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 Incoming links: Fourier vertex of a Schwinger boson chain

Take

L \geq 2

, so neighbouring operators act at distinct sites. The identity

S_{n}^{x} S_{n + 1}^{x} + S_{n}^{y} S_{n + 1}^{y} = \frac{1}{2} (S_{n}^{+} S_{n + 1}^{-} + S_{n}^{-} S_{n + 1}^{+})

H = - \frac{J}{2} n \sum (a_{n}^{†} b_{n} b_{n + 1}^{†} a_{n + 1} + b_{n}^{†} a_{n} a_{n + 1}^{†} b_{n + 1}) .

Operators at different sites commute, allowing normal ordering without a correction here. For the first quartic term, the inverse discrete Fourier transforms give

n \sum a_{n}^{†} b_{n + 1}^{†} b_{n} a_{n + 1} = \frac{1}{L ^{2}} n, p, q, r, t \sum e^{i (p + q - r - t) n} e^{i (q - t)} A_{p}^{†} B_{q}^{†} B_{r} A_{t} = \frac{1}{L} p, q, r, t \sum δ_{p + q, r + t}^{(L)} e^{i (q - t)} A_{p}^{†} B_{q}^{†} B_{r} A_{t} .

Here

δ^{(L)}

is one when its two momenta agree modulo

2 π

and zero otherwise; it expresses lattice momentum conservation. The other quartic term has the same ordered operators with phase

e^{i (p - r)}

p - r = t - q

2 π

, so the two phases sum to

2 cos (q - t)

H = - \frac{J}{L} p, q, r, t \sum δ_{p + q, r + t}^{(L)} cos (q - t) A_{p}^{†} B_{q}^{†} B_{r} A_{t} .

An equivalent three-sum expression sets

t = p + q - r

2 π

, so its coefficient is

cos (r - p)

. The factor

1/ L

comes from four factors

L^{- 1/2}

and one site sum, while the factor of two from the two exchange terms cancels the original

1/2

p

with

t

and

q

with

r

, and the coefficient is unchanged after imposing lattice momentum conservation. It also preserves each local total boson number in its real-space form, hence preserves the physical constrained subspace.