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

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- Solution b

Solution

Let

n_{n} = c_{n}^{†} c_{n}

and

P_{n} = \prod_{m < n} (1 - 2 n_{m})

. The Jordan–Wigner transformation gives

2 S_{n}^{x} = P_{n} (c_{n} + c_{n}^{†})

and

P_{n + 1} = P_{n} (1 - 2 n_{n})

. The canonical anticommutation relations imply

(c_{n} + c_{n}^{†}) (1 - 2 n_{n}) = c_{n}^{†} - c_{n} .

Consequently the bondwise cancellation of a Jordan-Wigner string yields

4 S_{n}^{x} S_{n + 1}^{x} = (c_{n}^{†} - c_{n}) (c_{n + 1}^{†} + c_{n + 1}), 2 S_{n}^{z} = 1 - 2 n_{n} .

Substitution therefore gives

H = - J n \sum [(c_{n}^{†} - c_{n}) (c_{n + 1}^{†} + c_{n + 1}) + g (1 - 2 n_{n})] .

The finite-ring parity qualification of an Ising quasiparticle gap matters here. For an exact periodic spin chain, the closing bond retains total fermion parity. The parity-dependent boundary condition of the Jordan-Wigner chain is

c_{N + 1} = - P c_{1}

, with

P = (- 1)^{N_{f}}

. Fixing that parity determines the allowed Fourier grid, and the resulting many-body states must still have the chosen parity. Dropping this boundary issue preserves the thermodynamic energy density, but can lose a degeneracy or the correct finite-ring excitation gap. ^[ref]

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