# Uniformly frustrated XY model: strengthening of the vortex lattice by
intrinsic disorder

I. Maccari  Department of Physics, Stockholm University, 106 91 Stockholm,
Sweden  L. Benfatto  ISC-CNR, Department of Physics, Sapienza University of
Rome, P. le A. Moro 5, 00185 Rome, Italy  C. Castellani  ISC-CNR, Department
of Physics, Sapienza University of Rome, P. le A. Moro 5, 00185 Rome, Italy

###### Abstract

In superconducting films, the role of intrinsic disorder is typically to
compete with superconductivity by fragmenting the global phase coherence and
lowering the superfluid density. Nonetheless, when a transverse magnetic field
is applied to the system and an Abrikosov vortex lattice forms, the presence
of disorder can actually strengthen the superconducting state against thermal
fluctuations. By means of Monte Carlo simulations on the uniformly frustrated
XY model in two dimensions, we show that while for weak pinning the
superconducting critical temperature $T_{c}$ increases with the applied field
$H$, for strong enough pinning the experimental decreasing dependence between
$T_{c}$ and $H$ is recovered with a resulting more robust vortex lattice.

##  I Introduction

The melting transition of two-dimensional (2D) Abrikosov vortex lattice (VL)
has always attracted a significant experimental and theoretical interest. The
interplay between magnetic field, random pinning and phase fluctuations makes
the phase diagram of the system rich of different phases of matter ranging
from the 2D Bragg glass to the 2D vortex glass Fisher  _et al._ (1991);
Blatter  _et al._ (1994); Klein  _et al._ (2001), from the isotropic vortex
liquid to the more recently observed hexatic liquid phase Guillamón  _et al._
(2009); Roy  _et al._ (2019). In the absence of disorder, at low temperatures
and low vortex densities, the vortex lattice exhibits a quasi-long range
order. As the temperature or the vortex density increases, topological defects
such as dislocations and disclinations form, eventually leading to the melting
of the vortex lattice according to the Berezinskii-Kosterlitz-Thouless theory
Berezinsky (1972); Kosterlitz and Thouless (1973); Kosterlitz (1974)
afterwards refined by Nelson, Halperin and Young Halperin and Nelson (1978);
Nelson and Halperin (1979); Young (1979). In this frame, the role of disorder
is on the one hand to favour the formation of such a lattice defects turning
the ground state from a quasi ordered Bragg glass Giamarchi and Le Doussal
(1995); Le Doussal and Giamarchi (2000) to a disordered vortex glass. On the
other hand, by acting as a pinning potential for vortices it also prevents the
VL from sliding throughout the system and destroying the global
superconducting (SC) phase coherence. Thus, differently from the Meissner
state, where disorder competes with superconductivity, in the mixed state its
role is less straightforward. Here, indeed, a true superconducting state
exhibiting zero resistivity in the limit of zero current can only settle if
certain degree of disorder is present within the system. Moreover, as recently
shown Ganguly  _et al._ (2017), vortex thermal fluctuations responsible for
the fast depletion of the superfluid stiffness can be even reduced by the
presence of a strong vortex pinning.

From a theoretical perspective, one possible way to investigate such interplay
is by studying the classical XY model in the presence of both disorder and
transverse magnetic field. As an effective model for superconducting phase
fluctuations, the 2D XY model has been over the years applied to the study of
disordered and inhomogeneous SC films Barabash  _et al._ (2000); Wysin  _et
al._ (2005); Erez and Meir (2013); Costa  _et al._ (2014); Maccari  _et al._
(2016); Kumar  _et al._ (2017); Maccari  _et al._ (2017, 2019) as well as to
the 2D vortex lattice melting for different values of the applied magnetic
field Teitel and Jayaprakash (1983); Franz and Teitel (1994, 1995); Hattel and
Wheatley (1995); Tanaka and Hu (2001); Hasenbusch  _et al._ (2005); Alba  _et
al._ (2008); Teitel (2013). The combined effect of quenched disorder and
transverse magnetic field has been so far addressed by including in the XY
model spatially random quenched magnetic fluxes, but with a resulting zero net
magnetic field Blatter  _et al._ (1994); Barabash  _et al._ (2000); Alba  _et
al._ (2009, 2010).

In this manuscript, we will instead consider the effect of quenched disorder
on the melting of the 2D vortex lattice which forms when a nonzero net
transverse magnetic field is included within the XY model. Starting from the
consideration that the presence of a lattice introduces a pinning effect at
low temperature even in the absence of disorder, our Monte Carlo simulations
show that apart from disordering the VL ground state, the presence of
inhomogeneities makes the vortex lattice stronger against thermal fluctuations
with respect the homogenous case according with the experimental observations
Ganguly  _et al._ (2017). The presence of a very week pinning can indeed lead
to a very counter-intuitive observation, namely the increase of the SC
critical temperature by the increase of the applied magnetic field. However,
when the disorder is sufficiently strong the experimentally observed
dependence between $T_{c}$ and $H$ is recovered.

##  II Materials and Methods

We consider a two-dimensional XY model in the presence of random pinning and
transverse magnetic field. Its Hamiltonian read:

$H_{XY}=-\sum_{\mu=\hat{x},\hat{y},i}J_{i}^{\mu}\cos(\theta_{i}-\theta_{i+\mu}+F^{\mu}_{i}),$
(1)

where $\theta_{i}$ is the SC phase at site $i$ of a $L_{x}\times L_{y}$
lattice, $J_{i}^{\mu}$ are the random couplings between the neighbouring sites
$i$ and $i+\mu$ and the phase shift $F^{\mu}_{i}$ accounts via the minimal
substitution for the presence of a finite transverse magnetic field
$H\hat{z}=\vec{\nabla}\times\vec{A}$. Being:

$F^{\mu}_{i}=\frac{2\pi}{\Phi_{0}}\int_{r_{i}}^{r_{i+\mu}}A_{i}^{\mu}\cdot
d\mu,$ (2)

with $\Phi_{0}=hc/2e$ the quantum flux, and $A_{i}^{\mu}$ is the vector
potential along the bond connecting two neighbours spins $i$ and $i+\mu$. The
sum of $F^{\mu}_{i}$ going counterclockwise around any closed path $C$ of
bonds on the lattice is $2\pi$ times the number of magnetic flux quanta
$f_{C}$ penetrating the path:

$\sum_{C}F^{\mu}_{i}=\frac{2\pi}{\Phi_{0}}\oint_{C}\vec{A}\cdot\vec{dl}=2\pi\frac{\Phi_{C}}{\Phi_{0}}=2\pi
f_{C}$ (3)

Since in the following we will always consider $H$ to be uniform in space, we
will refer to the intensity of the applied field in terms of the flux quanta
$f$ penetrating through an unitary plaquette $P$:

$2\pi
f=\frac{2\pi}{\Phi_{0}}\oint_{P}\vec{A}\cdot\vec{dl}=\frac{2\pi}{\Phi_{0}}\oint_{P}\vec{\nabla}\times\vec{H}\cdot\vec{dl}=\frac{2\pi}{\Phi_{0}}Ha^{2}$
(4)

with $a=1$, so that $f=\frac{H}{\Phi_{0}}$. In literature , one usually refers
to this case as the _uniformly frustrated_ XY model with frustration $f$.
Indeed, the phase shift $F^{\mu}_{i}$ in the cosine argument of (1) adds
_frustration_ to the system by rendering the ground state no longer
ferromagnetic: at $T=0$, the phases $\theta_{i}$’s instead to be all equal,
will vary from site to site trying to minimise the new gauge-invariant phase
$(\theta_{i}-\theta_{i+\mu}+F^{\mu}_{i})$. Consequently, the value of $f$ will
correspond to the level of such frustration, determining the inhomogeneous
space structure of the ground state itself. Specifically, the ground state of
the uniformly frustrated XY model will consist of a periodic configuration of
vortices in the phase angle $\theta_{i}$, whose number is directly
proportional to $f$. The number of vortices of the ground state for a given
value of $f$, can be easily derived by rewriting the charge neutrality
condition in terms of the new gauge invariant phase:

$\sum_{j}\oint_{P_{j}}(\vec{\nabla}\theta_{i}-\vec{A}_{i})=2\pi\sum_{i}(n_{i}-f)=2\pi\sum_{i}q_{i}=0$
(5)

so that:

$N_{v}=fL^{2}$ (6)

Since in the present study we will consider periodic boundary conditions, a
good gauge choice for the vector potential $\vec{A}$ is the Coulomb gauge:
$\vec{\nabla}\cdot\vec{A}=0$. For simplicity, we will consider:

$\vec{A}=B(0,x),$ (7)

so that:

$F^{\mu}_{i}=\begin{cases}0&\text{ if $\mu=x$}\\\ 2\pi fx_{i}&\text{ if
$\mu=y$}\end{cases}$ (8)

Finally, it is important to highlight that with this choice not all the values
of $f$ will be allowed. Indeed, periodic boundary conditions, together with
the Coulomb gauge $A_{y}=2\pi fx$, give rise to the constraint:

$L_{x}\cdot f=1,\,2,\,3,\,\dots$ (9)

Hence, for a given value of $L_{x}$ the smallest frustration we can introduce
within the system is: $f=1/L_{x}$.
In the present work, we have studied the model Eq.(1) on a square lattice with
periodic boundary conditions for a linear size of $L_{x}=L_{y}=L=64$ and
different values of $f$. In our Monte Carlo (MC) simulations, we have used a
local Metropolis algorithm, needed to probe the correct canonical distribution
of the system, combined with a micro-canonical Over-relaxation algorithm.
Specifically, each Monte Carlo step consists of $5$ Metropolis spin flips of
the whole lattice, followed by $10$ Over-relaxation sweeps of all the spins.
To help the correct thermalization of the system at lower temperatures, we
have used a Simulated-Annealing procedure. For each run we have made
50000-75000 MC steps, measuring the main observables with a frequency of 5
steps, after having discarded the firsts 25000. Finally, the averages have
been computed over 5 independent runs for the clean case and over 10 different
realization of quenched disorder for the disordered case.

In the present work, together with the ground state of the vortex lattice, we
have studied the SC transition as function of the frustration $f$ and the
level of disorder. To address this issue and measure the SC phase coherence of
the system, we have computed the superfluid stiffness $J_{s}^{\mu}$ which
accounts for the linear response of the system to an infinitesimal twist
$\Delta_{\mu}$ of the gauge invariant phase along a given direction $\mu$. As
such, it is defined as the second derivative of the free energy with respect
to $\Delta_{\mu}$ at $\Delta_{\mu}=0$:

$J_{s}^{\mu}\equiv-\frac{1}{L^{2}}\frac{\partial^{2}\ln{Z(\Delta_{\mu})}}{\partial\Delta_{\mu}^{2}}\large|_{\Delta_{\mu}=0}$

being it finite in the SC phase and zero in the normal phase. Its expression
for the model Eq.(1) reads:

$\begin{split}J_{s}^{\mu}=\frac{1}{L^{2}}&\big{\langle}\sum_{i}J_{i}^{\mu}\cos(\theta_{i}-\theta_{i+\mu}+F^{\mu}_{i})\big{\rangle}+\\\
-&\frac{\beta}{L^{2}}\Bigg{[}\big{\langle}\left(\sum_{i}J_{i}^{\mu}\sin(\theta_{i}-\theta_{i+\mu}+F^{\mu}_{i})\right)^{2}\big{\rangle}+\\\
-&\big{\langle}\sum_{i}J_{i}^{\mu}\sin(\theta_{i}-\theta_{i+\mu}+F^{\mu}_{i})\big{\rangle}^{2}\Bigg{]},\end{split}$
(10)

where $\beta$ is the inverse temperature and $\langle\dots\rangle$ stays both
for the Monte Carlo thermal average and for the average over the independent
runs.

##  III Results

In what follows, we will present the numerical results obtained via Monte
Carlo simulations both for the clean and for the disordered case, considering
different values of $f\in[0,\frac{1}{2}]$.

###  III.1 Clean case

Let us start with the clean case where $J_{i}^{\mu}=1$ $\forall i,\mu$ so
that:

$H_{XY}=-J\sum_{\mu=\hat{x},\hat{y},i}\cos(\theta_{i}-\theta_{i+\mu}+F^{\mu}_{i}).$
(11)

For $f=0$, the model (11) is the classical XY model which undergoes to the
well known Berezinskii-Kosterlitz-Thouless Berezinsky (1972); Kosterlitz and
Thouless (1973); Kosterlitz (1974) (BKT) transition at $T=T_{BKT}$. As already
mentioned, for finite values of $f$ the phase transition is no longer driven
by the unbinding of vortex-antivortex pairs, but rather by the melting of the
vortex lattice which naturally forms when a transverse magnetic field is
applied to the system. In the clean case, despite the absence of disorder, the
vortex lattice is pinned at low temperatures by the presence of the underlying
square grid that defines the array of Josephson junctions. By acting as a
periodic pinning potential for vortices, such a square grid can become
particularly relevant for large values of $f$ eventually determining the
symmetry of the vortex lattice itself. The most emblematic example is the
limiting case $f=1/2$, where in the ground state the vortex lattice assumes a
checkerboard ordered pattern. The model Eq.(11) with $f=1/2$ is also known as
the _fully frustrated_ XY model (FFXY) and as such it has been extensively
studied (see Teitel (2013) and reference therein) with particular focus on its
critical behaviour and the nature of the phase transitions it undergoes.

In the present work, we will instead focus on smaller value of $f$, where the
periodic pinning potential does not induce such peculiar vortex lattice
configuration. The resulting vortex lattice found numerically in the ground
state are shown in Fig.1. For all the values of $f$ considered, at low
temperatures the vortices form an ordered and pinned lattice whose symmetry
partially depends also on the commensurability with the underlying numerical
grid.

Figure 1: Vortex lattice ground state for different values of the frustration
$f=1/8;\,1/32;\,1/64$. Each vortex core is plotted as a blue point. The linear
size of the simulated system is $L=64$, while for the sake of clarity in the
figure only a portion of the whole VL is shown.

At finite but not maximal $f$ different kind of phase transitions can occur
Franz and Teitel (1995); Alba  _et al._ (2008), and their nature is still
unclear in most of the cases. Without pretending to address this issue, in the
present work we focus on the dependence of the critical temperature $T_{c}$,
at which the superfluid stiffness vanishes, on the applied magnetic field $f$.
The numerical results for different values of $f$ are reported in Fig.2.

Figure 2: Superfluid stiffness $J_{s}$ as a function of the temperature $T$
for different values of frustration $f$ in the absence of disorder. The linear
size of the simulated system is $L=64$.

We can immediately notice that the critical temperature $T_{c}$ at which the
system loses its phase coherence is strongly suppressed for smaller values of
$f$, completely at odd with the usual experimental observations Yazdani  _et
al._ (1993); Blatter  _et al._ (1994); Feigel’man  _et al._ (1994); Chen  _et
al._ (2007); Guillamón  _et al._ (2009); Misra  _et al._ (2013); Benyamini
_et al._ (2019); Roy  _et al._ (2019). The general trend seems, indeed, to be
a proportionality between $T_{c}$ and $f$: lower critical temperatures for
smaller frustration.
The observed trend has been already discussed in Alba  _et al._ (2008), where
Alba et al. have shown that in the limit of small frustration $f=1/n$ (and
$n\gg 1$), the critical temperature $T_{c}$ decreases with the increase of
$n$: $T_{c}\sim 1/n\to 0$ as $n\to\infty$.

Apart from specific cases where the commensurability with the underlying
square grid particularly strengthens or weakens the vortex-lattice structure,
by lowering the applied magnetic field the pinning of the vortex lattice
becomes weaker than in the case where the vortex density is higher, with a
resulting decrease of the critical temperature with $f$. By increasing the
pinning potential via the introduction of disorder, however, the scenario is
reversed and, in agreement with experimental observations, with increasing
applied magnetic field the critical temperature decreases. We will discuss the
numerical results of the disordered case in the following section.

###  III.2 Disordered case

We now consider the case of a transverse magnetic field applied to a
disordered SC film. In the present manuscript, we will use as disordered
coupling constants $J_{i}^{\mu}$ in Eq.(1) the inhomogeneous local stiffness
derived from the (quantum) XY pseudo-spin $1/2$ model in random transverse
field (RTF) Ma and Lee (1985); Cea  _et al._ (2014). This disorder has been
shown to be appropriate to model disordered superconductors with a non-trivial
spatial structure Ioffe and Mézard (2010); Lemarié  _et al._ (2013); Cea  _et
al._ (2014), as well as to account for the experimental observation of a
rather broad BKT transition around the critical temperature $T_{BKT}$ Maccari
_et al._ (2017); Ilaria Maccari  _et al._ (2018), at which a sharp jump of the
superfluid-density would be expected for zero disorder Nelson and Kosterlitz
(1977). Previous studies at zero magnetic fieldMaccari  _et al._ (2017);
Ilaria Maccari  _et al._ (2018) have shown that such a spatially-correlated
disorder with large enough low-stiffness puddles is crucial to induce an
anomalous vortex nucleation and, consequently, a smearing of the BKT
transition. On the other hand in the present case where vortices are induced
by a finite transverse magnetic field we do not expect that the results will
depend crucially on the choice of disorder, so our finding should be general
also for different disorder realizations. The level of disorder is here
labeled by $W/J$ (see Maccari  _et al._ (2017) for more details) and it is
taken to be quenched in temperature.

Let us start by considering a relatively weak disorder level $W/J=4$.
Consistently with the experimental observations of Ganguly  _et al._ (2017),
our MC numerical results reveal that the presence of disorder leads to a
modification of the ground-state vortex lattice, enlightening further its
underlying mechanism. As shown in Fig.3, the core of the vortices is indeed
pinned by the inhomogeneity of the local stiffness, which makes them moving
towards those regions with lower ${J}_{i}^{\mu}$ so to gain in energy by
minimising the Hamiltonian(1). To highlight the correlation in space between
low-couplings regions and the vortex lattice deformation, in Fig. 3 we have
superimposed the vortex lattice to the couplings map, obtained by computing
over each plaquette the average value of the local stiffness $J^{\mu}_{i}$.

The presence of disorder not only modifies the vortex lattice ground state,
but also impact the superfluid response of the system as function of the
applied magnetic field. In Fig.4, we report the temperature dependence of the
superfluid stiffness $J_{s}(T)$ obtained for different values of $f$. As one
can see, the presence of disorder restores the experimentally measured
dependence between $T_{c}$ and $f$ by rendering more robust the vortex lattice
against thermal fluctuations. In this regard, it is quite impressive to notice
that, compared to the homogeneous case (see Fig.3), the critical temperature
$T_{c}$ corresponding to the lowest value of $f=1/64$ has increased by a
factor of ten by the effects of the inhomogeneity.

The strengthening of the vortex lattice due to disorder is even more
pronounced when looking at stronger disorder regimes. In Fig.5, we report the
superfluid stiffness trend in temperature for the same values of $f$ but with
a disorder level of $W/J=10$. Looking for instance at the lowest value of the
field ($f=1/64)$, the critical temperature is reduced, because of the field,
just by half with respect to the zero-field value, while at weak disorder
($W/J=4$) it was five times smaller.

Figure 3: Vortex lattice ground state for different values of the frustration
$f=1/8;\,1/16;\,1/32;\,1/64$ superimposed to a map of a given disorder
realization with $W/J=4$. The gray scale refers to the the average value of
the local couplings constant $J_{i}^{\mu}$ around each single plaquette. Each
vortex core is plotted as a red point. The linear size of the simulated system
is $L=64$, while for the sake of clarity in the figure only a portion of the
whole VL is shown.  Figure 4: Temperature dependence of the superfluid
stiffness in the presence of weak disorder $W/J=4$, for different values of
the frustration $f$. The continuous gray line is the BKT critical line
$2T/\pi$ relative to the case $f=0$.  Figure 5: Temperature dependence of the
superfluid stiffness in the presence of strong disorder $W/J=10$, for
different values of the frustration $f$.The continuous gray line is the BKT
critical line $2T/\pi$ relative to the case $f=0$.

In order to highlight such increase of robustness, as effect of the increase
of the intrinsic disorder, we have reported in Fig.6 (a) the extrapolated
values of the critical temperature as function of the applied field, for the
two disorder regimes considered. In Fig.6(b), this effect is made even more
evident by rescaling the curves of $T_{c}$ by their value in absence of
magnetic field.

Figure 6: Magnetic field dependence of (a) the critical temperature $T_{c}(f)$
and (b) of the rescaled critical temperature $T_{c}(f)/T_{c}(f=0)$ for the two
regime of disorder considered: $W/J=4$ and $W/J=10$. For each values of $f\neq
0$ and disorder level, the critical temperature $T_{c}$ has been estimated
from the superfluid stiffness trend in temperature $J_{s}(T)$ as the
temperature at which $J_{s}$ vanishes starting from low temperatures. For the
case $f=0$, we used instead the Nelson-Kosterlitz universal relation Nelson
and Kosterlitz (1977).

##  IV Discussion

By means of Monte Carlo simulations on the uniformly frustrated $XY$ model, we
have shown that the presence of disorder, mimicked via random couplings,
modifies the ordered vortex lattice of the ground state. For strong enough
disorder, the energy of the system is no longer minimized by an ordered
vortex-lattice structure, but rather by a disordered structure in which the
vortex cores are located where the local stiffness of the superfluid is lower
so as to reduce the energy cost associated with a phase twist.

At the same time, we have shown that in the presence of a very weak pinning as
in the case with no disorder, where the only pinning potential is due to the
presence of the underlying square grid, the increase of the applied magnetic
field does not reduce the superconducting critical temperature, but in most
cases it contributes to its increase, in stark contrast with the experimental
observations Yazdani  _et al._ (1993); Blatter  _et al._ (1994); Feigel’man
_et al._ (1994); Chen  _et al._ (2007); Guillamón  _et al._ (2009); Misra  _et
al._ (2013); Benyamini  _et al._ (2019); Roy  _et al._ (2019). The presence of
disorder not only restores the experimentally observed dependence between
$T_{c}$ and $f$, but also acts by making the ground-state vortex lattice more
robust against thermal fluctuations. Comparing two different levels of
disorder, we have indeed shown that the superconducting critical temperature
is suppressed much less with respect to the zero-field case in the strong
disorder regime than in the weak disorder regime.

In conclusion, despite separately both disorder and magnetic field act on the
SC thin film by suppressing the superfluid stiffness, when a transverse
magnetic field is applied to the system, the presence of disorder help to
prevent the destruction of the SC state by thermal fluctuations.

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