What is the physical meaning of phase in a superconductor?

Question

I am trying to understand the physical meaning of phase in the order parameter of a superconductor. In particular,I was looking at this article that states the phase $e^{i\theta}$ in the BCS ground state

$$|\psi_G\rangle=\prod_k (u_k +v_k e^{i\theta}c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger)|\phi_0\rangle $$

is not a gauge invariance, but a global $U(1)$ phase rotation symmetry. However, the author (on page 6) then says that

The most significant difference between an antiferromagnet and a superfluid or superconductor with regard to the order parameter is that the broken rotational symmetry in the former case is much more evident to us, as all the macroscopic objects in our daily life experience violate rotational symmetry at one level or another...In the case a superconductor, we need a second superconductor to have a reference direction for the phase, and an interaction between the order parameter in both superconductors to detect a relative difference in the phases.

I can easily visualize this breaking of $U(1)$ symmetry in a antiferromagnet. However, I am still confused as to what, exactly, is the physical meaning of phase in the superconducting order parameter? In particular, how can I prescribe a physical meaning to the phase on the Cooper pairing term in the BCS ground state? As with my other post on a similar question I had, any explanations or resources at the level of Tinkham would be greatly appreciated.

Answer 1

I can offer one analogy between quantum mechanical phase and spatial orientation.

Suppose you are inside a spherical spaceship surrounded by nothingness. The interior of the spaceship is otherwise isotropic, but the presence of you breaks the rotational invariance.

You can tell left from right, and up from down, all relative to yourself. If you turn left by 90 degrees, then left is your new front, and front is your new right, etc, but everything just seems identical. If you forget how much you have rotated, then there is no way to tell what was your initial orientation. The absolute orientation loses all meaning. Yes, you are still facing some direction at any given instance, but you can't tell where you are facing because all directions are the same!

(Remembering your rotational history is comparing your present orientation to your past orientation: you are still measuring everything relative to a reference.)

We don't trap ourselves in deep space in everyday life, and there are plenty of surrounding objects relative to which we can orient ourselves. So we (falsely) feel that orientation has an intrinsic, absolute meaning. But the spaceship situation exactly describes a superconductor in isolation.

Answer 2

The phase of a superconductor is the phase of the condensate wave function. In the equation you have written down, $u_k$ and $v_k$ are probability amplitudes for paired and unpaired states. The paired states have a phase factor because they exhibit long-range phase coherence. You can think of the many-particle wave function as a macroscopically coherent wave with an amplitude that depends on the number of particles. This is in contrast to most many-particle wavefunctions that look like an incoherent mess.

Now if the phase of the superconductor is uniform throughout the sample, you could just define the phase to be zero, since it has no physical relevance. But if you have two superconductors with different phases close to each other, it lead to what is known as the Josephson effect, where current is driven by the phase difference. Alternatively, if phase gradients exist in the sample, it creates a current. This follows from the definition of quantum mechanical probability current (https://en.wikipedia.org/wiki/Probability_current). One can also have defects known as vortices, where the phase is undefined along a line and the phase winds by $2\pi$ around this defect. Due the the gradient there is a circular current around the line, hence the name vortex.

Reference: -"Superconductivity, Superfluids, and Condensates" by James F Annett is a helpful introduction.

Answer 3

In this answer, I consider a time-dependent Ginzburg Landau model of superconductivity with an order parameter which can be written (locally) $\psi = |\psi|e^{i\theta}$.

Being a gauge-dependent quantity, the phase $\theta$ has no intrinsic physical meaning. However, assuming that there is a path $\gamma$ between those two points along which the order parameter is non-vanishing (except possibly at isolated points), the gauge-invariant phase difference given by $$\Delta\varphi\equiv \int_\gamma\left(\nabla \theta - \frac{2e}{\hbar} \mathbf A\right)\cdot \mathrm d\mathbf r$$

does have a physical interpretation. This quantity measures the time-integrated EMF along the path $\gamma$ since it became superconducting, i.e.

$$\Delta \varphi = -\frac{2e}{\hbar}\int_0^t \mathrm dt'\ V(t')$$ $$ V = \int_\gamma \mathbf E \cdot \mathrm d\mathbf r$$

Strictly speaking, there are dissipative effects which can modify this relationship a bit if you are considering scenarios in which the order parameter magnitude has spatial variations which change over time. So for simplicity, imagine a scenario in which your superconductor starts in a static equilibrium state with no currents, which then evolves over time to some final state.

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In summary, the gauge-invariant phase along some path encodes the history of the EMF along that path. In that sense, superconductors "remember" the electric fields which exist within them, even after those fields are gone.

Answer 4

The phase of a superconductor is closely related to its quantum mechanical nature. Typically when people ask about the physical meaning of quantum mechanical notions they want a classical (i.e. non quantum mechanical) analogue.

For a phase the obvious analogue is waves. Optics is good, but one can even discuss water waves. Alternatively one can also think on a collection of harmonic oscillators (can be pendulums, masses on a spring, etc.). When there is no coherence all the components are moving uncorrelated. One cannot see any peaks or valley or notice any special feature at a specific time, so we call this cacophony "symmetry". Now imagine that some correlation appear, all oscillators reach the extrema point at the same time, or that the water at some point are lowest exactly when they are highest in another point and vice versa. Now we can see a structure and we say the symmetry is broken. The phase how we describe where the repeated motion start.

But, nothing is moving in the superconductor, so how does this apply? Well, as it is usually with classical analogues, they cannot be fully applied. Indeed the phase of a superconductor is a quantum mechanical phase and just like the phase of a quantum state can only be observed via comparison to another phase. In this case one would need another superconductor and observe the Josephson effect

Answer 5

Doing a search on the WEB, I get the following answer:

"Ordinary conductors, such as copper gradually get more conductive with a decrease in temperature but superconductors like metals mercury get conductive all at once, after the critical temperature. This is also called phase transition."

But the Mercury is a special case, because this metal has a melting temperature, that is relative low. Condensing to a solid state like a crystal, the electronic orbitals are interleaving to themselves.

This is beginning of a phase, where all the free electrons are placed together into a common layer. Maybe some people, those who studied Chemistry, may understand something else through this term of phase, which is more like a condensation.

Usually the negative electrons do not interact among themselves due to the electrostatic repulsion. But at distances, comparable with a size of the crystal network, the magnetic forces are stronger.

What is happening next, is a reorientation of all those elementary magnetic moments of the spin. They are orientating in a phase, which cannot be determined, it is more like a statistic effect.

There are pairs of Cooper electrons, which can have a different behaviour than usual electrons. This pair of Cooper electrons, would have a resultant number of spin, which is null:

$s = \frac{1}{2} - \frac{1}{2} = 0$

It is well known, that the electrons cannot exist too many in a space region, as they are fermions. The Bosons are allowed, to exist more in the same place, according to the Bose-Einstein statistics.

This can be quite a bosonic condensation, or another new phase, as it has been called sometimes.

A people who studied physics, or electro-technics, could understand something else through the phase. Can be that orientation of the magnetic moments of spin, which may vary after a wave function of probability.

It could be also a subjective denomination, which may differ from an author to another. Or it could be a term, which is used to define a certain member of an equation, if this can be associated with the "phase".

Maybe this paragraph will bring new light:

"Even for a fixed set of boundary conditions, we have an infinite set of solutions for the superfluid flow, corresponding via ... to all possible choices of the phase field ϕ (x). In a simply connected superfluid, the flow will be vortex-free,..."

And we have there the expression of an impulse with a wave function of probability:

$m {v_s} = \frac{h}{{2}{PI}} \nabla ϕ (x)$

What I understand from this paragraph, is that, the wave function, ϕ (x) is sometimes called a phase field. But the meaning could be also related to the context of the super-conductivity, next to the critical state of super-fluidity.

Actually this expression can be used to get a type of harmonic solution for the density of probability, which is quite a well known solution, if I'm not wrong too much.