b}{\sqrt{T/T_{c}-1}}\right).$ (17)

The resistivity vanishes at the transition temperature. It is important to
note that in realistic systems, there is a complex interaction between the
phase field and quasiparticle excitations. Nevertheless, the scaling behavior
described by Eq. (16) remains applicable and only the value of $T_{c}$ is
modified by this interaction. This assertion is based on the premise that the
interplay between these factors does not significantly alter the underlying
mechanisms governing the phase fluctuation and the BKT scaling behavior in Eq.
(16) is universal in the phase fluctuating regime of two-dimensional
superconductors.

Remarkably, as depicted in Figs. 2($\mathrm{a}$) and 2($\mathrm{b}$) , Eq.
(17) demonstrates a precise fit to the numerical data for both $s$\- and
$d_{x^{2}-y^{2}}$-wave pairing states. This accurate fitting yields an
accurate estimation of the transition temperature $T_{c}=0.089(2)$ for both
pairing cases, implying that the correlation length of the phase field indeed
dictates the electron scattering length. We have also conducted a scaling
analysis employing a formula proposed by Halperin and Nelson [65]. Their
formula accurately fits the numerical data for the $d_{x^{2}-y^{2}}$-wave
pairing state but fails in fitting the results of $s$-wave pairing. However,
in the $d_{x^{2}-y^{2}}$-wave pairing case, our scaling formula outperforms
theirs in precision, offering a superior fit. Contrary to our findings, a
recent study [66] suggested that the phase fluctuations may lead to a
linear-$T$ resistivity in the $s$-wave pairing state. This discrepancy likely
results from the approximation and the relatively small system sizes used in
their calculations.

Figure 3: Momentum dependence of the imaginary self-energy for (a) $s$-wave
and (b) $d$-wave pairing states at $T=0.11$, $\omega=0.1$. The self-energy is
calculated by taking an average of over $72$ samples with $L=70$ and
$\eta=0.008$. Panels (c) and (d) show the temperature dependency of the
scattering rate of electrons around the Fermi surface along the ($\pi,0$)
direction of the Brillouin zone. The solid lines are fitting curves using the
formula $1/\tau\sim\mathrm{exp}({-b}/{\sqrt{T/T_{c}-1}})$.

In the above analysis, we find that the scattering lifetime of electrons
follows the same temperature dependence of the correlation length of the
fluctuating superconducting order parameters. To gain a deeper understanding
of this observation, we calculate the self-energy of electrons using the Dyson
equation

$\Sigma(k,\omega)=G^{-1}(k,\omega)-G_{0}^{-1}(k,\omega),$ (18)

where $G_{0}^{-1}(k,\omega)$ is the single-particle Green’s function of free
electrons, described by ${\cal H}_{0}$ without the superconducting pairing.
The imaginary part of the self-energy, $\textrm{Im}\Sigma(k,\omega)$, measures
the damping effect induced by the fluctuating phases. Its inverse equals the
quasiparticle lifetime $\tau(\bm{q},\omega)$.

Figs 3$\left(\mathrm{a}\right)$ and 3$\left(\mathrm{b}\right)$ show the heat
maps of the quasiparticle scattering rate, $1/\tau({\bm{q}},\omega)$, in the
Brillouin zone for the $s$-wave and $d$-wave pairing states in a square
lattice of $L=70$ at $\omega=0.1$, respectively. The result indicates that the
quasiparticle scattering rates are the strongest around the Fermi surface
close to ($\pi,0$) and its equivalent points. The momentum anisotropy in the
scattering rate of $s$-wave pairing is presumably a consequence of the
anisotropy of the spectral weight crossing the Fermi surface originating from
the von Hove singularity in the band structure of electrons locating at
momenta $(\pm\pi,0)$ and $(0,\pm\pi)$. For the $d_{x^{2}-y^{2}}$-wave pairing
state, the anisotropy in the quasiparticle scattering rate is more pronounced.
Particularly, the scattering rate almost vanishes along the nodal direction of
the $d_{x^{2}-y^{2}}$-wave gap order parameter, indicating that the
quasiparticles are more coherent along the nodal direction. This is consistent
with the anisotropy of the $d_{x^{2}-y^{2}}$-wave pairing symmetry,

The lower panels of Fig. 3 show the temperature dependence of the scattering
rate $1/\tau({\bm{q}},\omega)$ at the points on the Fermi surface where the
scattering rates are largest at $\omega=0.1$. We average over eight equivalent
points on the Fermi surface to obtain the data. The quasiparticle lifetime
$\tau$ presumably relates to the scattering time. Therefore, we expect the
corresponding scattering rate to share the same temperature dependence as the
resistivity

$\tau^{-1}\sim\mathrm{exp}\left(-\frac{b}{\sqrt{T/T_{c}-1}}\right).$ (19)

This is indeed the case, as shown in Fig. 3. By fitting the data with the
above expression, we estimate the critical temperatures to be $T_{c}=0.088(2)$
for both the $s$-wave and $d_{x^{2}-y^{2}}$-wave states with $L=70$. These
values agree with the results of $T_{c}$ extracted from the resistivity data.

Figure 4: The resistivity coefficient $A$ defined in Eq. (17) versus the
pairing gap amplitude $\Delta$ for (a) $s$-wave and (b) $d$-wave pairing
states with $L=70$.

Above the critical temperature, the leading-order contribution to the self-
energy, $\Sigma(\mathbf{q},\omega)$, originates from the second-order
perturbation of the fluctuating phase fields. Consequently, the primary
contribution to the imaginary part of the self-energy is expected to exhibit a
quadratic dependence on the gap amplitude $\Delta$ such that
$A\propto\Delta^{2}$.

To confirm this quadratic scaling behavior, we evaluated the resistivity as a
function of temperature for various values of $\Delta$. Fig. 4 illustrates the
dependence of $A$ on $\Delta$, determined by fitting the numerical data to Eq.
(17). Our findings reveal that the equation $A=c\Delta^{2}$, where $c$ is a
coefficient independent of $\Delta$, accurately captures the numerical
results. It is important to note that this quadratic term solely represents
the lowest-order scattering process. The inclusion of higher-order
corrections, such as a $\Delta^{4}$ term, further refines the fit. These
results demonstrate a correlation between the resistivity coefficient $A$ and
the amplitude of the fluctuating field, paving the way for a deeper
understanding of the notable relationship between $A_{1}$ and $T_{c}$ observed
in cuprate [61] and other unconventional superconductors [62], as reported in
previous studies.

###  III.2 Superfluid density

From the current-current correlation function and the expectation value of the
diamagnetic current operator, we can also calculate the superfluid density in
the superconducting phase:

$\rho_{s}=-\langle K_{x}\rangle-\Lambda^{xx}(q_{x}=0,q_{y}\rightarrow
0,\omega=0),$ (20)

where $K_{x}$ is the $x$-component of the diamagnetic current operator

$K_{x}=-\frac{t}{N}\sum_{\bm{r},\sigma}\left(c_{\bm{r},\sigma}^{\dagger}c_{\bm{r}+\bm{e}_{x},\sigma}+c_{\bm{r}+\bm{e}_{x},\sigma}^{\dagger}c_{\bm{r},\sigma}\right).$
(21)

The formation of the phase coherence determines the superconducting
transition. For the two-dimensional XY model, the superfluid density exhibits
a universal jump at the critical point [67], where the correlation length
diverges. Exactly at the critical point, the superfluid density scales
linearly with $T_{c}$

$\rho_{s}\left(T_{c},L\right)\simeq\frac{2}{\pi J_{\theta}}T_{c}.$ (22)

Figure 5: Temperature dependence of the superfluid density for both $s$-wave
(left panel) and $d$-wave (right panel) pairing states. The black diagonal
line represents the universal scaling law
$\rho_{s}\left(T\right)\simeq{2T}/{\pi J_{\theta}}$, with its intersection
with the superfluid density curve identifying the superconducting transition
temperature $T_{c}$, denoted by the gray dashed line. The estimated $T_{c}$
from the superfluid density of $L=80$ is about $0.0835(4)$ for $s$-wave
pairing and $0.0822(4)$ for $d_{x^{2}-y^{2}}$ wave pairing. Inserts in the two
panels are extrapolations of the results with system sizes ranging from $46$
to $80$ by using quadratic polynomial, indicating $T_{c}\simeq 0.088(2)$ in
the thermodynamic limit.

Fig. 5 shows the superfluid density calculated in three different system
sizes. With increasing temperature, the superfluid density exhibits a
noticeable drop at a temperature close to the BKT transition temperature,
$T_{KT}\simeq 0.893J_{\theta}$, determined by the large-scale Monte-Carlo
simulation [68, 69], indicating that the superconducting coherence is
suppressed at high temperatures due to the phase fluctuation.

From the crossing point between the calculated superfluid density and the
linear scaling formula (22), we find $T_{c}=0.0835(4)$ and $T_{c}=0.0822(4)$
when the system size $L=80$ for $s$-wave and $d$-wave pairing symmetries,
respectively. Extrapolating the $T_{c}$ obtained from different system sizes,
shown in the inserts of Fig. 5, we estimate $T_{c}$ to be $0.088(2)$ in the
thermodynamic limit, close to the result of Monte Carlo
$T_{\mathrm{KT}}=0.0893$, independent of the pairing symmetry. It lends
support to the scaling hypothesis of resistivity shown in Eq. (17). The finite
superfluid density above $T_{c}$ is attributed to the finite-size effect.

##  IV summary

In this work, we systematically investigate the electrical resistivity induced
by the superconducting thermal phase fluctuation in two dimensions. Employing
the Monte Carlo approach, we numerically solve the BdG Hamiltonian in the
presence of classical pairing phase fluctuations governed by the ferromagnetic
XY model. For both $s$-wave and $d_{x^{2}-y^{2}}$-wave pairings, the DC
resistivity displays normal metallic behaviors, deviating from the
conventional Fermi-liquid scaling properties. The DC resistivity induced by
thermal phase fluctuations obeys the universal scaling dependence on the
temperature $\rho\left(T\right)=A\mathrm{exp}({-b}/{\sqrt{T/T_{c}-1}})$, which
is the same as critical scaling behavior of correlation length for BKT
transition. The subsequent calculations indicate that the quasi-particle
lifetime satisfies the same scaling relation, further substantiating the
results of DC resistivity scaling behavior. Further analysis shows that the
resistivity under superconducting fluctuations is related to the pairing
amplitude, and the coefficient $A$ of the resistivity is a quadratic function
of $\Delta$. The quadratic scaling relation potentially provides a direction
to understand the interplay between unconventional superconductivity and
anomalous dissipation in the normal states.

Our results show that superconducting thermal phase fluctuations lead to novel
transport properties in two dimensions. Since the temperature dependence of DC
resistivity arising from pairing phase fluctuation unveiled here is not
linear-T, our study cannot directly account for the scaling relation between
the strange-metal dissipation and superconducting $T_{c}$ as observed in
experiments. Nonetheless, as aforementioned, our results only apply to the
thermal phase fluctuations in two dimensions, where the superconducting
transition is a BKT transition. For the most unconventional superconductors,
we should consider a three-dimensional system and incorporate the effects of
inter-layer coupling. It is intriguing to study the DC resistivity arising
from pairing phase fluctuation in a three-dimensional system, and
particularly, investigate whether the strange-metal behavior could emerge or
not. Moreover, notice that we only consider the electronic dissipation arising
from pairing thermal phase fluctuations. Regarding realistic materials,
various scattering mechanisms affect the transport properties above
superconducting transition temperature $T_{c}$, including electron-electron
scattering, electron-phonon scattering and disorder effect. Considering the
cooperative effects on the electronic transport originating from pairing phase
fluctuation and these various ingredients is also an interesting direction,
which is left in our future study.

Acknowledgments: We thank Kui Jin, Kun Jiang, Chui-Zhen Chen and Haiwen Liu
for stimulating discussions. This work is supported by the National Natural
Science Foundation of China, (Grant No. 12488201, No. 12322403, No. 12347107,
No. 12488201, No. 11874095, and No. 11974396), the Strategic Priority Research
Program of Chinese Academy of Sciences (Grants No. XDB0500202, XDB33010100 and
No. XDB33020300), the National Key Research and Development Project of China
(Grants No. 2022YFA1403900 and No. 2017YFA0302901) and the Youth Innovation
Promotion Association of Chinese Academy of Sciences (Grant No. 2021004).

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