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Journal ArticleDOI

1D composite fermions : Bogoliubov-like mode in the Tonks-Girardeau gas

01 Jun 2006-EPL (EDP sciences)-Vol. 74, Iss: 5, pp 785-791
TL;DR: In this paper, the authors reformulated 1D boson-fermion duality in path-integral terms and obtained the long-wavelength asymptotics of the collective mode in 1D Boson systems at the Tonks-Girardeau regime.
Abstract: We reformulate 1D boson-fermion duality in path-integral terms. The result is a 1D counterpart of the boson-fermion duality in the 2D Chern-Simons gauge theory. The theory is consistent and enables, using standard resummation techniques, to obtain the long-wavelength asymptotics of the collective mode in 1D boson systems at the Tonks-Girardeau regime. The collective mode has the dispersion of Bogoliubov phonons: ω(q) = q(U(q)/m)1/2, where is the bosons density and U(q) is a Fourier component of the two-body potential.

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Summary

  • The theory is consistent and enables, using standard resummation techniques, to obtain the longwavelength asymptotics of the collective mode in 1D boson systems at the Tonks-Girardeau regime.
  • 1D boson systems attract much attention [1–20] in light of recent experiments on cigarshaped atomic traps [1–6] and gases exposed to linear carbon nanotubes [7–9].
  • There are two limiting cases for 1D systems: the high-density weak-interaction Thomas-Fermi (TF) regime [20], where the Bogoliubov energy functional [23, 24] and the thermodynamic limit of the Gross-Pitaevskii mean-field theory [25–27] apply; and the low-density strong interaction Tonks-Girardeau (TG) regime of impenetrable bosons [28– 33].
  • Unlike the TF regime, in the TG regime the boson wave function is Fermi-like and the fermion-boson duality method has been proposed for this regime [17,19,30,31].
  • The original Lieb-Liniger first-quantization approach enables to find analytically the lowenergy elementary excitations spectrum, i.e., the spectrum of the single particle-hole pair excitations, for all values of interaction constants [34].
  • III in ref. [34]), within this approach it is not possible to analyze the quasiparticles or collective modes in the system.
  • It turns out that in the TF regime the collective mode and the elementary excitations are similar.
  • As a result, the authors obtain that far inside the TG regime the situation is reminiscent of that in Fermi systems: the collective mode lies above the elementary excitation spectrum, which is Fermi-like, and the dispersion of the collective mode turns out to be of Bogoliubov form, i.e., sound with velocity proportional to the square root of the interaction constant.

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1D composite fermions: Bogoliubov-like mode in the Tonks-Girardeau gas
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Europhys. Lett., 74 (5), pp. 785–791 (2006)
DOI: 10.1209/epl/i2006-10034-8
EUROPHYSICS LETTERS 1 June 2006
1D composite fermions: Bogoliubov-like mode
in the Tonks-Girardeau gas
I. V. Ovchinnikov and D. Neuhauser
Department of Chemistry and Biochemistry, University of California at Los Angeles
Los Angeles, CA, 90095-1569, USA
received 24 February 2006; accepted 4 April 2006
published online 5 May 2006
PACS. 05.30.Jp Boson systems.
PACS. 71.10.Pm Fermions in reduced dimensions (anyons, composite fermions, Luttinger
liquid, etc.).
PACS. 71.45.Gm Exchange, correlation, dielectric and magnetic response functions, plas-
mons.
Abstract. We reformulate 1D boson-fermion duality in path-integral terms. The result
is a 1D counterpart of the boson-fermion duality in the 2D Chern-Simons gauge theory. The
theory is consistent and enables, using standard resummation techniques, to obtain the long-
wavelength asymptotics of the collective mode in 1D boson systems at the Tonks-Girardeau
regime. The collective mode has the dispersion of Bogoliubov phonons: ω(q)=q
¯ρU(q)/m,
where ¯ρ is the bosons density and U(q) is a Fourier component of the two-body potential.
1D boson systems attract much attention [1–20] in light of recent experiments on cigar-
shaped atomic traps [1–6] and gases exposed to linear carbon nanotubes [7–9]. The history of
theoretical studies of 1D bosons goes back to the celebrated work by Lieb and Liniger, who
found exact integrability of zero-range interacting bosons via the Bethe ansatz, for all values of
the interaction strength [21,22]. There are two limiting cases for 1D systems: the high-density
weak-interaction Thomas-Fermi (TF) regime [20], where the Bogoliubov energy functional [23,
24] and the thermodynamic limit of the Gross-Pitaevskii mean-field theory [25–27] apply; and
the low-density strong interaction Tonks-Girardeau (TG) regime of impenetrable bosons [28–
33]. Unlike the TF regime, in the TG regime the boson wave function is Fermi-like and the
fermion-boson duality method has been proposed for this regime [17, 19, 30,31].
The original Lieb-Liniger first-quantization approach enables to find analytically the low-
energy elementary excitations spectrum, i.e., the spectrum of the single particle-hole pair
excitations, for all values of interaction constants [34]. However, as first pointed out by Lieb
(see sect. III in ref. [34]), within this approach it is not possible to analyze the quasiparticles
or collective modes in the system. In the TF regime, the study of the collective modes can
be accomplished by using the Gross-Pitaevskii energy functional with the introduction of the
classical order parameter field (
1
). It turns out that in the TF regime the collective mode and
the elementary excitations are similar.
(
1
)See, e.g., ref. [35] and references therein.
c
EDP Sciences
Article published by EDP Sciences and available at http://www.edpsciences.org/eplor http://dx.doi.org/10.1209/epl/i2006-10034-8

786 EUROPHYSICS LETTERS
As to the TG regime, there is still no analytical tool to study the collective mode. In this
letter we propose an exact path-integral approach based on the boson-fermion duality idea.
The method is actually a 1D analogue of the 2D composite particles formalism, which enables
mapping fermions to bosons and vice versa in 2D by the coupling to the Chern-Simons gauge
field (
2
). As a result, we obtain that far inside the TG regime the situation is reminiscent
of that in Fermi systems: the collective mode lies above the elementary excitation spectrum,
which is Fermi-like, and the dispersion of the collective mode turns out to be of Bogoliubov
form, i.e., sound with velocity proportional to the square root of the interaction constant.
We start from a secondary quantized Hamiltonian of a homogeneous system of spinless 1D
bosons, which interact through a two-body potential U :
ˆ
H =
ˆ
K +
ˆ
U,
ˆ
K =
1
2m
dx
ˆ
ψ
b
(x)(i∂
x
)
2
ˆ
ψ
b
(x),
ˆ
U =
1
2
dxdx
δ ˆρ
b
(x)U(x x
)δ ˆρ
b
(x
),
where m is the mass;
x
= ∂/∂xρ
b
(x)=
ˆ
ψ
b
ˆ
ψ
b
(x) with
ˆ
ψ
b
being the boson operators and
δ ˆρ
b
(x) ˆρ
b
(x) ¯ρ
b
is the density fluctuations where ¯ρ
b
is the average boson spatial density.
The many-particle wave function in the TF regime experiences no crucial changes as one
particle passes another, whereas in the TG regime it falls down almost to zero as the coordinate
of one particle approaches the position of another one. The “fermionized” boson wave function
has zero value if the position of one particle coincides with that of another one. Such a
reduction of the Hilbert space to fermionized wave functions is an approximation, but is
justified far inside the TG regime.
A fermionized boson wave function can be constructed from a fermion antisymmetric wave
function in the following way [19, 30, 31]:
ψ
b
({x
i
})=
i<j
sign (x
i
x
j
) ψ
f
({x
i
}) . (1)
This relation is an approximation which is justified only far inside the TG regime.
In a secondary-quantized language eq. (1) corresponds to the introduction of the new
quasi-particle operators
ˆ
ψ
f
, which are related to
ˆ
ψ
b
by
ˆ
ψ
b
(x)=
ˆ
ψ
f
(x)exp
dx
θ(x x
ρ(x
)
, (2)
where θ is the Heaviside unit step function and
ˆρ(x)
ˆ
ψ
b
(x)
ˆ
ψ
b
(x)=
ˆ
ψ
f
(x)
ˆ
ψ
f
(x)
is the spatial particle density, which has the same form in terms of initial bosons and the new
quasi-particles. It is easy to prove that
ˆ
ψ
f
(x
1
)
ˆ
ψ
f
(x
2
) e
ˆ
ψ
f
(x
2
)
ˆ
ψ
f
(x
1
)=
ˆ
ψ
f
(x
1
),
ˆ
ψ
f
(x
2
)
= δ(x
1
x
2
),
where = θ(x
1
x
2
) θ(x
2
x
1
) = sign(x
1
x
2
)=±1. That is, the new operators satisfy
Fermi anti-commutation relations so that the quasi-particles are fermions. Let us call them
composite fermions (CF), after their predecessors in 2D.
(
2
)For a review see, e.g., ref. [36]. Our approach, however, is closer to that of ref. [37].

I. V. Ovchinnikov et al.: 1D composite fermions: Bogoliubov-like etc. 787
To see that the transformation (2) corresponds to eq. (1), note that if one starts creating
a boson wave function by repeatedly acting on a vacuum state with the operators
ˆ
ψ
b
from
eq. (2), then the CF operators will produce a fermionic wave function and the exponential
phase-factors will give the “sign” term in eq. (1).
The kinetic energy and two-body interaction in the CF operators’ representation take the
following forms:
ˆ
K =
1
2m
dx
ˆ
ψ
f
(x)(i∂
x
+ k
F
a
x
)
2
ˆ
ψ
f
(x),
ˆ
U =
1
2π
2
dxdx
ˆa
x
(x)U(x x
a
x
(x
),
with the constraint
ˆa
x
(x)=πδˆρ(x), (3)
where k
F
= π ¯ρ is the Fermi wave vector.
In a path-integral representation the constraint (3) is easily incorporated with the aid of
a Lagrange multiplier. The partition function has the following form so far:
Z(φ)=
Dψ
f
Dψ
f
Da
x
e
i
dtL
{t,x}
δ
a
x
π
ρ ρ
, (4)
L =
dx
ψ
f
(i∂
t
)ψ
f
+ ρφ
K(ψ
f
f
) U (a
x
),
where ρ ψ
f
ψ
f
and the constrained path integration is over the statistical field a
x
and the
Grassmann fields ψ
f
and ψ
f
which represent the CFs. An external potential φ has also been
added to the action. We will use it to probe the system, i.e., the density-density correlation
function is
ˆρ(t, xρ(t
,x
) = −Z(φ)
1
δ
2
Z(φ)
δφ(t, x)δφ(t
,x
)
φ=0
. (5)
The constraint in the partition function (4) can be rewritten through the introduction of
an auxiliary field a
t
as
{t,x}
δ
a
t
π
ρ ρ
=
Da
t
(t, x)e
i
dtdx
(
a
x
π
ρρ
)
a
t
.
As a result, the spatial density of the Lagrangian becomes
L = ψ
f
i∂
t
(i∂
x
+ k
F
+ a
x
)
2
2m
ψ
f
a
t
a
x
π
ρ ρ
+
+ρφ
1
2π
2
dx
a
x
(x)U(x x
)a
x
(x
),
and the integration in Z is assumed now over a time-space statistical field a =(a
t
,a
x
).
Before proceeding further, let us outline the connection of the proposed boson-fermion
transformation to existing theories. The transformation (2) is reminiscent of the inverse boson-
fermion transformation in Haldane’s bosonization approach for 1D Fermi liquids (
3
). Never-
theless, Haldane’s bosonization is developed for studies of low-frequency physics of fermion
(
3
)For review see, e.g., Chap. 5 of Ref. [38].

788 EUROPHYSICS LETTERS
systems in terms of bosons, which, in fact, represent the spatial density fluctuations of the
fermions. In our case, however, the transformation (2) involves operators of real bosons and
not density fluctuations. Density fluctuations, instead, are represented by the spatial statisti-
cal field component a
x
.
Physically, the proposed model is a 1D counterpart of Jain’s mechanism of attaching flux
quanta to 2D particles, which leads to the coupling of the composite objects to a Chern-
Simons gauge field [36]. There are, however, several aspects in which the theory proposed
differs from the Chern-Simons theory, apart from the different dimensionalities of the systems.
i) The proposed theory is not gauge invariant. ii) Time-reversal and space-reversal symmetries
are broken separately, though time-space-reversibility is present. iii) The coupling to Chern-
Simons gauge fields in 2D results in an additional magnetic field experienced by the composite
objects, whereas in our case the coupling to the statistical fields leads to a shift of the one-
particle kinetic energy dispersion by k
F
in momentum space. In a sense, it looks as though
the whole system starts moving. This fact is going to reveal itself later through a Doppler
shift in the response function.
The noninteracting part of the action becomes the sum of two Gaussian actions for the
CFs and the statistical fields governed by the following “bare” CFs’ and statistical fields’
propagators, respectively (in Fourier space):
G
1
0
(p)=ε
(p k
F
)
2
2m
,
ˆ
D
1
0
(q)=
1
π
01
1 v
F
u(q)
,
with u(q) U(q)/(v
F
π), where U (q) is the spatial Fourier transform of the two-body potential
and p =(ε, p), q =(ω, q). The interaction part of the action is
L
int
=(φ + a
t
)ρ a
x
j,
where j is the CF current density:
j =
1
2m
(ψ
f
((i∂
x
k
F
)ψ
f
)+((i∂
x
k
F
)ψ
f
)ψ
f
+ a
x
ρ).
Due to the coupling to the statistical fields the one-particle dispersion is shifted by k
F
in
momentum space. One can formally make a substitution p k
f
p and arrive at the
ordinary picture of 1D fermions at rest.
As the action is Gaussian in the CF fields, one can integrate them out. The integration
leads as usual to the fermion determinant in the effective statistical fields’ action. At this point
it is important to note that so far we have made no approximations beyond the assumption (1).
Now in order to obtain the density-density response of the system it suffices to leave in the
effective action only the terms quadratic in the statistical fields:
Z
eff
(φ)=
D
˜
aexp
i
1
2
d
2
q
(2π)
2
L
eff
. (6)
Here, the effective action has the following form:
L
eff
= a(q)(
ˆ
D
1
0
(q)
ˆ
Π
D
(q))a(q) (a + φ)(q)
ˆ
Π
P
(q)(a + φ)(q),
where Π
D
and Π
P
are the diamagnetic and paramagnetic polarization operators, respectively.
The renormalized statistical fields’ propagator is given by a Dyson equation:
ˆ
D(q)=
ˆ
D
1
0
(q)
ˆ
Π(q))
1
, (7)

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Q1. What are the contributions in "1d composite fermions: bogoliubov-like mode in the tonks-girardeau gas" ?

The original Lieb-Liniger first-quantization approach enables to find analytically the lowenergy elementary excitations spectrum, i. e., the spectrum of the single particle-hole pair excitations, for all values of interaction constants [ 34 ]. III in ref. [ 34 ] ), within this approach it is not possible to analyze the quasiparticles or collective modes in the system. In the TF regime, the study of the collective modes can be accomplished by using the Gross-Pitaevskii energy functional with the introduction of the classical order parameter field ( ). In this letter the authors propose an exact path-integral approach based on the boson-fermion duality idea. The method is actually a 1D analogue of the 2D composite particles formalism, which enables mapping fermions to bosons and vice versa in 2D by the coupling to the Chern-Simons gauge field ( ). The authors start from a secondary quantized Hamiltonian of a homogeneous system of spinless 1D bosons, which interact through a two-body potential U: