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Density fluctuations in the 1D Bose gas
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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Correlations in the weakly and
strongly interacting regimes
In this Appendix we will calculate the density-density correlation function for two simple models: free fermionic gas and weakly interacting bosonic gas. Let us start with the later.
C.1
Bogolyubov approximation
Let us start with the Bogolyubov approximation. As explained in the main text we write the field operator in the momentum space and single out the low momentum component
Ψ(x) = 1 LΨ0+ 1 L X k6=0 eikxΨk, (C.1)
and similarly for the Hermitian conjugated field. Substituting into the Lieb-Liniger Hamiltonian (2.3) we obtain H = ~ 2 2m 1 L X k6=0 k2Ψ†kΨk+ c L3 Ψ† 0Ψ † 0Ψ0Ψ0+ X k,q1,q26=0 Ψ†q 1+kΨ † q2−kΨq1Ψq2 (C.2) + c L3 X k6=0 4Ψ†0Ψ0Ψ†qΨq+ Ψ†0Ψ † 0ΨqΨ−q+ Ψ0Ψ0Ψ†qΨ † −q .
Thus we see that for small values ofc the energy is minimized by maximizing the occu-pancy of the lowest mode. To quantify this we project the Hamiltonian on a subspace where the occupancy of the lowest mode is of order of the total number of particlesN ,
Appendix C. Correlations in Ideal Gases 144 that is 1 L2Ψ † 0Ψ0 = N0 ∼ N. (C.3)
This allows to substitute the operatorsΨ(†)0 with c-numbers√LN0= L√n. Then in the
leading order we can neglect the quartic term involving only k6= 0 field operators and obtain the effective quadratic Hamiltonian (g = cn)
H = 1 L X k6=0 ~2 2mk 2+ 2g Ψ†kΨk+ g Ψ†kΨ†−k+ ΨkΨ−k + gn2. (C.4)
Diagonalization of the Hamiltonian is straightforward by means of the Bogolyubov trans-formation. To this end we define new operatorakgiven by a momentum dependent linear
combination of the operator Ψk
ak= ukΨk− vkΨ†−k, a † k= u ∗ kΨ † k− v ∗ kΨ−k. (C.5)
The coefficientsuk∈ C and vk ∈ C have to be chosen such that the new operators fulfill
the canonical commutation relations, we obtain the following constraints
|uk|2− |vk|2 = 1, (C.6)
ukv−k = vku−k, (C.7)
u∗kv−k∗ = vk∗u∗−k. (C.8)
This suggest that uk and vk can be conveniently parametrised as hyperbolic functions,
the first constraint is automatically fulfilled when
uk = eiαkcosh(γk), (C.9)
vk= eiβksinh(γk). (C.10)
Two other constraints require that all the functions are symmetric, that is αk = α−k,
βk= β−k and γk= γ−k.
The next step is to rewrite the Hamiltonian in terms of the new variables. To this end we invert Eq. C.5 to obtain
Ψk= u∗kak+ vka†−k. (C.11)
and then insert them in Eq. C.4. We get H = 1 L X k6=0 k2+ 2g |uk|2+|vk|2 + 2g (ukv∗k+ u∗kvk) a†kak (C.13) + 1 L X k6=0 k2+ 2gukvk+ g u2k+ v2k a†ka†−k + (h.c.) . (C.14)
The(h.c.) means that we add to the second line a term that is a hermitian conjugation of the first term in the same line. Now diagonalization of the Hamiltonian is equivalent to requiring that the second line is identical to zero which leads to two complex conjugated equations. Then only the first line is non-zero and the term multiplying a†kak plays a
role of the energy of quasi-particles as a function of momentum. Thus summarizing we have the following equations
(k) = k2+ 2g |uk|2+|vk|2+ 2g (ukv∗k+ u∗kvk) , (C.15)
0 = k2+ 2gukvk+ g u2k+ vk2
, (C.16)
0 = k2+ 2gu∗kv∗k+ gu∗k2+ v∗k2 (C.17) Combining the last two equations yields thatαk= βk= 0. This reduces the problem to
finding two real functions (k) and γk. It also allows for a convenient rewriting of the
set of equations
(k) = k2(u
k− vk)2, (C.18)
0 = g (uk+ vk)2+ k2ukvk. (C.19)
Solving the second equation forγk and substituting into the first one we obtain
(k) =pk4+ 4gk2, (C.20)
e4γk = k
2
4g + k2. (C.21)
Later it will turn out useful to know also the explicit form(uk+vk)2. It is straightforward
to show that
(uk+ vk)2= p |k|
Appendix C. Correlations in Ideal Gases 146
In the limit ofg→ 0 we find uk= 1 and vk = 0 what is expected from the definition of
operator ak (see Eq. C.5). The final Hamiltonian reads
H = ~ 2 2m 1 L X k6=0 (k)a†kak+ gn2. (C.23)
The ground state of it is the vacuum of the quasiparticles ak|0i = 0. Excited states are
created by the action ofa†k. Taking into account the bosonic nature of the quasiparticles we write the normalized excited states as
|nk1, nk2, . . .i = Y i a†k i nki p Lnki! |0i, (C.24)
where ni are eigenvalues of the momentum occupation operator nˆi = a†kiaki/L and the product extends over momenta present in the excited state.
Now let us move to the density operator (A.9) and its form factor. Recall its expression
ρk= 1 L X q Ψ†q+kΨq. (C.25)
Following the logic of the Bogoliubov approximation we expect that the main contri-bution to the density operator comes from k = 0, k + q = 0 or q = 0. Thus we can write ρk= N0δk,o+ (1− δk,0)n1/2 Ψ†k+ Ψ−k +O N0 0 . (C.26)
The first term that contributes only atk = 0 describes the disconnected part of the form factor. The second term is then the leading term and it describes process of creating/an-nihilating a quasi particle. For the the form factor we obtain
|hλ|ρk|0i|2= N02δk,0+ (1− δk,0) n|hλ| (uk+ v−k) a†k+ (u−k+ vk) a−k|0i|2
= N02δk,0+ (1− δk,0) N (uk+ v−k)2|hλ|1ki|2. (C.27)
Therefore we plug the form factor into the Lehmann representation of the density-density correlation function (A.10) and obtain
S0(k, ω) = 2πLn2δk,0δ(ω) + 2πnp |k|
k2+ 4gδ (ω− (k)) . (C.28)
Integrating over ω yields a static correlator
S0(k) = Z ∞ −∞ dω 2πS0(k, ω) = Ln 2δ k,0+ n|k| p k2+ 4g, (C.29)
and Fourier transform gives the spatial correlator S0(x) = 1 L X k e−ikxS0(k) = 1 2π Z ∞ −∞ dk e−ikxS0(k) +O(1/L) (C.30) = n2+ nδ(x) + n Z ∞ −∞ dk e−ikx p |k| k2+ 4g − 1 ! . (C.31)
The integral can be expressed through modified Struve functionL−1(x) and the modified
Bessel function of the first kindI1(x) and the final expression reads [31]
S0(x) = nδ(0) + n2(1−√γ (L−1(2√γnx)− I1(2√γx))) . (C.32)
Therefore we have computed the (static) density-density correlation function within the Bogolyubov approximation valid in the weakly repulsive regime.
C.2
Free Fermionic Gas
Now let us move to the free fermionic gas. We consider spinless fermions as these one are directly related to the Tonks-Girardeau gas. Consider the following Hamiltonian
HF F =
Z L 0
dxΨ†(x)∂x2Ψ(x), (C.33)
given in terms of operators obeying the fermionic anticommutation relations
{Ψ(x), Ψ†(y)} = δ (x − y) , {Ψ(x), Ψ(y)} = 0. (C.34) Hamiltonian becomes diagonal when written in terms of the Fourier transformed opera-tors H = 1 L X k k2Ψ†kΨk. (C.35)
The ground state for a fixed particle numberN is formed by a Fermi sea
|Ωi = Y
k≤kF Ψ†k √
L|0i. (C.36)
where |0i is again the vacuum state and hΩ|Ωi = h0|0i = 1. Fermi momentum kF is determined through the number of particles: kF = πn. All the other excited states with
Appendix C. Correlations in Ideal Gases 148
chosenk and q. For example the one particle-hole excited states are |Ω + 1k− 1qi = 1
LΨ
†
kΨq|Ωi, |k| > kF, |q| ≤ kF. (C.37)
Computation of the form factor of the density operator (A.9) is therefore trivial since the action of the density operator with k 6= 0 on the ground state creates simple one particle-hole excited states. Thus the only potentially non zero form-factors are of the following form
|hΩ + 1k1 − 1k2|ρk6=0|Ωi|
2. (C.38)
The density operatorρkchanges momentum byk and the momentum operator commutes
with the Hamiltonian thereforek1−k2= k. Moreover, from the Pauli principle,|k1| > kF
and |k2| ≤ kF. Finally we obtain
|hΩ + 1k1 − 1k2|ρk6=0|Ωi|
2 = δ
k,k1−k2Θ(|k1| − kF) (1− Θ(|k2| − kF)) . (C.39)
For k = 0 the form factor is non-zero only for λ = |Ωi and is equal to N2. Using the
Lehmann representation (A.10) for the correlation function we obtain
S0(k, ω) = 2πLn2δ(ω)δk,0 +2π L X k1,k2 |hΩ + 1k1 − 1k2|ρk6=0|Ωi| 2δ ω− k2 1 − k22 . (C.40)
The second line can still be greatly simplified by temporally introducing new variables q±= 1 2 ω k ± k , (C.41) in terms of which δk,k1−k2δ ω− k 2 1− k22 = δk,k1−k2 2|k| δ(k1− q+). (C.42) Therefore we have 2π L X k1,k2 |hΩ + 1k1− 1k2|ρk6=0|Ωi| 2δ ω−1 2 k 2 1− k22 = = 2π L X k1 Θ(|k1| − kF) (1− Θ(|k1− k| − kF)) 2|k| δ(k1− q+) = Θ(|q+| − kF) (1− Θ(|q−| − kF)) 2|k| 2π L X k1 δ(k1− q+). (C.43)
In the thermodynamic limit the last factor 2πL Pk
1δ(k1− q+) = 1. In the finite system this factor defines a set of delta peaks (at q+ = 2πm/L with m∈ Z). Upon increasing
the system size these peaks become denser and denser and finally for L = ∞ lead to a flat distribution, hence1. For simplicity we will set it to 1 already now as it should not lead to any confusion.
Since|q+| ≥ |q−| for ω > 0
Θ(|q+| − kF) (1− Θ(|q−| − kF)) = Θ(|q+| − kF)− Θ(|q−| − kF). (C.44)
and therefore the correlation function is non-zero only forω2(k) < ω < ω1(k) where
ω1(2)(k) =|k2± 2kk
F|. (C.45)
Finally the correlation function reads
S0(k, ω) = 2πLn2δ(ω)δk,0+ (1− δk,0)
Θ(ω− ω2(k))− Θ(ω − ω1(k))
2|k| . (C.46)
Integration over ω yields the static correlator
S0(k) = Ln2δ k,0+ n2k|k|F :|k| < 2kF, n :|k| ≥ 2kF. (C.47)
It is also straightforward to compute the spatial correlator, Fourier transform gives
S0(x) = nδ(0) + n2 1− 1 2 (kFx)2 +cos (2kFx) 2 (kFx)2 . (C.48)
