# Vanishing of Resistivity upon Freezing of Vortex Liquid in Clean
Superconductors

Naratip Nunchot and Ryusuke Ikeda Department of PhysicsDepartment of Physics
Graduate School of Science Graduate School of Science Kyoto University Kyoto
University Kyoto 606-8502 Kyoto 606-8502 Japan Japan

###### Abstract

Superconducting transition, defined as vanishing of the resistivity, under a
magnetic field in a clean bulk type II superconductor with weak sample
disorder is believed to be a reflection of freezing of the vortex liquid to a
kind of vortex solids. This fundamental issue on superconductivity is examined
in detail. Based on the Ginzburg-Landau fluctuation theory for a three-
dimensional (3D) system and through a supplementary study in 2D case, we find
that the resistivity in the weakly disordered 3D case vanishes in a nearly
discontinuous way, reflecting growth of the Bragg peaks on approaching the
vortex lattice melting transition. In contrast, such a sharp decrease of the
resistivity does not clearly appear in the corresponding 2D case. The
consequences of this difference in the vanishing behavior of the resistivity
between the 2D and 3D systems are discussed in relation to available
experimental facts.

##  1 Introduction

When a three-dimensional (3D) bulk type II superconductor is perfectly clean,
the resistivity $\rho_{\perp}$ in a finite magnetic field and under a current
perpendicular to the field remains finite even in the low temperature limit
because of the vortex flow [1]. Therefore, a small but nonvanishing amount of
disorder is inevitably assumed to be present in real clean superconducting
(SC) materials showing vanishing of $\rho_{\perp}$ at a finite temperature
$T_{\rho}$. A nearly discontinuous vanishing of $\rho_{\perp}$, observed in
high quality samples of such materials under much lower magnetic fields than
the zero temperature depairing field $H_{c2}(0)$, is regarded as a consequence
of a glass transition [2, 3]. As the ground state of the field-induced
vortices in lower fields, the Bragg-glass or the elastic glass was proposed
[2, 4]. It is believed that, on cooling to enter this glass phase, the
resistivity will vanish discontinuously reflecting the first order freezing
transition from the vortex liquid to the vortex solid in the perfectly clean
limit. However, the realization of a vortex glass (VG) in moderately clean
cases, where large bundles of positionally correlated vortices are formed, has
also been proposed earlier [3, 5]. In this picture, the vanishing of
$\rho_{\perp}$ does not have to discontinuously occur. Experimentally,
situations [6, 7] in which a nearly discontinuous decrease of $\rho_{\perp}$
does not directly reach complete superconductivity have been seen repeatedly
in clean samples. Therefore, an established theory on the nearly discontinuous
vanishing of $\rho_{\perp}$ in moderately clean 3D type II superconductors
still remains unavailable.

Similarly, a confirmed picture remains unavailable on the temperature
dependences of $\rho_{\perp}$ in relatively clean 2D systems of field-induced
point vortices. In such 2D systems, a nearly discontinuous vanishing of
$\rho_{\perp}$ is rarely seen even in relatively clean materials [8]. It is
unclear at present whether this fact implies that the 2D vortex lattice
melting in clean limit consists of two-step continuous transitions [9].

In this work, the effects of freezing of the 2D and 3D vortex liquids on
$\rho_{\perp}$ are studied in detail based on the high field Ginzburg-Landau
(GL) fluctuation theory for a superconductor with weak point-like disorder
included. It is well known that, in high fields, the SC fluctuation
contributing to thermodynamics is dominated by its lowest Landau level (LLL)
modes [10, 11, 12, 13, 14, 15]. On the Gaussian level, the VG fluctuation
formulated in a form including effects of the vortex solidification was
discussed [16] in the case with weak enough disorder, and a sudden enhancement
of the VG fluctuation accompanying the first order vortex lattice melting in
clean limit was argued to be the origin of the nearly discontinuous vanishing
of $\rho_{\perp}$ observed in high quality samples of high $T_{c}$ cuprates
[17]. However, no calculation results on $\rho_{\perp}$ based on a detailed
analysis including the interplay between the disorder and the mutual
interaction among the SC fluctuations have been presented previously.

In the present analysis, the parquet resummation technique [18, 19] is used to
evaluate the interplay between the fluctuation-interaction and the disorder
and to compute the renormalized vertex correction to the impurity strength
which forms the backbone of the VG correlation function. This treatment can be
fully performed in 2D, while the corresponding 3D results will be
approximately derived by utilizing the 2D results. It is shown that, in 3D
case, the resistivity shows a clear sudden drop, reflecting a sharp growth of
the Bragg peak of the vortex structural factor upon cooling [16], although the
assumed VG transition is a continuous one. In contrast, such a sharp drop of
the resistivity is not clearly seen in 2D case: There, it appears only faintly
close to the end of the vanishing resistivity curves. Both in 2D and 3D cases
with quite weak disorder, however, the temperature at which the resistivity
shows a sharp reduction is in good agreement with the vortex lattice melting
temperature estimated previously [21, 22] based on experimental results and
the numerical simulation in clean limit performed in the LLL approximation
[23, 24]. These calculation results will be discussed through comparison with
available experimental results [25].

The present paper is organized as follows. In sec.2, we introduce the GL model
and explain how the mass renormalization is performed in the model by focusing
on the 2D case. The parquet analysis in 2D case is explained in sec.3, and the
content in sec.2 and 3 is extended to the 3D case in sec.4. The VG fluctuation
in clean enough systems is discussed in sec.5, and the conductivity arising
from the VG fluctuation is calculated in sec.6. In sec.7, theoretical pictures
following from numerical results on the resistivity are presented, and
comments relevant to experimental data are given in sec.8. Some details on
theoretical analysis are given in Appendices A and B.

##  2 Model

The original model in studying a superconducting transition is conventionally
the Ginzburg-Landau (GL) Hamiltonian for the order parameter field
$\Delta({\bf r})$. In the case of a thin film with thickness $d$ shorter than
the zero temperature coherence length $\xi_{0}$, i.e., in 2D case, the GL
Hamiltonian is given by

${\cal H}_{\rm GL}=dN(0)\int d^{2}{\bf r}\biggl{[}(t-1)|\Delta|^{2}+u({\bf
r})|\Delta|^{2}+\frac{g_{0}}{2}|\Delta|^{4}+\xi_{0}^{2}\biggl{|}\biggl{(}-i\nabla+\frac{2\pi}{\phi_{0}}{\bf
A}\biggr{)}\Delta\biggr{|}^{2}\biggr{]},$ (1)

where $t=T/T_{c0}$ is the normalized temperature expressed by the zero field
SC transition temperature $T_{c0}$, $\phi_{0}$ is the flux quantum, and $N(0)$
is the density of states of the normal electron system. Further, a random
potential $u({\bf r})$ expressing a spatial variation of $T_{c0}$ has been
introduced for later convenience. By noting that the Boltzmann factor is
$\exp(-{\cal H}_{\rm GL}/k_{\rm B}T)$, however, we will choose to start from
the normalized form

${\cal H}[\psi]={\cal H}_{\rm GL}/(k_{\rm B}T)=\xi_{0}^{-2}\\!\\!\int
d^{2}r\,\biggl{[}(t-1)\rho({\bf r})+u({\bf r})\rho({\bf r})+\pi
tb^{(2)}_{G}|\rho({\bf
r})|^{2}+\xi_{0}^{2}\,\biggl{|}\biggl{(}-i\nabla+\frac{2\pi}{\phi_{0}}{\bf
A}({\bf r})\biggr{)}\psi({\bf r})\biggr{|}^{2}\biggr{]}$ (2)

where $\psi=\Delta\,\sqrt{d\xi_{0}^{2}N(0)/k_{\rm B}T}$, $\rho=|\psi|^{2}$,
and $b_{\rm G}^{(2)}=g_{0}k_{\rm B}T_{c0}/(2\pi\xi_{0}^{2}dN(0))$. The
randomness with respect to $u({\bf r})$ is defined by $\overline{u({\bf
r})}=0$ and

$\overline{u({\bf r})u({\bf
r}^{\prime})}=\xi_{0}^{2}\Delta^{(2)}\delta^{(2)}({\bf r}-{\bf r}^{\prime}),$
(3)

where the overbar implies that the random average was taken. Hereafter, all
the length scales will be normalized by $\xi_{0}$.

Consistently with the Abrikosov mean field theory [26], the order parameter
will be expanded in terms of the ortho-normalized LLL eigenfunctions
$U_{p}({\bf r})$. In the Landau gauge

${\bf A}=Hx{\bf e}_{y}$ (4)

with the unit vector ${\bf e}_{y}$ in the $y$-direction, they are expressed as

$\displaystyle\psi({\bf r})$ $\displaystyle=$
$\displaystyle\sum_{p}\varphi_{p}U_{p}({\bf r}),$ $\displaystyle U_{p}({\bf
r})$ $\displaystyle=$
$\displaystyle\biggl{(}\frac{h}{\pi{\tilde{L}}^{2}}\biggr{)}^{1/4}\exp\biggl{(}ipy-\frac{h}{2}\biggl{(}x+\frac{p}{h}\biggr{)}^{2}\biggr{)},$
(5)

where $\varphi_{p}$ is the LLL fluctuation field with the index $p$ expressing
the degree of freedom of the guiding center and counting the degeneracy in
LLL, ${\tilde{L}}$ is the system size normalized by $\xi_{0}$, $2\pi
r_{H}^{2}=\phi_{0}/H$ is the area per vortex, and
$h=H/H_{c2}(0)=2\pi\xi_{0}^{2}H/\phi_{0}$ is the dimensionless strength of the
magnetic field. Here, the number $N_{v}$ of degeneracy in LLL is given by

$N_{v}=\sum_{p}=\frac{HS}{\phi_{0}}=\frac{{\tilde{S}}h}{2\pi},$ (6)

where ${\tilde{S}}$ is the system area divided by $\xi_{0}^{2}$. Then, using
the Fourier transform of $\rho$, i.e.,

$\rho({\bf
k})=\frac{1}{{\tilde{S}}^{1/2}}\sum_{p}{}\exp(ipk_{1}/h)\,\varphi^{*}_{p-\frac{k_{2}}{2}}\,\varphi_{p+\frac{k_{2}}{2}},$
(7)

${\cal H}[\psi]$ becomes

${\cal H}[\psi]=\sum_{p}\,\mu_{0}|\varphi_{p}|^{2}+t\pi b_{\rm
G}^{(2)}\sum_{\bf k}v_{\bf k}\,\rho({\bf k})\rho(-{\bf k})+\sum_{\bf
k}u_{-{\bf k}}v_{\bf k}^{1/2}\rho({\bf k}),$ (8)

where ${\bf k}=(k_{1},k_{2})$, $\mu_{0}=-1+t+h$ is the bare mass, i.e., the
mass of the unrenormalized LLL fluctuation, and

$v_{\bf k}=\exp\biggl{(}-\frac{{\bf k}^{2}}{2h}\biggr{)}.$ (9)

Further, eq.(3) is rewritten as

$\overline{u_{\bf k}u_{-{\bf
k}^{\prime}}}=\Delta^{(2)}\delta_{k_{1},k_{1}^{\prime}}\delta_{k_{2},k_{2}^{\prime}}.$
(10)

Since, more or less, the disordered phase of a random system is treated in
this work, the replica trick [27] will be introduced for the free energy $F$
in the manner

$-\frac{F}{k_{\rm B}T}=\overline{{\rm
log}Z}=\lim_{n\to+0}\frac{\overline{Z^{n}}-1}{n},$ (11)

where

$\displaystyle\overline{Z^{n}}$ $\displaystyle=$
$\displaystyle\int\prod_{a=1}^{n}{\cal
D}(\varphi^{*a},\varphi^{a})\,\exp(-S_{0}-S_{I}),$ $\displaystyle S_{0}$
$\displaystyle=$
$\displaystyle\sum_{a=1}^{n}\sum_{p}\mu_{0}\,|\varphi_{p}^{a}|^{2},$
$\displaystyle S_{I}$ $\displaystyle=$ $\displaystyle\\!\frac{t\pi
b^{(2)}_{G}}{2{\tilde{S}}}\\!\\!\sum_{a,b,c,d=1}^{n}\sum_{p,p^{\prime},{\bf
k}}\exp(i(p-p^{\prime})k_{1}/h)[\delta_{a,c}\delta_{b,d}h_{bare,ab}({\bf k})$
(12) $\displaystyle+$
$\displaystyle\delta_{a,d}\delta_{b,c}{\hat{h}}_{bare,ab}({\bf
k})]\varphi^{*a}_{p_{-}}\varphi^{*b}_{p^{\prime}_{+}}\varphi^{c}_{p_{+}}\varphi^{d}_{p^{\prime}_{-}},$

where the indices $a,b,c,$ and $d$ denote the replica indices, $p_{\pm}=p\pm
k_{2}/2$, $p^{\prime}_{\pm}=p^{\prime}\pm k_{2}/2$,

$\displaystyle h_{bare,ab}({\bf k})$ $\displaystyle=$ $\displaystyle v_{\bf
k}\,(\delta_{a,b}-\theta(t)),$ $\displaystyle\theta(t)$ $\displaystyle=$
$\displaystyle\frac{\Delta^{(2)}}{2\pi tb_{G}^{(2)}},$ (13)

and

${\hat{h}}_{bare,ab}({\bf k})=\frac{1}{N_{v}}\sum_{{\bf
k}^{\prime}}\exp(i({\bf k}^{\prime}\times{\bf k})_{z}/h)\,h_{bare,ab}({\bf
k}^{\prime}).$ (14)

Here, for later convenience, ${\hat{h}}_{bare,ab}$ ($=h_{bare,ab}$) was
introduced to clarify the symmetry between the two ”particle-hole” channels.
In fact, it is found by examining [28] the perturbation series for the four-
point vertex function that this symmetric form is preserved in any order of
the perturbation series. That is, in the corresponding form to the bare
interaction action $S_{I}$, the fully renormalized interaction action takes
the form

$S_{R,I}=\frac{t\pi
b^{(2)}_{G}}{\tilde{S}}\sum_{a,b,c,d=1}^{n}\sum_{p,p^{\prime},{\bf
k}}\exp(i(p-p^{\prime})k_{1}/h)\,\Gamma_{ab,cd}({\bf
k})\,\varphi^{*a}_{p_{-}}\varphi^{*b}_{p^{\prime}_{+}}\varphi^{c}_{p_{+}}\varphi^{d}_{p^{\prime}_{-}},$
(15)

where

$\displaystyle\Gamma_{ab,cd}({\bf k})$ $\displaystyle=$
$\displaystyle\frac{1}{2}[\delta_{a,c}\delta_{b,d}h_{R,ab}({\bf
k})+\delta_{a,d}\delta_{b,c}{\hat{h}}_{R,ab}({\bf k})],$
$\displaystyle{\hat{h}}_{R,ab}({\bf k})$ $\displaystyle=$
$\displaystyle\frac{1}{N_{v}}\sum_{{\bf k}^{\prime}}\exp(i({\bf
k}^{\prime}\times{\bf k})_{z}/h)\,h_{R,ab}({\bf k}^{\prime}),$ (16)

and $h_{R,ab}({\bf k})$ is a quantity to be derived self-consistently and
takes the form

$h_{R,ab}({\bf k})=\delta_{a,b}f({\bf k})-\theta(t)\,w({\bf k}).$ (17)

We note that $f({\bf k})$ and $w({\bf k})$ are also dependent on $\theta(t)$.

Next, we explain how the physical quantities directly reflecting the
positional correlations between the vortices are defined in the present
replica formalism. First, let us consider the four-body correlation function
$\langle\varphi_{1}^{*}\varphi_{2}^{*}\varphi_{3}\varphi_{4}\rangle$ which can
be defined in clean limit with no randomness. To directly see the positional
correlation between the vortices, it is often convenient to focus on the
connected part of this correlation function in the manner

$\langle\varphi_{1}^{*}\varphi_{2}^{*}\varphi_{3}\varphi_{4}\rangle_{c}\equiv\langle\varphi_{1}^{*}\varphi_{2}^{*}\varphi_{3}\varphi_{4}\rangle-\langle\varphi_{1}^{*}\varphi_{3}\rangle\langle\varphi_{2}^{*}\varphi_{4}\rangle-\langle\varphi_{1}^{*}\varphi_{4}\rangle\langle\varphi_{2}^{*}\varphi_{3}\rangle.$
(18)

On the other hand, this quantity is replica-diagonal in the replica formalism
and is expressed in the form

$\overline{\langle\varphi_{1}^{*}\varphi_{2}^{*}\varphi_{3}\varphi_{4}\rangle}=\lim_{n\to+0}\frac{1}{n}\sum_{a=1}^{n}\langle\varphi_{1}^{a*}\varphi_{2}^{a*}\varphi_{3}^{a}\varphi_{4}^{a}\rangle.$
(19)

Further, in the case of focusing on the fully renormalized vertex function
$\Gamma({\bf k})$, it is convenient to use the relation between the connected
part of the correlation function and $\Gamma({\bf k})$

$\overline{\langle\varphi_{p_{-}}^{*}\varphi_{p^{\prime}_{+}}^{*}\varphi_{p_{+}}\varphi_{p^{\prime}_{-}}\rangle_{c}}=-\frac{4\pi
tb_{G}^{(2)}}{{\tilde{S}}\mu^{4}}\sum_{k_{1}}\,\exp(i(p-p^{\prime})k_{2}/h)\,\Gamma({\bf
k}),$ (20)

where $\mu=1/\langle|\varphi_{p}|^{2}\rangle$, and $\Gamma({\bf
k})=\Gamma_{aa,aa}({\bf k})$.

###  2.1 Structural Factor

The quantity measuring the vortex positional order is the two-body correlation
function on the density $\rho=|\psi|^{2}$ defined in the manner

$f({\bf R})\equiv\frac{\overline{[\langle(\rho({\bf r})-\rho_{0})(\rho({\bf
r}+{\bf R})-\rho_{0})\rangle]_{\bf
r}}}{\rho_{0}^{2}}=\rho_{0}^{-2}\overline{[\langle\rho({\bf r})\rho({\bf
r}+{\bf R})\rangle]_{\bf r}}-1,$ (21)

where $\rho_{0}=\overline{\langle\rho({\bf k}=0)\rangle}$, and, in LLL, it
takes the form

$f({\bf R})=\rho_{0}^{-2}\sum_{\bf k}e^{i{\bf k}\cdot{\bf R}}\,v_{\bf
k}\,\overline{\langle\rho({\bf k})\rho(-{\bf k})\rangle}-1.$ (22)

Further, using eqs.(7), (18), and (20), we find

$\displaystyle\overline{\langle\rho({\bf k})\rho(-{\bf k})\rangle}$
$\displaystyle=$
$\displaystyle{\tilde{S}}\sum_{p,p^{\prime}}\,\exp(i(p-p^{\prime})k_{1}/h)\,\overline{\langle\varphi_{p_{-}}^{*}\varphi_{p^{\prime}_{+}}^{*}\varphi_{p_{+}}\varphi_{p^{\prime}_{-}}\rangle}$
(23) $\displaystyle=$
$\displaystyle\rho_{0}^{2}[\delta_{k_{1},0}\delta_{k_{2},0}+1-2x\Gamma({\bf
k})],$

where $\rho_{0}={\tilde{S}}^{-1/2}N_{v}\mu^{-1}$, and

$x=\frac{t\,h\,b_{G}^{(2)}}{\mu^{2}}.$ (24)

In this way, we choose

$f({\bf k})=1-2x\Gamma({\bf k})$ (25)

as the quantity corresponding to the structural factor, because the original
correlation function is given by

$f({\bf R})=\sum_{\bf k}e^{i{\bf k}\cdot{\bf R}}v_{\bf k}f({\bf k}).$ (26)

Figure 1: Diagrams expressing the self-energy of the SC fluctuation in LLL.
Each line with an arrow indicates a LLL fluctuation propagator, the solid dot
denotes the bare interaction vertex, the open circle indicated as $\Gamma$
denotes the fully renormalized vertex, and the dashed curve is the pair-wised
random potential occurring after the random average. The hatched semicircle is
a vertex correction to the bare random potential, and its definition is
explained by Fig.2.  Figure 2: Details of the hatched semicircle which
includes the vertex correction to the bare random potential.

###  2.2 Renormalized Mass

Next, the diagrams expressing the mass renormalization will be examined. Since
these diagrams shown in Figs.1 and 2 are diagonal in the replica indices,
their expressions may be written in terms only of the notation used for the
clean limit.

As usual, the mass renormalization is expressed in terms of the self-energy
terms $\Sigma_{1}^{(2)}+\Sigma_{2}^{(2)}$ in the form

$\mu=t+h-1+\Sigma_{1}^{(2)}+\Sigma_{2}^{(2)}.$ (27)

The self-energy contribution $\Sigma_{1}^{(2)}$ appearing even in clean limit
is given by

$\Sigma_{1}^{(2)}=2\mu x[\,1-N_{v}^{-1}x\sum_{\bf k}v_{\bf k}\Gamma({\bf
k})\,].$ (28)

The first term corresponds to Fig.1 (a) which is the Hartree-Fock term. The
vertex correction (shaded hemisphere region) in Fig.1 (c) is given by Fig.2
and the vertex $\Lambda(p,p^{\prime})$ expressing Fig.2 is

$\displaystyle\Lambda(p,p^{\prime})$ $\displaystyle=$
$\displaystyle\langle\varphi_{p}\varphi_{p^{\prime}}^{*}\sum_{\bf k}u_{-{\bf
k}}v_{\bf k}^{1/2}\rho({\bf k})\rangle-\pi
tb_{G}^{(2)}\langle\varphi_{p}\varphi_{p^{\prime}}^{*}\sum_{{\bf
k}^{\prime}}\Gamma({\bf k}^{\prime})\rho({\bf k}^{\prime})\rho(-{\bf
k}^{\prime})\sum_{\bf k}u_{-{\bf k}}v_{\bf k}^{1/2}\rho({\bf k})\rangle$ (29)
$\displaystyle=$ $\displaystyle{\tilde{S}}^{-1/2}\mu^{-2}\sum_{\bf
k}\delta_{p^{\prime},p+k_{2}}u_{-{\bf k}}\,v_{\bf
k}^{1/2}\,\exp(ik_{1}(p+p^{\prime})/2h)$ $\displaystyle\times$
$\displaystyle\biggl{[}1-\frac{2x}{N_{v}}\sum_{{\bf k}^{\prime}}\Gamma({\bf
k}^{\prime})\exp(i(k_{1}^{\prime}(p^{\prime}-p)-k_{2}^{\prime}k_{1})/h)\biggr{]}.$

Using this expression, $\Sigma_{2}^{(2)}$ corresponding to Fig.1 (c) is simply
expressed as

$\displaystyle\Sigma_{2}^{(2)}$ $\displaystyle=$ $\displaystyle-\frac{\mu
x\theta(t)}{N_{v}}\sum_{\bf k}v_{\bf k}\biggl{(}1-\frac{2x}{N_{v}}\sum_{{\bf
k}^{\prime}}\Gamma({\bf k}^{\prime})\exp(i({\bf k}^{\prime}\times{\bf
k})_{z}/h)\biggr{)}$ (30) $\displaystyle=$ $\displaystyle-\,\frac{\mu
x\theta(t)}{N_{v}}\sum_{\bf k}v_{\bf k}(1-2x\Gamma({\bf k})).$

Using the fact that, since the bare interaction vertex $v_{\bf k}$ is
isotropic in ${\bf k}$ and a function of ${\bf k}^{2}$, $\Gamma({\bf k})$ is
also a function of ${\bf k}^{2}$ as far as no transition occurs, the mass
renormalization is finally defined by the self-consistent relation

$\mu=t-1+h+(2-\theta)\mu x-\frac{1-\theta}{h}\mu
x^{2}\int_{0}^{\infty}d(p^{2})v_{\bf p}\Gamma({\bf p}).$ (31)

Figure 3: Examples of the nonparquet diagrams. The solid dots represent the
bare vertices. For simplicity, the arrows of the fluctuation propagators are
not drawn.  Figure 4: Figure defining the parquet diagrams. The top row
consisting of the figures on $\Pi_{1}$ defines the particle-particle channel,
while the remaining two rows represent the two particle-hole channels.

##  3 Parquet Diagram Resummation

To obtain an expression of $\Gamma({\bf k})$ as accurately as possible, the
parquet resummation technique will be used hereafter following Ref.18. In this
approach [18, 19], the nonparquet diagrams illustrated in Fig.3 and any higher
order ones composed of them are neglected. Just like the diagrammatic analysis
for the Fermi liquid theory [29], the remaining diagrams, i.e., the bare
vertex carrying $v_{\bf k}$ and the parquet diagrams belonging to Fig.4, will
be taken into account to obtain a fully-renormalized interaction vertex
function $\Gamma$ as accurately as possible. It will be shown below how the
formal results derived in Ref.18 are reproduced in the present analysis using
the Landau gauge (4).

First, the clean limit with no disorder will be considered. As sketched in
Fig.4, the parquet diagrams are composed of one particle-particle channel
$\Pi_{1}$ and the two particle-hole channels $\Pi_{j}$ ($j=2$ and $3$).

The interaction term of the Hamiltonian expressed by the vertex $\Pi_{j}$
takes the form

$\sum_{k_{1}}\sum_{p_{1},p_{2},p_{3}}\exp(i(p_{2}-p_{3})k_{1}/h)\,\Pi_{j}(k_{1},p_{1}-p_{3})\,\varphi_{p_{1}}^{*}\varphi_{p_{2}}^{*}\varphi_{p_{3}}\varphi_{p_{1}+p_{2}-p_{3}}.$
(32)

According to this parametrization, the first row of Fig.4 expressing the
relation that $\Pi_{1}$ satisfies is expressed as

$\displaystyle\sum_{k_{1}^{\prime}}\exp\biggl{(}i\frac{(p_{2}-p_{3})k_{1}^{\prime}}{h}\biggr{)}\Pi_{1}(k_{1}^{\prime},p_{1}-p_{3})\\!$
$\displaystyle=$
$\displaystyle\\!\frac{x}{N_{v}}\\!\\!\sum_{k_{1}^{\prime},k_{1}^{{}^{\prime\prime}},p^{\prime}}\exp(i(p_{2}-p^{\prime})k_{1}^{\prime})/h)\,\Lambda_{1}(k_{1}^{\prime},p_{1}-p^{\prime})$
(33) $\displaystyle\times$
$\displaystyle\\!\\!\exp\biggl{(}i\frac{(p_{1}+p_{2}-p_{3}-p^{\prime})k_{1}^{\prime\prime}}{h}\biggr{)}\,\Gamma(k_{1}^{\prime\prime},p^{\prime}-p_{3}),$

where

$\displaystyle\Gamma({\bf k})$ $\displaystyle=$ $\displaystyle v_{\bf
k}+\sum_{j=1,2,3}\Pi_{j}({\bf k}),$ $\displaystyle\Lambda_{i}({\bf k})$
$\displaystyle=$ $\displaystyle v_{\bf k}+\sum_{j\neq i}\Pi_{j}({\bf k}).$
(34)

To change eq.(33) to a simpler form, the wave numbers $p_{1}-p_{3}$,
$p_{2}-p_{3}$, and $p^{\prime}-p_{3}$ are rewritten as $k_{2}$, $p$, and
$k_{2}^{\prime}$, respectively, and the resulting eq.(33) will be multiplied
by the factor $\exp(-ipk_{1}/h)$. Further, by summing over $p$, one obtains

$\Pi_{1}({\bf k})=-\frac{x}{N_{v}}\sum_{{\bf k}^{\prime}}\Lambda_{1}({\bf
k}-{\bf k}^{\prime})\Gamma({\bf k}^{\prime})\exp(i({\bf k}\times{\bf
k}^{\prime})_{z}/h)\equiv-x(\Lambda_{1}\bullet\Gamma)({\bf k}).$ (35)

It is much easier to obtain the corresponding expressions to be satisfied by
$\Pi_{2}$ and $\Pi_{3}$. Consequently, the second and third rows of Fig.4 are
expressed in the form

$\displaystyle\Pi_{2}({\bf k})$ $\displaystyle=$
$\displaystyle-2x\Lambda_{2}({\bf k})\Gamma({\bf k}),$
$\displaystyle\Pi_{3}({\bf k})$ $\displaystyle=$
$\displaystyle-\frac{2x}{N_{v}}\sum_{{\bf k}^{\prime}}\Lambda_{3}({\bf k}-{\bf
k}^{\prime})\Gamma({\bf k}^{\prime})$ (36) $\displaystyle\equiv$
$\displaystyle-2x(\Lambda_{3}\,*\,\Gamma)({\bf k}).$

The formulation explained above is straightforwardly extended to the
disordered case. For instance, one has only to assign replica indices $a$,
$b$, $c$, and $d$ to the external propagators $1$, $2$, $3$, and $4$ in Fig.4,
respectively. Then, we find

$\displaystyle\Pi_{1,{ab,cd}}({\bf k})$ $\displaystyle=$
$\displaystyle-x\sum_{e,f=1}^{n}(\Lambda_{1,{ab,ef}}\bullet\Gamma_{ef,cd})({\bf
k}),$ $\displaystyle\Pi_{2,{ab,cd}}({\bf k})$ $\displaystyle=$
$\displaystyle-2x\sum_{e,f=1}^{n}\Lambda_{2,{ae,cf}}({\bf
k})\Gamma_{fb,ed}({\bf k}),$ $\displaystyle\Pi_{3,{ab,cd}}({\bf k})$
$\displaystyle=$
$\displaystyle-2x\sum_{e,f=1}^{n}(\Lambda_{3,{ae,fd}}\,*\,\Gamma_{fb,ce})({\bf
k}),$ $\displaystyle\Gamma_{ab,cd}({\bf k})$ $\displaystyle=$
$\displaystyle\delta_{ac}\delta_{bd}(\delta_{ab}-\theta(t))v_{\bf
k}+\sum_{i=1}^{3}\Pi_{i,{ab,cd}}({\bf k}),$
$\displaystyle\Lambda_{i,{ab,cd}}({\bf k})$ $\displaystyle=$
$\displaystyle\Gamma_{ab,cd}({\bf k})-\Pi_{i,{ab,cd}}({\bf k}).$ (37)

Further, using symmetrical relations to be satisfied between $\Pi_{i,{ab,cd}}$
and $\Lambda_{i,{ab,cd}}$ (see Appendix A), these vertex functions are
represented in the form

$\displaystyle\Pi_{1,abcd}({\bf k})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\delta_{ac}\delta_{bd}[\delta_{ab}\Gamma_{1}({\bf
k})-\theta\Xi_{1}({\bf k})]$ $\displaystyle+$
$\displaystyle\frac{1}{2}\delta_{ad}\delta_{bc}[\delta_{ab}{\hat{\Gamma}}_{1}({\bf
k})-\theta{\hat{\Xi}}_{1}({\bf k})],$ $\displaystyle\Pi_{2,abcd}({\bf k})$
$\displaystyle=$ $\displaystyle{\hat{\Pi}}_{3,abdc}({\bf
k})=\frac{1}{2}\delta_{ac}\delta_{bd}[\delta_{ab}\Gamma_{2}({\bf
k})-\theta\Xi_{2}({\bf k})]$ $\displaystyle+$
$\displaystyle\frac{1}{2}\delta_{ad}\delta_{bc}[\delta_{ab}{\hat{\Gamma}}_{3}({\bf
k})-\theta{\hat{\Xi}}_{3}({\bf k})],$ $\displaystyle\Lambda_{1,abcd}({\bf k})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\delta_{ac}\delta_{bd}[\delta_{ab}I_{1}({\bf
k})-\theta J_{1}({\bf k})]$ $\displaystyle+$
$\displaystyle\frac{1}{2}\delta_{ad}\delta_{bc}[\delta_{ab}{\hat{I}}_{1}({\bf
k})-\theta{\hat{J}}_{1}({\bf k})],$ $\displaystyle\Lambda_{2,abcd}({\bf k})$
$\displaystyle=$ $\displaystyle{\hat{\Lambda}}_{3,abdc}({\bf
k})=\frac{1}{2}\delta_{ac}\delta_{bd}[\delta_{ab}I_{2}({\bf k})$ (38)
$\displaystyle-$ $\displaystyle\theta J_{2}({\bf
k})]+\frac{1}{2}\delta_{ad}\delta_{bc}[\delta_{ab}{\hat{I}}_{3}({\bf
k})-\theta{\hat{J}}_{3}({\bf k})].$

Then, these representations will be substituted into eq.(37). By performing
the summations on the replica indices and by taking the $n\to 0$ limit, we
obtain the following forms independent of the replica indices:

$\displaystyle\Gamma_{1}({\bf k})$ $\displaystyle=$
$\displaystyle-x(I_{1}\bullet f_{R}-\theta I_{1}\bullet v_{R}-\theta
J_{1}\bullet f_{R})({\bf k}),$ $\displaystyle\Gamma_{2}({\bf k})$
$\displaystyle=$ $\displaystyle-x(I_{2}({\bf k})f_{R}({\bf k})+I_{2}({\bf
k}){\hat{h}}_{R}({\bf k})$ $\displaystyle+$ $\displaystyle[{\hat{I}}_{3}({\bf
k})-\theta{\hat{J}}_{3}({\bf k})]f_{R}({\bf k})),$
$\displaystyle\Gamma_{3}({\bf k})$ $\displaystyle=$
$\displaystyle-x(I_{2}\,*\,f_{R}-\theta I_{3}\,*\,v_{R}-\theta
J_{3}\,*\,f_{R})({\bf k}),$ (39)

and

$\displaystyle\Xi_{1}({\bf k})$ $\displaystyle=$ $\displaystyle
x\theta(J_{1}\bullet v_{R})({\bf k}),$ $\displaystyle\Xi_{2}({\bf k})$
$\displaystyle=$ $\displaystyle-x(I_{2}({\bf k})v_{R}({\bf k})+J_{2}({\bf
k})[f_{R}({\bf k})+{\hat{h}}_{R}({\bf k})]$ $\displaystyle+$
$\displaystyle[{\hat{I}}_{3}({\bf k})-\theta{\hat{J}}_{3}({\bf k})]v_{R}({\bf
k})),$ $\displaystyle\Xi_{3}({\bf k})$ $\displaystyle=$ $\displaystyle
x\theta(J_{3}\,*\,v_{R})({\bf k}),$ (40)

where

$\displaystyle f_{R}({\bf k})$ $\displaystyle=$ $\displaystyle v_{\bf
k}+\sum_{j=1}^{3}\Gamma_{j}({\bf k}),$ $\displaystyle v_{R}({\bf k})$
$\displaystyle=$ $\displaystyle v_{\bf k}+\sum_{j=1}^{3}\Xi_{j}({\bf k}),$
$\displaystyle I_{i}({\bf k})$ $\displaystyle=$ $\displaystyle v_{\bf
k}+\sum_{j\neq i}\Gamma_{j}({\bf k}),$ $\displaystyle J_{i}({\bf k})$
$\displaystyle=$ $\displaystyle v_{\bf k}+\sum_{j\neq i}\Xi_{j}({\bf k}).$
(41)

##  4 Approximate treatment of 3D case

The analysis in sec.2 and 3 holds only in the 2D SC films. It is quite hard to
directly extend the approach in the preceding sections to the 3D case in a
numerically tractable way because of the additional appearance of the
component parallel to the magnetic field of the wavevector of the SC
fluctuation. Nevertheless, by assuming that the nature of freezing to the
vortex lattice is essentially the same as each other in both dimensions, it is
not difficult to give an approximate expression of the fluctuation
conductivity in 3D case. In this section, the method we have used to obtain
the 3D conductivity will be explained.

The simplest extension of the 2D GL Hamiltonian (1) is

${\cal H}^{(3)}[\psi]=\xi_{0}^{-3}\\!\\!\int d^{3}r\,\biggl{[}(t-1)\rho({\bf
r})+u({\bf r})\rho({\bf r})+\pi t\sqrt{b^{(3)}_{G}}|\rho({\bf
r})|^{2}+\xi_{0}^{2}\,\biggl{|}\biggl{(}-i\nabla+\frac{2\pi}{\phi_{0}}{\bf
A}({\bf r})\biggr{)}\psi({\bf r})\biggr{|}^{2}\biggr{]},$ (42)

where $\psi=\Delta\sqrt{\xi_{0}^{3}N(0)/k_{\rm B}T}$, $\rho=|\psi|^{2}$, and
$b_{\rm G}^{(3)}=[g_{0}k_{\rm B}T_{c0}/(2\pi\xi_{0}^{3}N(0))]^{2}$. In place
of eq.(3), the random-average is defined by $\overline{u({\bf r})}=0$ and

$\overline{u({\bf r})u({\bf
r}^{\prime})}=\xi_{0}^{3}\Delta^{(3)}\delta^{(3)}({\bf r}-{\bf r}^{\prime}).$
(43)

The order parameter field $\psi$ in 3D case is expanded via the product of the
LLL eigenfunction $U_{p}$ and the plain wave propagating along the field in
the manner

$\psi({\bf
r})={\tilde{L}}_{z}^{-1/2}\sum_{p,q}\varphi_{p,q}e^{iqz}U_{p}(x,y),$ (44)

where ${\tilde{L}}_{z}=L_{z}/\xi_{0}$. Then, the expression corresponding to
eq.(8) is

$\displaystyle{\cal H}^{(3)}[\psi]$ $\displaystyle=$
$\displaystyle\sum_{p,q}(\mu_{0}+q^{2})|\varphi_{p,q}|^{2}+\frac{t\pi\sqrt{b_{\rm
G}^{(3)}}}{{\tilde{S}}{\tilde{L}}_{z}}\sum_{\bf
K}\sum_{p,p^{\prime},q,q^{\prime}}\exp(i(p-p^{\prime})k_{1}/h)\,v_{{\bf
k}}\,\varphi_{p_{-},q_{-}}^{*}\varphi_{p^{\prime}_{+},q^{\prime}_{+}}^{*}\varphi_{p_{+},q_{+}}\varphi_{p^{\prime}_{-},q^{\prime}_{-}}$
(45) $\displaystyle+$
$\displaystyle\frac{1}{\sqrt{{\tilde{S}}{\tilde{L}}_{z}}}\sum_{p,q,{\bf
K}}u_{-{\bf K}}v_{{\bf
k}}^{1/2}\exp(ipk_{1}/h)\varphi_{p_{-},q_{-}}^{*}\varphi_{p_{+},q_{+}},$

where $q_{\pm}=q\pm k_{3}/2$. and ${\bf K}={\bf k}+k_{3}\,{\hat{z}}$. In
contrast to the 2D case, however, the coefficient of the quadratic term of the
renormalized Hamiltonian corresponding to the bare one eq.(45) is a
$q$-dependent function $\mu(q)$, and the renormalized vertex $\Gamma$
corresponding to the bare one $v_{{\bf k}}$ should also become dependent on
$q$ and $q^{\prime}$. It is not easy to follow such $q$-dependences entangled
with the ${\bf k}$-dependences. For this reason, a general analysis for the 3D
case becomes numerically formidable.

Here, as a numerically tractable approximation, the fully renormalized vertex
$\Gamma$ will be assumed hereafter not to be changed from the 2D result
obtained in the manner mentioned in the preceding section. This physically
means that the fluctuation effect on the vortex lattice formation is
overestimated in the present approximation for 3D systems. In this
approximation, eq.(35) is simply replaced by

$\Pi^{(3)}_{1}({\bf k})=-x^{(3)}(\Lambda^{(3)}_{1}\bullet\Gamma^{(3)})({\bf
k}),$ (46)

where $x^{(3)}=t\,h\,\sqrt{b_{G}^{(3)}}\sum_{q}[{\cal
G}(q)]^{2}/{\tilde{L}}_{z}$, and ${\cal G}(q)$ is the SC fluctuation
propagator in LLL. Here and below, the quantity $C^{(3)}$ implies the
corresponding expression in 3D case of the quantity $C$ in 2D case. Within the
present approximation, other relations associated with the parquet diagrams in
3D case are also obtained in the same way, i.e., simply by replacing $x$ and
$\theta$ in the corresponding ones in 2D case with $x^{(3)}$ and
$\theta^{(3)}$, respectively, where

$\theta^{(3)}=\frac{\Delta^{(3)}}{2\pi t\sqrt{b_{G}^{(3)}}}.$ (47)

Further, the $q$-dependence in ${\cal G}(q)$ will also be assumed not to be
renormalized in a consistent way with the above procedure for $\Gamma$. A
similar treatment for the SC fluctuation propagator ${\cal G}$ was used
previously [28]. Then, we have $[{\cal G}(q)]^{-1}=\mu^{(3)}+q^{2}$ with

$\mu^{(3)}=t-1+h+\Sigma^{(3)}.$ (48)

Then, the $q$-integrals appearing in the 3D counterpart of Fig.1 are easily
performed, and the self-energy $\Sigma^{(3)}$ becomes

$\Sigma^{(3)}=2\mu^{(3)}x^{(3)}(2-\theta^{(3)})-\frac{4\mu^{(3)}(x^{(3)})^{2}(1-\theta^{(3)})}{3h}\int_{0}^{\infty}d({\bf
k}^{2})v_{\bf k}\Gamma^{(3)}({\bf k}),$ (49)

where

$x^{(3)}=\frac{t\,h\,\sqrt{b_{G}^{(3)}}}{4(\mu^{(3)})^{3/2}},$ (50)

and $\Gamma^{(3)}$ is the fully-renormalized vertex function corresponding to
$\Gamma$ in 2D case and is determined from the parquet equations including
eq.(46) constructed in the manner explained above.

As already mentioned, it is anticipated that the present approximation
underestimates the 3D SC ordering, i.e., the growth of the lateral correlation
on the vortex position, compared with that to be obtained from a more
realistic method for 3D case. Nevertheless, it will be seen in sec.4 that the
resistivity in 3D case obtained numerically is strongly affected by the growth
of the positional correlation length and consequently that our view on the
vanishing of the resistivity in 3D case becomes clear.

##  5 Vortex Glass Correlation Function

In order to derive the VG correlation function $G_{\rm vg}$ and the resulting
term of the conductivity accompanied by $G_{\rm vg}$, the starting GL model in
sec.2 will be rewritten as a quantum action by giving a purely dissipative
dynamics to the order parameter. In the unit $\hbar=1$, the partition function
$Z$ in $D$-dimensional case is expressed as $Z={\rm Tr}\,\exp(-{\cal S}_{\rm
GL})$ using the quantum GL action [30, 20, 13]

${\cal S}_{\rm GL}=\xi_{0}^{-D}\int d^{D}{\bf
r}\sum_{\omega}\,\gamma|\omega|\,\psi^{*}({\bf r},\omega)\psi({\bf
r},\omega)+T\int_{0}^{1/T}d\tau\,{\cal H}[\psi(\tau)],$ (51)

where ${\cal H}[\psi]$ in $D=2$ case was given as eq.(2), $\omega$ is the
Matsubara frequency for bosons, $\tau$ is the imaginary time, $\gamma>0$, and
the Fourier transform $\psi_{\omega}$ of $\psi(\tau)$ is defined by

$\psi(\tau)=\sum_{\omega}\psi_{\omega}e^{-i\omega\tau}.$ (52)

It is expected, close to the VG transition, that the dominant frequency
dependence determining the time scale of the conductivity is carried by the VG
critical fluctuation and that detailed frequency dependences of the self-
energy of the noncritical SC fluctuation will play less important roles. For
this reason, the frequency dependence of the fully renormalized vertex part
$\Gamma$ representing the positional correlation between the vortices is
assumed to be negligible. This assumption should be valid in clean enough
cases of our interest. Next, the frequency dependence of the SC fluctuation
propagator ${\cal G}$ will be kept the bare one. That is, the bare form ${\cal
G}=1/(\mu+\gamma|\omega|)$ will be always used in 2D case.

Figure 5: Structure of the vortex-glass correlation function $G_{\rm vg}$
which is represented in the ladder approximation valid at weak disorder.
Figure (a) is the four-point vertex constructed based on the hatched
semicircles of Fig.2 after the random average. Here, the connected dashed line
denotes a pair-wised random potential occurring after the random-average and
carrying $\Delta^{(2)}$ in 2D case. As shown in (b), the VG correlation
function is expressed as a series of the figure (a).

The VG correlation function is given by

$G_{\rm vg}({\bf R};\tau_{1},\tau_{2})\propto\int d^{D}{\bf
r}\,\overline{|\langle\psi^{*}({\bf r},\tau_{1})\psi({\bf r}+{\bf
R},\tau_{2})\rangle|^{2}}.$ (53)

For a while, we focus on the 2D ($D=2$) case. The genuine VG transition, i.e.,
the 2D SC transition, does not occur in real systems at finite temperatures in
a field perpendicular to the plane [5, 31]. Nevertheless, the VG fluctuation
in 2D case will be considered here in the Gaussian approximation where the VG
transition line inevitably appears, in order to make a comparison between the
2D and 3D cases useful. The Fourier transform of eq.(53) with $D=2$ becomes

$G_{\rm vg}({\bf k};\omega_{1},\omega_{2})=\frac{1}{N_{v}}[{\cal
G}(\omega_{1}){\cal
G}(\omega_{2})]^{-1}\sum_{p,p^{\prime}}\exp(i(p-p^{\prime})k_{1}/h)\,{\overline{{\cal
G}_{p^{\prime},p}(\omega_{1}){\cal
G}_{p+k_{2},p^{\prime}+k_{2}}(\omega_{2})}}$ (54)

in a form normalized properly, where ${\cal
G}_{p,p^{\prime}}(\omega)=\langle\varphi_{p}(\omega)\varphi^{*}_{p^{\prime}}(\omega)\rangle$
is the LLL fluctuation propagator defined prior to taking the random average,
and ${\cal G}(\omega)=\overline{{\cal
G}_{p,p^{\prime}}(\omega)}=1/(\mu+\gamma|\omega|)$ is the random-averaged LLL
fluctuation propagator. First, the irreducible disorder vertex forming ${\cal
G}_{p,p^{\prime}}(\omega)$ will be explained. As far as the limit of weak
disorder is concerned, it is sufficient to regard the diagram with a hatched
semicircle of Fig.2 as the irreducible one for ${\cal
G}_{p,p^{\prime}}(\omega)$. By incorporating the frequency dependences in the
SC fluctuation propagators of Fig.2, the corresponding expression to eq.(29)
becomes

$\displaystyle\Lambda(p,p^{\prime};\omega_{1})$ $\displaystyle=$
$\displaystyle{\tilde{S}}^{-1/2}[{\cal G}(\omega_{1})]^{2}\sum_{\bf
k}\delta_{p,p^{\prime}+k_{2}}\,u_{-{\bf k}}\,v_{\bf
k}^{1/2}\exp(ik_{1}(p+p^{\prime})/2h)$ (55) $\displaystyle\times$
$\displaystyle\biggl{[}1-\frac{2x\mu^{2}}{N_{v}}\sum_{{\bf
k}^{\prime},{\omega_{1}}^{\prime}}[{\cal
G}(\omega^{\prime}_{1})]^{2}\,\Gamma({\bf
k}^{\prime})\exp(i(k_{1}^{\prime}(p^{\prime}-p)-k_{2}^{\prime}k_{1})/h)\biggr{]}.$

Since the dependences on the internal frequency $\omega^{\prime}_{1}$ are
unnecessary in obtaining the conductivity in the regime where the quantum SC
fluctuation is negligible, however, only the $\omega^{\prime}_{1}=0$ term will
be kept hereafter. Then, the expression in the square bracket of eq.(55)
simply becomes

$f(p-p^{\prime},k_{1})=1-2\,x\,\Gamma(k_{1},p-p^{\prime}),$ (56)

and the diagram of Fig.5 (a) arising after random -averaging takes the form

$\frac{\Delta^{(2)}}{\tilde{S}}[{\cal G}(\omega_{1}){\cal
G}(\omega_{2})]^{2}\sum_{\tilde{\bf
p}}\delta_{{\tilde{p}}_{2},p-p^{\prime}}\,v_{\bf{\tilde{p}}}\,\exp(-ik_{2}{\tilde{p}}_{1}/h)\,[f(p-p^{\prime},{\tilde{p}}_{1})]^{2},$
(57)

where ${\bf{\tilde{p}}}=({\tilde{p}}_{1},{\tilde{p}}_{2})$. Note that eq.(56)
is the same expression as eq.(25). Taking account of the fact that the VG
correlation function $G_{\rm vg}$ is expressed as a ladder series Fig.5 (b) by
regarding the expression (57) as its unit, it is found that

$G_{\rm vg}({\bf k};\omega,\omega+\Omega)=\frac{1}{1-X({\bf k}){\cal
G}(\omega){\cal G}(\omega+\Omega)},$ (58)

where

$X({\bf k})\equiv\frac{\mu^{2}x\theta}{N_{v}}\sum_{\bf p}v_{\bf p}|f({\bf
p})|^{2}\,\exp(i{\bf k}\cdot{\bf p}/h).$ (59)

Based on this expression of $G_{\rm vg}$, the VG correlation length $\xi_{\rm
vg}$ just above the VG transition is expressed by

$\xi_{\rm vg}=\frac{1}{2}\sqrt{\frac{c_{2}}{(1-c_{0})h}},$ (60)

where

$\displaystyle c_{0}$ $\displaystyle=$
$\displaystyle\frac{x\theta}{N_{v}}\sum_{\bf p}v_{\bf p}|f({\bf p})|^{2},$
$\displaystyle c_{2}$ $\displaystyle=$
$\displaystyle\frac{x\theta}{hN_{v}}\sum_{\bf p}{\bf p}^{2}v_{\bf p}|f({\bf
p})|^{2},$ (61)

and $\xi_{0}$ is the unit of the length scales. Of course, eq.(60) is valid
when $c_{0}<1$ and diverges on approaching the VG transition point at which
$c_{0}=1$.

###  5.1 3D case

As in 2D case, the 3D VG transition will be described below in the Gaussian
fluctuation. First, let us start from determining the Fourier transform of
eq.(53) properly. By choosing the prefactor of eq.(53) in a consistent manner
with eq.(54), the expression in 3D corresponding to eq.(54) is given by

$\displaystyle G_{\rm vg}({\bf K};\omega_{1},\omega_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{N_{v}}\biggl{(}{\sum_{q}{\cal G}(q,\omega_{1}){\cal
G}(q+k_{3},\omega_{2})}\biggr{)}^{-1}\sum_{p,p^{\prime};q,q^{\prime}}[\,\exp(i(p-p^{\prime})k_{1}/h)$
(62) $\displaystyle\times$
$\displaystyle\overline{\langle\varphi^{*}_{p,q}(\omega_{1})\varphi_{p^{\prime},q^{\prime}}(\omega_{1})\rangle\langle\varphi_{p+k_{2},q+k_{3}}(\omega_{2})\varphi^{*}_{p^{\prime}+k_{2},q^{\prime}+k_{3}}(\omega_{2})\rangle}\,],$

where ${\bf K}={\bf k}+k_{3}{\hat{z}}$, ${\bf k}=(k_{1},k_{2})$, and ${\cal
G}(q,\omega)$ is the random averaged fluctuation propagator in 3D case. The
expression in 3D case corresponding to eq.(51) is

$\displaystyle\Lambda(p,p^{\prime};q,q^{\prime};\omega_{1})\\!$
$\displaystyle=$ $\displaystyle\\!({\tilde{S}}{\tilde{L}}_{z})^{-1/2}[{\cal
G}(q,\omega_{1}){\cal
G}(q^{\prime},\omega_{1})]\sum_{\bf{\tilde{P}}}\delta_{p,p^{\prime}+{\tilde{p}}_{2}}\delta_{q,q^{\prime}+{\tilde{p}}_{3}}\,u_{-{\bf{\tilde{P}}}}\,v_{\bf{\tilde{p}}}^{1/2}\exp(i(p+p^{\prime})p_{1}/2h)$
(63) $\displaystyle\times$
$\displaystyle\biggl{[}1-\frac{8(\mu^{(3)})^{3/2}x^{(3)}}{{\tilde{L}}_{z}N_{v}}\sum_{{\bf
K}^{\prime},\omega^{\prime}_{1}}[{\cal
G}(q+k^{\prime}_{3},\omega^{\prime}_{1}){\cal
G}(q^{\prime}+k^{\prime}_{3},\omega^{\prime}_{1})]$ $\displaystyle\times$
$\displaystyle\Gamma^{(3)}({\bf
k}^{\prime})\exp(i[(p^{\prime}-p)k^{\prime}_{1}-{\tilde{p}}_{1}k^{\prime}_{2}]/h)\biggr{]},$

where ${\bf{\tilde{P}}}={\bf{\tilde{p}}}+p_{3}{\hat{z}}$, and
${\bf{\tilde{p}}}=({\tilde{p}}_{1},{\tilde{p}}_{2})$. By neglecting the
dependences on the internal frequency $\omega^{\prime}_{1}$ in
$\Lambda(p,q;p^{\prime},q^{\prime};\omega_{1})$, the vertex correction
corresponding to $f(p-p^{\prime},{\tilde{p}}_{1})$ in 2D case becomes

$f^{(3)}(p-p^{\prime},{\tilde{p}}_{1};q-q^{\prime})\equiv
1-\frac{8x^{(3)}\mu^{(3)}}{4\mu^{(3)}+(q-q^{\prime})^{2}}\Gamma^{(3)}(p-p^{\prime},{\tilde{p}}_{1}),$
(64)

and the expression representing the counterpart of Fig.5 (a) is

$\frac{\Delta^{(3)}}{{\tilde{S}}{\tilde{L}}_{z}}\sum_{{\bf
K}^{\prime}}\delta_{k^{\prime}_{2},p-p^{\prime}}\delta_{k^{\prime}_{3},q-q^{\prime}}v_{{\bf
k}^{\prime}}[f^{(3)}(k^{\prime}_{1},p-p^{\prime};q-q^{\prime})]^{2}e^{-i\frac{k_{2}k^{\prime}_{1}}{h}}{\cal
G}(q,\omega_{1}){\cal G}(q^{\prime},\omega_{1}){\cal
G}(q+k_{3},\omega_{2}){\cal G}(q^{\prime}+k_{3},\omega_{2}).$ (65)

Here, consistently with the approximation for $\Sigma^{(3)}$, the
$q-q^{\prime}$-dependence in eq.(64) will be neglected. Then, it is
straightforward to derive the expression on the Fourier transform of the VG
correlation function in 3D, and we have

$G^{(3)}_{\rm vg}({\bf K};\omega,\omega+\Omega)=\biggl{[}1-\frac{X^{(3)}({\bf
k})}{{\tilde{L}}_{z}}\sum_{q}{\cal G}(q,\omega){\cal
G}(q+k_{3},\omega+\Omega)\biggr{]}^{-1},$ (66)

where

$X^{(3)}({\bf
k})=\frac{4(\mu^{(3)})^{3/2}x^{(3)}\theta^{(3)}}{N_{v}}\sum_{{\bf
k}^{\prime}}v_{{\bf k}^{\prime}}[f^{(3)}({\bf k}^{\prime};0)]^{2}\exp(i({\bf
k}\times{{\bf k}^{\prime}})_{z}/h).$ (67)

As in 2D case, the vanishing behavior of the resistivity, which will be
discussed in the next section, is expected to be determined by the time scale
not of the SC fluctuation but of the VG correlation function. For this reason,
the frequency term of ${\cal G}^{-1}_{\omega}$ as well as the wave number
dependence in 3D case will be assumed to be unrenormalized by the randomness.
Thus, the expression

${\cal G}(q,\omega)=\frac{1}{\mu^{(3)}+q^{2}+\gamma|\omega|}$ (68)

for the SC fluctuation propagator with the wave number $q$ will be used
hereafter in 3D case. However, it will be discussed later that, in dirty
cases, this assumption has an unfavorable effect on the vanishing of the
resistivity.

Then, by focusing on the vanishing behavior of the r.h.s. of eq.(66), the VG
correlation lengths are defined in a similar manner to that in 2D case. The VG
correlation length $\xi_{{\rm vg},\perp}$ in perpendicular directions to the
magnetic field is given by

$\xi_{{\rm vg},\perp}=\frac{1}{2}\sqrt{\frac{c_{2}^{(3)}}{(1-c_{0}^{(3)})h}},$
(69)

while the corresponding correlation length in the direction of the magnetic
field is

$\xi_{{\rm vg},\parallel}=\sqrt{\frac{h}{c_{2}^{(3)}\mu^{(3)}}}\,\,\xi_{{\rm
vg},\perp},$ (70)

where

$\displaystyle c_{0}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{x^{(3)}\theta^{(3)}}{N_{v}}\sum_{\bf p}v_{\bf
p}|f^{(3)}({\bf p};0)|^{2},$ $\displaystyle c_{2}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{x^{(3)}\theta^{(3)}}{hN_{v}}\sum_{\bf p}{\bf p}^{2}v_{\bf
p}|f^{(3)}({\bf p};0)|^{2}.$ (71)

##  6 Vortex Glass Contribution to Conductivity

In general, the vortex pinning effect is formulated according to the two types
of modellings. One is the so-called random $T_{c}$ model corresponding to the
inclusion of the scalar potential $u({\bf r})$ in eq.(1), and what is pinned
in this case is a spatial variation of the amplitude $|\psi|=\sqrt{\rho}$
associated with the vortex core. However, this model is not useful in an
approach based on the London, or a phase-only model where $\rho$ is a
constant. The other model useful even in the phase-only approach is the
following one expressed in terms of a random gauge potential model: In 2D
case, the term

$\delta{\cal S}_{g}=T\int_{0}^{1/T}d\tau\int d^{2}{\bf r}\,\,\,{\bf j}({\bf
r},\tau)\cdot{\bf a}_{r}({\bf r}),$ (72)

should be added to eq.(51), where ${\bf j}=\psi^{*}(-i\nabla+2\pi{\bf
A}/\phi_{0})\psi+{\rm{c.c.}}$ is the supercurrent density. Using the fact
that, due to the continuity equation $\nabla\cdot{\bf j}=0$ to be satisfied in
equilibrium, ${\bf a}_{r}$ may be rewritten as $\nabla\times w({\bf
r}){\hat{z}}$ by introducing a scalar function $w({\bf r})$, the total random
potential term ${\cal S}_{r}$ may be reexpressed as

$\delta{\cal S}_{r}=\int_{0}^{1/T}d\tau\int d^{2}{\bf r}\,[w({\bf
r})[\nabla\times{\bf j}({\bf r},\tau)]_{z}+\xi_{0}^{-2}u({\bf r})\,\rho({\bf
r},\tau)].$ (73)

The random potential $w$ is assumed to yield ${\overline{w({\bf r})}}$=0, and

$\overline{w({\bf r})w({\bf
r}^{\prime})}=\Delta_{\Phi}\xi_{0}^{2}\delta^{(2)}({\bf r}-{\bf r}^{\prime})$
(74)

in 2D case. Since, under a fixed $\rho$, $w({\bf r})(\nabla\times{\bf j})_{z}$
is $-2\pi\rho w({\bf r})\xi_{0}^{2}(n_{v}({\bf r})-H/\phi_{0})$, $w({\bf r})$
can be regarded as the pinning potential for vortex cores [2, 4], where
$n_{v}({\bf r})$ is the vortex density. As in the previous works [16, 32, 33]
based on the GL approach, the randomness on $T_{c}$, i.e., $\Delta^{(2)}$ or
$\Delta^{(3)}$, is assumed to be much larger than $\Delta_{\Phi}$.

To derive the conductivity terms associated with the current due to the SC
fluctuations, we follow the method [12, 31, 22, 32, 33, 34] based on the use
of the Kubo formula. In the case under an applied current perpendicular to the
magnetic field, the second (or, $n=1$) LL modes are needed [12] in addition to
the LLL modes to obtain the fluctuation conductivity under a uniform current.
By keeping only the 0th and 1st order terms with respect to the SC fluctuation
$\varphi_{1}$ belonging to the $n=1$ LL, the action (73) is written as

$\displaystyle{\cal S}_{r}$ $\displaystyle=$
$\displaystyle{\tilde{S}}^{-1}\sum_{\omega,p,{\bf k}}v_{\bf
k}^{1/2}\exp(ipk_{1}/h)\biggl{[}(u_{-{\bf k}}-{\bf k}^{2}w_{-{\bf
k}})\,\varphi^{*}_{0,p_{-}}(\omega)\varphi_{0,p_{+}}(\omega)$ (75)
$\displaystyle-$ $\displaystyle\frac{i}{\sqrt{2h}}[u_{-{\bf k}}+(2h-{\bf
k}^{2})w_{-{\bf
k}}](k_{+}\varphi^{*}_{0,p_{-}}(\omega)\varphi_{1,p_{+}}(\omega)+k_{-}\varphi^{*}_{1,p_{-}}(\omega)\varphi_{0,p_{+}}(\omega))\biggr{]},$

where $p_{\pm}=p\pm k_{2}/2$, and $k_{\pm}=k_{1}\pm ik_{2}$. For simplicity,
we focus on a high field range in which the $n=1$ LL mode is so heavy and
hence, assume that the mass $\mu_{1}$ of the $n=1$ LL fluctuation remains its
renormalized value [22] $2h$ and that its dynamics is negligible.

The Kubo formula of the SC part of the dc conductivity under a current in the
$x$-direction is expressed in the form

$\sigma_{s,xx}=-k_{\rm
B}T\lim_{\Omega\to+0}\frac{\partial}{\partial\Omega}\,\frac{\delta^{2}\overline{{\rm
log}\langle\exp(-{\cal S}_{tot}({\bf A}^{\prime}))\rangle}}{\delta
A^{\prime}_{x}(0,i\Omega)\delta A^{\prime}_{x}(0,-i\Omega)}\biggl{|}_{{\bf
A}^{\prime}=0},$ (76)

where ${\cal S}_{tot}={\cal S}_{\rm GL}+\delta S_{g}$, and the gauge
substitution ${\bf A}\to{\bf A}+{\bf A}^{\prime}$ was performed to induce the
current vertices. By keeping only the terms consisting of the $n=0$ and $1$
LLs, the AL and VG terms of the SC part $\sigma_{s,xx}$ of the conductivity in
2D case arise from the expression

$\frac{dR_{q}\sigma_{s,xx}}{4h^{2}}=-\frac{k_{\rm
B}T}{N_{v}}\lim_{T\to\infty}\lim_{\Omega\to+0}\frac{\partial}{\partial\Omega}\sum_{\omega,\omega^{\prime}}\sum_{p,p^{\prime}}{\rm
Re}\overline{\langle\varphi_{0,p}(\omega)\varphi^{*}_{0,p^{\prime}}(\omega^{\prime})\varphi^{*}_{1,p}(\omega+\Omega)\varphi_{1,p^{\prime}}(\omega^{\prime}+\Omega)\rangle},$
(77)

where $R_{q}=\pi\hbar/(2e^{2})=6.45$ ($k\Omega$) is the quantum of resistance,
and, to simplify our analysis on $\sigma_{s,xx}$, the formal $T\to\infty$
limit is taken to neglect the quantum SC fluctuations. In the present high
field approximation, the VG fluctuation is formed by the LLL SC fluctuation,
while the spatially averaged current vertex is inevitably accompanied by the
$n=1$ LL fluctuation. The VG fluctuation term $\sigma_{{\rm vg},xx}$ of the
conductivity is accompanied by vertices due to random potentials unrelated to
the VG correlation function (see Fig.6). Below, such a random potential vertex
will be called an outer random potential.

Figure 6: Diagram expressing $\sigma_{1,{\rm vg},xx}^{(2)}$. The two double
lines with the frequency $\omega+\Omega$ are the fluctuation propagators in
the second LL. This conductivity term is accompanied by a single outer random
potential line forming an interaction between the LLL and the next lowest LL
fluctuations.

The lowest (i.e., the first) order term in the outer random potential in 2D
case, $\sigma_{1,{\rm vg},xx}^{(2)}$, is described by Fig.6 [16]. Its concrete
expression is

$dR_{q}\sigma_{1,{\rm vg},xx}^{(2)}=\frac{\Delta^{(2)}k_{\rm B}T}{4\pi
N_{v}}\sum_{\bf k}{\bf k}^{2}v_{\bf k}\,J_{1}^{(2)}({\bf k}),$ (78)

where

$\displaystyle J_{1}^{(2)}({\bf k})$ $\displaystyle=$
$\displaystyle\lim_{T\to\infty}\biggl{(}-\frac{\partial}{\partial\Omega}\biggr{)}\sum_{\omega}{\cal
G}(\omega){\cal G}(\omega+\Omega)$ (79) $\displaystyle\times$ $\displaystyle
G_{\rm vg}({\bf k};\omega,\omega+\Omega)\biggr{|}_{\Omega\to+0}.$

In obtaining eq.(78), the inequality $\Delta_{\Phi}\ll\Delta^{(2)}$ was
assumed because the main vortex pinning effect in the GL approach should be
due to the variation of the amplitude $|\psi|$ of the SC order parameter.
Using the relation

$J_{1}^{(2)}({\bf k})\simeq[G_{\rm vg}({\bf
k};0,0)]^{2}\lim_{T\to\infty}\biggl{(}-\frac{\partial}{\partial\Omega}\biggr{)}\sum_{\omega}{\cal
G}(\omega){\cal G}(\omega+\Omega)\biggr{|}_{\Omega\to+0}$ (80)

of which the validity is easily checked, it is found that $J_{1}^{(2)}({\bf
k})\simeq\gamma[G_{\rm vg}({\bf k};0,0)]^{2}/(2\mu^{3})$ (see Appendix B).
Thus, we obtain

$dR_{q}\sigma_{1,{\rm vg},xx}^{(2)}\simeq\frac{\Delta^{(2)}\gamma k_{\rm
B}T}{4h\mu^{3}}\int_{\bf k}{\bf k}^{2}v_{\bf k}[G_{\rm vg}({\bf k};0,0)]^{2}.$
(81)

Or, by using $\xi_{\rm vg}$ defined previously, $\sigma_{1,{\rm vg},xx}^{(2)}$
finally becomes

$dR_{q}\sigma_{1,{\rm vg},xx}^{(2)}=\frac{\Delta^{(2)}h\gamma k_{\rm
B}T}{16\pi\mu^{2}c_{2}^{2}}\biggl{[}\,{\rm log}\biggl{(}\frac{1+\xi_{\rm
vg}^{2}}{e}\biggr{)}+\frac{1}{1+\xi_{\rm vg}^{2}}\biggr{]},$ (82)

where $\sqrt{h}$ was chosen as the cutoff of the wave number $|{\bf k}|$.

Figure 7: Eight diagrams expressing $\sigma_{2,{\rm vg},xx}^{(D)}$ with $D=2$
and $3$. Here, the dashed curves are linear combinations of $\Delta^{(D)}$ and
$\Delta_{\Phi}$ .

The next lowest (i.e., the second) order term in the outer random potential,
$\sigma_{2,{\rm vg},xx}^{(2)}$, is more divergent close to the VG transition
point than $\sigma_{1,{\rm vg},xx}^{(2)}$ [32, 33]. This second order term is
composed of the contributions of the eight diagrams in Fig.7. As an example,
let us examine Fig.7 (a) and (b) which are expressed as

$\displaystyle dR_{q}\sigma_{2,{\rm vg},xx}^{(2)}|_{a+b}$ $\displaystyle=$
$\displaystyle\frac{k_{\rm B}T}{{\tilde{S}}N_{v}}\sum_{{\bf k},{\bf
k}^{\prime}}v_{\bf k}v_{{\bf
k}^{\prime}}\lim_{T\to\infty}\biggl{(}-\frac{\partial}{\partial\Omega}\biggr{)}\sum_{\omega}[{\cal
G}(\omega+\Omega)]^{2}{\cal G}(\omega){\cal G}(\omega+\Omega)G_{\rm vg}({\bf
k}-{\bf k}^{\prime};\omega,\omega+\Omega)$ (83) $\displaystyle\times$
$\displaystyle\biggl{[}[\Delta^{(2)}+\Delta_{\Phi}(2h-{\bf
k}^{2})^{2}][\Delta^{(2)}+\Delta_{\Phi}{{\bf k}^{\prime}}^{4}]{\bf
k}^{2}-[\Delta^{(2)}+\Delta_{\Phi}{\bf k}^{2}({\bf k}^{2}-2h)]$
$\displaystyle\times$ $\displaystyle[\Delta^{(2)}+\Delta_{\Phi}{{\bf
k}^{\prime}}^{2}({{\bf k}^{\prime}}^{2}-2h)]k_{-}k^{\prime}_{+}\exp(i({\bf
k}\times{\bf k}^{\prime})_{z}/h)\biggr{]}\biggl{|}_{\Omega\to+0}.$

Contributions of the remaining six diagrams can be expressed in a similar
manner [33]. Close to the VG transition, however, it is sufficient to keep
contributions of the lowest order in ${\bf k}-{\bf k}^{\prime}$ in the
coefficient corresponding to the second and third lines of eq.(83). By summing
these contributions of the eight diagrams in this way, the full expression of
$\sigma_{2,{\rm vg},xx}^{(2)}$ becomes

$dR_{q}\sigma_{2,{\rm vg},xx}^{(2)}=4\Delta^{(2)}\Delta_{\Phi}hk_{\rm
B}T\int_{\bf k}\int_{{\bf k}^{\prime}}\frac{{\bf k}^{2}}{2}(v_{\bf
k})^{2}J_{2}^{(2)}({\bf k}-{\bf
k}^{\prime})\simeq\frac{\Delta^{(2)}\Delta_{\Phi}}{2\pi}h^{3}k_{\rm
B}T\int_{\bf k}J_{2}^{(2)}({\bf k}),$ (84)

where

$J_{2}^{(2)}({\bf
q})=\lim_{T\to\infty}\biggl{(}-\frac{\partial}{\partial\Omega}\biggr{)}\sum_{\omega}[{\cal
G}(\omega)+{\cal G}(\omega+\Omega)]^{2}{\cal G}(\omega){\cal
G}(\omega+\Omega)G_{\rm vg}({\bf
q};\omega,\omega+\Omega)\biggl{|}_{\Omega\to+0}.$ (85)

Just like the procedure for $J_{1}^{(2)}$, we have evaluated $J_{2}^{(2)}$ in
terms of the series expansion of the VG correlation function (see Appendix B).
As the leading term of $J_{2}^{(2)}({\bf q})$, we obtain

$J_{2}^{(2)}({\bf q})\simeq\frac{43\gamma}{8\mu^{5}}[G_{\rm vg}({\bf
q};0,0)]^{2}.$ (86)

Using this and the result on $\xi_{\rm vg}$, $\sigma_{2,{\rm vg},xx}^{(2)}$ in
2D case becomes

$dR_{q}\sigma_{2,{\rm vg},xx}^{(2)}=\frac{43\Delta^{(2)}\Delta_{\Phi}\gamma
k_{\rm B}Th^{5}}{4\pi^{2}c_{2}^{2}\mu^{5}}\xi_{\rm vg}^{2}.$ (87)

###  6.1 3D case

As already mentioned, the fully renormalized vertex part in 3D case is assumed
to have the same structure as that in 2D case. Under this simplification, it
is straightforward to extend the derivation on the conductivity in 2D case
explained above to that in 3D case.

First, the corresponding lowest order term to eq.(78) simply becomes

$\xi_{0}R_{q}\sigma^{(3)}_{1,{\rm vg},xx}=\frac{\Delta^{(3)}k_{\rm B}T}{4\pi
N_{v}}\int_{\bf k}{\bf k}^{2}\,v_{\bf k}\int_{k_{3}}J^{(3)}({\bf K}),$ (88)

where ${\bf K}={\bf k}+k_{3}{\hat{z}}$, and

$J^{(3)}({\bf
K})=\lim_{T\to\infty}\biggl{(}-\frac{\partial}{\partial\Omega}\biggr{)}\sum_{\omega}\int_{q}{\cal
G}(q,\omega){\cal G}(q+k_{3},\omega+\Omega)G^{(3)}_{\rm vg}({\bf
K};\omega,\omega+\Omega)\biggr{|}_{\Omega\to+0}.$ (89)

As in 2D case, $J_{3}$ can be rewritten in the form proportional to
$[G^{(3)}_{\rm vg}({\bf K};0,0)]^{2}$ :

$\displaystyle J^{(3)}({\bf
K})\simeq\frac{3\gamma}{32(\mu^{(3)})^{5/2}}\,[G^{(3)}_{\rm vg}({\bf
K};0,0)]^{2}\,\frac{4\mu^{(3)}}{4\mu^{(3)}+k_{3}^{2}}.$ (90)

In 3D case, however, the resulting $\sigma_{1,{\rm vg},xx}^{(3)}$ remains
nondivergent on approaching the VG transition from above and thus, will be
neglected hereafter.

The 3D conductivity term of the next order in the outer random potential can
also be easily found because the wavevector dependences accompanying the outer
random potential lines are decoupled to the dependences on the wavevector
component parallel to the field in the fluctuation propagators. The
corresponding expression in 3D case to eq.(84) simply becomes

$\xi_{0}R_{q}\sigma_{2,{\rm
vg},xx}^{(3)}\simeq\frac{\Delta^{(3)}\Delta_{\Phi}}{2\pi}h^{3}k_{\rm
B}T\int_{\bf k}J_{2}^{(3)}({\bf k}),$ (91)

where

$J_{2}^{(3)}({\bf
k})=\lim_{T\to\infty}\biggl{(}-\frac{\partial}{\partial\Omega}\biggr{)}\sum_{\omega}{\cal
A}({\bf k};\omega,\Omega),$ (92)

with

$\displaystyle{\cal A}({\bf k};\omega,\Omega)\\!$ $\displaystyle=$
$\displaystyle\\!\int_{k_{3}}\int_{{k_{3}}^{\prime}}\int_{q}{\cal
G}(q,\omega){\cal G}(q+k_{3}-k^{\prime}_{3},\omega+\Omega){\cal
G}(q-k^{\prime}_{3},\omega)\,\biggl{[}\,{\cal G}(q-k^{\prime}_{3},\omega)$
(93) $\displaystyle+$ $\displaystyle\\!\\!G^{(3)}_{\rm vg}({\bf
k}+(k_{3}-k^{\prime}_{3}){\hat{z}};\omega,\omega+\Omega)\,X^{(3)}({\bf
k})\int_{q^{\prime}}{\cal G}(q^{\prime},\omega)$ $\displaystyle\times$
$\displaystyle{\cal G}(q^{\prime}+k_{3}-k^{\prime}_{3},\omega+\Omega){\cal
G}(q^{\prime}-k^{\prime}_{3},\omega)\,\biggr{]}.$

Here, by noting the fact that the wave numbers $q$ and $q^{\prime}$ appearing
in the ladder of the VG correlation function are small as well as
$k_{3}-k^{\prime}_{3}$, we may replace ${\cal G}(q-k^{\prime}_{3},\omega)$ and
${\cal G}(q^{\prime}-k^{\prime}_{3},\omega)$ by ${\cal G}(-k^{\prime}_{3},0)$.
Then, we have

$J^{(3)}_{2}({\bf K})\simeq\int_{k^{\prime}_{3}}[{\cal
G}(k^{\prime}_{3},0)]^{2}\int_{k_{3}}J^{(3)}({\bf
K})=\frac{1}{4\mu^{3/2}}\int_{k_{3}}J^{(3)}({\bf K}).$ (94)

By substituting eq.(90) into the above expression, $\sigma^{(3)}_{2,{\rm
vg},xx}$ becomes

$\xi_{0}R_{q}\sigma_{2,{\rm
vg},xx}^{(3)}=\frac{3\Delta^{(3)}\Delta_{\Phi}\gamma k_{\rm
B}Th^{5}}{128\pi^{2}(c_{2}^{(3)})^{2}(\mu^{(3)})^{4}}\,\frac{\xi_{{\rm
vg},\perp}^{2}}{\xi_{{\rm vg},\parallel}}.$ (95)

##  7 Numerical results of resistivity curves

Figure 8: $R/R_{N}$ vs $h=H/H_{c2}(0)$ curves in 2D case at $t=T/T_{c0}=0.637$
(red), $0.531$, $0.455$, $0.398$, $0.354$, and $0.318$ (violet) from left to
right. The used values of the material parameters $b_{G}^{(2)}$ and
$\Delta^{(2)}=10\Delta_{\Phi}$ are $0.01$ and $0.002$, respectively.

Using the formula on the vortex-glass (VG) contributions $\sigma_{{\rm
vg},xx}$ to the conductivity obtained in sec.6, eq.(95) in 3D and eqs.(82) and
(87) in 2D cases, the resulting resistivity curves will be examined in this
section. The total conductivity $\sigma_{\rm tot}$ in each dimension consists
of the normal conductivity $\sigma_{N}$, the so-called Aslamasov-Larkin (AL)
term $\sigma_{{\rm AL},xx}^{(D)}$ which is essentially the SC part of the
conductivity in the disorder-free case, and the VG terms. In 2D case, we set
$\sigma_{\rm tot}=\sigma_{N}+\sigma_{{\rm AL},xx}^{(2)}+\sigma_{1{\rm
vg},xx}^{(2)}+\sigma_{2{\rm vg},xx}^{(2)}$, while we set $\sigma_{\rm
tot}=\sigma_{N}+\sigma_{{\rm AL},xx}^{(3)}+\sigma_{2{\rm vg},xx}^{(3)}$ in 3D
case.

Regarding the normal term $\sigma_{N}$ of the conductivity, we simply assume
that $dR_{q}\sigma_{N}=1$ in 2D case, and $\xi_{0}R_{q}\sigma_{N}=1$ in 3D
case, respectively.

In 2D case, the AL term $\sigma_{{\rm AL},xx}^{(2)}$ is given by

$\displaystyle dR_{q}\sigma^{(2)}_{{\rm AL},xx}$ $\displaystyle=$
$\displaystyle-4Th^{2}\biggl{(}\frac{\partial}{\partial\Omega}\biggr{)}\sum_{\omega}{\cal
G}(\omega)\frac{1}{2h+\gamma|\omega+\Omega|}\biggr{|}_{\Omega\to+0}$ (96)
$\displaystyle\simeq$ $\displaystyle\frac{2\gamma k_{\rm B}Th}{\mu(\mu+2h)},$

where the quantum fluctuation terms were neglected in obtaining the second
line. The corresponding expression in 3D case is

$\xi_{0}R_{q}\sigma^{(3)}_{{\rm AL},xx}\simeq\frac{\gamma k_{\rm
B}T}{2\sqrt{\mu^{(3)}}}.$ (97)

Equation (31) is used for the renormalized mass $\mu$ in 2D case, while
eq.(48) is used for the renormalized mass $\mu^{(3)}$ in 3D case.

Below, examples of the normalized resistivity curves obtained numerically will
be discussed in a manner of comparing results in 2D case with those in 3D
case, where the normalized resistivity is given by
$R/R_{N}=\sigma_{N}/\sigma_{\rm tot}$. First of all, the normalized
resistivity vs magnetic field curves taken at various fixed temperatures are
examined. We note here that, throughout this work, the field dependences of
$R/R_{N}$ at a fixed temperature are presented for convenience of numerical
computations rather than its temperature dependences at a fixed field. Typical
curves in 2D case obtained under the fixed values of the fluctuation strength
$b_{G}^{(2)}=0.01$ and the disorder strength
$\Delta^{(2)}=10\Delta_{\Phi}=0.002$ are shown in Fig.8. There, since
$\theta(t)\simeq 0.1$ even at the lowest temperature $t=0.318$, the curves in
Fig.8 can be regarded as belonging to the category of moderately clean
systems. Over most of the field range with nonzero resistance, the normalized
resistivity just smoothly decreases with decreasing field. However, close to
the end of the smoothly vanishing resistance, a sign of a slight drop of the
resistance is seen at each temperature.

Figure 9: Normalized resistivity curves in 3D case corresponding to those in
Fig.8 taken at $t=0.637$ (red), $0.531$, $0.455$, $0.398$, $0.354$, and
$0.318$ (violet) from left to right. The values $\sqrt{b_{G}^{(3)}}=0.01$ and
$\Delta^{(3)}=10\Delta_{\Phi}=0.002$ are used here.

Here, we will show that this position of the faint sudden drop seen on each
curve in Fig.8 essentially corresponds to the vortex lattice melting field
$h_{m}$ in 2D case in clean limit obtained in the LLL approximation. According
to Refs.23 and 24, $h_{m}$ in clean limit satisfies

$1-t-h_{m}=c_{2}^{-1}\sqrt{h_{m}\,t\,b_{G}^{(2)}}$ (98)

with $c_{2}\simeq 0.0988$. By assuming $\Delta^{(2)}$ to be negligibly small
and substituting the values of $b_{G}^{(2)}$ and $t$ into eq.(98), the
obtained value $h_{m}\simeq 0.10$ for $t=0.637$ coincides with the field at
which the red (left) curve reduces to zero in Fig.8. Since, with decreasing
$t$, the $\theta$-value increases and the normalized resistivity value is
slightly lowered from that expected in clean limit, the $h_{m}$ value at lower
$t$-values becomes slightly lower than the field at which the corresponding
resistivity curve reduces to zero. Therefore, it is understood that, at least
at weak enough disorder, the 2D VG transition in the Gaussian approximation
occurs at the corresponding vortex lattice melting transition point. Although,
of course, the 2D VG transition is believed not to occur in a sophisticated
approach beyond the Gaussian approximation [5, 35], this result in the 2D
Gaussian approximation plays a valuable role in understanding the
corresponding result in 3D case. Hereafter, the field at which the resistivity
at weak enough disorder shows a sudden drop will be called $h_{g}$.

Figure 9 presents the normalized resistivity vs field curves in 3D case at the
same set of temperatures obtained using the comparable values of the material
parameters $\sqrt{b_{G}^{(3)}}=0.01$ and $\Delta^{(3)}=0.002$ with those in
Fig.8. Since the effective disorder strength $\theta^{(3)}$ increases with
decreasing $t$, the right (violet) curve corresponds to a case with stronger
disorder. In contrast to the curves in 2D case, each of the resistance curves
in Fig.9 shows a clear and nearly discontinuous drop at a field $h_{g}$ of the
type seen in experiments [7, 17]. As in 2D case, $h_{g}$ will be compared with
the melting field $h_{m}$ expected in clean limit. Evaluation of $h_{m}$ has
been performed based on the thermodynamic data [36] in optimally-doped
cuprates and on the LLL fluctuation theory. According to Refs.21 and 22, the
melting field $h_{m}$ in clean limit is given by

$1-t-h_{m}=c_{3}^{-1}(h_{m}\,t)^{2/3}\,(b_{G}^{(3)})^{1/3}$ (99)

with $c_{3}\simeq 0.27$. By assuming the disorder strength $\Delta^{(3)}$ to
be negligibly small, we find that the $h_{m}$-value obtained by choosing the
value $c_{3}=0.22$ in eq.(99) coincides with the $h_{g}$-value of the curve at
the highest temperature $t=0.637$ in Fig.9. This implies that the SC
transition field in Fig.9 is slightly lower than the melting field expected
based on eq.(99), or equivalently that the fluctuation effect in 3D case is
slightly overestimated in our calculation. As already suggested in sec.4, this
is a consistent result with our approximation used for the fully-renormalized
vertex and the self-energy in 3D case. Therefore, we can expect [5, 16] that,
in 3D systems with weak enough disorder, the freezing from the vortex liquid
to the vortex solid triggers a sudden vanishing of the resistivity, i.e., the
VG ordering.

Figure 10: (a1) Full: 3D normalized resistivity (red solid) curve at $t=0.637$
in Fig.9. AL: Corresponding (blue dashed) curve obtained by deleting the
$\sigma_{{\rm vg},xx}$ term from $\sigma_{\rm tot}$. The dashed vertical line
denotes the position of the $H_{c2}$-line, and
$h=H/H_{c2}(0)=\xi_{0}^{2}/r_{H}^{2}$. (a2) Data of the structure factor
$f(K_{\perp})$ (see eq.(25)) taken at $H_{c2}$ (black), at the SC transition
field $h_{g}$ (red), and at $h_{ma}$ (blue) slightly above $h_{g}$ indicated
on the Full curve of (a1). Here, $K_{\perp}=|{\bf k}|/\sqrt{h}$ denotes the
magnitude of the component perpendicular to the field of the wavevector
measured in the unit of $r_{H}^{-1}=\sqrt{2\pi H/\phi_{0}}$. (b1) Full: 2D
normalized resistivity (red solid) curve at $t=0.637$ in Fig.8. AL:
Corresponding (blue dashed) curve obtained by deleting the two $\sigma_{{\rm
vg},xx}$ terms from $\sigma_{\rm tot}$. (b2) Data of $f(K_{\perp})$ at
$H_{c2}$ (black) and at the field $h_{g}=0.103$ (red) corresponding to the 2D
VG transition field in the Gaussian approximation. (c1) Full: 3D normalized
resistivity (blue solid) curve at $t=0.637$ in the case with the larger
disorder strength $\Delta^{(3)}=0.016$ and the same value of $b_{G}^{(3)}$ as
in (a1). AL: Corresponding resistivity (blue dashed) obtained by deleting the
$\sigma_{{\rm vg},xx}$ term from $\sigma_{\rm tot}$. Explanation on the red
dotted curve is given in the text. (c2) Data of $f(K_{\perp})$ at the SC
transition field $h_{g}=0.328$ (blue) for the Full curve in (c1). This
depressed structure factor implies that the sharp vanishing of the resistivity
in (c1) is not due to the growth of the positional correlation of the
vortices. Further details are explained in the text. For comparison, the red
curve shown in the figure (a2) is also presented here.

In Fig.10 (a), the normalized 3D resistivity curve at the high temperature
$t=0.637$ in Fig.9 is compared with structure factor data taken close to
$h_{g}$ and at $h_{c2}=H_{c2}(t)/H_{c2}(0)$. The already-mentioned fact that
the field $h_{g}$ at which the resistivity suddenly vanishes coincides with
the melting field in clean limit implies that the $t=0.637$ case in Fig.9 is a
situation quite close to the clean limit. Therefore, the sudden vanishing of
the resistivity in the figure (a1) is a consequence of the growth of the
correlation on the vortex positions, as argued previously in Ref.16. It is
found in (a2) that, in this case with weak enough disorder, the Bragg peaks
just at $h_{g}$ and at a slightly higher field $h_{ma}$ than $h_{g}$ are
almost the same as each other. It suggests that there is another origin of the
sudden drop of the resistivity other than the growth of the positional
correlation. Judging from the fact that the melting field $h_{m}$ yields the
LLL scaling [10, 12, 15, 37], eq.(99), on the SC fluctuation in magnetic
fields, we argue that, after all, the nature of the SC fluctuation below
$H_{c2}$ plays essential roles, together with the growth of the correlation on
the vortex position, in determining the vanishing behavior of the resistance
at the SC transition point $h_{g}$. In fact, as seen in the figures (a1) and
(b1), there are differences in the vortex flow behavior of the resistance in
the fluctuation-induced vortex liquid between the 2D and 3D cases. The feature
in the figure (b1) that an, if any, sharp drop of the resistance in 2D case
merely appears at the end of the vanishing resistivity curve is in contrast to
the clear sudden drop of the 3D normalized resistivity in the figures (a1) and
(c1). This dimensionality dependence of the vanishing behavior at $h_{g}$ of
the resistivity is a consequence of the difference in the resistivity curve in
the vortex liquid, i.e., in the nature of the SC fluctuation properties,
between the 2D and 3D cases.

When compared with resistivity data in cuprates [7, 13, 17], one might feel
that the normalized values of the resistance $R/R_{N}$ in the vortex liquid
regime in Figs.9 and 10 (a1) are too small compared with those in the typical
resistance curves. In relation to this, we note that the value
$1/(R_{q}\xi_{0})$ we have assumed here for the normal part $\sigma_{N}$ of
the conductivity is too small compared with the typical values of $\sigma_{N}$
seen in cuprates. To understand the consistency between the result in the
figure (a1) and experimental data in clean samples of cuprates, we point out
that the jump value of the resistance in (a1) at $h=h_{g}$ is comparable with
the corresponding ones seen in cuprates. In fact, the resistivity curve in
(a1) indicates that the SC part of the conductivity, $\sigma_{\rm
tot}-\sigma_{N}$, at $h=h_{g}$ is about seven times larger than the
corresponding value at $h_{c2}=H_{c2}/H_{c2}(0)$. One can verify that this
ratio is almost the same as the corresponding value estimated from
experimental data on YBCO [7, 13, 17].

As can be seen by comparing the figures (a1), (c1), and (c2) with one another,
an increase of the disorder leads to an increase of $h_{g}$ and a shrinkage of
the Bragg peak near $h_{g}$. Accompanying this, the normalized resistivity vs
field curve changed from the concave vortex flow behavior like in (a1) to a
nearly straight one. The reasonable reduction of the Bragg peak due to the
enhanced disorder would imply that the manner of including the disorder effect
in the parquet approximation for the vertex function $\Gamma$ is satisfactory.
On the other hand, the nearly sharp vanishing of the resistance at $h_{g}$ in
the figure (c1) is a feature disagreeing with experimental data. In fact, it
is empirically known that the resistance in the cases with strong disorder,
where, as seen in (c2), the vortex positional correlation remains short-
ranged, vanishes continuously, i.e., accompanied by a vanishing slope $dR/dT$
[5, 31]. One origin of this discrepancy seems to consist in the use of eq.(68)
for the dynamical SC fluctuation propagator even in dirty cases. As a
consequence of the use of eq.(68), the dynamical critical exponent $z_{\rm
vg}$ for the 3D VG transition is two, and the resistivity in 3D case
inevitably vanishes like $\sim(h-h_{g})^{1/2}$, i.e., with a divergent slope
of $dR/dh$, irrespective of the magnitude $\Delta^{(3)}$ of the sample
disorder. However, it is usually expected that, for a continuous VG transition
in strongly disordered cases, the vanishing behavior of the resistance is
characterized by a dynamical critical exponent $z_{\rm vg}$ larger than four
[5]. In the Hartree approximation [31, 34] neglecting the vertex correction
due to the vortex positional ordering to the random potential, the relation
$z_{\rm vg}=4$ follows, and a qualitatively reasonable vanishing behavior of
the resistivity in a strongly disordered case on approaching the VG transition
from above is obtained. At present, it is unclear to us how the results in the
Hartree approximation can be approached by starting from the present approach
including the effect of the vortex positional ordering on the random
potential. For a reference, another resistivity (red dotted) curve is added in
the figure (c1). This dotted curve is obtained by assuming $z_{\rm vg}=5$ and
replacing the dimensionless quantity $(1-c_{0}^{(3)})^{-1/2}$ in the
expressions (69) and (70) of the VG correlation lengths $\xi_{vg,\perp}$ and
$\xi_{vg,\parallel}$ by $(1-c_{0}^{(3)})^{-2}$. Another origin of the
discrepancy in the strong disorder case seems to consist in our assumption of
weak disorder in forming the VG correlation function in Fig.5: As the disorder
is increased, the irreducible vertex shown in Fig.5 (a) should be formed by
not a single disorder line but multiple ones. If this treatment in Fig.5 is to
be improved, the resistivity value of the solid curve (indicated as Full) in
the figure (c1) should be lowered further in the vortex liquid regime. In any
case, if a reasonable interpolation between the clean and dirty cases becomes
possible, the disorder dependence of the resistivity curves of the type found
in Ref.38 and consequently, the resistive behavior suggesting the presence of
the vortex slush regime [6, 7] below the melting temperature in clean limit
could be explained theoretically.

##  8 Summary and Discussion

In the present paper, the resistive behavior in type II superconductors with
weak enough disorder under a magnetic field was studied for both 2D and 3D
systems. Empirically, a nearly discontinuous (sharp) vanishing of the
resistivity upon cooling is often observed in bulk clean materials and is
usually interpreted as an evidence of the first order freezing transition to a
vortex solid [7, 17], while the resistivity in 2D case decreases smoothly even
in relatively clean materials [8]. Further, no theory explaining the presence
of the vortex slush region found in moderately clean 3D materials [6, 7] was
available. To try to resolve these issues, we have studied the vortex-glass
(VG) ordering in type II superconductors with weak enough disorder and have
shown that the presence in 3D case of a sharp vanishing of the resistivity and
its absence in 2D case are consequences of an interplay of the proximity to
the vortex solidification and the nature of the SC fluctuation in the vortex
liquid regime. To the best of our knowledge, this is the first theoretical
study of realizing the nearly discontinuous vanishing of the resistivity in
moderately clean 3D case through detailed calculations.

On the other hand, the resistive behavior characteristic of the vortex slush
region, i.e., a sharp drop of the resistivity signaling the vortex
solidification followed by its continuous vanishing, cannot be explained
within the present approach because of our use of the Gaussian approximation
for the VG fluctuation. Our theory predicts that, in clean enough 3D systems,
the vortex lattice melting temperature in clean limit becomes the onset of the
VG fluctuation [16] upon sweeping the temperature. However, the use of the
Gaussian VG fluctuation with the exponent $0.5$ of the conductivity does not
lead to a tail of the resistivity following its sharp vanishing. Such a tail
becomes more visible in a dirtier system with a wider VG critical region.
Thus, a visible vortex slush region would be created by renormalizing the VG
fluctuation. Further, in 2D case, such renormalized VG fluctuations should
delete any vanishing of the resistivity [5] and would clarify the validity of
the picture [16, 39] that the vortex lattice melting would not be reflected in
the resistivity in relatively clean 2D systems [40].

It should be stressed that developing a theory on the vortex phase diagram in
a high field region is important in relation to the recent observation of new
fluctuation regions associated with the paramagnetic pair-breaking [41] and
the quantum fluctuation [42] in iron-based superconductors [43, 44], because
the approaches constructed based on the phase-only model [2, 3, 4], developed
in relation to the study on the vortex states in cuprates, are not applicable
to the vortex states in such a high field regime. In fact, a sharp vanishing
of the resistance accompanied by a broad vortex liquid regime has appeared
even in the high field regime in FeSe [43].

The present work was supported by a Grant-in-Aid for Scientific Research
[No.21K03468] from the Japan Society for the Promotion of Science.

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*

## Appendix A

To clarify symmetries in the relations of eq.(36), we note that the following
relations,

${\hat{\Gamma}}_{abcd}({\bf k})=\Gamma_{abdc}({\bf k}),$ (100)

and

${\hat{C}}({\bf k})={\hat{A}}({\bf k}){\hat{B}}({\bf k})$ (101)

for $C({\bf k})=(A\,*\,B)({\bf k})$, are satisfied together with the last
equation of (37). These two relations follow from their definitions. By
applying them to the relations of eq.(36), we find

$\displaystyle\Pi_{2,abcd}({\bf k})$ $\displaystyle=$
$\displaystyle{\hat{\Pi}}_{3,abdc}({\bf k}),$
$\displaystyle\Lambda_{2,abcd}({\bf k})$ $\displaystyle=$
$\displaystyle{\hat{\Lambda}}_{3,abdc}({\bf k}).$ (102)

It is straightforward to find

$\displaystyle\Pi_{1,abcd}({\bf k})$ $\displaystyle=$
$\displaystyle{\hat{\Pi}}_{1,abdc}({\bf k}),$
$\displaystyle\Lambda_{1,abcd}({\bf k})$ $\displaystyle=$
$\displaystyle{\hat{\Lambda}}_{1,abdc}({\bf k})$ (103)

in a similar manner.

## Appendix B

To evaluate the expressions $J_{1}^{(2)}$ and $J_{2}^{(2)}$ appearing in the
conductivities, the expression on a pair of the propagators

$\displaystyle p_{\mu_{1},\mu_{2}}$ $\displaystyle=$
$\displaystyle\biggl{(}-\frac{\partial}{\partial\Omega}\biggr{)}\sum_{\omega}{\cal
G}_{1}(\omega){\cal
G}_{2}(\omega+\Omega)\biggl{|}_{\Omega\to+0}=\frac{\gamma}{2}\sum_{\omega}\biggl{(}{\cal
G}_{1}(\omega){\cal G}_{2}(\omega)[{\cal G}_{1}(\omega)+{\cal G}_{2}(\omega)]$
(B.104) $\displaystyle-$ $\displaystyle[({\cal G}_{1}(\omega))^{2}+({\cal
G}_{2}(\omega))^{2}]\frac{{\cal G}_{1}(0){\cal G}_{2}(0)}{{\cal
G}_{1}(0)+{\cal G}_{2}(0)}\biggr{)}$

becomes useful, where ${\cal G}_{n}(\omega)=1/(\mu_{n}+\gamma|\omega|)$. Using
eq.(B.104), one obtains

$J_{1}^{(2)}\simeq[G_{\rm vg}({\bf
k};0,0)]^{2}\lim_{T\to\infty}p_{\mu,\mu}=\frac{\gamma}{2}[{\cal
G}(0)]^{3}[G_{\rm vg}({\bf k};0,0)]^{2},$ (B.105)

where the quantum fluctuations with nonzero $\omega$ were neglected by taking
the limit $T\to\infty$. This result on $J_{1}^{(2)}$ can also be found by
expanding $G_{\rm vg}({\bf k};0,0)$ in a power series in $X({\bf k})[{\cal
G}(0)]^{2}$ and using the relation

$\displaystyle p^{(mn)}_{\mu,\mu}|_{cl}$ $\displaystyle\equiv$
$\displaystyle\lim_{T\to\infty}\biggl{(}-\frac{\partial}{\partial\Omega}\biggr{)}\sum_{\omega}{\cal
G}^{m}(\omega){\cal
G}^{n}(\omega+\Omega)\biggl{|}_{\Omega\to+0}=\gamma(m+1){\cal G}^{m+n+3}(0)$
(B.106) $\displaystyle\times$
$\displaystyle\biggl{[}1-\frac{1}{2^{m+n+1}}\biggl{(}\frac{(2m+1)(m+n)!}{n!(m+1)!}-\frac{m-n}{m+1}\sum_{r=0}^{m-1}\frac{1}{2^{r}}\frac{(n+r)!}{n!r!}\biggr{)}\biggr{]}$

as the main term of an expression following from resumming up the power
series.

In a similar way, $J_{2}^{(2)}$ is evaluated using eq.(B.106).