Density fluctuations in the 1D Bose gas

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Panfil, M.K.

Publication date

2013

Link to publication

Citation for published version (APA):

Panfil, M. K. (2013). Density fluctuations in the 1D Bose gas.

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Appendix A

Notation and Convention

A.1 Notation

The following symbols have an unique meaning throughtout the dissertation N Number of particles,

L System length, n = N_L Particles density,

H The Lieb-Liniger Hamiltonian, c Interaction strength,

h Chemical potential,

H Hilbert space of the Lieb-Liniger Hamiltonian vs Sound velocty,

K Luttinger parameter, q Fermi rapidity.

A.2 Fourier Transform

We adopt the following convention for Fourier transform and its inverse in the distance domain f (k) = Z L 0 dx eikxf (x), (A.1) f (x) = 1 L X k e−ikxf (k), k∈ 2π Lm : m∈ Z . (A.2) 135

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Appendix A. Notation and Convention 136

In the time domain we use the opposite convention for_{± signs in the exponents}

f (ω) = Z ∞ −∞ dt eiωtf (t), (A.3) f (t) = Z ∞ −∞ dω 2π e −iωt_{f (ω).} _(A.4)

Therefore, for example, the Lieb-Liniger Hamiltonian (2.3) in the momentum space reads

H = ~ 2 2m 1 L X k k2Ψ†_kΨk+ 2c 1 L3 X k1,k2,q Ψ†_k 2+qΨ † k1+qΨk2Ψk1, (A.5)

where the fields in momentum space are given accordingly to the above definition and fulfill the bosonic commutation relations

h Ψk, Ψ†q

i

= Lδk,q, (A.6)

and zero otherwise. The particle number operator and the total momentum follow the same way ˆ N = 1 L X k Ψ†_kΨk, (A.7) ˆ P = 1 L X k k Ψ†_kΨk. (A.8)

For a convenience we give here also an expression for the density operator ρ(x) = Ψ†_{(x)Ψ(x) in the momentum space.}

ρk= 1 L X q Ψ†_q+kΨq, (A.9)

and for the density-density correlation function in the Lehmann representation

S(k, ω) = 2π L

X

λ∈H

|hλ|ρk|Ωi|2δ (ω− (Eλ− EΩ)/~) . (A.10)

Finally we note that due to the definition of the Fourier transform fields in the real space have dimension L−1/2 while in the momentum space L1/2. Thus the density operator in the momentum space is dimensionless. In general keeping in mind this different dimensionality allows to easily trace the appearing and disappearing powers of L in various expressions.

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Appendix A. Notation and Convention 137

A.3 Principal Value Integrals

We define three distinct types of principal value integrals as follows,

P Z b a dx f (x) x− c = limδ→0 Z c−δ a dx f (x) x− c + Z b c+δ dx f (x) x− c , (A.11a) P− Z b a dx f (x) x− a = limδ→0 Z b a+δ dx f (x) x− a+ f (a) log(δ) , (A.11b) P+ Z b a dx f (x) x_{− b} = limδ→0 Z b−δ a dx f (x) x_{− b} − f(b) log(δ) , (A.11c) P± Z b a dx f (x)(b− a) (x− a)(x − b) = P+ Z b a f (x) x− b − P− Z b a f (x) x− a. (A.11d)

Note that we consider all quantities being integrated to be dimensionless, and the bounds of integration to be real numbers with no physical units. All such special principal value integrals evaluated in the text are first made dimensionless by mapping the region of integration to the interval(_{−1, 1).}