Condensate of electron quadruplets as a new phase of condensed matter

The formation of fermion quadruplet condensates is a new exotic condensed matter phenomenon discovered by Prof. Egor Babaev almost 20 years ago and 8 years after publishing a paper predicting it. Recently Babaev and collaborators presented in Nature Physics evidence of fermion quadrupling in a series of experimental measurements on the iron-based material, Ba_1-xK_xFe₂As₂.

The abstract of the article summarizes the finding.

The most well-known example of an ordered quantum state superconductivity is caused by the formation and condensation of pairs of electrons. Fundamentally, what distinguishes a superconducting state from a normal state is a spontaneously broken symmetry corresponding to the long-range coherence of pairs of electrons, leading to zero resistivity and diamagnetism.

Here we report a set of experimental observations in hole-doped Ba_1-xK_xFe₂As₂. Our specific-heat measurements indicate the formation of fermionic bound states when the temperature is lowered from the normal state. However, when the doping level is x ∼ 0.8, instead of the characteristic onset of diamagnetic screening and zero resistance expected below the superconducting phase transition, we observe the opposite effect: the generation of self-induced magnetic fields in the resistive state, measured by spontaneous Nernst effect and muon spin rotation experiments. This combined evidence indicates the existence of a bosonic metal state in which Cooper pairs of electrons lack coherence, but the system spontaneously breaks time-reversal symmetry. The observations are consistent with the theory of a state with fermionic quadrupling, in which long-range order exists not between Cooper pairs but only between pairs of pairs.

Fermion quadruplets are proposed to be formed as pairs of Cooper pairs are formed somewhat above the critical temperature T_c for a transition to superconductivity. Breaking of the time reversal symmetry T is involved.

The question is why quadruplets are stable against thermal noise above the critical temperature. Superconductivity is thought to be lost by the thermal noise making the bound states of electrons in Cooper pair unstable. Is the binding energy for quadruplets larger than for Cooper pairs so that quadruplet condensate is possible below higher critical temperature. What is the mechanism of binding?

The discovery is highly interesting from the TGD point of view.

1. TGD leads to a model of super-conductivity involving new physics predicted by TGD.
2. Adelic physics number theoretic view about dark matter as h_eff >h phases h_eff proportional to the order of the Galois group. This leads to the notion of Galois confinement. Galois confinement could serve as a universal mechanism for the formation of bound states including also Cooper pairs and even quadruplets. In quantum biology triplets of protons representing genetic codons and even their sequences representing genes would be formed by Galois confinement.
3. The finding also allows to develop more preices view of TGD view concerning discrete symmetries and their violation.

Time reversal symmetry in TGD

What do time reversal symmetry and its violation mean in TGD.

1. The presence of magnetic field causes violation of T in condensed matter systems.
2. Second, not necessarily independent, manner to violate T in TGD framework is analogous to that in strong CP breaking but different from it many crucial aspects. Vacuum functional is exponent of Kähler function but exponent can contain also an instanton term I, which is equal to a divergence of topological instant current which is axial. so that non-vanishing I suggests parity violation. The fact that exponent of I is imaginary while exponent of Kähler action is real, means C violation. If instanton current is proportional to conserved Kähler current its divergence is vanishing and M⁴ projection is less than 4-D.

   I is non-vanishing only if the space-time sheet in X⁴\subset M⁴\times CP₂ has 4-D CP₂ or M⁴ projection. The first case corresponds to CP₂ instanton term I(CP₂) and second case to I(M⁴) present since twistor lift forces also M⁴ to have an analog of Kähler structure. The two Kähler currents are separately conserved.

3. These two mechanisms of T violation might be actually equivalent if the T violation is caused by the M⁴ part of Kähler action. Consider a space-time surface with 2-D string world sheet as M⁴ projection carrying Kähler electric field but necessarily vanishing Kähler magnetic field B_K. If it is deformed to make M⁴ projection 4-D, B_K is generated and T is violated. Therefore generation of B_K in M⁴ can lead to a T violation.

Generalized Beltrami currents

Generalized Beltrami currents are nother key notion in TGD based view about superconductivity (see this).

1. The existence of a generalized Beltrami current j= Ψ dΦ implies the existence of global coordinate Φ varying along the flow lines of the current. Also the condition dj∧ j=0 follows. The 4-D generalization states that Lorentz force and electric force vanish. In effectively 3-D situation, j could correspond to magnetic field B and dj to current as its rotor and the Beltrami condition fof B implies that Lorentz force vanishes.
2. The proposal is that for the preferred extremals CP₂ resp. M⁴ Kähler current is proportional to instanton current I(CP₂) resp. I(M⁴) and therefore topological for D(CP₂)=3 resp. D(M⁴)=3. For D=2 the contribution to instanton current vanishes. In this case the Lorentz force vanishes so that the divergence of the energy momentum tensor is proportional to I and vanishes so that dissipation is absent. One can verify this result using the effective 3-dimensionality of the projection and using 3-D notations: in this formulation the vanishing of Lorentz force reduces to Beltrami property for B as 3-D vector. With this assumption, dissipation for the preferred extremals of Kähler action just as it is absent in Maxwell's theory. An open question is whether this situation is true always so that dissipation and the observed loss of quantum coherence would be due to the finite size of space-time sheet of the system considered.
3. Beltrami property would serve as a classical space-time correlate for the absence of dissipation and presence of quantum coherence. Beltrami property allows defining of a supra current like quantity in terms of Ψ and Φ. Usually the superconducting order parameter Ψ is actually not an order parameter for a coherent state as a superposition of states with a varying number of Cooper pairs. Now the geometry of the space-time sheets (magnetic flux tube carrying dark Cooper pairs) allows the identification of this order parameter below the quantum coherence scale. The TGD interpretation is that the coherent state is an approximation, which does not take into account the fact that the system is not closed. There is exchange of electron pairs between ordinary and dark space-time sheets with h_eff>h (see this). Dark Cooper pairs would form bound states by Galois confinement.
4. In the superconducting state space-time regions would have at most 3-D M⁴ projection at fundamental level and T would not be violated. There is no dissipation and pairs are possible below critical temperature.

   One can also understand the Meissner effect. According to the TGD view, the monopole flux tubes generate the analog of the field H perhaps serving as an approximate average description for the field of monopole flux tubes. This field induces the analog of magnetization M involving non-monopole flux tubes. Also M would be an average field. For superconductors in the diamagnetic phase, the sum would be zero: B= H+M=0. If the Cooper pairs have spin, the supracurrents of Cooper pairs at monopole flux tubes could generate the compensating magnetization.

TGD view about quadruplet condensate

How could one understand quadruplet condensate in the TGD framework?

1. T violation could be accompanied by the presence of Kähler instanton term I(M⁴) or I(CP₂) requiring 4-D M⁴ or CP₂ projection: this would also generate M⁴ magnetic fields. The M⁴ option would bring in new physics for which also the Magnus effect of hydrodynamics suggesting Lorentz force serves as an indication this).

   For 4-D M⁴ projection, the divergence of the axial instanton current would be non-vanishing and the proportionality of Kähler current and instanton current implying a vanishing classical dissipation would be impossible. The instanton number can be expressed as instanton flux over 3-D surfaces, which would be "holes".

2. For the quadruplet condensate M⁴ projection is 4-D and T is violated. Kähler magnetic fields originating from M⁴ part of Kähler action would be present as also dissipation. For quadruplet condensate M would not compensate for H so that net magnetic fields B would be generated and correspond to space-time sheets with 4-D M⁴ projection.
3. Dark matter as phases with h_eff>h would however be present and quadruplets would correspond to bound states of 4 electrons formed by Galois confinement (see this and this) stating that the total momentum of the bound state as sum of momenta, which are algebraic - possibly complex - integers, is a rational integer in accordance with the periodic boundary conditions.
4. What prevents the formation of Cooper pairs? Above T_c thermal energy exceeds the gap energy so that Cooper pairs are thermally stable. If the binding energy for quadruplets is larger, they are stable.
5. In what sense the quadruplets could be regarded as bound states of Cooper pairs? Since the ordinary Cooper pairs are Galois singlets, bound state formation does not look plausible since Cooper pairs themselves are unstable. A more plausible option is that Cooper pairs involved are "off-mass-shell" in that they have momenta, which are non-trivial algebraic integers and that the sum of these momenta is a rational integer in the bound state.

Remark: Four-momenta as algebraic integers are in general complex. Usual charge conjugation involves complex conjugation in CP₂ degrees of freedom. Is it accompanied by conjugation of the complex 4-momenta. Kähler currents of M⁴ and CP₂ are separately conserved: should one regard complex conjugations in M⁴ and CP₂ as independent charge conjugation like symmetries. C(M⁴) would however leav Galois singlets invariant.

See the article TGD and condensed matter physics and the book TGD and Condensed Matter".

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.