Meccanismo di Higgs per sistemi "non convenzionali":

teoria ed applicazione ai superconduttori

S. EspositoDipartimento 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 .

The Lagrangian density for a scalar field interacting with the electromagnetic field is given by ( ):

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.

(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 (non-zero) 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:

Examples:

We assume that such vev corresponds to .

The most general representation of the Higgs field is the following:

Field reparametrization: general theory

General constraints: 1) non-vanishing vev for ; 2) diagonal kinetic terms. These imply:

The Higgs mass takes the form:

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).

3. The Higgs mass squared is a positive quantity, then:

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 .

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 ordered Cooper pairs in BSCCO cuprates

differently formed Cooper pairs in a Fe-based superconductors

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 firstSSB at : the medium becomes superconducting. By futher lowering thetemperature, at the condensation involving the second order parameter isenergetically 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

Rotational degrees of freedom in superconductivity may be accounted for in the previous model by introducing two mutually interacting order parameters:

Wavefunction and spin directions for electrons is the Cooper pairs

spin-triplet model in (TMTSF)2PF6 organic superconductor

Spin-triplet one-phase systems

By expanding around the vevs, with a redefinition of the fields,

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.

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

Only one superconducting phase is present below the critical temperature .

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:

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.

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).

n=0 n=1

n=1 n=2

Applications:

This gives an estimate of the ratio which is independent of (though in agreement with) specific heat measurements.

Moreover, from the expressions of the parameters for the different Cooper pairs:

the ratio of the two slopes for n=2 can be deduced:

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).

ConclusionsThe 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.)

• 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 P-WAVE SUPERCONDUCTORS.E. Di Grezia, S. Esposito, G. Salesi Mod.Phys.Lett.B22:1709-1716,2008

• GENERALIZED GINZBURG-LANDAU MODELS FOR NON-CONVENTIONAL 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

References