Paper 1 (Superconductivity and Quantum Coherence, Printed paper 255)

Solution

In BCS theory, pairing mixes an electron state with a hole in its opposite-momentum partner. For a uniform spin-singlet gap, the uniform Bogoliubov--de Gennes Hamiltonian gives the momentum-pair eigenproblem

(ξ_{k} Δ^{*} Δ - ξ_{k}) (v _{k} u _{k}) = E_{k} (v _{k} u _{k}), E_{k} = ξ_{k}^{2} + ∣Δ ∣^{2} .

Here

ξ_{k} = ε_{k} - μ

. A fermionic Bogoliubov quasiparticle is a coherent particle–hole combination, with

∣ u_{k} ∣^{2} = \frac{1}{2} (1 + \frac{ξ _{k}}{E _{k}}), ∣ v_{k} ∣^{2} = \frac{1}{2} (1 - \frac{ξ _{k}}{E _{k}}) .

Far above the Fermi surface the excitation is mostly particle-like; far below it is mostly hole-like; at

ξ_{k} = 0

the two weights agree. The transformation preserves the fermionic canonical anticommutation relations. Its mean-field particle-number mixing does not mean that the full physical theory violates charge conservation.

For a fully gapped isotropic state, the minimum single-quasiparticle energy is

∣Δ∣

, while creating a pair of quasiparticles costs at least

2∣Δ∣

. This produces suppressed low-temperature thermal populations, an activated electronic heat capacity and a gap feature in superconducting tunneling. Gap nodes in an unconventional state instead allow low-energy excitations, changing the temperature dependence. These fermionic excitations are distinct from collective phase and amplitude oscillations of the paired condensate.

c

Solution

A neutral superfluid's broken global

U (1)

phase symmetry produces a Goldstone mode. Coupling phase to density fluctuations generally gives a long-wavelength phonon with

E (p) ≃ c p

, rather than the quadratic dispersion of a noninteracting condensate. In a dilute interacting Bose gas, the Bogoliubov quasiparticle dispersion is

E (p) = ε_{p} (ε_{p} + 2 g n), ε_{p} = \frac{p ^{2}}{2 m}, c = \frac{g n}{m} .

For a heavy weak obstacle moving at speed

v

, emitting an excitation of momentum

p

changes the available energy by

E (p) - v \cdot p

. The Landau criterion therefore gives

v_{L} = p > 0 in f \frac{E ( p )}{p} .

The dilute-gas spectrum has

v_{L} = c

. In liquid helium, a roton minimum can make the minimum ratio smaller than the sound speed; the slope of a line from the origin tangent to the full dispersion determines the kinematic bound. A neutral paired Fermi superfluid also has a fermionic pair-breaking bound of order

Δ/ p_{F}

in weak coupling. The smallest relevant branch bound must be considered. Actual flow can fail earlier through vortex nucleation, surface defects or other instabilities: the excitation-energy criterion is not a complete dynamical guarantee of dissipation-free flow.

d

Solution

In weak-coupling conventional superconductivity, a useful guide is

k_{B} T_{c} \sim ℏ ω_{ph} exp [- 1/ λ_{eff}]

: an appreciable phonon scale and an effective attraction after Coulomb effects are favorable. A large density of states at the Fermi level can strengthen pairing, while light constituent masses can raise relevant phonon frequencies. Increasing electron–phonon coupling without limit is not a general strategy: lattice instabilities, competing order and changes to the phonon spectrum constrain it, and stronger coupling requires treatment beyond the simple exponential expression.

For unconventional high-temperature superconductivity, strong electronic interactions and magnetic fluctuations can support sign-changing pairing, as in d-wave superconductivity. Layered structures and proximity to magnetic order may enhance useful correlations, but can also promote competing insulating or density-wave states. Sufficient mobile-carrier phase stiffness is needed to turn pairing into coherent superconductivity; strong pairing alone does not guarantee a large transition temperature. Disorder that is relatively harmless to an isotropic conventional pair state can be pair-breaking for sign-changing gaps. There is no universal recipe established by these qualitative conditions: the pairing interaction, symmetry, carrier density, competing phases and phase coherence must all be considered.

2

Solution

In three dimensions an ideal Bose gas can saturate its excited-state population at chemical potential zero, with the remaining particles forming a Bose-Einstein condensate. The transition follows from the Bose-Einstein distribution integrated against its density of states; below the ideal-gas transition,

n_{0} / n = 1 - (T / T_{c})^{3/2}

. Condensation alone does not supply the finite Landau critical velocity of an interacting superfluid: the ideal quadratic spectrum has

in f_{p} E (p) / p = 0

By contrast, a weak attractive interaction near a Fermi surface produces the Cooper instability and the BCS gap equation. Its logarithmic enhancement permits pairing at arbitrarily weak attraction in the ideal model. The low-temperature excitations have

E = ξ^{2} + Δ^{2}

, and in weak coupling

k_{B} T_{c} = (e^{γ_{E}} / π) Δ (0) ≃ 0.567Δ (0)

. This transition is controlled by pair formation when phase fluctuations are small; deeply bound pairs instead motivate a Bose-condensation description with their own density and effective mass.

For the given ellipsoidal dispersion, set

q_{x} = k_{x}

q_{y} = k_{y}

and

q_{z} = k_{z} / γ

. Then

ε = ℏ^{2} q^{2} / (2 m)

while the integration measure is

d^{3} k = γ d^{3} q

. Thus the anisotropic ideal-gas density of states per unit volume and per internal state is

N (ε, γ) = γ N (ε) = \frac{γ}{4 π ^{2}} (\frac{2 m}{ℏ ^{2}})^{3/2} ε .

The full continuum momentum range is assumed; this is an anisotropic three-dimensional gas, not yet a discrete stack of two-dimensional bands.

At the Bose transition, substitute

x = ε / (k_{B} T)

n_{B} = \frac{γ}{4 π ^{2}} (\frac{2 m}{ℏ ^{2}})^{3/2} (k_{B} T)^{3/2} \int_{0}^{\infty} \frac{x ^{1/2} d x}{e ^{x} - 1} .

The integral is

Γ (3/2) ζ (3/2) ≃ 2.315

. Solving for the Bose-Einstein condensation temperature for an anisotropic effective mass gives

k_{B} T^{BEC} = \frac{2 π ℏ ^{2}}{m} [\frac{n _{B}}{γ ζ ( 3/2 )}]^{2/3} ≃ \frac{3.31 ℏ ^{2} n _{B}^{2/3}}{m γ ^{1/3}} .

(3)

The heavier

z

-direction effective mass increases the available excited-state count and lowers the condensation temperature at fixed number density.

For a balanced two-spin Fermi gas, let

n_{F}

denote total fermion number density and take

N_{s}

to be the single-spin density of states used in the conventional gap equation. Counting the filled ellipsoid gives

ε_{F} (γ) = \frac{ℏ ^{2}}{2 m} (3 π^{2} n_{F})^{2/3} γ^{- 1/3} \equiv ε_{F 0} γ^{- 1/3},

(4)

N_{s} (ε_{F}, γ) = \frac{m ( 3 π ^{2} n _{F} ) ^{1/3}}{2 π ^{2} ℏ ^{2}} γ^{1/3} \equiv N_{s 0} γ^{1/3} .

(5)

If a spin-summed density of states is used instead, the coupling convention in the supplied integral must change accordingly; inserting a factor two into only one side would change the exponent incorrectly.

For

ε_{c} ≪ ε_{F}

, replace the density of states across the interaction window by

N_{s} (ε_{F}, γ)

. Writing

λ_{0} = ∣ g ∣ N_{s 0}

, the zero-temperature integral becomes

1 = λ_{0} γ^{1/3} arsinh (\frac{ε _{c}}{Δ}),

(6)

and therefore

Δ (γ) = \frac{ε _{c}}{sinh [ 1/ ( λ _{0} γ ^{1/3} )]} ≃ 2 ε_{c} exp [- \frac{1}{λ _{0} γ ^{1/3}}] .

(7)

The second expression additionally requires

λ_{0} γ^{1/3} ≪ 1

. The explicit density dependence is

λ_{0} = ∣ g ∣ m (3 π^{2} n_{F})^{1/3} / (2 π^{2} ℏ^{2})

. This mass-anisotropy enhancement of a weak-coupling BCS gap initially raises the pairing scale, while the ideal Bose condensation scale decreases as

γ^{- 1/3}

Figure 1.
Formal anisotropy-dependent gap and Bose scales, with a schematic coherence-limited transition
. **Details**.

Large anisotropy weakens transport and coherence in the heavy direction. The separation of pair formation and phase ordering then matters, so a large mean-field gap need not imply a large actual

T_{c}

. A qualitative sketch has

T_{c}

initially rising with the pairing scale and eventually turning down when coherence is the smaller scale. If preformed fermion pairs behave as bosons, use

n_{B} = n_{F} /2

and pair mass about

2 m

; in this continuum model this gives

k_{B} T^{BEC} ≃ 0.218 ε_{F 0} γ^{- 1/3}

The turnover cannot be located reliably by taking the weak-coupling formulas to

γ \to \infty

ε_{F}

falls,

λ_{0} γ^{1/3}

grows, the narrow-window assumption fails and the pair chemical potential and mass require a self-consistent treatment. The smooth curve in the figure is explicitly an illustrative interpolation of two extrapolated scales, not such a calculation. If actual decoupled two-dimensional layers are approached, ordinary ideal-gas finite-temperature condensation is absent and interacting phase order requires a different analysis, including the Berezinskii–Kosterlitz–Thouless transition when appropriate.

The cuprates make the distinction between pairing and phase coherence relevant: they are layered and have anisotropic motion, but their strong correlations, lattice band structure and predominantly d-wave superconductivity are outside this attractive continuum

s

-wave model. Anisotropy by itself is therefore not an explanation or quantitative fit of cuprate transition temperatures.

3

Solution

Use a complex pair order parameter

Ψ = ∣Ψ∣ e^{i θ}

with signed charge

q

∣ q ∣ = 2 e

. A local isotropic Ginzburg-Landau theory of superconductivity has free energy

F = \int d^{3} x [α ∣Ψ ∣^{2} + \frac{β}{2} ∣Ψ ∣^{4} + \frac{∣ ( - i ℏ\nabla - q A ) Ψ ∣ ^{2}}{2 m _{*}} + \frac{B ^{2}}{2 μ _{0}}] .

For

α < 0

β > 0

, the uniform amplitude is

∣ Ψ_{0} ∣^{2} = - α / β

. Variation gives the static field equation and supercurrent

\frac{( - i ℏ\nabla - q A ) ^{2}}{2 m _{*}} Ψ + α Ψ + β ∣Ψ ∣^{2} Ψ = 0, j_{s} = \frac{q}{m _{*}} Re [Ψ^{*} (- i ℏ\nabla - q A) Ψ] .