Quantum part of phenomenon:T_C from singular response to pair field

N. Prokof’ev

Simons FND school, July 2025

Spontaneous symmetry breaking:

Suppose we have some quantity Q, and by symmetry of the problem H(Q)=H(-Q).

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Then, . How can one get in a physical system?

Ising model:

at any T

Symmetry:

N. Bogoljubov:

1. Add symmetry-breaking term to Hamiltonian: , now

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2. Take the thermodynamic limit in the following sense:

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3. Observe that (i) for response to symmetry-breaking field is weak:

(ii) for response to symmetry-breaking field is singular: when

! 4. Q₀ cannot be transformed to -Q₀ via local dynamics without overcoming macroscopic barriers 🡪

order-parameter related defects: domain walls, vortexes, skyrmions, …

Spontaneous symmetry breaking = singular response to vanishingly small symmetry-breaking field at

Superconductivity

Symmetry:

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(particle number conservation)

Apply:

and compute to find divergent response at/below T_C

This is not just a concept talking point, it’s a practical technical tool to determine T_C !

(Anomalous average)

* I will introduce/discuss “conventional” s-wave symmetry. Daniel and Andrey will take you further …

U(1) symmetry

Diagrammatic representation:

Full Green’s function

(interaction effects included)

Cooper-channel irreducible

vertex function

This is how the formalism goes …

Re-define:

(Dyson type equation)

Diagrammatic representation:

In vector-matrix notations:

Solution in terms of eigenvalues and eigenvectors of ,

Singular response corresponds to the largest eigenvalue reaching unity at :

Gap function equation

BCS (weak-coupling) theory:

- ideal Green’s function with

- attractive coupling constant at energies and zero otherwise

FS density of states (per spin)

BCS (weak-coupling) theory:

with

Electron-phonon coupling:

(more realistic modeling)

attractive and decays to zero for

Any coupling based on exchange of bosonic excitations/fluctuations (including collective modes):

spin waves/paramagnons, two-level systems, etc.

One more step beyond original BCS:

Green’s function in an interacting system is always renormalized!

At least consider the lowest-order diagram based on the same

vertex function, say this one for electron-phonon coupling:

The net result within the Fermi liquid

theory at low temperature

smooth across

FS even at T=0

important given exponential dependence of T_C on

This setup is the essence of the Migdal-Eliashberg theory

All good but does not explain superconductivity in metals where repulsive Coulomb forces dominate!

Bogoljubov, Tolmachev, Shirkov:

Take repulsive bare coupling and consider effective low-energy interaction

external lines are at low

energy/freq. below

internal lines have high

energies/freq. above

In the particle-particle channel

& near the Fermi surface g>0

is renormalized to a smaller value

Metals:

Exact solution of the gap equation:

Rietschel-Sham model (exactly solvable):

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Frequency-dependent repulsion, just weaker at low frequency, f < 1

Define , ,

Piece-wise solution with Low- and High-frequency values

(i) Consider separately two s-wave superconducting mechanisms and compute the corresponding

T_C values for each of them using the “frequency only” gap function equation (assume

Mechanism A:

Mechanism B:

(ii) Compute analytic expressions for the largest eigenvalues for mechanisms A and B

and plot them as functions of ln(T)

(iii) Compute the largest eigenvalue and T_C for the sum of two mechanisms:

Mechanism C:

Homework assignment

Emergent BCS from purely repulsive frequency-dependent interaction g>0:

(i) , i.e. “attractive-type” for any f < 1 !

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(ii) is increasing monotonously as T 🡪 0 !

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(iii) But is may not reach unity ever …

1. If f is small enough, the system will go superconducting. Large makes it pretty “easy”

2. , no superconductivity (in this symmetry channel) despite properties (i) and (ii).

3. The largest eigenvalue solution changes sign at frequency Ω !

Repulsion should strong enough (!) @ fixed f, E_F/Ω

is a non-linear function of ln(Ω/T) and cannot be used to predict T_C from normal state calculations

(i) Consider separately two s-wave superconducting mechanisms and compute the corresponding

T_C values for each of them using the “frequency only” gap function equation (assume

Mechanism A:

Mechanism B:

(ii) Compute analytic expressions for the largest eigenvalues for mechanisms A and B

and plot them as functions of ln(T)

(iii) Compute the largest eigenvalue and T_C for the sum of two mechanisms:

Mechanism C:

Homework assignment

All good but does not explain superconductivity in metals where repulsive Coulomb forces dominate!

Bogoljubov, Tolmachev, Shirkov:

Take repulsive bare coupling and consider effective low-energy interaction

external lines are at low

energy/freq. below

internal lines have high

energies/freq. above

In the particle-particle channel

& near the Fermi surface g>0

is renormalized to a smaller value

Metals:

Back to effective low-energy/frequency interaction

For Rietschel-Sham model:

instead of g

Sign change from repulsive to attractive for small enough f !

repulsion

attraction

Alternative formulation of the T_c problem

original

alternative

Different/renormalized eigenvalue problem

trivial dependence on L: “miracle” of implicit renormalization

General “advice”: the best treatment is to use altetrnative formulation with

Is this what happens in metals?

Screened Coulomb repulsion

e-ph attraction

No! It’s a long story which starts with dynamic screening of Coulomb interactions, Green’s function renormalization, and high-order vertex corrections …

Screening in Coulomb systems:

Screening:

RPA approximation: (Lindhard function at T=0)

At we have

At we have

SC for r_S>2

Thomas-Fermi

Screening momentum

- Before 1957: Electrons cannot form bosonic pairs because of Coulomb repulsion!

- Phonon mediated interactions are attractive (Friedel)! This may lead to paring if we ignore Coulomb pot. (BCS)!

- Coulomb repulsion is still there and will prevent phonon-mediated pairing …

- Renormalization weakens Coulomb potential (Bogoljubov, Tolmachev, Shirkov, Anderson, Morel)!

Take ; in RPA

• This is radical mistreatment and false picture! Screening is dynamic and can induce paring all by itself (Takada, Rietschel, Sham, …, )!

- How to explain the success of phenomenological in a vast number of superconductors then?

as observed experimentally!

Ironic history of the dominant mechanism behind superconductivity:

Recall

What else is missing?:

And this is the precise result for jellium = homogeneous electron gas

And vertex corrections …

+

+ …

+

+

1. Mass renormalization is tiny at r_S=4

6-th order VMC calculation by K. Chen, K. Haule ’21

< 1% change

Ultimate solution

2. product is close to unity

Density Functional

Perturbation Theory

Too fancy and complicated?

Theor. physics community

“physics of scenarios”

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- Unconventional mechanisms

- New materials

- Strong correlations, non-Fermi liquid

• Flat bands & quantum geometry
• …

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“physics of real materials - computations”

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None of this exists!

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If

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Need higher-order treatment of Coulomb in material science codes!