Anisotropic Vortices in High-Temperature Superconductors and the Onset of
Vortex-like Excitations above the Critical Temperature

O. Hudak111e-mail<EMAIL_ADDRESS>
Department of Theoretical Physics, Faculty of Mathematics and Physics,
Comenius University, Mlynska dolina F2, SK - 845 01 Bratislava

## Abstract

In our recent papers we have found that a three-dimensional superconducting
state with anisotropic vortices localized at the vortex-lattice points is a
stable state in zero and nonzero external magnetic field for the layered high-
temperature superconductivity materials. There exists a phase transition at
the temperature $T_{c}^{v}$ from the normal phase to the vortex
superconducting state which is of the first order. Note that the transition is
in zero magnetic field. The first order phase transition shows overheating and
overcooling effects. Nucleation of the superconducting phase in the normal
phase thus may occur at temperatures higher than the transition temperature
$T_{c}^{v}$. Then the onset of the vortex-like excitations above the
transition temperature $T_{c}^{v}$ occurs in our theory. The onset of the
vortex-like excitations in Nerst signal and some other experimental evidence
for these excitations above the transition temperature $T_{c}^{v}$ in LSCO,
YBCO and in other similar high-temperature materials may be explained thus by
our theory. The vortex-like excitations above and below the transition
temperature $T_{c}^{v}$ in LSCO, YBCO and in other similar high-temperature
materials continuously evolve. This fact may be explained within our theory.

##  1 Introduction

Sine-Gordon vortices may exist in superconductors [1] and [2]. These vortices
may be present in superconducting states also in zero magnetic fields. Their
topological charge is preserved. We studied a magnetic-field-induced
superconductive state in heavy fermion systems [3] and [4]. The strength of
the coupling between the order parameter and magnetic field drives a
superconductive state induced by a magnetic field, [3] and [4]. Thus such a
state may be induced increasing the magnetic field. Experimentally it was
reported that in $CePb_{3}$ there exists a ferromagnetic phase and
simultaneously there is an evidence for presence of superconductivity. We
studied the Kondo-lattice superconductivity in this connection, [4] and [5].
Superconducting states of different symmetry s, p and d were found to exist in
the Kondo-lattice systems. A phase transition from the normal phase to the
superconducting phase in $CePb_{3}$ type materials occurs in our
phenomenological model through creation of a vortex lattice, in which vortices
are superconducting regions induced by an external magnetic field. Increasing
the magnetic field the density of such vortices and their shape becomes larger
and the superconducting phase increases its volume in the material. In high-
temperature superconductors, which are layered materials, the interlayer
distance is larger than the in-layer lattice constant. The interlayer distance
is comparable with the in-layer penetration length. Our results [3] \- [5]
lead to study of formation of a vortex in the plane, and to study of
interaction of such vortices in neighbouring layers. In the presence of an in-
layer vortex in-plane and out-of-plane magnetic field structures exist due to
currents which are present. The vortex state which we described for a plane
superconductor in 1982 may exist in zero magnetic field [6]. This fact is very
important difference from usual vortices studied in superconductors, which are
present due to an external magnetic field. The stability of the vortex in zero
magnetic field [6] is due to the topological charge of this vortex which is
preserved. The magnetic field generated by this stable vortex is influencing
neighbouring planes and the plane in which it is localized . This field is
coupled with the superconducting order parameter due to symmetry. As we
discussed above the strength of the coupling between the magnetic field and
the order parameter may drive a superconductive state with vortices induced by
the magnetic field of this vortex, [3] and [4]. Thus in layered high-
temperature superconductors coupling between the magnetic field generated by
this stable vortex and the order parameter may lead to generation of other
vortices (antivortices). This is a selfconsistent process. The field of these
vortices influences this stable vortex. Thus then we may study vortices and
their in-plane and out-of-plane interactions due to the inductance coupling.
Thus the strength of the coupling between the order parameter and magnetic
field drives a superconductive state induced by a magnetic field as mentioned
above, [3] and [4], and in layered high-temperature superconductors drives a
superconducting state with vortices. Note that also in the zero external
magnetic field such a selfconsistent process may exists. Creation of a vortex
in one plane leads to magnetic fields which influence the neighbouring planes,
in these neighbouring planes these fields may induce creation of vortices the
magnetic fields of which influence the original vortex in the plane: its
existence and properties. Creation of a vortex in the plane leads also to
magnetic fields which influence the same plane, in this plane these fields
again may induce creation of vortices the magnetic fields of which influence
the original vortex in the same plane: again its existence and properties.

Thus it follows that in layered materials one may expect that formation of
such vortices may stabilize the superconducting state of different from
classical symmetry in zero external magnetic field. Note that in nonzero
external magnetic fields the self-consistent process may be modified but still
present.

The vortex in the superconductor plane which we studied [6] is anisotropic
vortex. This anisotropy (symmetry different from symmetry of isotropic
vortices) then leads to coupling to other quantities of the system for which
an invariant contributing to free energy expansion may exist. Integrating out
these other quantities from the free energy of the superconducting material we
obtain the effective free energy expansion for the superconducting order
parameter with anisotropy terms present.

Recently we described and studied the mentioned mechanism and formulated a
general problem: to study behaviour of such anisotropic vortices in zero and
non-zero magnetic fields. The vortices of the in- and inter- layer type should
be compared as concerning their theoretical properties in zero and non-zero
fields with existing experiments concerning study of vortices in high-
temperature superconductors, [7], [8], [9] and [10]. It was found
experimentally that the anisotropy of the interplane-vortices in these
materials is intrinsic. This is natural, the vortex is parallel to layers,
layers are anisotropic. Planes are involved in such vortex configuration, we
are speaking about the inter-plane vortex induced by an external magnetic
field. As far as we know the anisotropy of such vortices is interpreted as due
to anisotropy of layers.

The vortices which we were studied theoretically are intrinsically
anisotropic. However this anisotropy is present for vortices which are present
within an isotropic plane. Their anisotropy is not generated due to anisotropy
of the material. Thus it is important to compare experimental results
concerning anisotropy properties of the vortices in the layered materials with
those properties of vortices which we found theoretically. This is the reason
why we reviewe different superconducting states with anisotropic vortices: the
superconducting states in which single vortex exists, the superconducting
states in which lines and lattices of vortices exist, the superconducting
states in which pairs of vortex-antivortex type are present and form a kind of
liquid. At larger distances between vortex-antivortex in the pair the energy
is linearly dependent on this distance. This is a kind of confinement
phenomena known from quarks forming baryons. For smaller distances there
exists a minimum distance for which the vortex-antivortex pair exist in a
ground state. The pair vortex-antivortex thus does not annihilate. Further we
considered in our papers the superconducting states in which solitons and
strings of vortices, which represent some kind of structures induced by
internal mechanisms /twinning of the crystals, etc./ or by surface, may exist.

Intrinsic Josephson effect in layered systems was described in [11] and a
summary of the Josephson effect in the high-temperature superconductors is
given in [12]. The time dependent equations for the gauge invariant phase
differences of the layered order parameters for superconductive states and for
a general vector potential of the electromagnetic field we discussed in [1]
and [2]. The interlayer coupling within the mentioned model equations was
studied with its influence on the properties of vortices in layers. These
vortices in neighbouring layers are coupled. They are forming a string of
vortices perpendicular to the layers in which the vortices exist. The string
goes through the centers of these vortices.

We used in our study a small parameter expansion method, and an exact method
of solution of the corresponding Lagrange-Euler equations to compare different
macroscopic superconducting states. We have found their free energy and
discussed their properties. The corresponding Lagrange-Euler equations are
nonlinear coupled sine-Gordon equations in the approximation of the constant
amplitude of the order parameter. Let as note that the nonlinearity describing
the vortex configurations in nonzero /and in zero/ magnetic field is neglected
due to difficulties of the mathematical origin and mainly due to expectation
that the nonlinearity is negligible around the core of the vortex, which is
assumed to be large. However we have found that nonlinearities in order
parameter interactions are physically essential, [6]. The Lagrange-Euler
equations for the superconducting state order parameter and for the vector
potential of the electromagnetic field in the constant amplitude approximation
for the scalar (s - type) superconducting order parameter reduced in the model
studied in our papers cited above to equations which are similar to the two-
dimensional sine-Gordon equation. They are generalized coupled two-dimensional
sine-Gordon equation. p and d symmetric superconducting states were not
studied until now.

The two-dimensional sine-Gordon equations occur in other physical systems, in
quantum antiferromagnets, in classical two-dimensional XY model with the in-
plane magnetic field [13] \- [14], in models describing crystal growth [15]
and [16], and in models describing defects in incommensurate systems [17] and
[18]. It is useful to use analogy and mathematical properties of solutions of
these equations for study of anisotropic vortices in superconductors.

In our paper from 1982 we used a new method for solving a two-dimensional
sine-Gordon equation [6]. We transformedg this equation into a set of two
second order ordinary nonlinear non-coupled equations. Multi - (Resonant-
Soliton) solutions and vortex-like solutions of the two- and three -
dimensional sine-Gordon equations were studied in [19]. The structure of a
vortex in terms of the solitons was studied in [20]. The vortex is formed from
solitons which are intersecting. The multi-vortex solutions of the sine-Gordon
equation were studied in [21]. Exact rational-exponential solutions of this
equation were studied in [22]. Vortices and the boundary value problem were
studied by the inverse scattering transform method for this type of equation
in [23]. The singular solutions of the elliptic sine-Gordon equation were
studied numerically in [24]. Numerical studies of dynamical isoperimeter
pattern are in [25].

Vortex-like excitations in $La_{2-x}Sr_{x}CuO_{4}$ at temperatures
significantly above the critical temperature were found in the Nerst effect
signal [26]. In overdoped $La_{2-x}Sr_{x}CuO_{4}$ (LSCO) the upper critical
field $H_{c2}$ does not end at $T_{c0}$ but at a much higher temperature.
These results imply, according to [27], to a loss in phase rigidity rather
than in a vanishing of the pairing amplitude. Nerst measurements in
$YBa_{2}Cu_{3}O_{4}$ (YBCO) and in $Bi_{2}Sr_{2-y}La_{y}CuO_{6}$, [28], and in
LSCO in fields up to 33 T show the existence of vortex-like excitations high
above $T_{c0}$. This anomaly is related to the key question of whether the
Meissner transition in zero field is caused by the collapse of long-range
phase coherence or by the vanishing of the pairing amplitude. In zero magnetic
field there is equal number of vortices with plus and minus topological
charge. In nonzero fields there is nonequal number of vortices with plus and
minus topological charge.

Preformed pairs as superconductor fluctuations may exist in Bose condensation
of localized Cooper pairs with short coherence length, [29], [30] and [31].
Pairing correlations without phase coherence are described in [32] and [33].

In our papers [1] and [2]. new results were found as concerning the influence
of neighbouring planes on the vortex state in a given plane and new exact
solutions of the coupled Lagrange-Euler sine-Gordon equations were found,
their properties and free energy were compared. We have found that the three-
dimensional superconducting state with vortex lattice, which is a stable state
in zero external magnetic field for the layered materials. There exists a
phase transition at the temperature $T_{c}^{v}$ from the normal phase to the
vortex superconducting state which is of the first order. Depending on the
parameters of the material the transition temperature $T_{c}^{v}$ may be
higher than zhat in the case of the second order phase transition from normal
phase to the homogeneous superconducting phase without vortices. The
transition temperature $T_{c}^{v}$ would be zero for zero topological charge
of vortices. Note that the transition to vortex lattice state is present also
in zero magnetic field. The first order phase transition shows in experiments
usually overheating and overcooling effects. Nucleation of the superconducting
phase in the normal phase occurs at temperatures higher than the transition
temperature $T_{c}^{v}$. And vice-versa. Nucleation of the normal phase in the
superconducting phase occurs at temperatures lower than the transition
temperature $T_{c}^{v}$. Thus the onset of the vortex-like excitations above
the transition temperature $T_{c}^{v}$ occurs in our theory due to nucleation
of the superconducting phase with vortex lattice (or parts of this lattice) in
the normal phase at temperatures higher than the transition temperature
$T_{c}^{v}$. The onset of the vortex-like Nerst signal above the transition
temperature $T_{c}^{v}$ in LSCO, YBCO and in other similar high-temperature
materials may be thus explained by our theory.

In this paper we review our results on vortex configurations in two-
dimensional sine-Gordon systems in connection with high-temperature
superconductors. We will use this results in our discussion concerning
anisotropic vortices and the onset of Nernst signal and of some other signals
corresponding to presence of vortex-like excitations above the critical
temperature. The superconducting planes are coupled by the Josephson effect.
The Lawrence-Doniach functional leads to the Lagrange-Euler equations for the
order parameter. In [1] and [2]. We discussed the vortex states using exact
solutions of these coupled Lagrange-Euler sine-Gordon equations. Properties of
the exact solutions and the free energy of the corresponding order parameter
configurations weres compared with the normal phase state and the homogeneous
superconducting state. Stable vortex configurations in zero external magnetic
field for the layered materials exist, the same holds for a lattice of
vortices case. The first order phase transition overheating and overcooling
effects are present here. Nucleation of the superconducting phase in the
normal phase, which occurs at temperatures higher than the transition
temperature $T_{c}^{v}$, exists. The onset of the vortex-like excitations
above the transition temperature $T_{c}^{v}$, which occurs in our theory, is
discussed. The onset of the vortex-like Nerst signal above the transition
temperature $T_{c}^{v}$ in LSCO, YBCO and in other similar high-temperature
materials is not well understood in experiments. The vortex-like excitations
above the transition temperature $T_{c}^{v}$ in LSCO, YBCO and in other
similar high-temperature materials continuously evolve into vortex states
below the transition temperature $T_{c}^{v}$, this fact may be explained
within our theory. Our theory may explain this fact and some of other
experimentally observed properties of high-temperature superconductors
described above.

##  2 Superconducting Planes Coupled by the Josephson Effect

Superconducting layers in layered superconductors of the high-temperature type
are coupled due to intrinsic Josephson effect. The Lawrence - Doniach free
energy functional for the order parameter for such a system has the form [1]

$F(\Psi_{n}(\bf r),{\bf A}({\bf R}))=\frac{H^{2}_{c}s}{4\pi}\sum_{n}\int d{\bf
r}[\zeta^{2}_{ab}\mid(-i\nabla+\frac{2\pi}{\Phi_{0}}{\bf
A}_{n})\Psi_{n})\mid^{2}-$ (1)
$-\mid\Psi_{n})\mid^{2}+\frac{1}{2}\mid\Psi_{n})\mid^{4}+$
$+r(\mid\Psi_{n})\mid^{2}+\mid\Psi_{n+1})\mid^{2}-\Psi_{n}\Psi^{*}_{n+1}\exp(-i\chi_{n,n+1})-\Psi^{*}_{n}\Psi_{n+1}\exp(+i\chi_{n,n+1}))$
$+\int d{\bf R}\frac{B^{2}}{8\pi}$

The order parameter in the n-th layer is $\Psi_{n}({\bf r})=\mid\Psi_{n}({\bf
r})\mid\exp(i\Phi_{n}](\bf r))$, here z=ns, and s is an interlayer distance, R
= (r, z), $\nabla=\frac{\partial}{\partial{\bf r}}$, $H_{c}$ is the bulk
critical field, $\zeta_{ab}$ is the coherence length in the ab plane which is
perpendicular to the c-direction. $\Phi_{0}=\frac{\pi\hbar c}{\mid e\mid}$,
${A}_{n}(\bf r)=(A_{nx},A_{ny})$ is an average value of the vector potential
over the distance $[(n-\frac{1}{2})s,(n+\frac{1}{2})s$ along the c-axis, B =
rot (A), and $\chi_{n,n+1}=\frac{2\pi}{\Phi_{0}}\int^{(n+1)s}_{ns}dzA_{z}$.

If we denote the Josephson coupling constant between neighbouring layers order
parameters as r, where $r<<1$,, the coherence length $\zeta_{c}$ in the
c-direction is given by

$\zeta^{2}_{c}=\frac{s^{2}r}{2}$ (2)

The in-plane penetration depth $\lambda_{ab}$ is given by

$\lambda^{2}_{ab}=\frac{\Phi^{2}_{0}}{8\pi^{2}H^{2}_{c}\zeta^{2}_{ab}}$ (3)

The anisotropy ratio $\frac{\lambda_{ab}}{\lambda_{c}}$ is given by

$\frac{\lambda_{ab}}{\lambda_{c}}=\frac{\zeta_{ab}}{\zeta_{c}}$ (4)

The interlayer coupling is weak, $\zeta_{c}<<s$. The general case and the
limit of strong interlayer coupling $s<<\zeta_{c}$ will not be discussed here.

Let $\phi_{n}(\bf r)$ be the change of the order parameter phase around a
vortex. The constant amplitude approximation $\mid\Psi\mid^{2}=1$ we consider
first. The constant amplitude approximation in which the density of
superconducting pairs is temperature dependent will be considered at the end
of the paper. There is experimental evidence that the constant amplitude
approximation is quite a good approximation to understand vortex state
(phase), see discussion below.

##  3 Lagrange-Euler Equations for the Order Parameter

The Lagrange-Euler equations for the phase difference $\phi_{m,m+1}$ are the
sine-Gordon coupled equations

$-\sum_{m}L_{n,m}\nabla^{2}\phi_{m,m+1}+\frac{1}{\lambda^{2}_{J}}\sin(\phi_{n,n+1})=0$
(5)

where the summ over m is over layers.

The interlayer inductance $L_{n,m}$ between the layers n and m has the form

$L_{n,m}=\int^{2\pi}_{0}\frac{dq}{2\pi}\frac{\cos(n-m)q}{2(1-cos(q))+\frac{s^{2}}{\lambda^{2}_{ab}}}=\frac{\lambda_{ab}}{s}(1-\frac{s}{\lambda_{ab}})^{\mid
n-m\mid}$ (6)

The boundary conditions for the Lagrange - Euler equations (5) are

$(\nabla_{x}\nabla_{y}-\nabla_{y}\nabla_{x})\phi_{n,n+1}(\bf
r)=2\pi\sum_{n}(\mid\delta({\bf r}-{\bf r}_{n\nu})-\delta({\bf r}-{\bf
r}_{n\nu})\mid)$ (7)

The equations (5) are thus a system of the sine-Gordon equations in two
dimensions (planes ab) coupled in the perpendicular direction c.

##  4 The Distance s is Comparable with the Penetration Depth $\lambda_{ab}$

When the distance s is comparable with the penetration depth $\lambda_{ab}$ as
it is the case in high-temperature superconductors the mutual interlayer
inductance $L_{n,n}$ (which is in this case when m=n the inlayer inductance in
fact) is nonzero and much larger than the other inductances (interlayer,
because m and n are different) $L_{n,m}$

$L_{n,n}=\frac{\lambda_{ab}}{s}$ (8)

Thus inductances $L_{n,m}$ with n different from m are not taken into account
here in this section.

For the phase difference $\phi_{m,m+1}$ the equations have in this case the
form

$-L_{n,n}\nabla^{2}\phi_{n,n+1}+\frac{1}{\lambda^{2}_{J}}\sin(\phi_{n,n+1})=0$
(9)

where the lower order terms in $\frac{L_{n,m}}{L_{n,n}}$ for n and m different
are neglected. They would lead to multi-Sine-Gordon equations.

Let us denote by

$\phi=\phi_{n,n+1}$ (10)

an angle variable. We assume that this angle is the same for every plane,
which may be done due to expected homogeneity in the sense of the same states
present in all layers.

Let us denote further

$\lambda_{p}=\lambda_{J}.L_{n,n}$ (11)

The two-dimensional sine-Gordon equation (9) takes the form

$\triangle\phi=\frac{1}{\lambda^{2}_{p}}\sin(\phi)$ (12)

This form of the two-dimensional sine-Gordon equation was studied by the
author in [6].

##  5 The Distance s is Comparable with the Penetration Depth $\lambda_{ab}$
\- Influence of Next Layers

In the previous section we described lagrange-Euler equations for the case
when the distance s is comparable with the penetration depth $\lambda_{ab}$.
We neglected the influence of neighbouring layers. Let us consider now the
effect of nearest neighbour layers. The interlayer inductance for neighbouring
layers in this case $L_{n,n\pm 1}$ is

$L_{n,n\pm 1}=\frac{\lambda_{ab}}{s}(1-\frac{s}{\lambda_{ab}})$ (13)

The equation for the phase differences $\phi_{m,m+1}$ has the form

$-L_{n,n}\nabla^{2}\phi_{n,n+1}-L_{n,n+1}\nabla^{2}\phi_{n+1,n+2}-L_{n-1,n}\nabla^{2}\phi_{n-1,n}+\frac{1}{\lambda^{2}_{J}}\sin(\phi_{n,n+1})=0$
(14)

Let us denote

$\lambda_{p}=\lambda_{J}.\sqrt{L_{n,n}}$ (15)

The two-dimensional sine-Gordon equation (14) takes now the form

$-\nabla^{2}\phi_{n,n+1}-l\nabla^{2}\phi_{n+1,n+2}-l\nabla^{2}\phi_{n-1,n}+\frac{1}{\lambda^{2}_{p}}\sin(\phi_{n,n+1})=0$
(16)

where the parameter l is small, and it is defined as

$l=(1-\frac{s}{\lambda_{ab}})^{1}$ (17)

We left the exponent 1 in order to indicate how the next to the nearest layers
will influence the Lagrange-Euler equations. In this form of the two-
dimensional sine-Gordon equation three neighbouring planes are coupled
explicitly. The solution of the equation (16) is again looking in the
homogeneous form, in which we again introduce the variable

$\phi=\phi_{n,n+1}$ (18)

for every n. This again may be done due to expected homogeneity in the sense
mentioned above as concerning the states in planes in this limit.

The equation (16) takes now the form

$-\nabla^{2}\phi+\frac{1}{\lambda^{{}^{\prime}2}_{p}}\sin(\phi)=0$ (19)

where we defined the renormalized Josephson length

$\lambda^{{}^{\prime}}_{p}=\lambda_{J}\sqrt{1+2l}$ (20)

which is larger than $\lambda_{p}$ due to the fact that l is positive. Note
that the larger l the larger renormalised Josephson length.

This homogeneous solution considered in this section has the characteristic
length $\lambda^{{}^{\prime}}_{p}$ larger than in the one-layer case length
$\lambda_{p}$ due to the inter-layer coupling via interlayer inductance.

The free energy of the single vortex in every layer and for the same
homogeneous state (homogeneous as concerning dependence on the the number n of
the layer, the vortex state is off course nonhomogeneous state) in all layers
depends linearly on L, the system linear dimension

$2^{\frac{3}{2}}N\pi
J2^{2}\frac{L}{\lambda^{{}^{\prime}}_{p}}-2k_{B}TN\ln(\frac{L}{a_{0}})$ (21)

where N is the number of layers, the cutoff constant $a_{0}$ is a constant of
the order of lattice constant.

We see that if the coupling between layers is taken into account then the free
energy is lower due to smaller energy of the vortex. The interlayer coupling
decreases the free energy and stabilizes the state with vortices the cores of
which are localizes on the same line in the direction c perpendicular to the
planes ab. Taking into account even higher order inductance couplings it is
expected that further lowering of the free energy will be found and that this
will further stabilize the vortex state.

Let us consider the solution of the equation (16) in which the sign of the
angle $\phi$ changes in neighbouring layers. This solution describes a
nonhomogeneous sate in the sense that the order parameter angle changes its
sign in neighbouring planes , e.i. it is shifted by the angle $\pi$

$\phi_{n,n+1}=(-1)^{n}\phi$ (22)

for every n. The equation (16) takes in this case the form

$-\nabla^{2}\phi+\frac{1}{\lambda^{{}^{\prime\prime}2}_{p}}\sin(\phi)=0$ (23)

where the Josephson characteristic length has now the form

$\lambda^{{}^{\prime\prime}}_{p}=\lambda_{J}\sqrt{1-2l}$ (24)

This Josephson length is smaller than the Josephson length $\lambda_{p}$, we
still assume l to be a small parameter. Thus the solution in which the sign of
angle $\phi$ changes in neighbouring layers has the length
$\lambda^{{}^{\prime\prime}}_{p}$ smaller than the length
$\lambda^{{}^{\prime}}_{p}$ and smaller than the original in-plane Josephson
length $\lambda_{p}$. The free energy of the state with a single vortex in a
given layer and with minus sign in neighbouring layers depends again linearly
on L, the system linear dimension

$2^{\frac{3}{2}}N\pi
J2^{2}\frac{L}{\lambda^{{}^{\prime\prime}}_{p}}-2k_{B}TN\ln(\frac{L}{a_{0}})$
(25)

where again N is the number of layers, the cutoff constant $a_{0}$ is a
constant of the order of the lattice constant.

If the inductance coupling l between layers is taken into account then we have
found that the free energy of the state in which the sign of the angle $\phi$
changes in neighbouring layers is higher with respect to the state in which
the sign of the angle $\phi$ does not change in neighbouring layers. The free
energy of the state in which the sign of the angle $\phi$ changes in
neighbouring layers is also higher with respect to the state in which the
layers are not coupled at all.

The small parameter l may be taken as an expansion parameter when solving the
equation (16). One finds in that case that the homogeneous in above sense
state with a vortex in one layer and with other vortices in other layers with
their core center in the same point in the plane ab /a line of vortices in the
direction c/ is more stable. Single-vortex state in one layer and the same
state in all layers give lower free energy due to the interlayer inductance
coupling. The interlayer inductance coupling plays stabilisation role for
these vortices. One can ask now that if many-vortex states in planes are
realized, which distance between vortices is such that the vortex
superconducting state becomes more preferable than the normal state, or than
the homogeneous superconducting state without vortices (usual superconducting
state in classical superconductors). The free energy of the single vortex in a
given layer and the same (homogeneous case as concerning the signs of the
phase variable) state with a vortex in neighbouring layers depends linearly on
L, the system linear dimension and the (critical) value of the parameter l. At
a given temperature it is given by the condition that the free energy with the
vortex at the same place in every layer is the same as the free energy of the
superconducting state without vortex

$F(vortex)=2^{\frac{3}{2}}N\pi
Jq^{2}\frac{L}{\lambda^{{}^{\prime\prime}}_{p}}-2k_{B}TN\ln(\frac{L}{a_{0}})=F(hom)=0$
(26)

For a given temperature and other parameters of the system increasing the
parameter l decreases the energy in the free energy, it may happen then that
the vortex state becomes the most stable state. It is clear that the
transition to the superconducting state occurs at higher temperature than for
the state without vortex. The transition to the single-vortex state with the
inter-layer coupling occurs at higher temperature than for the state in which
the single-vortex superconducting state does not take into account the inter-
layer inductance coupling.

If the density of superconducting pairs depends on temperature, we introduce
$\alpha=\alpha_{0}(T-T_{c})$, $T_{c}$ is the bulk transition temperature,
$\beta$ is a constant, the free energy $F(hom)$ is found from (1) for
$\mid\Psi\mid^{2}=\frac{\mid\alpha\mid}{\beta}$ . The Lawrence-Doniach
functional for the order parameter with $\Phi$ constant gives

$F(\Phi_{hom})=-\frac{H^{2}_{c}sL^{2}N\alpha^{2}}{8\pi\beta}$ (27)

where $H_{c}$ is the bulk critical magnetic field. The constant J and the bulk
critical magnetic field value $H_{c}$ are related

$J=\frac{H_{c}^{2}s\zeta^{2}_{ab}\alpha^{2}}{4\pi\beta}$ (28)

Here we consider the constant amplitude approximation
$\mid\Psi\mid^{2}=\frac{\mid\alpha\mid}{\beta}$ instead of
$\mid\Psi\mid^{2}=1$, we assume that temperature is near the transition
temperature. The gradient term in the phase in this case gives also
contribution to the free energy. It is of the order of $\frac{4s}{\lambda}$ as
a contribution to the bulk critical temperature $T_{c}$. Note that the case
with non-zero magnetic field may be easily described and will not be
considered here.

The normal state is given by $\Psi=0$ and its free energy is 0. The
superconducting state with the homogeneous (in the classical sense)
superconducting order parameter without vortices is given by
$\mid\Psi\mid^{2}=\frac{\mid\alpha\mid}{\beta}$ and its free energy is

$F(hom)=-\frac{NsL^{2}\zeta^{2}_{ab}H^{2}_{c}}{8\pi}=-\frac{JNL^{2}\alpha^{2}}{2\beta\zeta^{2}_{ab}}$
(29)

The vortex state with the $N_{v}$ vortices cores of which are localized in the
square /for simplicity/ lattice has the free energy

$F(N_{v}-vortex)=N_{v}N[2^{\frac{3}{2}}\pi
Jq^{2}\frac{L_{v}}{\lambda^{{}^{\prime}}_{p}}-2k_{B}T\ln(\frac{L_{v}}{a_{0}})]$
(30)

The inter-vortex distance $L_{v}=\frac{L}{N_{v}}$ can be found minimizing the
free energy $F(N_{v}-vortex)$ with respect to $L_{v}$. We have found the
minimal value $L_{v}$

$L_{v}=\frac{\sqrt{2}\lambda^{{}^{\prime}}_{p}k_{B}T}{\pi
Jq^{2}}ln(\frac{\sqrt{2}\lambda^{{}^{\prime}}_{p}k_{B}T}{\pi Jq^{2}a_{0}})$
(31)

The inter-vortex distance /vortex lattice constant/ is increasing with
temperature T and is decreasing with temperature T depending whether
temperature is above or below the critical temperature $T^{v}_{c}$,
respectively. The critical temperature $T^{v}_{c}$ is given by

$T^{v}_{c}=\frac{\pi Jq^{2}a_{0}}{\sqrt{2}\lambda^{{}^{\prime}}_{p}k_{B}e}$
(32)

The free energy for the vortex lattice state with $N_{v}$ vortices in the
plane has the form

$F(N_{v}-vortex)=NL^{2}[\frac{2\pi^{2}J^{2}q^{4}ln(\frac{\sqrt{2}\lambda^{{}^{\prime}}_{p}k_{B}T}{\pi
Jq^{2}a_{0}})}{\lambda^{{}^{\prime}2}_{p}k_{B}T}-\frac{\pi^{2}J^{2}q^{4}}{\lambda^{{}^{\prime}2}_{p}k_{B}Tln^{2}(\frac{\sqrt{2}\lambda^{{}^{\prime}}_{p}k_{B}T}{\pi
Jq^{2}a_{0}})}]$ (33)

The free energy of the $N_{v}$ \- vortex state should be compared with the
free energy $F(N)=0$ of the normal (metal) state, and with the free energy of
the homogeneous (in the classical sense) superconducting state without
vortices given by

$F(sc)=-\frac{JNL^{2}\alpha^{2}}{2\beta\zeta^{2}_{ab}}$ (34)

As can be seen the transition at the transition temperature $T_{cv}$ from the
normal phase to the $N_{v}$ \- vortex superconducting state occurs, it is of
the first order.

Depending on the parameters of the material the transition temperature
$T_{cv}$ may be higher for this transition than the transition temperature in
the case of the second order phase transition from the normal phase to the
homogeneous superconducting phase without vortices. It is given by

$T_{cv}=\frac{\pi Jq^{2}a_{0}}{\sqrt{2}\lambda^{{}^{\prime}}_{p}k_{B}}$ (35)

Note that the transition temperature $T_{cv}$ would be equal zero for zero
topological charge q. Note further that the transition temperature $T_{cv}$
increases when the interlayer inductance increases.

We will now discuss several solutions of the two-dimensional Sine-Gordon
equations.

##  6 Single-Vortex Configurations

In [6] we have found the single vortex configuration as a solution of the two-
dimensional sine-Gordon equation, and thus of the order parameter. It has the
form

$\phi=\pm\tan^{-1}(\frac{\sqrt{\frac{a}{(2-a)}}\sinh(\sqrt{\frac{2-a}{a}}\frac{(x-x_{0})}{\lambda})}{\sinh(\sqrt{\frac{a}{2}}\frac{(y-y_{0})}{\lambda})})$
(36)

here $\lambda$ equals $\lambda_{p}$. The vorticity of this vortex is plus
minus four. In the limit of $\lambda>>L$ we obtain a usual isotropic vortex
configuration. The configuration (36) breaks the rotational symmetry. The free
energy of the configuration given by (36) is linear in L. This is in
difference to isotropic vortices. In the limit of $\lambda>>L$ where a usual
isotropic vortex appears the logarithmic dependence of the free energy of the
configuration on L is found. Thus in systems in which anisotropic vortices are
present and which are small, e.i. in which $\lambda>>L$, the anisotropy of the
vortex will be almost undetectable.

In the limit of $\lambda_{p}\gg L$ the anisotropic vortex takes an isotropic
form is the zeroth approximation.

$\phi=\tan^{-1}(\frac{(x-x_{0})}{(y-y_{0})})$ (37)

In isotropic material systems in which anisotropic vortices are present and
which are large enough, e.i. in which $\lambda<<L$, the anisotropy of the
vortex will be detectable.

##  7 Multi-Vortex Configurations

Multi - (Resonant-Soliton) - Soliton solutions and vortex-like solutions in
two and three dimension for the sine-Gordon equation were studied in [19]. For
the two- and three- dimensional sine-Gordon equations there exist exact multi
- (resonant-soliton) - soliton solutions and vortex-like solutions, in
addition to exact multi-soliton and resonant - soliton solutions. In [20] it
was shown that the quasi-vortex type solutions from [6] can be derived from
the multiple soliton solutions by the proper procedure. Thus it follows from
that there exist multiple vortex-like solutions. Moreover an anisotropic
vortex configuration far from the core of the vortex has the form of almost
unchanging order parameter regions separated by soliton-like walls.

##  8 Static Lattice of Vortices, Multi-vortex Solutions

Let us briefly overview solutions of the sine-Gordon equations which represent
static lattice of vortices and multi-vortex solutions. These solutions and
space periodic vortex chains and lattice and breather solutions were studied
in [21]. The boundary problem for two-dimensional sine-Gordon equation was
studied by the inverse scattering method in [34]. Relation between vortex
solutions of the two-dimensional sine-Gordon equation and solitons of the same
equations were studied in [20]. Single vortex solution [6] can be derived from
the known multiple soliton solutions by a proper procedure. This fact shows
possibility to find multi-vortex solutions of the two-dimensional sine-Gordon
equations due to existence of multisoliton solutions.

Vortex-type lattice solutions found by Hirota method and by Backlund
transformation represent alternating vortices with $q=\pm 4$ and form a
tetragonal or square lattice. Interaction of the spin-wave and vortex lattice
are similar to the soliton-wave interactions.

Vortex-antivortex pair with the vorticity $q=\pm 4$ is given by

$\phi=\tan^{-1}(\frac{1-x+\sinh(y)\exp(-x)}{\sinh(y)-(1+x)\exp(-x)})$ (38)

The total topological charge of the vortex-antivortex pair is zero. Such pairs
may be in pre-vortex-lattice states, as a kind of liquid.

A vortex array was found with topological charges $q=\pm 4$. As far as the
author is aware the vortex arrays were not observed directly, however several
vortex arrays which are parallel may have an equilibrium distance between
arrays and thus may be difficult to distinguish them from the tetragonal phase
at first sight. However one may expect that stiffness of such configuration in
the direction of arrays will be substantialy different from that in the
perpendicular direction. Numerical singular solutions of the elliptic sine-
Gordon equation [24] were studied. A semi-infinite $2\pi$ \- kink starting
from a singular point of the vortex type was found. Ring-like $2\pi$ \- kinks
with a common center in the singular point were found. The singular point with
a linear region of the logarithmic spiral was found. Exact solutions of the
sine-Gordon equations as rational-exponential solutions are described in [35].
Exact rational-exponential solutions of the two-dimensional sine-Gordon
equations are constructed by a method based on the formal perturbation method.
These solutions represent the nonlinear superposition of two interacting
$2\pi$ kinks, at the point of intersection we have a vortex with topological
charge plus and minus 4. Numerical analysis shows that the minimum distance
between centers of vortices with opposite topological charges in dipole state
exists and is given by approximately 1.1 in lambda units. These two vortices
cannot annihilate in spite of zero total topological charge of the dipole.

##  9 Dynamic Properties of Two-Dimensional Sine-Gordon Vortices

The two-dimensional sine-Gordon equation with time dependence was studied in
[25], and it was found that a particular dynamical pattern in a perturbed two
-dimensional sine - Gordon systems exists. The stability of vortex-like
solutions were checked. The main topological invariant of the vortex-like
solutions is due to the total length of the $\pm 2\pi$ wave fronts entering as
elementary kink-like patterns in the constitution of the whole configuration.
This corresponds to the energy conservation law. In [25] it was also shown
that a particular dynamical pattern in a perturbed or not two-dimensional
sine-Gordon system exists. Its stability with respect to perturbations and in
time was studied. Their main topological invariant is the total length of the
kink wave fronts forming the pattern and the invariant is consistent with the
conservation of energy in all situations.

In [19] the time dependent sine-Gordon equation without damping was studied.
The procedure like that used in [20] leads to a two-soliton solution, thus
intersection of these solitons corresponds with a moving vortex. On the other
hand the author [19] shows that three or higher number soliton solutions are
known to exist only with limited parameter range. Thus time dependence without
damping here does not lead to a vortex-lattice in general, this may exist only
in a limited range of parameters.

##  10 Summary and Discussion

In our papers [1] and [2] we studied layered superconducting materials.
Vortex-like states were described and compard with normal metalic phase and
superconducting state without vortices. Results were found as concerning the
influence of neighbouring planes on the vortex state in a given plane and new
exact solutions of the coupled Lagrange-Euler sine-Gordon equations were
found, their properties and free energy were compared. Three-dimensional
superconducting state with a vortex in every plane with cores localized in the
same point on a line perpendicular to planes is a stable state in zero
external magnetic field for the layered materials. The same holds for a
lattice of vortices case. There exists a phase transition at the temperature
$T_{c}^{v}$ between the normal phase and the vortex superconducting state
which is of the first order. Depending on the parameters of the material the
transition temperature $T_{c}^{v}$ may be higher than in the case of the
second order phase transition from normal phase to the homogeneous
(conventional) superconducting phase without vortices. The transition
temperature $T_{c}^{v}$ is zero for zero topological charge of vortices. Note
that the transition is in zero magnetic field. The first order phase
transition has its properties. It shows usually overheating and overcooling
effects. Moreover nucleation of the superconducting phase in the normal phase
occurs at temperatures higher than the transition temperature $T_{c}^{v}$. And
vice-versa. Nucleation of the normal phase in the superconducting phase occurs
at temperatures lower than the transition temperature $T_{c}^{v}$. Thus the
onset of the vortex-like excitations above the transition temperature
$T_{c}^{v}$ occurs in our theory due to presence of nucleation of the
superconducting phase with vortices. The onset of the vortex-like Nerst signal
above the transition temperature $T_{c}^{v}$ in experiments in LSCO, YBCO and
in other similar high-temperature materials which we discussed in the
Introduction may be explained by our theory. Vortex-like excitations in
$La_{2-x}Sr_{x}CuO_{4}$ at temperatures significantly above the critical
temperature were found in the Nerst effect signal [26]. Such excitations may
correspond to formation of nuclei of the vortex-superconducting phase above
the transition temperature. In overdoped $La_{2-x}Sr_{x}CuO_{4}$ (LSCO) the
upper critical field $H_{c2}$ does not end at $T_{c0}$ but at a much higher
temperature. The upper critical field $H_{c2}$ would correspond in our theory
to a magnetic field at which pairs in nuclei are destroyed, however firstly
vortex-antivortex dipols and prelattices should be destroyed. Nerst
measurements in $YBa_{2}Cu_{3}O_{4}$ (YBCO) / and in
$Bi_{2}Sr_{2-y}La_{y}CuO_{6}$ /, [28], and in LSCO, in fields up to 33 T show
the existence of vortex-like excitations high above $T_{c0}$. For these
materials the same as above holds as concerning nuclei of superconducting
vortex phase and vortices. The high field phase diagram of cuprates derived
from the Nernst effect implies that $T_{c0}$ corresponds to a loss in phase
rigidity rather than a vanishing of the pairing amplitude, [27]. Phase
rigidity should be present for the phase as the macroscopic order parameter.
In nuclei this macroscopic character is preserved, but nuclei have smalller
volume than a bulk superconducting phase. Thus there is a loss in phase
rigidity in the sense that nuclei are not correlated, macroscopic phase in
nuclei is not correlated too. In zero magnetic field there is equal number of
vortices with plus and minus topological charge. Vortex-antivortex pairs or
parts of a vortex lattice are strengthening presence of pairing amplitude due
to selfconsistent process described above. In nonzero field there is nonequal
number of vortices with plus and minus topological charge. Preformed pairs as
superconductor fluctuations may exist in Bose condensation of localised Cooper
pairs with short coherence length, [29], [30] and [31]. Pairing correlations
without phase coherence are described in [32] and [33]. Our theory is
different from just mentioned theories.

Some of recent reviews of the high-temperature superconductors properties can
be found in [36], [37], [38], [39], [40] and [41]. The experimental
spectroscopic data are more consistent. It is not quite clear what is the
nature of connection between antiferromagnetism and superconductivity.
Antiferromagnetic phase is the best understood part of the phase diagram.
ARPES methods lead to better understanding of the quasiparticle dispersion
[42]. Inelastic X-ray Raman scattering [43] give evidence for broken particle-
hole symmetry. There is present an incommensurate phase in antiferromagnetic
insulator phase [44]. The corresponding Hamiltonian contains nearest
neighbour, next-nearest neighbour magnetic interactions and the interaction of
spins of the fourth order. Magnetic Raman scattering [45] shows that $A_{2}$
is very weak in $Gd_{2}CuO_{4}$. In the phase diagram there is a pseudogap
region (phase) in YBCO, LSCO and BSCCO (2212). DC resistivity [46] in BSCCO
shows contribution of the pseudo-gap to resistivity at high temperatures (180K
- 200K). Pseudogap is seen also in ARPES and tunneling experiments. While
there are several possible interpretations of the pseudo-gap region presence,
like stripes, antiferromagnetic fluctuations, spin-charge separation and
proximity to the QC point, the preformed pairs in superconducting fluctuations
above the critical temperature is also one of such possibilities [41]. These
contributions may be due to contribution of vortex-like excitations which we
described theoretically above. Aharonov-Bohm effect is present in YBCO at 30K
above the critical temperature. Oscillations with flux period h/2e are seen
[47] . This may correspond to the presence of the vortex-like excitations
(vortex-antivortex pairs, etc.) above the transition temperature. Phase
coherence in BSCCO was studied. Phase stiffness as a function of temperature
at frequencies of 100 GHz and 600 GHz was determined [48]. See also [49]. The
Kosterlitz-Thouless superconducting transition is predicted when the stiffness
intersect this line increasing temperature. It is interesting that high-
frequency stiffness is non-vanishing above the transition (short-length
scales), and it is vanishing at low frequencies (long-length scales). We have
found [2] that the inter-vortex distance (vortex lattice constant) for the
vortex state is increasing with temperature T and decreasing with temperature
T depending whether the temperature T is above or below the critical
transition temperature, respectively. Fluctuations of the vortex state above
the critical transition temperature are bubbles which may contain pairs or
remnants of the vortex state. The decrease of the inter-vortex distance with
increasing temperature in our theory is consistent with these phase stiffness
measurements. High-frequency superfluid stiffness is non-vanishing (short-
length scales) which would correspond to smaller inter-vortex distance in our
theory. Note that we used $\phi_{n}(\bf r)$, a change of the order parameter
phase around a vortex, for description of the system. The constant amplitude
approximation $\mid\Psi\mid^{2}=1$ was considered, then we considered the
constant amplitude approximation in which the density of superconducting pairs
is temperature dependent. The constant amplitude approximation is quite a good
approximation due to the fact that phase coherence and pairing amplitude are
present above the critical temperature in experiments: the high field phase
diagram of cuprates implies that $T_{c0}$ corresponds to a loss in macrosocpic
(bulk) phase rigidity rather than a vanishing of the pairing amplitude, [27].

The inter-vortex distance is decreasing above this temperature in our theory
as it is observed in experiments with phase stiffness, see above. The
properties of $\phi_{n}(\bf r)$, a change of the order parameter phase around
a vortex, are more important than the changes of the density of pairs for
description of the vortex state (phase), and thus changes of the amplitude of
the order parameter are less important.

In this paper we reviewed our results on vortex configurations in two-
dimensional sine-Gordon systems in connection with high-temperature
superconductors. In [1] and [2]. we discussed the vortex states using exact
solutions of these coupled Lagrange-Euler sine-Gordon equations, their
properties and their free energy was compared with the normal phase state and
the homogeneous (conventional) superconducting state. Stable vortex
configurations in zero external magnetic field for the layered materials
exist, the same holds for a lattice of vortices case. The first order phase
transition overheating and overcooling effects are present in the system.
Nucleation of the superconducting phase in the normal phase, which occurs at
temperatures higher than the transition temperature $T_{c}^{v}$, then occurs.
The onset of the vortex-like excitations above the transition temperature
$T_{c}^{v}$, which occurs in our theory, may correspond to the onset of the
vortex-like Nerst signal above the transition temperature $T_{c}^{v}$ in LSCO,
YBCO and in other similar high-temperature materials. Thus the vortex-like
excitations not well understood in experiments may be compared with properties
of vortices from our theory. It was found experimentally that the vortex-like
excitations above the transition temperature $T_{c}^{v}$ in LSCO, YBCO and in
other similar high-temperature materials continuously evolve as concerning
their properties to vortex states below the transition temperature
$T_{c}^{v}$. This fact may be explained within our theory. Thus our theory may
explain some of the experimentally observed properties of high-temperature
superconductors described above. It will be interesting to study and compare
other properties of vortex-like excitations and vortex states in high-
temperature materials and compare these properties with those described by
anisotropic vortices.

## Acknowledgements

The author wishes to express his sincere thanks to the Abdus Salam
International Center for Theoretical Physics for the invitation to present
some of the results presented in this paper at the 12th Workshop on Strongly
Correlated Electron Systems, 2000 and at the Workshop Emergent Materials and
Highly Correlated Electrons, 2002. The papers contains some of the results
presented in the invited talk given at the Second International School on
Condensed Matter in Debrecen, September 2000, the author especialy wishes to
express his sincere thanks to Zs. Gulaczi for his interest. The author would
like to express his thanks for invitation to present part of the results
presented in this paper on the Second International Workshop on Electron
Correlations and Materials Properties, Rhodos, Greece, 25th Jun - 29th Jun
2001 to organizers, especially his sincere thanks are due to Dr. A. Gonis. The
paper was supported by the grant VEGA No. 1/0250/03, Department of Theoretical
Physics, Faculty of Mathematics, Physics and Informatics, Comenius University,
Bratislava.

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