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

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.

## Summary (1 min read)

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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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2006 Europhys. Lett. 74 785

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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

= ∂/∂x;ˆρ

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

−iπ

dx

θ(x − x

)ˆρ(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

−iπ ∆

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

+ˆa

x

)

2

ˆ

ψ

f

(x),

ˆ

U =

1

2π

2

dxdx

ˆa

x

(x)U(x − x

)ˆa

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

,ψ

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, x)ˆρ(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

−ρ+¯ρ

)

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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