Transverse-field Ising model

The transverse-field Ising model is a quantum spin model with competing Ising exchange and a perpendicular field, for example

H = - J \sum_{i} X_{i} X_{i + 1} - h \sum_{i} Z_{i}

. In one dimension its ordered and paramagnetic phases meet at

∣ h / J ∣ = 1

Table of contents
- Ising-chain quantum critical point Transverse-field Ising model
- Ising-chain Bogoliubov diagonalization Transverse-field Ising model
  - Ising-chain ground-state energy Ising-chain Bogoliubov diagonalization
  - Ising-chain quasiparticle dispersion Ising-chain Bogoliubov diagonalization

Ising-chain quantum critical point ( $g_{c} = 1$ )

 0  0

For positive ferromagnetic exchange and nonnegative transverse field, the Ising-chain quasiparticle dispersion closes its gap at

g = 1

, with

ϵ_{k} \sim 2 J ∣ k ∣

. This is the quantum phase transition between the model's ordered and field-polarized regimes. The exact dispersion identifies the gapless scale; computing its order parameter additionally requires ground-state correlations.

Ising-chain Bogoliubov diagonalization ( $ϵ_{k} = 2 J 1 + g^{2} - 2 g cos k$ )

 0  0

For a paired non-self-conjugate momentum, rotate the Ising-chain Nambu spinor by an angle with

cos θ = (g - cos k) / r_{k}

sin θ = - sin k / r_{k}

and

r_{k} = 1 + g^{2} - 2 g cos k

. Opposite angles at

k, - k

preserve the canonical anticommutation relations. The result is

H = \sum_{k} ϵ_{k} (γ_{k}^{†} γ_{k} - 1/2)

. Self-paired momenta have no pairing and are handled directly as number levels. Periodic spin-chain sectors additionally constrain fermion parity and allowed mode quantization.

Ising-chain ground-state energy ( $E_{0} = - \frac{1}{2} \sum_{k} ϵ_{k}$ )

 0  0

The vacuum of the diagonal Bogoliubov quasiparticles has the displayed energy. In the thermodynamic bulk,

E_{0} / N = - J (2 π)^{- 1} \int_{- π}^{π} 1 + g^{2} - 2 g cos k d k

. The Hellmann–Feynman theorem then gives the field magnetization

\sum_{n} ⟨ S_{n}^{z} ⟩ = \sum_{k} J (g - cos k) / ϵ_{k}

. Sector-dependent finite-chain boundary terms must be restored when exact finite-size levels are needed.

Ising-chain quasiparticle dispersion ( $ϵ_{k} = 2 J 1 + g^{2} - 2 g cos k$ )

 0  0

For nonnegative field

g

, the bulk quasiparticle gap is

2 J ∣ g - 1∣

. At

g = 0

the dispersion is flat,

2 J

; at large

g

it is

2 J g - 2 J cos k + O (J / g)

. At

g = 1

it becomes

4 J ∣ sin (k /2) ∣

, linear near the gapless point. The finite periodic spin chain has additional global excitation constraints; a single isolated domain wall is not an allowed closed-chain spin configuration.

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The transverse-field Ising model (TFIM) is a fundamental model in statistical mechanics and quantum physics used to study phase transitions and quantum critical phenomena. It is an extension of the classical Ising model, which describes spins on a lattice that can take on values of +1 or -1, interacting with their neighbors. In the TFIM, in addition to the nearest-neighbor interactions, a transverse magnetic field is applied, which influences the spins in a direction perpendicular to the Ising interactions.

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