Download Document

Meccanismo di Higgs per sistemi
"non convenzionali":
teoria ed applicazione ai
superconduttori
S. Esposito
Dipartimento di Scienze Fisiche – Università di Napoli “Federico II”
& I.N.F.N. – Sezione di Napoli
(Standard) Higgs mechanism
The Higgs mechanism is the basic ingredient of theories describing phenomena
where spontaneous symmetry breaking takes place:
- massive gauge theories;
- superconductors;
- superfluids; etc.
(Standard) Higgs mechanism
The order parameter of the given system is assumed to be a scalar field
For superconductors, for example, it is interpreted
as the wavefunction of the Cooper pairs in their
center-of.mass frame.
In a superconductor the electromagnetic gauge field
becomes short-ranged, that is the photon becomes
massive.
The Lagrangian density for a scalar field
electromagnetic field
is given by (
interacting with the
):
.
(Standard) Higgs mechanism
If the scalar field develops a non-vanishing vev
, corresponding to
the condensation value of the field below a critical temperature,
the photon becomes massive below that (nonzero) critical temperature, thus explaining the
phenomenology of standard superconductivity.
Indeed, the thermodynamical free energy of the system takes the form
and a non-zero critical temperature is predicted:
Different representations
What happens if the two (real) degrees of freedom of the system are described
by another representation of the complex scalar field? For example:
The Lagrangian is invariant under a reparametrization of the scalar field, but
this is not the only basic ingredient. In fact, the expression of the free energy
comes out to be different:
with a different critical temperature:
(Ni, Xu & Cheng, 1984)
Different representations
At the classical (tree) level, the various representations of the complex scalar
field describe the same physical reality, but the dynamics ruled by different
degrees of freedom could lead, in principle, to different predictions when
considering also the radiative, non-tree corrections.
However, such a difference is purely formal if only one condensation occurs
since, in this case, the different dependence of the critical temperature on the
model parameters is not observable, being such parameters not directly
observable.
The situation is, instead,
just different if two (or
more) condensations
take place in the same,
peculiar physical
system…
Field reparametrization: general theory
Basic (simple) assumptions: two different degrees of freedom, described by two
real scalar fields,
and
, and only one non-vanishing vev:
We assume that such vev corresponds to
.
The most general representation of the Higgs field is the following:
Examples:
Field reparametrization: general theory
General constraints: 1) non-vanishing vev for
kinetic terms. These imply:
The Higgs mass takes the form:
; 2) diagonal
Field reparametrization: general theory
The Higgs mass does depend on the representation chosen.
Instead the photon mass:
obviously comes out to be independent of the representation chosen.
By evaluating the quantum, temperature-dependent, radiative corrections to the
scalar potential, the critical temperature of the system turns out to be given by:
The observable
does depend on the representation chosen for the scalar
field through the Higgs mass.
The term parameterizes the relative strength between the self-interaction of
the Cooper pairs (ruled by ) and the electromagnetic interaction (ruled by ).
Field reparametrization: general theory
1.
do not depend on
coefficients:
Representations that differ only for the imaginary part give the same
(but the constraint on the coefficients prevent that the system be described only
by one real field).
2.
depend only on the
coefficients:
Representations whose real parts differ in their expansion around the vev only
for odd power terms in the field or for
even power terms give the
same
.
3. The Higgs mass squared is a positive quantity, then:
which is a further constraint on the field coefficients.
Field reparametrization: general theory
By changing the representation of the Higgs field, the critical temperature of the
system cannot assume any arbitrarily large value but is bounded in the interval:
corresponding to the (reverse of the) interval
.
1. For a Cooper pair self-interaction much stronger than the electromagnetic
interaction among electrons
, we have
(that is, no
superconductivity),
2. For a Cooper pair self-interaction much weaker than the electromagnetic
interaction among electrons
,
approaches its maximum value.
A possible representation for that is:
Field reparametrization: general theory
The maximum critical temperature has an impressive physical interpretation in
terms of the entropy of the system:
The maximum critical temperature corresponds, for given temperature, to the
minimum of the entropy of the system (different from zero for a non-vanishing
vev) or, in other words, to the maximum possible order of the system.
Indeed, higher temperatures correspond to smaller Higgs masses which, in turn,
advantages the transition to the more ordered broken phase.
Scalar two-phase systems
Systems with two (or more) phase transitions (two-phase superconductors, etc.)
are usually taken into account by introducing two (or more) complex scalar
fields describing, in superconductivity, the wavefunctions of differently ordered
Cooper pairs or even differently formed Cooper pairs (electrons in the pairs
experience different phonon interactions):
and the different values of the constants lead to obviously different critical
temperatures, thermal and magnetic properties, etc.
differently formed Cooper pairs in a Fe-based superconductors
differently ordered Cooper pairs in BSCCO cuprates
Scalar two-phase systems
However, based on the previous findings, one can obtain similar results without
involving new unknown parameters, just requiring that the two scalar fields are
endowed with equal bare masses and self-interaction coupling sonstants:
Let us assume that two different condensations occur in the same system:
1. the first one is described by a non-vanishing vev of the modulus of
;
2. the second one is described by a non-vanishing vev of the real part of
.
Two different critical temperatures then arise:
Starting from high values and then lowering the temperature we meet a first
SSB at
: the medium becomes superconducting. By futher lowering the
temperature, at
the condensation involving the second order parameter is
energetically favored and a new phase transition starts:
Scalar two-phase systems
This term is negative, so that the free energy decreases with respect to the
previous phase: the system becomes “more” superconducting.
Since no additional unknown parameters are present, this model is fully
predictive: all the relevant parameters can be deduced from the directly
observable critical temperatures.
Scalar two-phase systems
Thermal properties. with respect to the
standard one-phase case we have:
The pressure is predicted to be larger for two-phase superconductors:
The latent heat absorbed during the formation of the SC phase: the difference
between one- and two-phase systems reaches its maximum at
.
Whilst the free energy and the entropy are continuous at , a finite jump in the
specific heat is predicted at : this is a distinguishing feature of a first order
phase transition.
Scalar two-phase systems
Magnetic properties.
The London penetration depth δ of the magnetic field is predicted to be smaller
(even of about 70%) with respect to the one-phase case below :
Due to the different coherence lengths ξ of the two different Cooper pairs in the
two phases, two distinct behaviors of the crittical magnetic fields are as well
predicted, leading to peculiar properties.
For
, the system is a type-II
(δ/ξ >1/√2) superconductor, while below
it could behave as a type-I (δ/ξ <1/√2)
if:
upper critical fields versus temperature for different domains
Spin-triplet one-phase systems
Wavefunction and spin directions for electrons is the Cooper pairs
spin-triplet model in (TMTSF)2PF6 organic superconductor
Rotational degrees of freedom in superconductivity may be accounted for in the
previous model by introducing two mutually interacting order parameters:
Spin-triplet one-phase systems
The potential
is definite-positive, so that it describes a repulsion
between the two fields. It corresponds to the main term for small phase
difference of the Leggett interaction.
By expanding around the vevs, with a redefinition of the fields,
the lagrangian of the system becomes (up to second order terms):
Of the original 4 d.o.f. embedded into two complex scalar fields, only one of
them is disappeared giving rise to a massive photon while, by virtue of the
interaction potential, the remaining 3 d.o.f. all have the same mass, and can be
combined to form a triplet field
.
Spin-triplet one-phase systems
The very peculiar interaction breaks the isotropy of the medium and allows pair
of electrons to arrange into possible S=1 Cooper pairs.
The main physical properties follow from the free energy of the system:
Only one superconducting phase is present below the critical temperature
.
In this model, the main thermodynamical and magnetic properties of the present
p-wave system turn out to be essentially the same as for conventional s-wave
superconductors.
Applications
For
, the observed behavior of the specific heat, the
peculiar temperature dependence of the upper and lower
critical fields, and the pressure effects arising from the
competition between the two bands result to be very
similar to the ones predicted by the two-phase model
described above..
Some problems, however, still remains due to the large
difference between the two critical temperatures
measured (39K and 13K).
The intriguing properties of
are, instead,
well described by the models above.
It is a superconductor with a very low critical
temperature and long coherence length.
Its unconventional magnetic properties strongly suggest
a spin-triplet superconductivity, while a firm
thermodynamic evidence for a second superconducting
phase exists.
Applications:
Two jumps in the specific heat are
observed in the presence of a given
magnetic field. The exact values of the
two transition temperatures depend on
this magnetic field, the effect being
suppressed at very low temperatures.
The ratio of the critical temperatures,
however, is always close to the predicted
upper limit
.
The different effective masses of two
different Cooper pairs are responsible of
the two observed peaks.
In our model, the value of the ratio of the critical temperatures close to the
upper limit points out the extremely large value of the scalar self-interaction
with respect to the electromagnetic coupling.
Applications:
The “splitting” of the
curve:
For sufficiently high magnetic
fields and low temperatures,
higher eigenvalues (n>1) in the
Landau level solutions of the
linearized Ginzburg-Landau
equation come into play in the
equation for the depression in
transition temperature as a
function of the magnetic field.
n=0
n=1
n=1
n=2
In our model this is parametrized as:
where
and
depend on the type of Cooper pair condensed.
This behaviour is in good agreement with the experimental fits (see figure).
Applications:
Moreover, from the expressions of the
pairs:
parameters for the different Cooper
the ratio of the two slopes for n=2 can be deduced:
This gives an estimate of the ratio
which is independent of (though in
agreement with) specific heat measurements.
The “weaker” bound Cooper pairs are expected to start the transition from n=1
to n=2 at a lower value of the magnetic field applied and at a higher
temperature, as effectively observed (see figure).
Conclusions
The physics of spontaneous symmetry breaking may effectively depend on the
representation of the complex Higgs fields: different representations leads to
different Higgs masses, transition temperatures, etc. depending on what degree
of freedom (or combination of them) effectively condenses during the
transition.
Physical consequences come out only if two or more order parameters describe
the system (for superconductors, different Cooper pairs od different ordering of
them coexist). In this case, two or more different phases show themselves at
different temperatures, with peculiar thermodynamic and magnetic properties.
If the two (degenerate) Higgs fields describing the system interact between
them with a particular potential, only one phase is present showing itself as a
spin-triplet superconductivity with, however, thermodynamical and magnetic
properties similar to those for spin-sunglet systems.
These features are apparently observed in
and especially in
non-conventional superconductivity, but they likely should apply to other
research fields (particle physics, cosmology, etc.)
References
•
•
•
•
•
•
•
SUPERCONDUCTORS WITH TWO CRITICAL TEMPERATURES.
E. Di Grezia, S. Esposito, G. Salesi, Physica.C451:86,2007
SECOND DISCONTINUITY IN THE SPECIFIC HEAT OF TWO-PHASE
SUPERCONDUCTORS.
E. Di Grezia, S. Esposito, G. Salesi, Physica C467:4,2007
MAGNETIC PROPERTIES OF TWO-PHASE SUPERCONDUCTORS.
E. Di Grezia, S. Esposito, G. Salesi, Physica C468:883,2008
A GENERALIZATION OF THE GINZBURG-LANDAU THEORY TO PWAVE SUPERCONDUCTORS.
E. Di Grezia, S. Esposito, G. Salesi Mod.Phys.Lett.B22:1709-1716,2008
GENERALIZED GINZBURG-LANDAU MODELS FOR NONCONVENTIONAL SUPERCONDUCTORS.
S. Esposito, G. Salesi, Horiz.World Phys.264:121,2009
DESCRIBING SR2RUO4 SUPERCONDUCTIVITY IN A GENERALIZED
GINZBURG-LANDAU THEORY.
E. Di Grezia, S. Esposito, G. Salesi, Phys.Lett.A373:2385,2009
DEPENDENCE OF THE CRITICAL TEMPERATURE ON THE HIGGS
FIELD REPARAMETRIZATION.
E.
Di Grezia, S. Esposito, G. Salesi, J.Phys.G36:115001,2009