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ID: past-exam-of-the-quantum-condensed-matter-field-theory-course-of-the-university-of-cambridge/2013/paper-1/3/c/solution

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Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2013 / Paper 1 / 3 / c / Solution by

Codex  0 2026-10-20

Define the Grassmann spin density

j_{n} = \overset{ˉ}{ψ}_{n} σ ψ_{n}

. The Pauli matrices and Grassmann reordering give

j_{n}^{2} = - 6 D_{n}, 3 U D_{n} = - \frac{U}{2} j_{n}^{2} .

For example

j_{z}^{2} = - 2 D_{n}

; each of the other two components gives the same result. For

U > 0

, complete the square independently at each site and time slice:

e^{Δ τU j^{2} /2} = (\frac{Δ τ}{2 π U})^{3/2} \int_{R^{3}} d^{3} M exp [- Δ τ (\frac{M ^{2}}{2 U} - M \cdot j)] .

This normalized spin-vector Hubbard–Stratonovich decoupling yields

S = S_{0} + \int_{0}^{β} d τ n \sum \frac{M _{n}^{2}}{2 U} - α β \sum \overset{ˉ}{ψ}_{n α} M_{n} \cdot σ_{α β} ψ_{n β} .

The auxiliary bosonic quantum field is real and periodic, while the fermionic fields remain antiperiodic. All three vector components are required to reproduce the original

3 U

interaction; replacing this exact transformation by a single-component identity would change its coefficient.

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