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Density fluctuations in the 1D Bose gas
Panfil, M.K.
Publication date
2013
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Citation for published version (APA):
Panfil, M. K. (2013). Density fluctuations in the 1D Bose gas.
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Exact prefactors of the
density-density correlation function
in the (non-) linear Luttinger liquid
model
This chapter is based on [56].
In this chapter we study the 1D Bose gas in the low energy limit. This limit can be described by an effective field theory, the Luttinger liquid. The Luttinger liquid theory is a universal theory describing one-dimensional liquids, systems with gapless excitations (c.f. section 2.1.2). We start with a schematic “derivation” of the Luttinger liquid (also in its non-linear version). This serves two aims, firstly it makes the discussion more complete as no a priori knowledge of Luttinger liquid is required to understand the results of this chapter. Secondly it is in this introductory part that we will show how microscopic information (those that come from the 1D Lieb-Liniger model) shapes the physical behavior of the effective field theory. In order to be more concrete, let us write down the density-density correlation function as predicted by the Luttinger liquid theory
S(x) = n2 1− K 2 (kFx)2 + X m>0 Amcos (2kFmx) (nx)2m2K ! . (4.1) Here in accordance with our conventionn is the density of the gas, kF = πn is the Fermi
momentum and K is the Luttinger parameter, a dimensionless phenomenological num-ber roughly speaking representing strength of interactions. Moreover Am are constant
prefactors. This formula is valid only in the large distance limit, x n−1. 65
Eq. (4.1) is valid for a wide class of 1D liquids, such like the 1D Fermi gas or 1D spin chains in gapless regime, the model-specific features enters this formula only through the prefactors of the oscillatory terms Am. Thus the first objective of this chapter is to
provide a relation between Am and microscopic quantities of the Lieb-Liniger model.
Beside the prefactors of the real-space correlation function we also compute prefactors of the momentum space correlation function in the vicinity of the single particle (hole) excitation threshold ω1(2)(k) (3.134). To achieve this we have to extend the Luttinger liquid description towards high energies. This is done by immersing an impurity in the liquid and studying the resulting, effective theory: the non-linear Luttinger liquid. The theory predicts that the density-density correlation function in a momentum space and at energies close to the lower excitation threshold ω2(k) takes the following form
S(k, ω) = θ(ω− ω2(k))
2πS2(k) (ω− ω2(k))µ˜R+˜µL−1
Γ (˜µR+ ˜µL) (vs+ vd)µ˜L|vs− vd|µ˜R
, (4.2) wherevd = ∂ω2(k)/∂k1 is the impurity velocity,vs is the sound velocity andµ˜L(R) are
the exponents fixed by the Luttinger parameterK. The momentum dependent prefactor S2(k) is not universal and cannot be determined within the effective theory. Similar
expression holds for the correlation function in the vicinity of the upper thresholdω1(k)
only with a different prefactor S1(k). As for Am we will relate both functions to the
microscopic quantities of the Lieb-Liniger model.
The second part of this chapter contains the main results, namely we compute the ther-modynamic limit of the form factor (3.67) of the density operator. This allows to compute Am and S1(2)(k) knowing the particle density n, the Luttinger parameter K and in the
latter case alsoω1(2)(k).
In the introduction presented below we follow the spirit of the original paper of Hal-dane [57] and the review by Cazalilla [58]. Of course every writing on the Luttinger liquid benefits from consulting the book of Giamarchi [59].
4.1
Luttinger Liquid Theory
In order to obtain an effective, low energy description of the Lieb-Liniger model it is con-venient to represent the field operatorsΨ(†)(x) in terms of two self-conjugated operators
1
Note that the ω1(2)as defined in (3.134) is a function of rapidity and in order to write it as a function of momentum we need equation (3.104) relating a momentum of an excited state to the rapidities.
describing density and phase fluctuations
Ψ(x) = ei ˆφ(x)pρ(x),ˆ (4.3) Ψ†(x) =pρ(x)eˆ −i ˆφ(x). (4.4) The bosonic canonical commutation relation implies that
ei ˆφ(y)ρ(x)e−i ˆφ(y)− ˆρ(x) = δ(x− y). (4.5) Moreover the periodic boundary conditions restrict values ofφ(x)
ˆ
φ(x + L)− ˆφ(x) = πJ, J ∈ 2Z. (4.6) We are going to find now a convenient representation of the density operator.
Since the particles are confined to one spatial dimension, there is a sense of ordering (recall solution 2.1.1 to the Lieb-Liniger model through the Coordinate Bethe Ansatz method.). We can define an operator whose eigenvalues increase monotonically as we go through the system. Let us call it ˆΘ(x). In the system with N particles and periodic boundary conditions we expect that (factor π is introduced just for convenience)
ˆ
Θ(L + x)− ˆΘ(x) = πN. (4.7) We can now relate the positionxj of the j-th particle to the equality ˆΘ(xj) = jπ. This
allows us to rewrite the particle density operator as ˆ ρ(x) = N X j=1 δ (x− ˆxj) = ∂xΘ(x)ˆ X j∈Z δΘ(x)ˆ − jπ, (4.8) If the particles were to develop a perfect crystalline order then ˆΘ(x) would equal jπ for x = j/n. However, in reality, particles positions fluctuate around these positions and to quantify these fluctuations we introduce a new fieldθ(x)2
ˆ
Θ(x) = πnx− ˆθ(x). (4.9) Employing the Poisson’s summation formula the density operator becomes
ˆ ρ(x) = n− 1 π∂xθ(x)ˆ X m∈Z e2im(πnx−ˆθ(x)). (4.10) 2
Beside sharing the same symbol θ, the field ˆθ(x) has nothing to do with the phase-shift θ(λ) intro-duced in the previous chapters.
In what follows we assume that the low-energy sector is described by a long-wave fluc-tuations of density and we can perform expansion in derivatives of ˆθ(x). We find, using eq. (4.5), that ˆθ(x) and ˆφ(x) are canonically conjugated fields
h ˆ
φ(x), ∂yθ(y)ˆ
i
= iπδ(x− y). (4.11) Having a convenient representation of the density operator, we take its square root and arrive at the field operators. The only technical detail is that the square root of the δ-function is proportional to theδ-function. The proportionality constant, which depends on the regularization of the δ-function, we leave unspecified. Physically speaking the proportionality constant depends on the short-wave physics which in turn is not universal and depends on the microscopic form of interaction. Focusing here on the universal aspects of 1D liquids we do not attempt to fix it.
Ψ†(x)∼ r n− 1 π∂xθ(x)ˆ ∞ X m=−∞ e2im(πnx−ˆθ(x))e−i ˆφ(x), (4.12)
Now let us see what is the structure of the low-energy Hamiltonian. Consider equa-tions of motion for the fieldsθ(x) and φ(x) with dynamics governed by the Lieb-Liniger Hamiltonian. Forθ(x) we easily obtain
i∂θ(x)
∂t = [θ(x), H] = 2nπiφ(x) + . . . , (4.13) where . . . stands for higher order terms in the derivatives. For the field φ(x) the com-mutator is not easy to compute but, in analogy to (4.13), we can guess the leading term to be
i∂φ(x)
∂t = [φ(x), H] = 2απiθ(x) + . . . , (4.14) where α is some constant that cannot be determined3. This approach can be justified
in the logic of the renormalization group: the terms that we kept are the most relevant ones. Construction of the Hamiltonian boils down to inverting equations of motion (4.13) and (4.14). Again the simplest (and containing the most relevant operators) Hamiltonian that we can write is
H = Z
dxn (∂xφ(x))2+ α (∂xθ(x))2
. (4.15) It is useful to rewrite it in a slightly different form by explicitly putting in it the sound velocity vs. To do this we also introduce the Luttinger parameter K, a dimensionless
3
number setting the relative strength of the phase fluctuations to the density fluctuations. We write H = ~vs 2π Z L 0 K (∂xφ(x))2+ 1 K (∂xΘ(x)) 2 dx, K = 2kF vs . (4.16) These are both phenomenological parameters determining the low-energy physics. We see that the Luttinger parameter controls the relative importance of phase fluctuations to density fluctuations with the former one dominating in theK → ∞ limit. Luttinger parameter K can be connected to the value of the interaction parameter c through the expression for the sound velocity (3.133) as
K = ρ2(q). (4.17) Now we are in a position to discuss the correlation functions. The one relevant for us is the density-density correlation function which in the leading order reads
S(x, t) =hρ(x, t)ρ(0, 0)i = = n2+ 1 π2h∂xθ(x, t)∂xθ(0, 0)i + X m6=0 e2imkFxhe2im(−θ(x,t)+θ(0,0))i + . . . , (4.18) where. . . represents terms that decay faster with x or t and therefore are less important at large distances and/or long times.
The Luttinger liquid Hamiltonian (4.16) describes essentially two free bosonic fields in 1 + 1 dimensions4. Therefore it is a conformally invariant theory and one can use the
apparatus of CFT to compute expectations values appearing in (4.18). This can be done quite generally for finite systems at finite temperatures and also for time dependent correlations. However for the brevity of the discussion these somehow standard compu-tations are omitted here. The resulting expressions for the expectation values appearing in (4.18) are as follows [58] h∂xθ(x, t)∂xθ(0, 0)i = − K 4 h d (x + vst|L)−2+|d (x − vst|L)−2 i , (4.19a) he2im(−θ(x,t)+θ(0,0))i = Am 2 n2d (vst + x|L) d (vst− x|L)−m 2K , (4.19b)
4To see this express the Luttinger liquid Hamiltonian (4.16) in terms of new fields φ
±(x) = φ(x)/ √
K± θ(x)√K. The resulting Hamiltonian is a sum of quadratic Hamiltonians for the + and the − fields. It is also easy to see that [φ+(x), φ−(y)] = 0, and thus the new fields are independent of each other.
whered(x|L) is a function defined as
d(x|L) = L sin (πx/L) π ie
−iπx/L. (4.20)
We will refer to d(x|L) as a chord function, however strictly speaking chord function is given by|d(x|L)|. In the thermodynamic limit, L → ∞, we have |d(x|L)| → x.
We note that the second expectation value (4.19b) cannot be determined completely within the Luttinger liquid approach. The constantAm depends on the microscopic
the-ory underlying the hydrodynamic approximation and in order to be determined we need to understand how the microscopic theory (in our case the Lieb-Liniger Hamiltonian) is submerged into the effective description. To this end let us first write down the Lut-tinger liquid prediction for the equal-time correlator in the infinite system. From (4.18) and (4.19) we obtain (4.1) S(x) = n2 1− K 2 (kFx)2 + X m>0 Am cos (2mkFx) (nx)2m2K ! , (4.21) The prefactors Am can be found in the following way [60]. Let us consider the Fourier
series of the chord function to the power −m2K as it appears in (4.19b). We write first d(vst± x|L) =
L 2π
1− e2iπ(vst±x)/L, (4.22)
and use the binomial expansion to arrive at d(vst± x|L)−m 2K = 2π L m2K X nL(R)≥0 C nL(R), m2Ke2πinL(R)(vst±x)/L, (4.23) where C nL(R), µ = Γ nL(R)+ µ Γ nL(R)+ 1Γ (µ). (4.24) NumbersnL(R)describe the number of excitations in the vicinity of the left (right) Fermi edge. Considering the lowest lying state nL(R) = 0, the term proportional to Am in the
correlator (4.21) is given by n2A m 2 2π nL m2K/2 e2imkFx. (4.25)
On the other hand using a resolution of the identity in the density-density correlation function we get
S(x, t) =X
k,ω
ei(kx−ωt)|hk, ω|ρ(0)|Ni|2, (4.26)
where|hk, ω|ρ(0, 0)|Ni|2is the form factor of the density operator between the N-particle ground state and an excited state with the momentum k and the energy ω. Upon comparing with (4.25) we easily recognize anm-th order umklapp state (c.f. 2.1.2) and obtain Am = 2 n2 nL 2π m2K/2 |hN, m|ρ(0)|Ni|2. (4.27) Thus we have computed the large distance asymptote of the density-density correlation function (4.26) with prefactors given by (4.27). This description has however a few weak points. First of all the predictions are only valid in the limitx n−1. Connected with
it there is also a rather trivial form of the correlation function in the momentum space. From the very construction, the momentum space information that is contained in the Luttinger liquid theory is restricted to a linear spectrum around the2mkF points with a
sound velocityvsand a competition between the phase and density fluctuations encoded
in the value of K. This is clearly not enough to determine even the static correlator S(k).
4.2
Nonlinear Luttinger Liquid
The problem we have encountered can be easily understood from the Fig. 4.1 showing S(k, ω) for different interaction strengths. The search for an effective theory that is capable of encompassing the structure ofS(k, ω) turned out to be quite cumbersome. The most natural starting point, that is considering the curvature in the dispersion relation and therefore going beyond the linear approximation of the Luttinger liquid turns out to be rather difficult to control mathematically. Reason being that the additional term lifts up the linear approximation of the spectrum and therefore is not appropriate for a perturbative analysis.
Much better idea is to treat a high energy excitation5 as an impurity in an otherwise usual Luttinger liquid. Here by the impurity we mean a particle which is identical to all the other particles in the gas and thus interacts with the gas as well. The name impurity comes from the assumption that it is a high energy particle and in this sense is
5
0.00 0.25 0.50 0.75 1.00 1.25 1.50 ω[k2 F] 0.0 0.5 1.0 1.5 2.0 2.5 S kF 2 ,ω k− 1 F c→ ∞ c = 4 c→ 0
Figure 4.1: Plot of the fixed momentum cut (k = kF/2) through the density-density correlation function at different interaction strengths. Forc→ 0, ∞ the data is taken from the AppendixC. Thec = 4 case was taken from [38]. As the interaction strength is decreased the initial rectangular shape gets distorted. The singularity develops at the upper threshold ω1(k) while the correlation at the lower threshold ω2(k) vanishes continuously. Further decrease of the interaction strength causes a collapse of the correlation function. In the weekly interacting limit the correlation follows a single particle dispersion line as predicted by Bogolyubov theory (see App. C).
distinguishable from the remaining (low-energy) particles. We can also consider a deep hole (that is a hole far from the Fermi edges) in the Fermi sea as an impurity. This allows us to understand the evolution of the density-density of the correlation function in the vicinity of the excitation thresholdsω1(2)(k).
Following this idea we can almost immediately write down a Hamiltonian for such a theory. Let us write the field operatorΨ(x) as a sum of two terms
Ψ(x) = ΨLL(x) + eikxd(x), (4.28)
where ΨLL(x) describes the low energy part and d(x) is the impurity operator. The
impurity has momentumk and its dispersion relation follows the dispersion of excitation ω1(2)(k) in the full theory. Linearizing the dispersion around k leads to a new velocity
vd= ∂k±(k)/∂k. The effective Hamiltonian reads [61,62]
HnLL = H0+ Hd+ Hint (4.29) Hd= Z L 0 dx d†(x) ((k)− ivd∂x) d(x), (4.30) Hint= Z L 0 dxVR(k)∂x φ(x)− θ(x) − VL(k)∂x φ(x) + θ(x) d†(x)d(x), (4.31)
whereH0 is the usual Luttinger Liquid Hamiltonian (4.16),Hddescribes the kinetics of
the impurity andHint describes the interaction of the impurity with the low energy part
of the gas through the momentum dependent potentialsVR(L)(k).
The whole power of this description comes from the observation that there is an unitary transformation that decouples the impurity from the Luttinger liquid [62,63]
U†(H0+ Hd+ Hint) U = H0+ Hd. (4.32)
It is useful to consider first the case K = 1, the generalization to arbitrary K then follows easily. Let us supplement the fields θ(x), φ(x) and potentials VL(R) at K = 1
with a tilde to distinguish them from a general case. For K = 1 the unitary operator that diagonalizes the Hamiltonian is
UK=1= exp i Z L 0 dx ˜δ+ 2π ˜ φ(x)− ˜θ(x)− ˜δ− 2π ˜ φ(x) + ˜θ(x) ! d†(x)d(x) ! , (4.33) with ˜ δ−(k) =− ˜ VL(k) vd+ vs , δ˜+(k) =− ˜ VR(k) vd− vs . (4.34) Straightforward but quite lengthy computations yield that
UK=1† H˜nLLUK=1= ˜H0+ Hd+ const. (4.35)
In order to generalize to an arbitraryK one just notices that the following transformation of fields bringsHLL with an arbitrary K into the K = 1 form
˜
θ(x) =√Kθ(x), φ(x) =˜ √1
Kφ(x). (4.36) Thus the general transformation diagonalizing HnLL reads
U = exp i Z L 0 dx δ+ 2π(φ(x)− θ(x)) − δ− 2π(φ(x) + θ(x)) d†(x)d(x) , (4.37) with the phase shifts δ± related to the tilted ones through
δ+− δ+= ˜ δ+− ˜δ− √ K, δ++ δ− = ˜ δ++ ˜δ− /√K. (4.38) We could also find the relations between the potentials VL(R) and the phase shifts δ±.
It can be shown using quite general arguments that VL(R) relates to the derivatives of ω1,2(k) [62]. However we should not follow this way and instead employ once again the
directly related to the back-flow function [18,61,64] ˜
δ±(k) = 2πF (±kF|λ, q). (4.39)
This follows from an observation that within linear approximation the impurity scatters only with particles at Fermi rapidity ±q. On the other hand the phase shift acquired through scattering are proportional to the back-flow function [18] what yields (4.39). Finally the last missing piece that we need to know for the computation of the correla-tion funccorrela-tion are transformacorrela-tion rules for the impurity operator. Again straightforward computations yield
U†d(x)U = e−iδ+2π[φ(x)−θ(x)]+i δ−
2π[φ(x)+θ(x)]d(x). (4.40)
Now we are in the position to consider the correlation function. Let us start first with the representation of the density operatorρ(x). The process that we want to describe is a creation of a high momentum particle accompanied by creation of a hole next to the right Fermi point thus we write6
ρ(x) = Ψ†(x)Ψ(x)∼ d†(x)eiφ(x)−iθ(x). (4.41) Then the correlation function in the momentum space reads
S(k, ω)∼ Z
dx Z
dt eiωthei(θ(x,t)−φ(x,t))d(x, t)d†(0, 0)e−i(θ(0,0)−φ(0,0))iHnLL, (4.42)
where the expectation values are calculated with respect to the non linear Luttinger liquid Hamiltonian (4.29). Applying now the unitary transformationU decouples impurity from the Luttinger liquid and the expectation value becomes equal to
hei(θ(x,t)−φ(x,t))d(x, t)d†(0, 0)e−i(θ(0,0)−φ(0,0))iHnLL =hd(x, t)d
†(0, 0)
iHd
×heiµ(θ(x,t)−φ(x,t))e−iν(θ(0,0)−φ(0,0))iH0, (4.43)
where µ = 1 + δ++ δ− 2π , ν = 1− δ+− δ− 2π , (4.44) 6
Factor e−iθ(x)was added for a convenience. From the definition of ρ(x) (4.10) it is easy to show that e−iθ(x) corresponds to the Jordan-Wigner string and thus the projection holds only for the fermionic particles. However for the density-density correlation function the statistics of the particles does not play any role and since calculations with an extra θ(x) field are simpler we use the fermionic projection.
The propagator of the impurity is simple to compute and yields hd(x, t)d†(0, 0)iHd= e −itω1(k)1 L X nd∈Z e2πind(x−vdt)/L. (4.45)
Computation of the second expectation value follows again from the conformal symmetry of the Luttinger liquid Hamiltonian (4.16). Just like before we only give the final answer (note that again the expectation value cannot be fully determined) [58]
heiµ(θ(x,t)−φ(x,t))e−iν(θ(0,0)−φ(0,0))iH0 ∼ [d (vst + x|L)] ˜ µL[d (v st− x|L)]µ˜R, (4.46) where ˜ µL(R)= µ√K/2± ν/(2√K)2. (4.47) Expressing the r.h.s first in terms ofδ±, then using eq. (4.38) in terms of ˜δ± and finally
in terms of the back-flow function through relation (4.39) we obtain µL(R)= µ√K/2± ν/(2√K)2 = F (∓q|λ) + √ K 2 ± 1 2√K !2 , (4.48) Thus the correlation function in the vicinity of the upper edge reads
S(k, ω) = S1(k) L X nd∈Z Z L 0 dx Z ∞ −∞ dt eit(ω−ω1(k))e2πind(x−vdt)/L × [d (vst + x|L)]µ˜L[d (vst− x|L)]µ˜R, (4.49)
where we fixed the proportionality by introducing a momentum dependent function S1(k)7. We will find this function by comparing (4.49) with the Lehmann representation
of the correlation function (2.33) S(k, ω) = 2πL X
λ∈HN
|hλ|ˆρ(0)|Ni|2δ (ω− (Eλ− EGS)) δk,Pλ (4.50)
where the matrix element is between the N -particle ground state and an excited state. Using the expansion of the cord function (4.23), the Fourier transform in (4.49) can be
7The proportionality factor must be momentum dependent because the phase shifts δ
±entering the expectation value A(x, t, µ, ν) as well as the momentum of the impurity depend on the total momentum k.
directly computed S(k, ω) = 2π 2π L µ˜L+˜µR S1(k) X nR,nL≥0 C(nR, µr)C(nL, µL) ×δ ω− ω1(k) + 2πnR L (vs− vd) + 2πnL L (vs+ vd) . (4.51) Upon comparing (4.51) with Lehmann representation (4.50) and focusing on the simplest case ofnL(R)= 0 we obtain S1(k) = L L 2π µ˜L+˜µR |hk; N|ρ(0)|Ni|2, (4.52)
where the matrix element is computed between the N -particle ground state and the excited state of momentumk formed by a single particle excitation above the Fermi see. Thus we have fixed the prefactors of the correlation in the vicinity of the upper edge. Taking the thermodynamic limit of the correlation function (4.49) we obtain a slightly more transparent expression for the correlator. First of all the impurity propagator simplifies tohd(x, t)d†(0, 0)iHd= e
−itω1(k)δ(x− v
dt), second the chord functions become
simple algebraic functions d(x|L) → 1/ix and we easily arrive at (δω = ω − ω1(k))
S(k, ω) = θ(δω) sin (π ˜µL) + θ(−δω) sin (π˜µR) Γ (˜µL+ ˜µR) sin (˜µL+ ˜µR)
2πS1(k) (δω)µ˜L+˜µR−1
(vs+ vd)µ˜L|vs− vd|µ˜R
. (4.53) Similar calculations performed in the vicinity of the lower edge ω2(k) lead to
S(k, ω) = θ(δω) Γ (˜µL+ ˜µR)
2πS2(k) (δω)µ˜L+˜µR−1
(vs+ vd)µ˜L|vs− vd|µ˜R
. (4.54) and the momentum dependent prefactor is connected with the matrix element in the following way S2(k) = L L 2π µ˜L+˜µR |hk; N|ρ(0)|Ni|2, (4.55)
and the matrix element is between the N-particle ground state and an excited state of momentumk formed by a single hole in the Fermi sea.
With this we conclude the introductory part of this chapter. We have shown how the low energy sector of the Lieb-Liniger model can be described using an effective field theory. We have also computed the density-density correlation function. We were able to compute it in two limiting cases. First in a real space we obtained the large distance, long time asymptote of the correlation. Second in a momentum space we computed it in
the vicinity of the excitation thresholds. In both cases the field theoretical description suffers from lack of explicitness. Only the shape of the correlation functions is predicted but not their relative weight. We fix this by matching the prefactor of the correlation function with matrix elements of the Lieb-Liniger model.
In the rest of this chapter we compute the thermodynamic limit of matrix elements of the density operator and then apply them to the prefactors of the correlation function.
4.3
Thermodynamic Limit of the Density Operator
As demonstrated in the previous sections, full knowledge of matrix elements calculated from the microscopic theory is sufficient to obtain the associated prefactor of a correlation function. Thus, we wish to obtain the thermodynamic limits of various form factors between ground state and excited states specified by the field theory. The final expression for the limit of the form factors is expected to have a power law behavior as a function of the system length L, see eqs. (4.27), (4.52) and (4.55). Our task is to separate the power law from the prefactor in the form factor when faced with terms that are badly divergent (O(NN)).
The form factor of the density operator between normalized eigenstates |{µ}Ni and |{λ}Ni is given by eq. (3.57): FFρ {λi}Ni=1,{µi}Ni=1 = h{λi} N i=1|ˆρ(0)|{µi}Ni=1i q h{λi}Ni=1|{λi}Ni=1i q h{µi}Ni=1|{µi}Ni=1i , (4.56) where according to the results of Chapter 3 the matrix element of the density operator equals (3.67) hµ|ρ(0)|λi = (Pµ− Pλ) N Y j=1 Vj+− Vj− YN j,k=1 λj− λk+ ic µj− λk detN(δjk+ Ujk) Vp+− Vp− , (4.57) with Vj±= N Y m=1 µm− λj± ic λm− λj± ic , (4.58a) Ujk = µj− λj Vj+− Vj− K (λj− λk)− K (λj− λp). (4.58b)
The norm of the Bethe state follows formula (3.60). Later we have performed a partial thermodynamic limit and arrived at the following expression (3.98)
h{λj}Nj=1|{λj}Nj=1i = (2πLc)N N Y j=1 ρ(λj) N Y j6=k λj− λk+ ic λj − λk det [−q,q] 1− ˆ K 2π ! . (4.59)
We will frequently use the notion of such partial limits because individual terms in the algebraic expressions for the form factors may not have good thermodynamic limits -they may often diverge in a non polynomial way. However, -they can be regrouped to obtain expressions that have good (finite or power law divergent) limits. Thus we want the final expression for the form factor to be well defined in the limit, on the other hand the intermediate steps leading up to this will be a mixture of thermodynamically well defined quantities and the aforementioned partial thermodynamic limits. We will take partial limits of simple groupings occurring in the form factors in the following sections and later collect them and express the final answer for the form factors.
We expect the final answer to depend on the density of ground state rapiditiesρ(λ) and on the back-flow functionF (λ) specifying the excited state. We found that both quantities fulfill some integral equations (eq. (3.124) and eq. (3.102) respectively). However to successfully account for all constant factors in the thermodynamic limit, we will need to consider finite size corrections (in orders of 1/L) to these functions and collect any corrections that sum to finite values in the thermodynamic limit.
Let us start with the ground state distribution. Recall formula (3.86) for ρt(λ) and
that at the zero temperatureρ(λ) = ρt(λ) for λ ∈ [−q, q] and zero otherwise. Thus for
λ∈ [−q, q] we can write ρ(λ) = 1 2π 1 + 1 L N X i=1 K(λ− λi) ! . (4.60) The summation can be written now as an integral. In order to keep the finite-size corrections we use the Euler-Maclaurin formula and arrive at
ρL(λ) = 1 2π 1 + Z λN −λN K(λ− µ)ρ(µ) dµ + ρ(λN) 2L (K(λ + λN) + K(λ− λN)) − 1 24L2 K 0(λ − λN)− K0(λ + λN) +O(1/L3), (4.61)
where we used that the ground state rapidities are symmetric: λ1 = -λN. The first order
correction can be absorbed in the integral by defining the Fermi rapidityq = λN+2Lρ(λ1 N).
of order1/L3) ρL(λ) = 1 2π 1 + Z q −q K(λ− µ)ρ(µ) dµ − 1 24L2 K 0 (λ− q) − K0(λ + q) . (4.62) In the similar way we want to characterize now the excited state. Let us follow again the notation introduced in section 3.2.3. We characterize the excited state by a set of excited rapidities (particles) {µ+j }nj=1 and holes {λ−j }nj=1. We also define artificial rapidities{λ−j }nj=1 which are shifted with respect to their counterparts{λ−j}nj=1by order 1/L. We also use the prime and double prime notation for sums (see. section3.2.3). In what follows we denote the back-flow function by F (λ) keeping in mind that in fact it also depends on particles and holes rapidities.
Starting from the Bethe equation for the particle excitation we proceed as follows: Lµ+ = 2πI+− N X k=1 θ(µ+− µk), Lµ+ = 2πI+−X0θ(µ+− µ k)− θ(µ+− µ−) + θ(µ+− µ−), Lµ+ = 2πI+− N X k=1 (θ(µ+− λk)− K(µ+− λk)(µk− λk)) + θ(µ+− λ−) − K(µ+− λ− )(µ−− λ−) + O(1/L2), µ+ = 2πI + L − N X k=1 θ(µ +− λ k) + K(µ+− λk)LρF (λL(λ)k) L +θ(µ+− λ−) L + O(1/L 2), µ+ = 2πI + L − Z q −q dλ θ(µ+− λ)ρL(λ) + 1 L − Z q −q dλ K(µ+− λ)F (λ) + θ(µ+− λ− ) + O(1/L2). (4.63) Note that in the above derivation we have used the fact that we expectµk− λk, µ−− λ−
to be O(1/L), and the relation, (λj − µj) = Lρ(λF (λjj)) + O(1/L2). Using the last step
of the above as our starting point, we may now take the thermodynamic limit. The prescription we use is to keep n+/L constant as we send N, L → ∞. Furthermore
let us separate the O(1) and O(1/L) contributions to µ+ for the sake of clarity, i.e.
µ+= µ+ 0 + µ + 1/L+ O(1/L2). We obtain µ+0 + µ+1/L = 2πI + L − Z q −q dλ θ(µ+0 − λ)ρ(λ) − µ+1/L Z q −q dλ K(µ+0 − λ)ρ(λ). + 1 L − Z q −q dλ K(µ+0 − λ)F (λ) + θ(µ+ 0 − λ − ) + O(1/L2), (4.64)
where on the right hand side we have substitutedρ for ρL, etc. because the lowest order
difference between such terms is higher order in1/L than we are keeping. Thus we obtain as the thermodynamic limit and the first order correction
µ+0 = 2πI + L − Z q −q dλ θ(µ+0 − λ)ρ(λ), (4.65a) µ+1/L= 1 L − Rq −qdλK(µ+− λ)F (λ) − θ(µ+− λ−) 1 +R−qq dλ K(µ+0 − λ)ρ(λ) ! =−F (µ + 0) Lρ(µ+0). (4.65b) In the last step above, we have used eqs. (3.124), (3.102).
Similarly, let us consider the Bethe equation for the ground state rapidity, λ−, λ−= 2π LI − −L1 N X k=1 θ(λ−− λk). = 2πI − L − Z q −q dλ θ(λ−− λ)ρ(λ) + O(1/L2). (4.66) We wish to now obtain µ−. Since we expect µ− = λ−+ O(1/L), and the lowest order
correction comes from the usual shift in the excited state rapidities, we can quantify this exactly as µ−0 = 2πI − L − Z q −q dλ θ(µ−0 − λ)ρ(λ), (4.67a) µ−1/L=−F (µ − 0) Lρ(µ−0). (4.67b) We are now ready to obtain the lowest order finite size corrections to the back-flow function. We may obtain an equation for FL by subtracting the Bethe equations forµj
and λj for µj not in {µ+, µ−}:
0 = L(λj− µj) + N X k=1 (θ(λj − λk)− θ(µj− µk)) = L(λj− µj) + N X k=1 0 0 (θ(λj− λk)− θ(µj − µk)) + θ(µj − µ−)− θ(µj− µ+). (4.68)
The summation can be simplified by expanding the second term aroundλj− λk N X k=1 0 0 (θ(λj− λk)− θ(µj− µk)) =− N X k=1 0 0 (K(λj− λk) (µj − µk− λj+ λk) + K0(λj− λk) (µj− µk− λj+ λk)2 +O(1/L2). (4.69)
Also terms containingµ± can be expanded using eqs. (4.65) and (4.67) θ µj− µ±= θ λj − µ±+ µj − λj = θ λj − µ±0 + K λj− µ±0 µ±1/L+ µj− λj +O(1/L2) = θ λj − µ±0 − 1 LK λj− µ ± 0 F µ±0 ρ µ±0 + F (λj) ρ (λj) ! (4.70) Thus, after rearranging and expressing sums as integrals, eq. (4.68) can be written as
FL(λ) = θ(λ− µ+0)− θ(λ − µ−0) 2π + 1 2π Z q −q dµ K(λ, µ)FL(µ) − 1 2πL K(λ, µ+0) FL(λ) ρL(λ) − FL(µ+0) ρL(µ+0) − K(λ, µ−0) FL(λ) ρL(λ) − FL(µ−0) ρL(µ−0) + 1 4πL Z q −q dµ ρL(µ)K0(λ− µ) FL(λ) ρL(λ) − FL(µ) ρL(µ) 2 + O(1/L2), (4.71) where we defined FL(λ) = 2πL (λj− µj) ρL(λj). (4.72)
Notice that in the thermodynamic limit this definition coincides with the earlier one (3.100).
It is also convenient to define an analog of the ground state momentum density function for the excited state. This will help in calculating the normalization of the excited state that appears in (3.67). Before proceeding, let us write down the integral equations for the derivatives of ρL, FL as these will be used in the simplifications to follow.
2πFL0(λ) = Z q −q dµ K0(λ− µ)FL(µ) + K(λj− µ+0)− K(λj − µ − 0) + O(1/L), (4.73a) 2πρ0L(λ) = Z q −q dµ K0(λ− µ)ρL(µ) + O(1/L2). (4.73b)
To obtain the density function for the excited state, we start with the Bethe equation for a generic rapidity from the excited state (3.26) and differentiate once with respect to
µj: L = 2πLρex,L(µj)− N X k=1 0 K(µj− µk)− K(µj − µ−) + K(µj− µ−)− K(µj− µ+) L = 2πLρex,L(µj)− N X k=1 0 0 K(µj − µk)− K(µj− µ+) + K(µj− µ−) L = 2πLρex,L(µj)− N X k=1 K(λj− λk)− K0(λj− λk) FL(λj) LρL(λj) − FL(λk) LρL(λk) − K(λj− µ+0) + K(λj − µ − 0) + O(1/L). (4.74)
The first two steps of the above set of equations are exact. From the third step on we retain only terms up to O(1/L). In going to the third step we have expanded all terms dependent onµj in terms of λj as this is what is useful for us in our final expression (see
the following sections). Expressing the sum as an integral yields 2πρex(µj) = 1− K(λ− µ+ 0)− K(λ − µ − 0) L + Z q −q dν K(λ− ν)ρL(ν)− 1 LK 0 (λ− ν) ρL(ν) FL(λ) ρL(λ) − FL (ν) + O(1/L2), (4.75) and using integral equations for ρL(λ) (4.62),ρ0(λ) (4.73b) andF0(λ) (4.73a) we obtain
ρex,L(µj)− ρL(λj) = 1 L FL0(λj)− FL(λj)ρ0L(λj) ρL(λj) + O(1/L2). (4.76) Thus we have characterized ground state and excited states above it by the thermody-namic functions including the leading order corrections in the system size. We have now enough information to take the thermodynamic limit of the form factor. We perform this rather lengthy calculations in a number of steps. The first step consists of rewriting the form factor as a product of a few terms, each having a proper thermodynamic limit. We will take then the thermodynamic limit of each of these terms separately and in the end combine them into a complete expression.
Let us consider first the term Vj+/Vj− withVj± as defined in (4.58a) log V + j Vj− ! = n X i=1 log µ+i − λj+ ic µ+i − λj− ic − n X i=1 log λ−i − λj+ ic λ−i − λj− ic + N X m=1 log 1− FL(λm) LρL(λm)(λm− λj+ ic) − log 1− FL(λm) LρL(λm)(λm− λj− ic) − n X i=1 log µ−i − λj + ic λ−i − λj+ ic + n X i=1 log µ−i − λj− ic λ−i − λj − ic = n X i=1 log µ+i − λj+ ic µ+i − λj− ic − n X i=1 log µ−i − λj+ ic µ−i − λj− ic + N X m=1 i 2cFL(λm) LρL(λm)((λm− λj)2+ c2) + i 2 4c(λm− λj)FL2(λm) L2ρ2 L(λm)((λm− λj)2+ c2)2 +O(1/L2) = n X i=1
ihθ(λj − µ+i,0)− K(λj− µ+i,0)µ+i,1/L− θ(λj− µ−i,0) + K(λj − µ−i,0)µ−i,1/L
i + N X m=1 iFL(λm)K(λm− λj) LρL(λm) − N X m=1 i 2 K0(λm− λj)FL2(λm) L2ρ2 L(λm) + O(1/L2) = 2πiFL(λj) + i L K(λj− µ+0) FL(λj) ρL(λj) − K(λj− µ − 0) FL(λj) ρL(λj) − i 2L Z q −q dµK0(λj− µ)ρL(µ) FL(λ) ρL(λ) − FL(µ) ρL(µ) 2 + i 2L Z q −q dµF 2 L(µ) ρL(µ) K0(λj− µ) + O(1/L2) = 2πiFL(λj) + i L K(λj− µ+0) FL(λj) ρL(λj) − K(λj− µ − 0) FL(λj) ρL(λj) + i L Z q −q dµK0(λj− µ) FL(λj) ρL(λj) FL(µ)− FL(λj)ρL(µ) 2ρL(λj) + O(1/L2). (4.77) Now using the relations given by (4.73a) we have
K(λj− µ+0)− K(λj− µ − 0) + Z q −q dµ K0(λj− µ) FL(µ)− FL(λj)ρL(µ) 2ρL(λj) = 2π FL0(λj)− FL(λj)ρ0L(λj) 2ρL(λj) . (4.78) Substituting in (4.77) we obtain log V + j Vj− ! = 2πiFL(λj) 1 + 1 LρL(λj) FL0(λj)− FL(λj)ρ0L(λj) 2ρL(λj) + O(1/L2). (4.79)
Thus we may handle the following term appearing in the form factor (4.57), as follows: N Y j (Vj+− Vj−) = N Y j Vj+ 1− exp −2πiFL(λj) 1 + 1 LρL(λj) FL0(λj)− FL(λj)ρ0L(λj) 2ρL(λj) + O 1 L = N Y j Vj+eiφ(j)2 sin πFL(λj) 1 + 1 LρL(λj) FL0(λj)− FL(λj)ρ0L(λj) 2ρL(λj) + O(1/L), (4.80) where in the above expression the termφ(j) is purely a phase. Such terms will drop out in the final answer because we only need the absolute square of the form factor. The product of sine terms can be expanded in the following way
V = N Y j=1 sin πFL(λj) 1 + 1 LρL(λj) FL0(λj)− FL(λj)ρ0L(λj) 2ρL(λj) = N Y j=1 sin[πFL(λj)] exp N X j πFL(λj)cos(πFL(λj)) LρL(λj)sin(πFL(λj)) FL0(λj)− FL(λj)ρ0L(λj) 2ρL(λj) = N Y j=1 sin[πFL(λj)] exp π Z q −q dλ FL(λ) cot(πFL(λ)) FL0(λ)−FL(λ)ρ 0 L(λ) 2ρL(λ) . (4.81) The other terms inhµ|ρ(0)|λi may also be regrouped - let us write Θ to denote det(δjk+Ujk)
Vp+−Vp− , we have hµ|ρ(0)|λi = (Pµ− Pλ) Θ× V N Y j=1 2ieiφ(j)Vj+ N Y j,k λjk + ic µj− λk = (Pµ− Pλ) Θ× V N Y j=1 eiφ(j)+3iπ/2 2V + j λj− µj ! N Y j,k (λjk+ ic) N Y j6=k 1 µj− λk = (Pµ− Pλ) Θ× V N Y j=1 eiφ0(j)2LρL(λj)V + j FL(λj) ! N Y j,k (λjk+ ic) N Y j6=k 1 µj− λk × n Y i=1 FL(λ−i ) LρL(λ−i )(λ−i − µ+i ) . (4.82)
With this form forhµ|ρ(0)|λi we may combine it with the norm terms : |FFρ| = i (Pµ− Pλ) Θ× V N Y j=1 eiφ0(j)Vj+2LρL(λj) FL(λj) N Y j,k (λjk + ic) N Y j6=k 1 µj− λk × n Y i=1 FL(λ−i ) LρL(λ−i )(λ−i − µ+i ) YN j6=k λ2 jkµ2jk
(λjk+ ic)(λjk− ic)(µjk+ ic)(µjk − ic)
!1/4 × N Y j=1 1 (2πLcρL(λj)) 1 +ρex(µj)− ρL(λj) ρL(λj) −1/2 × det 1− 1 2πK −1 = i (Pµ− Pλ) Θ× V N Y j=1 Vj+ 1 πFL(λj) N Y j,k (λjk+ ic) N Y j6=k 1 µj − λk × N Y j6=k λ1/2jk µ1/2jk
(λjk+ ic)1/4(λjk− ic)1/4(µjk+ ic)1/4(µjk− ic)1/4
det 1− 1 2πK −1 × n Y i=1 FL(λ−i ) L(ρL(λ−i )ρex(µi+))1/2(λ−i − µ+i ) × N Y j=1 1 + F 0 L(λj)−FL(λj)ρ 0 L(λj) ρL(λj) LρL(λj) −1/2 = i (Pµ− Pλ) Θ× V N Y j=1 1 πFL(λj) N Y j6=k λ1/2jk µ1/2jk µj − λk n Y i=1 FL(λ−i ) L(ρL(λ−i )ρex(µi+))1/2(λ−i − µ+i ) × N Y j,k (µk− λj+ ic) (λjk+ ic)1/2(µjk+ ic)1/2 det 1− 1 2πK −1 × N Y j,k (λjk+ ic)1/4(µjk+ ic)1/4 (λjk− ic)1/4(µjk− ic)1/4 × exp −1 2 Z q −q dλ FL0(λ)−FL(λ)ρ 0 L(λ) ρL(λ) . (4.83)
In going to the last line we have collected a purely phase term in the last product. After substituting the expression forV and regrouping terms we obtain the final form |FFρ| = Y j,k (λj− µk+ ic)(λj− µk+ ic) (λjk + ic)(µjk+ ic) 1/2 × Y j sin(πFL(λj)) πFL(λj) Y j6=k λjkµjk (µj− λk)2 1/2 × n Y i=1 FL(λ−i ) L(ρL(λ−i )ρex(µ+i ))1/2(λ − i − µ+i ) i (Pµ− Pλ) det(δjk+ Ujk) (Vp+− Vp−) det 1− Kˆ 2π × exp Z q −q dλ πFL(λ) cot(πFL(λ)) FL0(λ)−FL(λ)ρ 0 L(λ) 2ρL(λ) −12 FL0(λ)−FL(λ)ρ 0 L(λ) ρL(λ) . (4.84)
It is more convenient to separately evaluate the thermodynamic limits of specific groups of terms appearing in the expressions eqs. (4.84). We begin with the terms in the square brackets.
4.3.1 Evaluation of M1
We will denote byM1 the first group of terms appearing in square brackets in Eq. (4.84)
M1= N Y j,k=1 (λj− µk+ ic)(λj − µk+ ic) (λjk+ ic)(µjk + ic) 1/2 . (4.85) We expect the first group of terms to be finite in the thermodynamic limit because there are no obvious singularities in the individual terms. We start by breaking the term into four parts as follows
M1 = p1× p2× p3× p4, (4.86) where p2 1 = Y j,k 0 0λj− µk+ ic λjk+ ic , (4.87a) p22 = Y j,k 0 0λj− µk− ic µjk − ic , (4.87b) p3 = n Y h=1 Y j (λj− µ+h + ic)(λj− µ+h + ic) (λj− λ−h + ic)(µj− µ+h + ic) , (4.87c) p4 = n Y h1,h2 (λ−h 1− µ + h2 + ic)(λ − h1 − µ + h2 + ic) (λ−h 1− λ − h2+ ic)(µ + h1 − µ + h2 + ic) !1/2 , (4.87d) and the prime notation is the same as used earlier. Note that because we are only interested in the absolute value when evaluating the termM1, we can freely replace terms
by their complex conjugate in order to avoid generating unbounded imaginary parts in the various terms, see e.g. p2
2. The n quasimomenta corresponding to excitations are
treated separately and are indexed by h.
We start with the first two pairs of numerator and denominator terms p21= Y j,k 0 0λj− µk+ ic λjk+ ic = Y j,k 0 0λjk+ FL(λk) LρL(λk)+ ic λjk + ic = Y j,k 0 0 1 + FL(λk) LρL(λk)(λjk + ic) . (4.88)
Similarly, p2 2 = Y j,k 0 0λj− µk− ic µjk− ic = Y j,k 0 0 1 − FL(λj) Lρ(λj)(λjk− ic + FLρ(λL(λkk))) −1 = Y j,k 0 0 1 + FL(λk) Lρ(λk)(λjk+ ic−FLρ(λL(λjj))) −1 = Y j,k 0 0 1 + FL(λk) Lρ(λk)(λjk + ic) + FL(λj)FL(λk) L2ρ(λ j)ρ(λk)(λjk + ic)2 −1 . (4.89) To evaluate the products above we do a logarithmic expansion:
log(p1p2) = X j,k log(p1) + log(p2) =− 1 2 Z q −q dλ Z q −q dµ F (λ)F (µ) (λ− µ + ic)2 + O(1/L). (4.90)
Similarly, the termp3 can be reduced to the following form
p3 = Y j n Y h=1 µj− λ−h + ic λj− λ−h + ic µj− µ+h + ic λj − µ+h + ic −1 = Y j n Y h=1 1 + FL(λj) LρL(λj) µ+h − λ−h (λj − λ−h + ic)(λj− µ+h + ic) + O 1 L2 = exp " n X h=1 Z q −q dλ F (λ) µ + h − λ − h (λ− λ−h + ic)(λ− µ+h + ic) # + O 1 L . (4.91) Thus, the overall factorM1 takes the following form
M1= exp −1 2 Z q −q dµ Z q −q dλ F (λ)F (µ) (λ− µ + ic)2 Yn h1,h2 " (µ−h 1 − µ + h2 + ic) 2 (µ−h 1,h2 + ic)(µ + h1,h2+ ic) #1/2 × exp " n X h=1 Z q −q dλ F (λ)(µ + h − µ − h) (λ− µ−h + ic)(λ− µ+h + ic) # . (4.92) 4.3.2 Obtaining term M2
Here we will focus on the term M2 = Y j sin(πFL(λj)) πFL(λj) Y j6=k λjkµkj (µk− λj)2 1/2 . (4.93) This term is expected to contain a power law divergence and a prefactor, and appears in all form factors. We require a consistent and physically clear method of extracting the divergence and all contributions to the prefactor which are finite in the thermodynamic
limit. The intuition we will use to do this comes from the fact that we know the final answer is expected to scale as a power law ofL. In the thermodynamic limit, the infor-mation about L is contained in the quantization of λ, µ. Consequently it is reasonable to expect that terms which collect to give this power law divergence must appear as a difference in λj− λk whenλj andλk are near one another. Therefore it is important to
isolate such terms on the basis of the nearness of theλ0s. Thus we propose to use a cutoff in order to control the nearness of these parameters and perform controlled expansions to extract the divergence and the prefactor. The cutoff will drop out of the final answer. This is made more precise below.
We will consider first a single particle-hole pair, with quasimomenta,µ−, µ+and indices,
i−, i+, and lastly generalize the calculation to include multiple excitations.
It is easier to calculate the thermodynamic limit of M2 after regrouping terms in a
convenient way. Our procedure will be the following.
First we rewrite the terms in round brackets in (4.93) as T0 as shown below T0 =Y j6=k 0 λjkµkj (µk− λj)2 1/2 = Y j6=k 0λj− µk λj− λk −2 µj − µk λj − λk 1/2 =Y j6=k 0 1 + FL(λk) LρL(λk)(λj− λk) −1/2 1 + FL(λk) LρL(λk)(λj− λk)− FL(λj) LρL(λj)(λj − λk) 1/2 × 1− FL(λj) LρL(λj)(λj− λk) −1/2 . (4.94)
The prime notation is the same as used earlier - i.e. we leave off the quasimomenta corresponding to particles and holes.
It is more convenient to evaluate a “temporarily incorrect" term, which we will call T00, where we include µ− instead ofµ+ and then correct the mistake with a separate term.
Thus to obtain the correct T0, we will calculate a correction term T
hole that must be
multiplied toT00 in order to remove the extra terms:
The terms that we need to remove are exactly those in which the difference, µ−k − λ−k
appears. Consequently, we obtainThole in the following way:
Thole= Y j6=i− 1 + FL(λ −) LρL(λ−)(λj − λ−) Y j6=i− 1 + FL(µ −) LρL(λ−)(λj− λ−) − FL(λj) LρL(λj)(λj − λ−) −1/2 × Y k6=i− 1 + FL(λk) Lρ(λk)(λ−− λk) − FL(λ−) LρL(λ−)(λ−− λk) −1/2 = Y j6=i− 1 + FL(λ −) LρL(λ−)(λj − λ−) Y j6=i− 1 + FL(λ −) LρL(λ−)(λj− λ−) − FL(λj) LρL(λj)(λj − λ−) −1 = Y j6=i− 1 − FL(λj) LρL(λj)(λj− λ−) 1 + FL(λ−) LρL(λ−)(λj−λ−) −1 = Y j6=i− 1 − FL(λj) LρL(λj)(λj− λ−) +FL(λ −)ρ L(λj) ρL(λ−) −1 . (4.96)
In the second line, we have relabeled index k as j to obtain the third line in (4.96). Finally, the terms that were removed must be replaced by the correct terms. Let us call the product of these correct termsTparticle. The way to obtain this term is similar to the
process used to remove the incorrect terms as in (4.96). We get
Tparticle = Y j6=i− 1 +λ −− µ+ λj− λ− −1 Y j6=i− 1 +λ −− µ+ λj − λ− − FL(λj) LρL(λj)(λj− λ−) 1/2 × Y k6=i− 1 + FL(λk) Lρ(λk)(λ−− λk) − λ−− µ+ λ−− λ k 1/2 = Y j6=i− 1 − FL(λj) ρL(λj)(λj− λ−) 1 +λλ−−µ+ j−λ− = Y j6=i− 1− FL(λj) LρL(λj)(λj− µ+) . (4.97)
Thus, the final answer will be given by
We start with the expression T00=Y j6=k 0 0 λjkµkj (µk− λj)2 1/2 = Y j6=k 0 0λj− µk λj − λk −2 µj− µk λj− λk 1/2 =Y j6=k 0 0 1 + FL(λk) LρL(λk)(λj− λk) −1/2 1 + FL(λk) LρL(λk)(λj− λk) − FL(λj) LρL(λj)(λj− λk) 1/2 × 1− FL(λj) LρL(λj)(λj− λk) −1/2 . (4.99)
At this stage there are two levels of approximations that can be made. One, we can express differences in λ using
j− k = L (x(λj)− x(λk)) , j− k = L x(λj)− x(λj)− x0(λj)(λk− λj)− x00(λj)(λk− λj)2/2+ O(λ3jk), j− k = LρL(λj)(λj− λk)− LρL0 (λj)(λj− λk)2/2 + O(λ3jk), LρL(λj− λk) = (j− k) + ρ0L(λj) 2LρL(λj)2 (j− k)2+ O(1/L2). (4.100)
Two, the products in (4.99) above can be exponentiated, and the resulting logarithms expanded order by order in1/L.
Both these procedures have a region of validity. For instance, it is incorrect to perform the logarithmic expansion whenj and k are near one another. On the other hand, when j and k are far away, the approximation in (4.100) breaks down. We calculate T00 by combining these two approximations. The idea behind this is to distinguish two regions of the range of λj, λk using a cutoff - a region when they are near one another, and a
region when they are far away. Then the two approximations outlined above can be used in the region in which they are valid.
This is illustrated in Fig. 4.2. The blue region I, corresponds to the case when j and k are far enough from each other for the logarithmic expansion to become valid. The red region II corresponds to when j and k are near enough for (4.100) to become valid. The third, yellow region III will be treated more carefully since (4.100) is valid here, but proximity to the quasi-Fermi points plays a role in how terms are treated in this region. Moreover, the denotations a and b are used to differentiate the region where j > k and j < k, respectively, to make the calculation more convenient.
The separation of the regions described above is not arbitrary. To do this we introduce a cutoff in the following way - in region I, we only allow the indexk to get within n∗ 1
of the index j for all products/sums. In the figure this corresponds to the lines at the interface of the red and blue regions. Since we will be working in the continuum limit in region I, we would like to know what this condition about indices means for the quasi momenta. Since n∗ is at the crossover where the approximation (4.100) just starts to break down, we will use this approximation to define theν∗ corresponding ton∗
ν∗(λ) =|λj− λk| ≈ j− j ± n∗ LρL(λj = λ) = n∗ LρL(λ) . (4.101) Note that we have a choice of either keeping n∗ or ν∗ constant - fixing one parameter endows the other with a dependence on its position in the range [-q,q] or [1,N] respectively. Our particular choice keepsn∗ fixed, and makesν∗(λ) dependent on λ.
Using the two approximations in the relevant regions should produce, first, a (principal value) integral which captures the contribution to the prefactor from the “far-away” region (I a,b in Fig.4.2). Second, a constant from the entire region whereλj is near λk,
but when they are not both at the same quasi-Fermi point (II a,b in Fig. 4.2). Lastly, a constant and a power law divergence from the region where bothλ are near the same quasi-Fermi point (III a,b in Fig. 4.2). While the intermediate steps may produce cutoff dependent terms, these terms should mutually cancel. The final answer will be shown to be independent of the cutoff parameter, n∗ 1.
Let us denote the value of T00 in region I as T
I. To treat the product in region I we
may expand the products in T00 logarithmically in orders of 1/L. Only terms of order 1/L2 and higher survive after the double products are evaluated. Without any additional
approximation, we obtain the following term TI= Y j6=k 1 + FL(λk) LρL(λk)(λj− λk) −1/2 1 + FL(λk) LρL(λk)(λj− λk) − FL(λj) LρL(λj)(λj− λk) 1/2 × 1− FL(λj) LρL(λj)(λj− λk) −1/2 ≈ exp 1 2 X j6=k FL(λj)FL(λk) L2ρ L(λj)ρL(λk)(λj− λk)2 . (4.102)
Figure 4.2: Schematic of range of λ’s. Cutoff parameter n∗is used to separate regions I, II and III. In region I the indicesj, k are far enough to perform logarithmic expansions as in, for example, (4.102). In region II and III, the approximation given by (4.100) is valid, but the logarithmic expansion fails. Consequently products must be carefully and explicitly evaluated discretely. In region III proximity to the quasi-Fermi point demands greater care in evaluating products as demonstrated in (4.117), for example.
With the definition of the cutoff in λ, (4.101), we may write log(TI) = 1 2 X j6=k FL(λj)FL(λk) L2ρ L(λj)ρL(λk)(λj− λk)2 = 1 2 Z q −q+ν∗(−q) dλ Z λ−ν∗(λ) −q dµFL(λ)FL(µ) (λ− µ)2 +1 2 Z q−ν∗(q) −q dλ Z q λ+ν∗(λ) dµFL(λ)FL(µ) (λ− µ)2 = 1 2 Z q −q+ν∗(−q) F2 L(λ)dλ Z λ−ν∗(λ) −q dµ 1 λ− µ 2 +1 2 Z q−ν∗(q) −q FL2(λ) dλ Z q λ+ν∗(λ) dµ 1 λ− µ 2 −1 4 Z q −q+ν∗(−q) dλ Z λ−ν∗(λ) −q dµ FL(λ)− FL(µ) λ− µ 2 −14 Z q−ν∗(q) −q dλ Z q λ+ν∗(λ) dµ FL(λ)− FL(µ) λ− µ 2 . (4.103) In the last step we have isolated a straightforwardly integrable (lines 3 and 4) and a cutoff dependent part (lines 1 and 2).
Let us first analyze the term 1 2 Z q −q+ν∗(−q) F2 L(λ) dλ Z λ−ν∗(λ) −q dµ 1 (λ− µ)2 = = 1 2 Z 1 −1+ν∗(−q)/q dx F2 L(qx) 1 ν∗(qx)/q − 1 x + 1 =−1 2F 2 L(−q) log(qL) − 1 2 " FL2(−q) log ρL(q) n∗ − Z q −q+ν∗(q) dλF 2 L(λ) ν∗(λ) # −1 2P− Z 1 −1 dxF 2 L(qx) x + 1 , (4.104)
where we introduced the generalized principal value integrals denoted by P−, P+, P±,
defined in Appendix B. Note that such integrals can only be well defined when the integrand is made dimensionless. This is achieved by mapping the range of integration from(−q, q) → (−1, 1).
Similarly we may obtain the second cutoff dependent term of (4.103). The combined answer in region I becomes
TI = ν∗(q) q 1 2(FL2(q)+FL2(−q)) × exp ( P± Z 1 −1 dxF 2 L(qx) x2− 12 − 1 4 Z q −q dλ Z q −q dµ FL(λ)− FL(µ) λ− µ 2) × exp ( L n∗ Z q−ν∗(q) −q dλ ρL(λ)FL2(λ) + L n∗ Z q −q+ν∗(−q) dλ ρL(λ)FL2(λ) ) . (4.105) In Region II we know that we can no longer expand the log of the products like in the previous section because here all terms are O(1) or bigger. We will consider the two sets of products in this region, products in IIa, and, IIb, separately. In both these products we can approximateλj− λk as in (4.100). The second thing is to treat terms
likeFL(λj)− FL(λk) with the same precision with which we treat differences in λ in this
region. Thus we need to evaluate the difference in FL to O(1/L) as follows
FL(λj) ρL(λj) − FL(λk) ρL(λk) = d dλ FL(λ) ρL(λ) |λ=λj(λj− λk). (4.106)
Let us define the following notation, Fj = FL(λj), Cja =
F
L(λj) ρL(λj)
0
, where the prime denotes that the derivative has been taken with respect to λ and evaluated at λ = λj.
Similarly, defineCjb =− ρ0L(λj) 2ρL(λj)2.
Thus we have from (4.99) TIIa= N Y j=n∗+1 j−1 Y k=j−n∗ 1 + FL(λk) LρL(λk) λjk −1/2 1 + FL(λk) LρL(λk) λjk − FL(λj) LρL(λj) λjk 1/2 1 − FL(λj) LρL(λj) λjk −1/2 = N Y j=n∗+1 j−1 Y k=j−n∗ (LρL(λj)λjk)1/2(LρL(λj)λjk(1− Cja/L))1/2 (LρL(λj)λjk− Fj)1/2(LρL(λj)λjk(1− Cja/L) + Fj)1/2 = N Y j=n∗+1 j−1 Y k=j−n∗ (j− k)(1 + 1 LCjb(j− k))(1 + L1Cjb(j− k))−1(1− Cja/L)1/2 (j− k − Fj 1+L1Cb j(j−k) )1/2(j− k + Fj (1+L1Cb j(j−k))(1−Caj/L) )1/2(1− Ca j/L)1/2 = N Y j=n∗+1 j−1 Y k=j−n∗ (j− k) (j− k − Fj 1+L1Cb j(j−k) )1/2(j− k + Fj (1+L1Cb j(j−k))(1−Caj/L) )1/2 = N Y j=n∗+1 j−1 Y k=j−n∗ (j− k)(1 + 1 LFjCjb)−1/2(1−L1FjCjb)−1/2 (j− k − Fj 1+1 LFjC b j )1/2(j− k + Fj 1−1 LC a j−L1FjC b j )1/2 = N Y j=n∗+1 Γ(n∗+ 1)Γ(1− Fj+L1Fj2Cjb)1/2Γ(1 + Fj+L1(Fj2Cjb+ FjCja))1/2 Γ(n∗+ 1− F j+L1Fj2Cjb)1/2Γ(n∗+ 1 + Fj+L1(Fj2Cjb+ FjCja))1/2 . (4.107) In the above expressions, onlyO(1/L) terms will survive and consequently terms of higher order in1/L have been dropped in the intermediate steps. Similarly for the region IIb
TIIb = N −n∗ Y j=1 j+n∗ Y k=j+1 (j− k)(1 + 1 LFjCjb)−1/2(1−L1FjCjb)−1/2 (j− k − Fj 1+1 LFjC b j )1/2(j− k + Fj 1−1 LC a j−L1FjC b j )1/2 = N −n∗ Y j=1 Γ(n∗+ 1)Γ(1 + F j−L1Fj2Cjb)1/2Γ(1− Fj− L1(Fj2Cjb+ FjCja))1/2 Γ(n∗+ 1 + F j−L1Fj2Cjb)1/2Γ(n∗+ 1− Fj−L1(Fj2Cjb+ FjCja))1/2 . (4.108) We may expand the log of the cutoff dependent terms in then∗ 1 limit. This leaves the following terms
log(Tcutof f) =− N −n∗ X j=1 F2 j 2n∗ − N X j=n∗ F2 j 2n∗ +O 1 n∗ =−L n∗ Z q−ν∗(q) −q dλFL2(λ)ρL(λ) + Z q −q+ν∗(−q) dλFL2(λ)ρL(λ) ! +O 1 n∗ . (4.109) The first two terms exactly cancel the cutoff dependent term from region I in (4.105). The last term can be ignored with the precision to which we are considering expansions
in the parameter n∗.
Now we focus on the terms that are not cutoff dependent. First we note that all the Cb j
terms drop out to O(1/L). We would like to treat the Cja terms. Using the notation, ψ(x) = dxd log(Γ(x)), we expand the terms to precision O(1/L) as follows
Γ 1± Fj 1− Cj/L = exp log(Γ(1± Fj 1− Cj/L )) ≈ exp [log(Γ(1 ± Fj± FjCj/L))] ≈ exp [log(Γ(1 ± Fj))± ψ(1 ± Fj)FjCj/L] = Γ(1± FL(λj)) exp ±ψ(1 ± FL(λj)) FL(λj) L FL(λj) ρL(λj) 0 . (4.110) We will need to evaluate the product over j for such terms. This can be done by first noting as a consequence of the chain rule
∂ ∂xlog(Γ(x)) = ∂ ∂λlog(Γ(x(λ)))× 1 ∂x(λ) ∂λ . (4.111) We will evaluate the product of such terms as follows
N −n∗ Y j=n∗ exp CjFj L ∂ ∂xlog(Γ(x))|x=1+FL(λj)− ∂ ∂xlog(Γ(x))|x=1−FL(λj) = exp X j CjFj LFL0(λj) ∂ ∂λlog [Γ(1− FL(λj))Γ(1 + FL(λj))] = exp X j CjFj LF0 L(λj) ∂ ∂λ log πFL(λj) sin(πFL(λj)) = exp "Z q−ν∗ −q+ν∗ dλ ρL(λ) FL(λ) F0 L(λ) ∂ ∂λ FL(λ) ρL(λ) ∂ ∂λ log πFL(λ) sin(πFL(λ)) # = exp Z q −q dλ ρL(λ) FL(λ) FL0(λ) ∂ ∂λ FL(λ) ρL(λ) ∂ ∂λ log πFL(λ) sin(πFL(λ)) . (4.112) Note that we have let ν∗ go to zero. In so doing, we have added in some terms that occur at the edges, i.e. form 1 to n∗ and from N − n∗ to N . However, these terms
only make an order 1/L contribution and can be safely treated. Moreover, we treat the Γ(1 + Fj)Γ(1− Fj) terms by extending the product from 1 to N and dividing by the
“extra terms” N −n∗ Y j=1 (Γ(1 + Fj)Γ(1− Fj))1/2 N Y j=n∗+1 (Γ(1 + Fj)Γ(1− Fj))1/2 = N Y j=1 πFL(λj) sin(πFL(λj)) sin(πFL(q)) sin(πFL(−q)) π2F L(−q)FL(q) n∗/2 , (4.113) where, in the last step we have set all theFj within n∗ of the edges to be FL(±q) since
this only generates a 1/L error that can be ignored. We have to carry out the above procedure since terms involving Γ(1± Fj) are O(1) and any terms added or removed
may result in a finite constant in the final answer. We combine all the terms obtained in region II as follows TII = Y j πFL(λj) sin(πFL(λj)) × exp 1 2 Z q −q dλ ρL(λ) FL(λ) FL0(λ) ∂ ∂λ FL(λ) ρL(λ) ∂ ∂λ log πFL(λ) sin(πFL(λ)) × exp " −L n∗ Z q−ν∗(q) −q dλ ρL(λ)FL2(λ)− L n∗ Z q −q+ν∗(−q) dλ ρL(λ)FL2(λ) # × 1 Γ(1− FL(q))Γ(1 + FL(q))Γ(1 + FL(−q))Γ(1 − FL(−q)) n∗/2 . (4.114) Finally we move to the third region (see Fig. 4.2). In this region bothλj, λk are near the
same quasi-Fermi point. Moreover, because there are only at mostn∗ terms to consider in this region, we do not require the precision we used in the rest of region II. This is becausen∗/L can be neglected in the thermodynamic limit. Thus, by ignoring 1/L terms in the near edge region, and, simply sendingν∗ to zero in the integral in (4.114), we will
only be makingO(1/L) errors in the final answer.
This simplifies the analysis considerably, since we will be setting Fj = Fk = FL(±q) in
this region. We obtain two sets of products TIIIa= n∗ Y j=1 j−1 Y k=1 (j− k) (j− k + FL(−q))1/2(j− k − FL(−q))1/2 , (4.115) TIIIb= N Y j=N −n∗+1 N Y k=j+1 (j− k) (j− k + FL(q))1/2(j− k − FL(q))1/2 . (4.116)
Starting with the first expression and expanding the answer for n∗ 1, we obtain TIIIa= n∗ Y j=1 Γ(j)Γ(1 + FL(−q))1/2Γ(1− FL(−q))1/2 Γ(j + FL(−q))1/2Γ(j− FL(−q))1/2 = G(n ∗+ 1)G(1 + F L(−q))1/2G(1− FL(−q))1/2Γ(1 + FL(−q))n ∗/2 Γ(1− FL(−q))n ∗/2 G(n∗+ F L(−q) + 1)1/2G(n∗− FL(−q) + 1)1/2 ≈ [Γ(1 + FL(−q))Γ(1 − FL(−q))]n ∗/2 [G(1− FL(−q))G(1 + FL(−q))]1/2(n∗)− F 2L(−q) 2 , (4.117) whereG(x) denotes the Barnes G function, (see Eqs.(A1) - (A4)).
Similarly, for the other product
TIIIb=Γ(1 + FL(q))Γ(1− FL(q)) n∗/2 [G(1− FL(q))G(1 + FL(q))]1/2(n∗)− F 2L(q) 2 . (4.118) The final result for T00 is given by
T00= (qLρL(q))− 1 2(F 2 L(q)+FL2(−q)) N Y j=1 πFL(λj) sin(πFL(λj)) × (G(1 + FL(q))G(1 + FL(−q))G(1 − FL(q))G(1− FL(−q)))1/2 × exp P± Z 1 −1 dxF 2 L(qx) x2− 1 − 1 4 Z q −q dλ Z q −q dµ FL(λ)− FL(µ) λ− µ 2! × exp 1 2 Z q −q dλ [1− πFL(λ) cot(πFL(λ))] FL0(λ)−FL(λ)ρ 0 L(λ) ρL(λ) . (4.119) Having obtained the expression for T00, we need to remove the “incorrect” hole terms,
(4.96), to obtain T0 as in (4.95). The treatment of this term depends on the distance of the hole from the quasi-Fermi points. To quantify the notion of the distance from the nearest quasi-Fermi point, we define a quantityqk in the following way
q+k = Ik−− IN − 1. (4.120)
Thus if the kth hole is at the right quasi-Fermi point, qk = −1, if it is in the middle of
distribution then qk ≈ −N/2 etc. Moreover, we say that a hole is near the right
quasi-Fermi point, if qk/L is 0 in the thermodynamic limit, and conversely we say it is “deep”,
Similarly for holes near the left quasi-Fermi point, we define qk− to be
qk−= Ik−− I1+ 1. (4.121)
Now we wish to evaluate Thole= Y j6=i− 1 − FL(λj) LρL(λj)(λj− λ−) + FL(λ −)ρ L(λj) ρL(λ−) −1 . (4.122) We would again like to use the cutoff procedure used to obtain T00. In the region where |j − i−| ≥ n∗, we expandT
hole logarithmically as follows
TholeI = exp − X |j−i−|≥n∗ log 1 − FL(λj) LρL(λj)(λj− λ−) +FL(λ −)ρ L(λj) ρL(λ−) = exp X |j−i−|≥n∗ FL(λj) LρL(λj)(λj− λ−) + O 1 L . (4.123) Meanwhile, in the region where |j − i−| < n∗, we use the approximation in (4.100), to evalute discrete products as follows
TholeII = Y
|j−i−|<n∗
j− i−+ F (λ−)
j− i− . (4.124)
Case I : Hole is deep inside the distribution
When the hole is deep inside the distribution, we obtain from Eqs. (4.123), (4.124) log(Tholek,deep) =
Z λ−k−ν∗(λ− k) −q dλ FL(λ) λ− λ−k + Z q λ−k+ν∗(λ− k) dλ FL(λ) λ− λ−k − log Γ(1 + FL(λ−k))Γ(1− FL(λ−k))Γ(n∗)2) Γ(n∗+ F L(λ−k))Γ(n∗− FL(λ−k)) . (4.125) We get cancellation between divergent and cutoff dependent terms from the integrals due to the symmetric cutoff. Thus in the thermodynamic limit we have
Tholek,deep = sin[πF (µ
− k)] πF (µ−k) exp PZ 1 −1 dx F (qx) x−µ−k q . (4.126)
Meanwhile when the hole is near either ±q, we obtain from Eqs. (4.123), (4.124), the following expression
Tholek,edge= (qLρ(q))∓F (±q)sin[πF (±q)]
πF (±q) Γ(∓qk±± F (±q)) Γ(∓q±k) exp P± Z 1 −1 dxF (qx) x∓ 1 , (4.127) where we use qk± to refer to the location of holes near a quasi-Fermi point as defined in Eqs. (4.120), (4.121).
In order to replace the “correct” term in the evaluation of M2, we require the following
term: Tparticle = Y j6=i− 1 − FL(λj) LρL(λj)(λj− µ+0) + F (µ+0)ρ(λj) ρ(µ+0) . (4.128) Again, we distinguish the case where the particle excitation is far from either edge and the case where it is near the edge. For this purpose we define a means of quantifying nearness to a quasi-Fermi point. Let us define,p+
k = Ik+− IN− 1, for a particle near the
right quasi-Fermi point, and p−k = Ik+− I1+ 1, for a particle near the left quasi-Fermi
point. We treat two distinct cases, Case I: Particle far from edge
If we findpk/L is finite in the thermodynamic limit, then we have the following convergent
term Tparticlek,f ar = exp −Z 1 −1 dx F (qx) x−µ+k q . (4.129)
Case II: Particle near quasi-Fermi Point
On the other hand ifp±k/L vanishes in the thermodynamic limit, we need to use our cutoff procedure to separately evaluate the contribution to Tparticle from the region where λj
is near µ+, and where they are far. We let the j product extend as near as ν∗(±q) to
the relevant quasi-Fermi point, and treat the terms within ν∗ of the quasi-Fermi point separately Tparticlek,near = qLρ(q) n∗ ±F (±q)Γ(n∗)Γ(±p± k + 1∓ F (±q)) Γ(±p±k + 1)Γ(n∗∓ F (±q)) = (qLρ(q))±F (±q)Γ(±p ± k + 1∓ F (±q)) Γ(±p±k + 1) exp −P± Z 1 −1 dxF (qx) x∓ 1 . (4.130)
Now we consider how the term M2 needs to be modified when there are n
particle-hole pairs. Most obviously, the shift function used, F (λ), becomes the sum of the shift functions with the different particle-hole contributions. The termT00is unaltered since it is evaluated withµ−i substituted forµ+i , making the definition ofF (λ) in (4.72) valid for all λj. In order to “correct” for this convenient substitution, we include the terms Thole
associated with each of the holes, and termsTparticle associated with each of the particles
as described in the previous subsections. However, when there are multiple particle-hole pairs, this is no longer sufficient to correctly evaluateM2. There will be another group
of terms which we shall call Tcross whose origin can be understood in the following way.
Let us start by writing down an expression forM2 and show how to arrive at the correct
final answer M2= T00× n Y k=1 Tparticle(k) × Thole(k) = N Y j6=k 0 0 λjkµkj (µk− λj)2 1/2 × n Y k=1 Y j<n−k µ j − µ−k λj− µ−k −1 µj− µ+k λj− µ+k = N Y j6=k 0 0 λjkµkj (µk− λj)2 1/2 × n Y k=1 N Y j 0 0µj− µ−k λj− µ−k −1 µj− µ+k λj− µ+k × n Y k16=k2 (µ−k 1 − µ − k2)(µ + k1− µ + k2) (µ−k 1 − µ + k2) 2 !1/2 = T00× n Y k=1 Tparticle(k) × Thole(k) ! × Tcross, (4.131)
where the notationThole(k) (Tparticle(k) ) means the contribution from the hole (particle) term specific to the details of the excitation,µ−k(µ+k), and n−k refers to the index corresponding to the excitation pair µ+
k, µ − k.
Thus we obtain the expression
Tcross= n Y k16=k2 (µ−k 1− µ − k2)(µ + k1 − µ + k2) (µ−k 1− µ + k2) 2 !1/2 . (4.132) We expect that the contributions toTparticle, Tholefrom the excitations{µ+1, ..., µ+n|µ−1, ..., µ−n}
fall into one of the cases discussed in previous subsections. Thus, the form of these terms is always known in principle.
The termTcross is sensitive to the details of the excitations - for instance it will contain
divergences when some number of the particles, or holes, are clustered near each other. For our purposes, this is only relevant to form factors of states containing high order Umklapp excitations, i.e. several particles and holes near the left (right) quasi-Fermi
