Time and temperature-dependent correlation function of an impurity in one-dimensional Fermi and Tonks-Girardeau gases as a Fredholm determinant.

PAPER

Time and temperature-dependent correlation function of an impurity

in one-dimensional Fermi and Tonks

–

Girardeau gases as a Fredholm

determinant

Oleksandr Gamayun1,5

, Andrei G Pronko2

and Mikhail B Zvonarev3,4

1 _{Instituut-Lorentz, Universiteit Leiden, PO Box 9506, 2300 RA Leiden, The Netherlands} 2 _{Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191023, Russia}

3 _{LPTMS, CNRS, University of Paris-Sud, Université Paris-Saclay, F-91405 Orsay, France} 4 _{ITMO University, 197101, Saint-Petersburg, Russia}

5 _{Author to whom any correspondence should be addressed.}

E-mail:[email protected],[email protected]@gmail.com

Keywords:impurity dynamics, Fredholm determinants, one-dymensional systems, quantum gases, Bethe ansatz

Abstract

We investigate a free one-dimensional spinless Fermi gas, and the Tonks

–

Girardeau gas interacting

with a single impurity particle of equal mass. We obtain a Fredholm determinant representation for

the time-dependent correlation function of the impurity particle. This representation is valid for an

arbitrary temperature and an arbitrary repulsive or attractive impurity-gas

δ

-function interaction

potential. It includes, as particular cases, the representations obtained for zero temperature and

arbitrary repulsion in

(Gamayun

et al

2015

Nucl. Phys.

B

892

83

–

104

), and for arbitrary temperature

and in

fi

nite repulsion in

(Izergin and Pronko 1998

Nucl. Phys.

B

520

594

–

632

).

1. Introduction

Quantum many-body interacting systems in one spatial dimension solved by the Bethe ansatz are unique in that their complete set of eigenfunctions and eigenstates are known explicitly[1]. The use of this property constitutes a whole researchfield by now. Describing the time evolution of the observables is one of the most challenging problems in thefield, remaining largely open.

The present paper is devoted to a particular Bethe ansatz-solvable model: a single mobile impurity

interacting with a free Fermi gas in one spatial dimension through aδ-function potential of an arbitrary(positive or negative)strengthg.Its eigenfunctions and spectrum have been found in[2,3]. The eigenstates were

represented as a sum of a product of plane waves with special coefficients. Such a representation is common for the Bethe ansatz-solvable models. The solution[2,3]may be obtained from the one for the Gaudin–Yang model

[4–6]as a particular case. However, the model we consider here has its own specificity: there exists a

representation for its eigenstates through a single determinant resembling the Slater determinant for the free Fermi gas[7–9]. We use this representation as a starting point to investigate the time-dependent two-point impurity correlation function. We also argue that this function is the same as the one for a single mobile impurity interacting with the Tonks–Gurardeau gas[10,11].

Our main result is an exact expression for the aforementioned correlation function in the limit of infinite system size,L ¥,and for arbitrary chemical potential(or density). This representation is obtained for the repulsive as well as attractive impurity-gasδ-function interaction potential of arbitrary strengthg,and for arbitrary temperature. The key ingredient is the Fredholm determinant of a linear integral operator of integrable type(see, e.g, section XIV.1 of[1]). The present solution encompasses two particular cases known so far(i) infinite repulsiong ¥[12,13](ii)arbitrary repulsiong0at zero temperature[14].

The paper is organized as follows: in section2we define the model under consideration. In section3we summarize our main results. In section4we explain the transformation from the laboratory to the mobile impurity reference frame. In section5we write the eigenfunctions in the mobile impurity reference frame. They OPEN ACCESS

RECEIVED

3 December 2015

REVISED

14 February 2016

ACCEPTED FOR PUBLICATION

10 March 2016

PUBLISHED

7 April 2016

Original content from this work may be used under the terms of theCreative Commons Attribution 3.0 licence.

Any further distribution of this work must maintain attribution to the author(s)and the title of the work, journal citation and DOI.

are represented as a determinant differing from the Slater determinant representation of the free Fermi gas by a single row only. Using this representation we obtain the form-factors(the overlaps of the eigenfunctions of the model under consideration with the eigenfunctions of the free Fermi gas)in the form of the determinants of somefinite-dimensional matrices. In section6we perform a form-factors summation in the caseg0which leads us to the Fredholm determinant representation of the impurity correlation function. In section7we extend the results obtained in section6to the caseg<0.Some properties of form-factors and Fredholm determinants, which we use to derive our results, are given in the appendices.

2. Model

In this section we define our model. We consider a free one-dimensional spinless Fermi gas at temperatureTin the presence of a single distinct quantum particle(impurity)propagating through it. We investigate the correlation function

( )= áy( )y†( )ñ ( )

G x t

Z x t

, 1 , 0, 0 T. 2.1

The canonical creation(annihilation)operatorsy†s(ys)carry the subscripts= for the Fermi gas, ands=  for the impurity. The canonical anticommutation relations read

( ) † ( ) † ( ) ( ) ( ) ( )

ys x y_s¢ x¢ +y_s¢ x¢ ys x =d dss¢ x- ¢x . 2.2

The temperature-weighted averageá ñTin equation(2.1)is defined as

∣ ∣ ∣ ∣ ( )

{ }

( )



å



á ñ = á Ä á b m ñ Ä ñ

 N - - N 

0 e 0 . 2.3

T N

EN N

Here

∣Nñ =c_p†_c†_{p  }∣0 ñ (2.4)

N

1

is the free Fermi gas state containingNfermions with the momentap₁,¼,p_N.The gas is on a ring of circumferenceL,and periodic boundary conditions are imposed. We have

( ) ( )

†

å

†

y_{s x} = - _s = p =   ¼

L c p

n

L n

1

e , 2 , 0, 1, 2, . 2.5

p px

p i

The vacuum∣0sñis the state with no particles,cs s∣0 ñ = 0.The sum in equation(2.3)runs through all possible values ofp₁,¼,p_N,andN:

( )

{ }



å

=

å å

å

= ¥

. 2.6

N N 0p₁ p_N

The parameterβis the inverse temperature,b =1 T.The Boltzmann constant,kB, and the Planck constant,ÿ,

are equal to one in our units. The Boltzmann–Gibbs weight_e-b(EN-mN)_{is de}fi_{ned by the value of the free Fermi}

gas energyENin the state(2.4):

∣ ñ = ∣ ñ º ( ¼ ) ( )



H N E NN , EN EN p1, ,pN 2.7

and by the chemical potentialμ. The Hamiltonian of the free Fermi gas reads

( ) ( ) ( )

†

ò

y y

= - ¶

¶

  ⎛ 

⎝

⎜ ⎞_⎠⎟

H x x

m x x

d 1

2 , 2.8

L

0

2

2

wheremis the particle mass. Therefore

( ) ( )



å

=

=

EN p , 2.9

j N

j 1

where

( ) ( )

 q = q

m

2 . 2.10

2

The grand partition functionZin equation(2.1)is

( ) ( )

{ }

( ) [ ( ) ]

å



= -b -m = + -b -m

Z e 1 e . 2.11

N

E N

p

p

N

Here, the product is taken over all possible values of the single-particle momentum, p

= =   ¼

p 2 n L n, 0, 1, 2, .The operatoryin equation(2.1)evolves with timetas

( ) ( )

( )

= +

H H Himp 2.12

is the Hamiltonian of the entire system, and

( ) ( ) ( ) ( ) ( ) ( ) ( )

† † †

ò

y y y y y y

= - ¶

¶ +

     

⎡ ⎣

⎢ ⎛_⎝⎜ ⎞

⎠

⎟ ⎤

⎦ ⎥

H x x

m x x g x x x x

d 1

2 . 2.13

L imp

0

2

2

The Hamiltonian(2.12)defines the fermionic Gaudin–Yang model[4–6], in which the number of impurity particles

( ) ( ) ( )

†

ò

y y

=

  

N dx x x , 2.14

L

0

is arbitrary. However, only states of the Hamiltonian(2.12)withN=0, andN=1are relevant for the dynamics described by the correlation function(2.1). The parametergin equation(2.13)gives the strength of the impurity-gas interaction. We consider the case of both repulsive,g0, and attractive,g <0, interactions. The

first-quantized form of the Hamiltonian(2.12)withN=1is

( ) ( )

å

å

d

= - ¶

¶

-¶

¶ +

-=  =

 H

m x m x g x x

1 2

1

2 . 2.15

j N

j j

N

j 1

2

2

2

2 1

Here,x1,¼,xNare coordinates of the gas particles, andxis the coordinate of the impurity.

Let us now consider the Tonks–Girardeau gas[10,11]. It consists of bosons interacting with each other through aδ-function repulsive potential of infinite strength. The eigenfunctions of the Tonks–Girardeau gas,

( )

YB x1,¼,xN , and those of the free Fermi gas,YF(x1,¼,xN), are in one-to-one correspondence:

( ) ( ) ( ) ( )







Y ¼ = Y ¼

-<

x, ,x x, ,x sign x x . 2.16

F N B N

j l N

j l

1 1

1

Here,sign( )x stands for the sign function, equal to one(minus one)forxpositive(negative). Adding a single mobile impurity to the Tonks–Girardeau gas leads to the model whose eigenstatesYB(x x, 1,¼,xN)satisfy

( ) ( ) ( ) ( )







Y  ¼ = Y  ¼

-<

x x, , ,x x x, , ,x sign x x , 2.17

F N B N

j l N

j l

1 1

1

whereYF(x x, 1,¼,xN)is an eigenstate of the Hamiltonian(2.15). We stress thatYB(x x, 1,¼,xN)

(YF(x x, 1,¼,xN))is symmetric(antisymmetric)with respect to any permutation ofx1,¼,xN. The impurity is

distinguishable from the gas particles, therefore no symmetry restriction needs to be enforced when exchanging 

x withx1,¼,xN. Equations(2.16)and(2.17)imply that the correlation function(2.1)is the same whether the

impurity is injected into the free Fermi gas or the Tonks–Girardeau gas. Note that this statement holds true for certain other observables(for example, the average impurity momentum[15]). We present our calculations for an impurity injected into a free Fermi gas, bearing in mind that our results, which are summarized in section3, are applicable to the case of the impurity injected into the Tonks–Girardeau gas as well.

3. Results

In this section we show the main result of our paper: the Fredholm determinant representation of the correlation function(2.1)at a given temperatureTand chemical potentialm.The structure of the representation crucially depends on the sign of the impurity-gas interactiong.

In section3.1we give the representation for the repulsive impurity-gas interaction,g0.The result for the attractive interaction,g<0,is given in section3.2. The particular cases of the infinitely strong repulsion,

 ¥

g , and attraction,g -¥, are discussed in a separate section3.3.

3.1. Impurity-gas repulsion,g0

The Fredholm determinant representation for the correlation function(2.1)in the case of the repulsive impurity-gas interaction,g0,reads

( )

ò

[( ) ( ˆ ˆ ) ( ˆ ˆ ˆ )] ( )

p

= L - + + +

--¥ ¥

G x t, 1 h I V I V W

2 d 1 det det . 3.1

rep

Here

[ ( ) ( )]

∣ ∣ ( )

( ) ( )

ò

ò

p n n p

= - =

-t t

-¥ ¥

-- +

-¥

¥

-+

h q q q g q

k q

1

2 i d e

1

4 d

e

, 3.2

q q

i 2 i

2

where

( ) ( ) ( )

the function( )q is defined by equation(2.10),

( ) ( )



n =

- q g

q k

2 1

, 3.4

and

( ) ( )

= L 



k g

2 i . 3.5

The identity operator is denoted byIˆ. The kernels of the linear integral operators_Vˆ_andWˆ, on the domain

[-¥ ¥ ´ -¥ ¥, ] [ , ],are defined by

( ¢ =) ( ) ( )¢ - ( ) ( )¢ ( )

- ¢

+ - - +

V q q e q e q e q e q

q q

, , 3.6

and

( ¢ =) +( ) +( )¢ ( )

W q q, e q e q , 3.7

respectively. The functionseare defined as

( ) ( ) ( ) ( ) ( ) ( ) ( )

p J

= = t

+ -

-e q e q e q, e q 1 q ei q 2, 3.8

where

( )

ò

( ) ( ) ( ) ( )

p

n n

= ¢ ¢

¢ - -

-¢ ¢ - +

t -¥

¥

- ¢ ⎛ - +

⎝

⎜ ⎞

⎠ ⎟

e q q q

q q

q

q q

1

2 i d e i0 i0 , 3.9

q i

and

( ) ( )

[ ( ) ]

J =

+

b -m

q 1

e q 1 3.10

is the Fermi weight. The determinant—‘det’symbol in equation(3.1)—stands for the Fredholm determinant of the corresponding linear integral operator. For completeness, we discuss the properties of Fredholm

determinants, which we use to derive our results, in appendixA.

The kernel(3.6)belongs to a class of integrable kernels[1,16]. Due to this property the resolvent operatorRˆ, defined as

ˆ_- ˆ_{=( ˆ + ˆ )}- _{( )}

I R I V 1, 3.11

is also integrable:

( ¢ =) ( ) ( )¢ - ( ) ( )¢ ( )

- ¢

+ - - +

R q q f q f q f q f q

q q

, . 3.12

The functionsf_are the solutions to the integral equations

( )+

ò

¢ ( ¢) ( )¢ = ( ) ( )



-¥ ¥

 

f q dq V q q f, q e q. 3.13

Using the fact that the operatorWˆ, with the kernel(3.7), has rank one, we recast the representation(3.1)into the following form:

( )

ò

( ) ( ˆ ˆ ) ( )

p

= L - +

-¥ ¥

++

G x t, 1 h B I V

2 d det , 3.14

rep

where

( ) ( ) ( )

ò

=

++ -¥

¥

+ +

B dq e q f q . 3.15

3.2. Impurity-gas attraction,g<0

The Fredholm determinant representation for the correlation function(2.1)in the case of the attractive impurity-gas interaction,g<0,reads

( )= ( )+ ( ) ( )

G x t, G x t, G x ti , . 3.16

attr rep

( )

ò

[ ( ) ( )][ ( ˆ ˆ ˆ ) ( ˆ ˆ ˆ ˆ )] ( )

p

= L t t + + - + +

--¥ ¥

- ++

-G x t, g I V V I V V V

4 1

2 d e det det . 3.17

i 2 i k k

1 1 2

The kernels of the operators entering equation(3.17)are defined on[-¥ ¥ ´ -¥ ¥, ] [ , ].The kernel ofVˆis given by equation(3.6), and those ofVˆ1andVˆ2are given by

( ) ( ) ( )

( )( )

( ) ( ) ( ) ( )

( )

¢ = ¢

- - ¢

-¢

-

-¢ - ¢

-

-+

-- +

-+

-+

V q q he q e q

k q k q

e q e q

k q

e q e q

k q

, , 3.18

1

and

( ) ( ) ( )

∣ ∣ ∣ ∣ ( )

¢ = ¢

- - ¢

-

-+ +

V q q g e q e q

k q k q

, , 3.19

2 2 _{2 2}

respectively. The functions entering equations(3.18)and(3.19)are defined in section3.1. Note thatGiaccounts for the bound states formed by the impurity and the gas particles, which exist for the Hamiltonian(2.15)with

<

g 0. This is explicitly shown in section7.

3.3. The limiting cases of infinitely strong repulsion and attraction,g ¥

In this section we consider the Fredholm determinant representation for the correlation function(2.1)in the limiting cases of infinitely strong impurity-gas repulsion,g ¥, and attraction,g -¥. In the case

 ¥

g our representation coincide with the one given in[12,13]. In the previously unexplored caseg -¥ the representation is shown to be identical with the one obtained forg  ¥.

In theg ¥limits the function(3.2)reads as follows:

( ) ( )

h

=

¥ ⎜ ⎟

⎛ ⎝ ⎞⎠

h G x t

lim 2 sin

2 , . 3.20

g

2

0

Here, the variableηis related toΛby the formula

( )

h

L = - ⎜⎛ ⎟

⎝ ⎞⎠ cot

2 , 3.21

andG0is the Green’s function of a free particle

( )

( )

( )

( )

ò

p _d

= = ¹

=

t p

-¥ ¥

-

-p

⎪

⎪

⎧ ⎨ ⎩

G x t k t

x t

, 1

2 d e

e e 0,

0.

3.22

k

m t

0 i

i 2

imx t

4

2 2

Let us consider the functionGrep(x t, ), which is defined by equation(3.1)for both repulsive,g0, and

attractive,g<0, interactions. Theg ¥limits of this equation read

( )

ò

{[ ( ) ] ( ˆ ˆ ) ( ˆ ˆ ˆ )} ( )

p h

= - + + + -  ¥

p p ¥

- ¥ ¥ ¥

G x t, 1 G x t I V I V W g

2 d 0 , 1 det det , . 3.23

The kernels of the operators_Vˆ_¥andWˆ_¥are

( ¢ º) ( ¢) ( )

¥

¥

V q q, lim V q q, , 3.24

g

and

(

)

( ¢ º) ( ¢) ( )

h ¥

¥

W q q, 1 W q q

2 sin

lim , , 3.25

g 2

2

whereV q q( , ¢)andW q q( , ¢)are defined by equations(3.6)and(3.7), respectively. In theg ¥limits the function(3.9)reads as follows:

( ) ( ) ( )

( )

( )

ò

h

p

h h

º = ¢

¢ - +

t

t ¥

¥ -¥

¥ - ¢

-⎜ ⎟

⎛ ⎝ ⎞⎠

e q e q q

q q

lim sin

2 1

p.v. d e sin

2cos2e , 3.26

g

q

q

2 i

i

where the symbol‘p.v.’indicates that the integral has to be interpreted as the Cauchy principal value. Note that the representation(3.23)coincides with equation(5.63)of[13]in the limitB ¥andh -¥while keepingh+B=const=m,whereμis the chemical potential in our model.

The functionG x ti( , )_{entering equation(3.16), and defined by equation(3.17), vanishes in theg -¥}

limit

( )=⎛  -¥ ( )

⎝ ⎜ ⎞

⎠ ⎟ G x t

g g

, 1 , . 3.27

i

( )

ò

[ ( ) ( )] ( ) ( )

p

= L t t L  -¥

-¥ ¥

- ++

-G x t, 1 u g

2 d e , , 3.28

i i k k

where

( )

( ) ( ˆ ˆ ) ( )

L =

L + + ¥

--u 4 I V B

1 det . 3.29

2 2

Here, the kernel of the operatorVˆ¥is defined by equation(3.24), and

( ) ( ) ( )

ò

=

---¥

¥

-

-B dq e q f q , 3.30

wheref_-is defined by equation(3.13). We have

( ) ( ) ( ) ( )

t k+ +t k- = t L - - L

m g

x

4 1 2 , 3.31

2 2

whereτis defined by equation(3.3), andkby equation(3.5). Therefore, the right-hand side of equation(3.28) decays at least as fast as1 gif

∣ ( )∣u 0 < ¥. (3.32)

The analytic properties ofu( )L discussed in section 2.3 of[17]imply the validity of equation(3.32), thus completing the proof of equation(3.27).

We found that the correlation function(2.1)is given by the same expression(3.23)for bothg  ¥and

 -¥

g limits. This implies, in particular, that the results forG¥(x t, )obtained by the technique developed in

[18](which is specific to theg ¥limit)are valid in theg -¥limit.

4. Mobile impurity reference frame

In this section we take our model defined in the laboratory reference frame, and write it in the mobile impurity reference frame, following[8]. We obtain the Hamiltonian which does not contain the impurity coordinate and contains the total momentum(a good quantum number)explicitly. The calculations performed in the rest of our paper are based on this representation of the Hamiltonian.

Let us introduce the operator

( )

=_eiP X _{, 4.1}

where

( ) ( ) ( ) ( ) ( )

† †

ò

y y

ò

y y s

= = - ¶

¶ =  

s s s s s ⎜⎛_⎝ ⎟⎞_⎠ s

X x x x x P x x

x x

d , d i , , . 4.2

L L

0 0

The transformation of an arbitrary operatorfrom the laboratory to the mobile impurity reference frame reads

( )

=-1. 4.3

It is often referred to as a polaron, or the Lee–Low–Pines transformation[19]. For single-particle operators, we get

( ) ( ) ( )



y x =e-iP x y x 4.4

( ) ( ) ( )



y x =y x-X, 4.5

and for the Hamiltonian(2.12)subjected to the conditionN=1, we get

( )

= +  =

H H Himp , N 1. 4.6

Here

( )

=

 

H H 4.7

and

( )

( ) ( )

= r

-+ =

 

 

H P P

m g N

2 0 , 1. 4.8

imp

2

The total momentumPis conserved in the model(2.12):

[P H, ]=0, P=P+P. (4.9)

Equation(4.9), written in the mobile impurity reference frame reads

[P H, ]=0, P=P, N=1. (4.10)

The state(2.4)is an eigenfunction of the momentum operator for the free Fermi gas

∣ ñ = ∣ ñ =

å

( )



=

P N P NN , PN p. 4.11

j N

j 1

Using thatX ∣0ñ =0we get for the operator(4.1)

∣ ∣ ∣ ∣ ( )

Nñ Ä 0ñ = Nñ Ä 0 .ñ 4.12

We note that

∣ ñ Ä∣ñ = ∣ ñ Ä∣ñ ( )

H N 0 E NN 0 , 4.13

where the energyENof the free Fermi gas in the state∣Nñis defined by equations(2.7)and(2.9). Applying the

transformation(4.3)to the operators entering equation(2.1)and using equations(4.4)and(4.11)–(4.13)we get

( ) ∣ ∣ ( ) ( )∣ ∣ ( )

{ }

( )  †

å

y y

= - -b -m á Ä á ñ Ä ñ

  -  

G x t

Z N x N

, 1 e e e 0 e 0 0 . 4.14

N

E t P x E N tH

iN i N N i

Using the momentum space decomposition(2.5)for the operatorsyandy†entering equation(4.14), we get

( ) ∣ ∣ ( )

{ }

( ) ( ) ( )

å

å

= -b -m - - á - ñ

G x t

Z L N N

, 1 e e 1 e e . 4.15

N

E N E t

p

P p x tH p

i i i

N N N

Here,H( )p is obtained fromH,equation(4.6), by projecting the operatorPentering equation(4.8)onto the statec†_{p }∣0ñhaving momentump:

( ) ( ) ( ) ( ) ( ) ( )

 = +   = -  + r

H p H H p H p p P

m g

,

2 0 . 4.16

imp imp

2

By using the mobile impurity reference frame we reduce the problem of calculating the correlation function

(2.1)to the analysis of the matrix elementsá_N∣_e-itH( )p∣_Nñ_{entering equation}(_4.15)_{. The impurity-gas}

interaction term,gr_( )0, in equation(4.16)has the form of a static potential scattering off the gas particles. Using the completeness condition

∣ ∣ ( )

å

f ñáf =1 4.17

f

p p

p

for the eigenfunctions∣f_pñof the HamiltonianH( )p at a givenp,

( )∣ ∣ ( )

 ñ = ñ

H p f_p E ff p , 4.18

we write

∣ ( )∣

å

∣ ∣ ∣ ( )

á_Ne-tH p _Nñ = á_{N f} ñ _e- _{. 4.19}

f

p tE

i 2 i

p

f

The overlapsáN f∣ _pñ, often called form-factors, vanish in the largeNlimit for anyg¹0(the decay rate for some

∣

áN f_pñis found in[8]). This vanishing is an example of the Anderson orthogonality catastrophe[20]. We give an explicit expression foráN f∣ _pñin section5. The sum overfpin equation(4.19)contains infinitely many terms, and

does not vanish asN ¥.We calculate this sum in sections6and7.

5. Bethe ansatz

In section5.1we present the eigenfunctions and spectrum of the Hamiltonian(4.16). In section5.2we discuss the solutions of the Bethe ansatz equations. In section5.3we calculate the overlaps between the eigenfunctions of the Hamiltonian(4.16)and the eigenfunctions(2.4)of the free Fermi gas.

5.1. Eigenfunctions and spectrum

The eigenfunctions and spectrum of the mobile impurity model(2.15)were found exactly in[2,3]. The eigenfunctions in the coordinate representation could be written as a sum running over all permutations ofN

We let the particle mass

( ) =

m 1 5.1

through the rest of the paper in order to lighten notations. We write the eigenfunctions∣f_pñ, defined by equation(4.18), in the coordinate representation as the determinant of the(N+1)´(N+1)matrix

( )

!

( ) ( )

( )

  

n n

=

- - +

+

+

f x x Y

N L

k k

,...,

e ... e

e ... e

...

. 5.2

p N

f

N

k x k x

k x k x

N 1

i i

i i

1 1

N

N N N

1 1 1 1

1 1

Here,n-is defined by equation(3.4). The factorYfensures the normalization condition

∣ d ( )

áf f_{p p}¢ñ = _ff¢. 5.3

Note thatáf f_p∣ _p¢ ñ ¹¢ d_ff¢forp¹ ¢p.We calculateYfin section5.3. The set of quasi-momentak1,¼,kN+1

satisfies a system of nonlinear equations(Bethe equations)

( )

= - L = ¼ +

k L k

g j N

cot 2

2

, 1, 2, , 1, 5.4

j j

and

( )

å

=

= +

p k. 5.5

j N

j 1

1

Recall thatpis the total momentum of the system in the laboratory reference frame, and it is quantized as follows

( )

p

= =  

p

L n n

2

, 0, 1, 2 ,.... 5.6

The energyEfof the state∣f_pñis

( )

å

=

= +

E 1 k

2 . 5.7

f j N

j 1 1

2

The determinant in equation(5.2)differs from the Slater determinant for the eigenfunctions of a free Fermi gas by the last row only. Note that the explicit form of the matrix under the determinant can be changed by a unitary transformation; its size can also be reduced by decomposing the determinant. This explains the variety of the representations equivalent to the one given by equation(5.2), found in the literature[7–9,14,15].

5.2. Solutions of the Bethe equations

In this section we discuss the properties of the solutions of the Bethe equations(5.4)and(5.5)that we use to derive our results.

Any solutionk1,¼,kN+1,Lof the Bethe equations(5.4)and(5.5)has the following properties[3]:(i)Λis

real.(ii)Ifg0allkjʼs are real.(iii)Ifg <0either allkjʼs are real, ork1,¼,kN-1are real, whilekNandkN+1have

a non-zero imaginary part, andkN=kN*+1. Furthermore, ifkNandkN+1are complex, they read as follows in the

largeLlimit:

( ∣ ∣ ) ( ∣ ∣ ) ( )

 

= ++ - + = -+

-kN k e g L, kN 1 k e g L, 5.8

wherekare defined by equation(3.5).

We will often use the following representation of the Bethe equations(5.4):

( )

( ) ( )

n n

= - = ¼ +

+ k

k j N

ek L j , 1, , 1, 5.9

j ij

wherenare given by equation(3.4). Taking the derivative of equation(5.9)with respect toΛwe get

( ) ( )

( ) ( )

( )

n n

n n

¶

¶L = + = ¼ +

- +

- +

k L

k k

k k j N

2

1 , 1, , 1. 5.10

j j j

Lg j j

4

Substituting equation(5.8)into(5.10)we obtain

( ∣ ∣ ) _* ( )



¶

¶L =

¶

¶L = + =

+

-+

k k g

k k

2 e , . 5.11

N N g L

N N

1

Let us introduce the real-valued functionR(x)as a solution of the equation

( )= + ( ) ( )

R x x

gLR x

cot 4 . 5.12

This function is single-valued for-¥ < < ¥x if

( )



-gL 4

1. 5.13

For the rest of the paper we assume that equation(5.13)holds true. We note that the results obtained using this condition are valid for arbitrarygin theL ¥limit.

The functionR(x)decays monotonously

( )

( )

¶

¶ < -¥ < < ¥

R x

x 0, x 5.14

and takes the values between 0 andπ:

(-¥ =) p (¥ =) ( )

R , R 0. 5.15

Using this function, any real-valued quasi-momentakjentering equation(5.4)can be parametrized as follows

( )

p d

p

= ⎛ - =   ¼

⎝

⎜ ⎞_⎠⎟ k

L n n

2

, 0, 1, 2, , 5.16

j j

j

j

where

( )



d = ⎛L - p d <p

⎝

⎜ ⎞

⎠ ⎟ R

gLn 4

, 0 . 5.17

j j j

Combining equations(5.16),(5.17), and(5.14)one can immediately see that

( )

¶

¶L > -¥ < L < ¥ = ¼ +

k

j N

0, , 1, , 1 5.18

j

for any real-valued quasi-momentumkj.

Evidently, real quasi-momentakjʼs, are in one-to-one correspondence withnjʼs. Thekjʼs should be different

from each other, otherwise the function(5.2)vanishes. This implies that thenjʼs should be different as well.

Therefore, each eigenstate(5.2)is uniquely characterized by an ordered setn1<<nN+1(or by



< <

-n1 nN 1ifkN =k*N+1)for a givenp.We also stress that taking the limitsL  ¥andL  -¥one

gets the same subset of eigenstates(5.2). The conditiondj<pimposed in equation(5.17)ensures that this

subset is taken into account only once.

5.3. Determinant representation foráN f∣ _pñ

In this section we present an explicit formula for the overlapsáN f∣ _pñbetween the eigenfunctions(2.4)of the free Fermi gas and the eigenfunctions(5.2)of the mobile impurity model. We also present an explicit formula for the normalization constantYfin equation(5.2).

Straightforward calculations(see appendixB.1for details)lead us to the following result:

∣



n ( ) ( )

á ñ =

= +

-⎜ ⎟

⎛ ⎝ ⎞⎠

N f Y

L D k

2i

det , 5.19

p f

N

f j N

j 1 1

where

( )

  

=

-

--

-+

+

D det

...

...

1 ... 1

5.20

f

k p k p

k p k p

1 1

1 1

N

N N N

1 1 1 1

1 1

is the determinant of the(N+1)´(N+1)matrix, andn-is defined by equation(3.4). We get from equation(5.19)

∣á ∣ ñ =∣ ∣ ∣ ∣ ∣



∣n ( )n ( )∣ ( )

= +

- +

⎜ ⎟

⎛ ⎝ ⎞⎠

N f Y

L D k k

2

det . 5.21

p f

N

f j N

j j

2 2

2

2

Now, using equation(B16)from appendixB.2we write an explicit expression for∣ ∣Yf2:

∣ ∣ ( ) ( ) ( ) ( )

( ) ( )

( )



n n

å

n n

n n

= +

+

-=

+

- +

= +

- +

- +

Y

gL k k

k k

k k

1 4

1 . 5.22

f j N

j j

j N

j j

gL j j

2

1 1

1 1

4

Substituting this expression into equation(5.21)and using equation(5.10)we get

∣á ∣ ñ =∣ ∣ ∣

å

¶



( )

¶L

¶ ¶L

=

+

-= +

⎜ ⎟

⎛ ⎝ ⎞⎠ N f

L D

k k

2

det . 5.23

p

N

f j N

j

j N

j

2 2

1

1 1

1 1

Recall thatk1,¼,kN+1,Lsolve the Bethe equations(5.4)and(5.5).

In sections6and7we employ the representation(5.23)to perform the summations in equations(4.19)and

(4.15)and obtain the result expressed in terms of the Fredholm determinants.

6. Summation in case of impurity-gas repulsion,

g



0

In this section we perform the summations in equations(4.19)and(4.15). This results in the Fredholm determinant representation for the correlation function(2.1). We assume that the impurity-gas interaction is repulsive,g0.The case of the attractive interaction,g<0,is considered in section7.

In section6.1we perform the summation in equation(4.19)using the explicit formula(5.23). In section6.2

we transform the representations obtained in section6.1, using the fact that the system sizeL ¥.In section6.3we perform the summations in equation(4.15).

6.1. Summation overf_p

In this section we perform the summation overfpin equation(4.19).

The quasi-momentak1,¼,kN+1characterizing the eigenfunctionsfpfor a givenpare coupled to each other

by the condition(5.5). There, the total momentumpis quantized, as given by equation(5.6). The quantization is due to afinite system size,L,and should play no role for the observables asL ¥.We can therefore write the following identity

(

)

_{( )}

ò

å

d

p

- = å  ¥

= +

-¥ ¥

-= +

⎛ ⎝

⎜⎜ ⎞

⎠ ⎟⎟

k p 1 z L

2 d e , . 6.1

j N

j k p z

1 1

i _jN₁ j

1

The argumentåN_j=+11kj-pof the Diracδ-function on the left-hand side of this expression is equal to zero when equation(5.5)is satisfied. Using equation(6.1)to calculate the correlation function(2.1)produces results which are exact in theL ¥limit.

Let us now introduce the function

( )

ò

å

d

å

( )

D ¼ = L ¶

¶L

-+

-¥ ¥

= +

= + ⎛ ⎝

⎜⎜ ⎞_⎠⎟⎟

n, ,n d k k p . 6.2

p N

j N

j

j N

j

1 1

1 1

1 1

Here,n1,¼,nN+1are integers which definek1,¼,kN+1as given by equations(5.16)and(5.17). We stress thatkjʼs

depend onΛthroughdjʼs as given by equation(5.17). Therefore,Dp(n1,¼,nN+1)=1if there exists a solution

to the Bethe equations(5.4)and(5.5)for a given setn1,¼,nN+1, andp,and is equal to zero otherwise:

( ) ( ) ( ) ( )

D ¼ + =⎧⎨ ¼ +

⎩

n, ,n 1 n, ,n , andp solve equations 5.4 and 5.5 ,

0 otherwise. 6.3

p 1 N 1 1 N 1

Defining

( ¼ + )= ∣á ∣ ñ∣ ¼ + ( ) ( ) ( )

⎧ ⎨ ⎩

F n, ,n N f n, ,n , andp solve equations 5.4 and 5.5 ,

0 otherwise, 6.4

p N p

N

1 1

2

1 1

and using equations(5.23)and(6.2)we get

( ¼ )=

ò

L



¶ ∣ ∣ d

å

( )

¶L

-+

-¥ ¥

= +

= +

⎜ ⎟

⎛ ⎝ ⎞⎠

⎛ ⎝

⎜⎜ ⎞_⎠⎟⎟

F n n

L

k

D k p

, , d 2 det . 6.5

p N

N

j N

j

f j N

j

1 1

1 1

2

1 1

We stress thatFpis symmetric with respect to the permutations ofn1,¼,nN+1, and vanishes if any two integers

∣ ∣

( )! ( ) ( )

( )



å

á ñ =

+ ¼ -¼ + -+ N N

N F n n

e 1

1 , , e . 6.6

tH p

n n

p N tE

i

, ,

1 1 i

N

f

1 1

On the right-hand side of this expression the summation over eachnjgoes independently over all integers. The

factor1 (N+1)!compensates the multiple counting of the Bethe ansatz solutions stemming from the removal of the ordering conditionn1<<nN+1.Substituting the identity(6.1)into equation(6.6)we get

∣ ∣ ( )! ∣ ∣ ( ) ( ) ( ) 

ò

ò

å

_p



á ñ =

+ L ´ ¶ ¶L t --¥ ¥ ¼ -¥ ¥ -= + -+ ⎜ ⎟ ⎛ ⎝ ⎞⎠ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ N N N z L D k e 1 1 d d 2 e 2

det e , 6.7

tH p n n pz N f j N k j i , , i 2 1 1 i N j 1 1 where ( ) ( )

t k =tk -zk

2 . 6.8

2

We stress that eachkjdepends onnjas given by equations(5.16)and(5.17). Sincek1,¼,kN+1are real we have

∣detDf∣2 =(detDf)2 (6.9)

and equation(5.18)is applicable. We can, therefore, rewrite equation(6.7)as follows

∣ ∣ ( )! ( ) ( ) ( ) ( ) 

ò

ò

å

_p



á ñ =

+ L ´ ¶ ¶L t --¥ ¥ ¼ -¥ ¥ -= + -+ ⎜ ⎟ ⎛ ⎝ ⎞⎠ ⎡ ⎣⎢ ⎤ ⎦⎥ N N N z L D k e 1 1 d d 2 e 2

det e . 6.10

tH p n n pz N f j N k j i , , i 2 1 1 i N j 1 1

We now transform the summation overn1,¼,nN+1in equation(6.10)using a technique explained in

appendixB.3. Employing equation(B23)we get

( )! ( ) ( ) ( ) ( ) ( )

å



+ ¶ ¶L = - + -t ¼ = + -+ ⎜ ⎟ ⎛ ⎝ ⎞⎠ ⎡ ⎣⎢ ⎤ ⎦⎥

L N D

k

h A A B

2 1

1 det e

1 det det . 6.11

N n n f j N k j , , 2 1 1 i N j 1 1 Here ( ) ( )

å

= ¶

¶L -t

h ke , 6.12

n k i ( )( ) ( ) ( )

å

= ¶

¶L - - = ¼

t -A

L k

k p k p j l N

2 e

, , 1, , , 6.13

jl n k j l i and ( ) ( ) ( )

= = ¼

B

Le p e p j l N

2

, , 1, , , 6.14

jl _{j l}

where

( ) ( )

( )

å

p

= ¶

¶L - = =   ¼

t

-e q k

k q q L n n

e

, 2 , 0, 1, 2, . 6.15

n

k i

We stress that the summation indexnin equations(6.12),(6.13), and(6.15)is related tokby equation(5.16). We rewrite equation(6.13)in a form used in our subsequent calculations

( ) ( ) ( ) ( ) ( ) ( )

å

d d = ¶

¶L - +

-- = ¼

t -⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ A L k k p

e p e p

p p j l N

2 e

1 , , 1, , . 6.16

jl jl n k j jl j l j l i 2

Substituting equation(6.11)into(6.10)we get

∣ ( )∣

ò

ò

[( ) ( )] ( )

p

á - ñ = L - +

--¥ ¥

-¥ ¥

-Ne N d dz h A A B

2 e 1 det det . 6.17

tH p pz

i i

6.2. LargeLlimit

We introduce the function

( )= - L - ( )

b k k

g

kL 2

cot

2 6.18

whose zeroesb k( ) =0are the solutions to the Bethe equations(5.4). It satisfies the following identity

( )

( ) ( )

¶

¶ =

¶

¶L =

-⎜ ⎟ ⎛ ⎝

⎞ ⎠ b k

k

k

b k

at 0. 6.19

1

Using this identity we transform the summation in equations(6.12),(6.16), and(6.15)as follows

∮

( )

( ) ( ) ( )

å

_¶L¶ = _p

g k

f k k

b k f k 1

2 i d

. 6.20

n

The contourγis oriented counter-clockwise. It is chosen in such a way that the zeroes ofb(k)are inside the domain bounded byγ, and the poles of the functionf(k)are outside.

We now take the largeLlimit. This will remove the effects of the boundary conditions, and further simplify the formulas. The functioncot(kL 2)enteringb(k)is periodic with the period2p L.The variation over the period of the other terms enteringb(k)can be neglected in the integrals containingb(k)to the leading order of the1 Lexpansion. Taking into account that

( ) ( )

ò

p - = + ¹

p k

z k z z z

1

d 1

cot

1

isign Im , Im 0 6.21

0

we can replacecot(kL 2)withiifImg>0and with-iifImg<0in the functionb(k)on the right-hand side of equation(6.20).

We begin with applying the arguments from the previous paragraph to the representation(6.20)of the function(6.12). We take the contourγconsisting of two straight lines,g+andg-. Here,g+runs from¥ +i0 to-¥ +i0, andg-runs from-¥ -i0to¥ -i0. Therefore

( )

( ) ( )

ò

p

=

- L - - - L +

t t

-¥

¥ ⎛ -

-⎝

⎜ ⎞

⎠ ⎟

h k

k g k g

1

2 i d

e

2 i

e

2 i . 6.22

k k

i i

We now perform the same procedure with the functione q( ),defined by equation(6.15). We take the contourγconsisting ofg g+, -,andc.Here,cis a clockwise-oriented closed contour around the pointq.We have

∮

_{( )} p ( )

- = = =   ¼

k

b k k q q L n n

d 1

0 at 2 , 0, 1, 2, . 6.23

c

Therefore

( )

( )

( ) ( ) ( )

( ) ( )

( ) ( )

ò

ò

p

p

=

- L - - - L + - +

= - L

- L + + - L +

-t t

t t

-¥

¥ -

--¥

¥

-⎛ ⎝

⎜ ⎞

⎠ ⎟

e q k

k g k q k g k q

q g

q g k k g k q

1

2 i d

e

2 i

1 i0

e

2 i

1 i0

2 e

2 1

1

p.v. d e

2 1

1

, 6.24

k k

q k

i i

i

2

i

2

where p.v. stands for the Cauchy principal value of the integral. Now we transform the diagonal entries of the matrix(6.16):

∮

( ) ( ) ( )

( ) ( )

å

_p

º ¶

¶L - = _{- L -}

-t

g

t

-

-A L

k

k p

k

k p

2 e 1

2 i

d cot

e

. 6.25

jj n

k

j k

g

kL k

j i

2 2

2 i

2

We take the same contourγas fore q( ).The contourcgives now a non-vanishing contribution. We have

( )∣ ( )

( )

= -t + ¶

= A

L e q

e 2 . 6.26

jj i pj q q p

j

Using

( )

( )



t =

=

- - +

e e , 6.27

j N

p tE P z

1

i j i N i N

we arrive at the following expressions for the determinants enteringequation(6.17):

( ˜ ) ( )

= - + +

A I V

and

(A-B)= - + (I+V˜ -W˜ ) ( )

det e itEN iP zN det . 6.29

Here

˜ ₌ p ( ) ( ) - ( ) ( ) _{( )}

- = ¼

+ - - +

V L

E p E p E p E p

p p j l N

2

, , 1, , 6.30

jl

j l j l

j l

and

˜ ₌ p _{+( ) +( ) = ¼ ( )}

W

L E p E p j l N

2

, , 1, , . 6.31

jl j l

The functionsEare defined as

( ) ( ) ( ) ( ) ( ) ( )

p

= = t

+ -

-E q e q E q , E q 1 ei q 2, 6.32

and the functionsh e q, ( ), andt( )q by equations(6.22),(6.15), and(6.8), respectively. Substituting equations(6.28)and(6.29)into equation(6.17)we get

∣ ∣

[( ) ( ˜ ) ( ˜ ˜ )] ( )

( )

( )



ò

ò

_p

á ñ = L

´ - + + +

--¥ ¥

-¥ ¥

-

-N N

z

h I V I V W

e d

d

2 e e 1 det det . 6.33

tH p

P p z E t i

i N iN

6.3. Summation overpand{ }N

In this section we perform the summation overpand{ }N in equation(4.15)and obtain the Fredholm determinant representation for the correlation function(2.1).

We replace the sum overpwith an integral

( )

ò

å

 _p

-¥ ¥

L p

1 1

2 d , 6.34

p

while taking theL ¥limit of equation(4.15), and get

( ) ∣ ∣ ( )

{ }

( )

ò

( ) ( )

å

_p

= -b -m á ñ

-¥ ¥

- -

-G x t Z

p

N N

, 1 e e d

2 e e . 6.35

N

EN N iE tN iPN p x itH p

The right-hand side of equation(6.33)depends onpthrough the functione-ipz_{only. Taking this into account}

and substituting equation(6.33)into(6.35)we perform the integration overp,

( ) ( )

( )

ò

_p =d

--¥ ¥

-p

x z d

2 e , 6.36

p x z i

and then the integration overz, to obtain the following expression

( ) [( )

( ˜ ) ( ˜ _{˜ )]} ( )

{ }

( )

ò

å

p

= L

-´ + + +

-b m -¥

¥

-

-G x t

Z h

I V I V W

, 1

2 d

1

e 1

det det . 6.37

N

EN N

Here,h V, ˜,andW˜ are given by equations(6.22),(6.30), and(6.31), respectively.

Equation(6.37)is the desired Fredholm determinant representation for the correlation function(2.1). This representation may be written in the following equivalent form(see appendixAfor a proof)

( )

ò

[( ) ( ˆ ˆ ) ( ˆ ˆ ˆ )] ( )

p

= L - + + +

--¥ ¥

G x t, 1 h I V I V W

2 d 1 det det . 6.38

Here,Iˆis the identity operator, andVˆandWˆ are linear integral operators with the kernels

( ) ( ) ˜ ( ) ( )

( ) ( ) ˜ ( ) ( ) ( )

J J

J J

¢ = ¢ ¢

¢ = ¢ ¢

V q q q V q q q

W q q q W q q q

, , ,

, , , 6.39

where

( ) ( )

[ ( ) ]

J =

+

b -m

q 1

is the Fermi weight, and( )q =q2 2_{is the single-particle energy. The kernels}

˜ ( ¢ =) ( ) ( )¢ - ( ) ( )¢ ( )

- ¢

+ - - +

V q q E q E q E q E q

q q

, 6.41

and

˜ ( ¢ =) +( ) +( )¢ ( )

W q q, E q E q 6.42

are obtained from equations(6.30)and(6.31), respectively.

7. Summation in case of impurity-gas attraction,

g

<

0

In the present section we adapt the approach of section6to the case of attractive impurity-gas interaction,

<

g 0. We make our analysis forLandgsatisfying equation(5.13). Our results are applicable for all values ofgin theL ¥limit.

The quasi-momentakNandkN+1could be complex forg<0(this is discussed in section5.1). We split the

sum on the right-hand side of equation(4.19)into two parts:

∣ ( )∣

å

∣ ∣ ∣

å

∣ ∣ ∣ ( )

á_Ne-tH p _Nñ = á_{N f} ñ _e- + á_{N f} ñ _e- _{. 7.1}

f

p tE f

p tE

i 2 i 2 i

p r f p i f

The sum overfprruns over the real setsk1,¼,kN+1, while the sets withkN =kN*+1constitute the sum overfp. i

The approach of section6can be applied to the sum overfprdirectly. On substituting this sum into equation(4.15)

it leads to the same Fredholm determinant representation forG x t( , ), as given by equation(6.38)in the caseg0.

Let us now consider the sum overf_pi.Proceeding as in section6we get the analogue of equation(6.10):

∣ ∣ ∣ ( )! ∣ ∣ ( ) [ ( ) ( )] ( )

ò

ò

å

å



p

á ñ =

- L ´ ¶ ¶L ¶ ¶L ¶ ¶L t t t --¥ ¥ -¥ ¥ - - + + ¼ = -+ -⎜ ⎟ ⎛ ⎝ ⎞⎠ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ N f N z k k L D k e 1 1 d d

2 e e

2

det e . 7.2

f

p tE pz k k

N N N

n n

f j N

k j

2 i i i

1 , , 2 1 1 i p i f N j 1 1 Sincek1,¼,kN-1are real, andkN=kN*+1, we have

∣detDf∣2 = -(detDf)2. (7.3)

Using equation(5.18)fork1,¼,kN-1, equation(5.11)forkNandkN+1, and taking into account equation(7.3),

we transform equation(7.2)into

∣ ∣ ∣ ( )! ( ) ( ) [ ( ) ( )] ( )

ò

ò

å

å



p

á ñ =

-- L ´ ¶ ¶L t t t --¥ ¥ -¥ ¥ - - + ¼ = -+ -⎜ ⎟ ⎛ ⎝ ⎞⎠ ⎡ ⎣⎢ ⎤ ⎦⎥ N f N g z L D k e 1

1 4 d

d

2 e e

2

det e . 7.4

f

p tE pz k k

N n n f j N k j

2 i 2 i i

, , 2 1 1 i p i f N j 1 1

We now transform the summation overn1,¼,nN-1in equation(7.4)using the technique explained in

appendixB.3. Employing equation(B30)and the asymptotic formula(5.8)we get

( )! ( ) ( ) ( )

( )

å



-¶

¶L = - + +

t ¼ = -⎜ ⎟ ⎛ ⎝ ⎞⎠ ⎡ ⎣⎢ ⎤ ⎦⎥

L N D

k

C C D

2 1

1 det e det det . 7.5

N n n f j N k j , , 2 1 1 i N j 1 1 Here ( )( ) ( )( )( )( ) ( ) ( )

å

= ¶ ¶L -

-- - - - = ¼

t -- + - + C L

k k k k k

k p k p k p k p j l N

2 e

, , 1, , , 7.6

jl n

k

j l j l

i

and

( ) ( ) ( )

= = ¼

D

Lv p v p j l N

2

, , 1, , , 7.7

jl _{j l}

where

( )

( )( ) ( )

p

=

-- - = =   ¼

- +

- +

v q k k

k q k q , q L n n

2

, 0, 1, 2, . 7.8

( ) _{( )}

( )( ) ( )

=

-- - - + - - = ¼

+ - - +

C A

L e p

k p L

e p

k p L

h

k p k p j l N

2 2 2

, , 1, , . 7.9

jl jl

j

l

l

j j l

Here,h A, , andeare defined by equations(6.12),(6.13), and(6.15), respectively. Substituting equation(7.5)into

(7.4)we get

∣ ∣ ∣

[ ( )] ( )

[ ( ) ( )]

ò

ò

å

á ñ = L _p

´ - +

t t

--¥ ¥

-¥ ¥

- - ++

-N f g z

C C D

e

4 d

d

2 e e

det det . 7.10

f

p 2 itE pz k k

2

i i

p i

f

Further transformations of this sum into the Fredholm determinant representation of the function(2.1)can be performed using steps similar to those performed in sections6.2and6.3, and we arrive at the results

summarized in section3.2.

8. Discussion and outlook

The main result of the present paper is the Fredholm determinant representation for the time-dependent two-point impurity correlation function(2.1). We consider a particular Bethe ansatz solvable model: A mobile impurity interacting with a free Fermi gas(or the Tonks–Girardeau gas[10,11])through aδ-function potential of arbitrary strengthg.We extend the approach of[14]to the case where the impurity-gas interaction can be attractive,g<0,and the temperature can befinite.

Let us discuss how our results can be used to investigate the mobile impurity dynamics. Various asymptotic formulas for a Fredholm determinant can be obtained by formulating the matrix Riemann–Hilbert problem and solving it asymptotically[1,22–25]. In model(2.12)Fredholm determinant representations are known in the

 ¥

g limit[12,13], and corresponding asymptotic solutions of the matrix Riemann–Hilbert problem have been discussed in[17,26–28]. However, these solutions are specific to certain regimes in which the temperature-weighted average is different from the one defined by equation(2.3) (see[29,30]for further description of these regimes). The results obtained in[14]and in the present paper make it possible to formulate the matrix

Riemann–Hilbert problem at arbitraryg,in the regime where the temperature-weighted average is defined by equation(2.3). The large time and distance asymptotic solution of the matrix Riemann–Hilbert problem should

be compared with the predictions of[18](see equation(6)therein)valid wheng ¥and the temperature is zero. The Fredholm determinant representation is also promising for investigating the spectral function

(defined by the Fourier transform of the correlation function(2.1)from space and time to momentum and frequency variables). All existing approaches based on a mapping between microscopic interacting theories and effective freefield theories describe a shape of the spectral function only in some vicinity of a singularity(see

[31,32]and references therein). In contrast, a numerical evaluation of the Fredholm determinant, combined with asymptotic expansions, has been shown for several models to give precise data for correlation and spectral functions everywhere in space–time and momentum–frequency domains[28,33,34]. We expect that this approach will give precise data for the impurity spectral function in our model as well. Having such data may help verify the phenomena predicted in[35].

The approach developed in the present paper could be applied to other observables in the model. Of particular interest is the time-dependent average momentum of an impurity particle injected with some initial momentum into a free Fermi gas. This problem has been investigated using the Bethe ansatz[15]. However, the form-factors summation(which we do analytically in section6for the impurity correlation function)was done numerically in[15]. Later, the same problem was addressed with other techniques[36–40]. The existing results were obtained for particular time scales(for example, short time in the case of simulations using time-dependent density matrix renormalization group)and values of the external parameters(for example, weak impurity-gas coupling in the case of calculations done by diagrammatic methods). We expect that, using the Fredholm determinant representation, the average momentum of the impurity particle could be precisely calculated at all times and for all values of the external parameters.

equation(2.1), exists for anyg. Whether its existence is due to some mapping between our model and a non-interacting quantumfield theory remains an open problem.

The linear integral operators entering the Fredholm determinant representation given in the present paper are of a special, integrable type(see, e.g, section XIV.1 of[1]). Exploiting the properties of operators of this type, helped represent correlation functions of several models as solutions to nonlinear differential equations

[16,45,46]. Our results make it possible to try applying this approach to the mobile impurity model considered here.

In the caseg ¥, Fredholm determinant representations for the correlation function(2.1)of the free Fermi gas and the Tonks–Girardeau gas on the lattice are found in[47,48], respectively. A straightforward modification of the approach given in the present paper would lead to a Fredholm determinant representation for the same models with arbitraryg.

Acknowledgments

We thank AMaitra, SMajumdar, GSchehr, and ASykes for careful reading of the manuscript. The work of OG and MBZ is part of the Delta ITP consortium, a program of the Netherlands Organisation for Scientific Research

(NWO)that is funded by the Dutch Ministry of Education, Culture and Science(OCW). The work of AGP is partially supported by the Russian Science Foundation, Grant No. 14-11-00598.

Appendix A. Fredholm determinants for the free Fermi gas

In this appendix we recall how a Fredholm determinant is defined and prove an identity used in sections6and7. Take an arbitraryM×MmatrixV.We have

( )

!  ( )

 det d +V =

å

₌ N

å

₌

å

₌   V 1

det . A1

j l M jl jl _N M

a M

a M

j l N a a

1 , _{0 1 1 1} ,

N

j l

1

The limitM ¥in this expression defines the Fredholm determinant of the operatorIˆ+Vˆ:

( ˆ ˆ )

!    ( )

å

å

å

+ =

= ¥

= ¥

= ¥

I V

N V

det 1 det . A2

N a a j l N

a a

0 1 N 11 ,

j l

1

The limitM ¥may not exist, and even if it does, the expression on the right-hand side of equation(A2)may diverge for someV. We assume, however, that the necessary existence and convergence conditions are fulfilled for the operators encountered in our paper. These operators are linear integral operators, and we therefore use the following rigorous formula to define the Fredhom determinant(see, e.g.,[49], vol IV, p 24):

( ˆ ˆ )

!

( ) ( )

( ) ( )

( )

   

ò

ò

å

+ =

= ¥

-¥ ¥

-¥ ¥

I V

N k k

V k k V k k

V k k V k k

det 1 d d

, ... ,

, ... ,

. A3

N

N

N

N N N

0

1

1 1 1

1

Let us now prove an identity which we use in sections6and7. We take

( ˜ ) ( )

{ }

( )

å

= -b -m +

Z

Z I K

1

e det . A4

K N

EN N

Here

( )

{ }

( )

å

= -b -m

Z e A5

N

EN N

andK˜ is an arbitraryN×Nmatrix whose entries_K˜_{jl ºK}˜_{p p}

j lare functions ofp1,¼,pN.The energyENis the

sum of single-particle energies

( ) ( )



å

=

=

EN p . A6

j N

j 1

The momentap₁,¼,p_Nare quantized

( )

p

= =   ¼

p

L n n

2

, 0, 1, 2, A7