Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2012 / Paper 1 / 1 / d

Table of contents
- Solution d
- i d
  - Solution i
- ii d
  - Solution ii
- iii d
  - Solution iii

Solution

Put

d_{k} = (g - cos k)^{2} + sin^{2} k

. The matrix in the Ising-chain Nambu spinor has eigenvalues

\pm d_{k}

. A canonical Bogoliubov transformation, with paired angles satisfying

cos ϑ_{k} = (g - cos k) / d_{k}

and

sin ϑ_{k} = - sin k / d_{k}

, diagonalizes it. Accounting for full-zone fermionic Nambu energy counting gives

H = k \sum ϵ_{k} (γ_{k}^{†} γ_{k} - \frac{1}{2}), ϵ_{k} = 2 J d_{k} .

Thus the bulk Ising-chain ground-state energy is

E_{0} = - \frac{1}{2} k \sum ϵ_{k}, \frac{E _{0}}{N} ⟶ - \frac{J}{2 π} \int_{- π}^{π} 1 + g^{2} - 2 g cos k d k .

Exact finite-chain states additionally obey their fermion parity sector.

The Ising-chain quasiparticle dispersion has bulk gap

2 J ∣1 - g ∣

, closing at the Ising-chain quantum critical point. Finite-system parity splittings and the bulk gap are distinct scales. The three requested limits are illustrated below.

Figure 1.
Ising quasiparticle bands
. Source. Details.

i

Solution

At g=0,

ϵ_{k} = 2 J

is flat. A domain wall costs 2J, but walls occur in pairs on a periodic spin ring. A single bulk quasiparticle is therefore not automatically the first physical finite-ring spin excitation.

ii

Solution

At large positive g,

ϵ_{k} = 2 J g - 2 J cos k + O (J / g)

. The gap grows as

2 J (g - 1)

while the leading bandwidth remains 4J. The field-polarized vacuum is unique.

iii

Solution

At g=1,

ϵ_{k} = 4 J ∣ sin (k /2) ∣ \sim 2 J ∣ k ∣

. The gap closes at zero momentum and

E_{0} / N = - 4 J / π

. With unit lattice spacing the critical velocity is

2 J /ℏ

.

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