The two particle-hole pairs contribution to the dynamic correlation functions of quantum integrable models
TThe two particle-hole pairs contribution to the dynamiccorrelation functions of quantum integrable models
Mi(cid:32)losz Panfil
Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland
August 21, 2020
Abstract
We consider a problem of computing dynamic correlation functions of quantum integrablemodels employing thermodynamic form-factors approach. Specifically, we focus on correla-tions of local operators that conserve the number of particles and consider the 2-particle-holecontribution to their two-point functions. The method developed is in principle applicable toany finite energy and entropy state, our primary focus lies on the thermal states. To exem-plify this approach we choose the Lieb-Liniger model and study the dynamic density-densitycorrelation function and two-point functions of higher local conserved densities and currentspresent in integrable theories.
In this work, we consider a problem of computing dynamic correlation functions in quantum in-tegrable models. Our focus lies in thermodynamically large systems at finite energy and entropydensity - a canonical example is the state of thermal equilibrium at non-zero temperature. Re-cent years witnessed various developments in this direction. These include microscopic approachbased on the ABACUS method [1–3]; computation of long-distance and large time asymptoticsin and out of the equilibrium [4–9] in quantum many-body systems; computation of correlationsfunctions in the setup of Integrable Quantum Field Theories at finite temperature [10–15] andout of the equilibrium [16–20]. These includes closed-form expression for one-point functions inan arbitrary state of the system [21–23].In this work, we approach the correlation functions using the thermodynamic form-factors [24–28] and focus on the Lieb-Liniger model. Besides a general interest in computation of correlationfunctions in strongly correlated systems, the finite temperature dynamic correlation functions inLieb-Liniger model are important for e.g. the Bragg spectroscopy in cold atomic gases [29–32].Another incentive to develop general methods of computing correlation functions comesfrom experimental progress allowing to probe the nonequilibrium dynamics with cold atomicgases [33–38]. In that respect the thermodynamic form-factors approach has already shownsome utility, leading for example to the concept of generalized detailed balance [25, 39, 40] andto the prediction of edge singularities in correlation functions of certain nonequilibrium steadystates [41].Paralleling these various developments was an introduction of Generalized Hydrodynamics(GHD) [42–44], an effective theory for integrable models in inhomogeneous setups. It led to aburst of activity in formulating GHD for various models and setups including the Lieb-Linigergas [6, 45–50] and related continuum models [51, 52] with recent experimental confirmation ofits validity [53]. Originally, formulated at the ballistic level, the diffusion effects were includedin [54–56], partially with an input provided by the thermodynamic form-factors.1 a r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug inally, the thermodynamic bootstrap program [27, 28] hints at universal structure behindthermodynamic form-factors. It generalizes the vacuum form-factors program [57] of IntegrableQuantum Field Theories to finite density states. Recently, the predicted universal structure ofthe thermodynamic form-factors, combined with the quench action approach [58, 59], led to areconstruction of the GHD [60].To introduce the main aim of this work, we start with an outline of the thermodynamicform-factors approach applied to the repulsive Lieb-Liniger model. This approach relies on thespectral representation and involves the particle picture of an integrable theory: the eigenstateof the system is characterised, once its particle content is specified. Denoting | ϑ (cid:105) a state ofthe system with a particle content labelled by ϑ we are interested in the computation of the(connected) two-point function S ( x, t ) = (cid:104) ϑ |O ( x, t ) O (0) | ϑ (cid:105) , (1)of some hermitian operator O ( x ), or its Fourier transform S ( k, ω ). For local operators conservingthe number of particles, the correlation function, due to the presence of an effective Pauliprinciple, is a sum of contributions with a fixed number of particle-hole excitations, S ( k, ω ) = ∞ (cid:88) m =1 S m ph ( k, ω ) , (2)where the m particle-hole contribution ( m ph in short) is S m ph ( k, ω ) = 1( m !) d p m d h m |(cid:104) ϑ |O (0) | ϑ, p , h (cid:105)| δ ( ω − ω ( p , h )) δ ( k − k ( p , h )) . (3)The sum is performed over all possible states containing m pairs of excitations created ontop of | ϑ (cid:105) . We denote the corresponding state by | ϑ, p , h (cid:105) , with the notation p m = { p j } mj =1 abbrievated to p when the cardinality of the set is clear from the context or irrelevant. Thepositions of particles p and holes h are parametrized by real numbers. The particles and holescarry momentum and energy. We denote ω ( p , h ) and k ( p , h ) the total energy and momentumof the state | ϑ, p , h (cid:105) with respect to | ϑ (cid:105) . Finally, |(cid:104) ϑ |O (0) | ϑ, p , h (cid:105)| are thermodynamic formfactors of the operator O (0)The form-factors in (3) have simple poles, called annihilation or kinematic poles, whenever h i = p j . These singularities are regularized by adopting the Hadamard regularization for inte-grals over particles (holes) positions. Integration over the holes (particles) position is performedafterwards and is regular. The Hadamard regularization assigns a finite value d xf ( x ) = lim (cid:15) → + (cid:18) ˆ d xf ( x )Θ( | x | − (cid:15) ) − (cid:15) lim x → (cid:0) x f ( x ) (cid:1)(cid:19) , (4)to a function f ( x ) having a double pole (here assumed to be at x = 0) in the region of theintegration. We use the subscript below the integration symbol to denote the position of thepole when necessary. Unless otherwise stated, all the integrals are along the real line.The applicability of this approach to the dynamic correlation function relies on access tothe ingredients of (3). In the context of integrable models functions ω ( p , h ) and k ( p , h ) followfrom the standard construction of Thermodynamic Bethe Ansatz [61–65]. On the other hand,the thermodynamic form-factors are known only in a few cases. The notable examples arethe thermodynamic form-factor of the density operator in the Lieb-Liniger model [24, 26], thesmall particle-hole number form-factors in the small momentum limit of conserved charges andcurrents for a generic integrable model [6, 7], and form-factors of the vertex operators in theSinh-Gordon Integrable Quantum Field Theory [27].From these limited number of examples arose hints on the universal structure of the ther-modynamic form-factors whenever particles and holes are close to each other. In the TBP these2niversal structure is captured by the annihilation axiom [27], and states that form-factors in-volving two or more pairs of particle-hole excitations have simple poles. These singularities havea deep physical meaning, signalling large contribution of those weakly excited particle-hole pairsto the spectral sum. On the other hand, when the particle and hole are exactly on top of eachother, there is actually no particle-hole excitation and potential contribution to the correlationfunction should not be included. The solution to this problem comes from adopting a regu-larization scheme [24, 27, 55] which can be recast in the form of Hadamard integrals presentedabove.Given the universal pole structure of the form-factors, the question then arises how to, inpractice, perform the spectral sum. In this work, we address this problem by focusing on thesimplest non-trivial case of 2ph contributions. To be specific, we consider the Lieb-Linigermodel and density-density correlation functions (with a straightforward generalization to two-point functions of higher conserved charges and currents). However, the methods developed areapplicable to other quantum integrable models including Integrable Quantum Field Theories.In Section 2 we recall the ingredients of the solution to the Lieb-Liniger. In the followingSection 3, we formalize the concept of small particle-hole excitations. In Section 4 we presentthe main result, the computation of the 2ph contribution for small excitations. In the followingtwo sections we apply this result in two contexts: first to compute the same type of contributionto two-point functions of higher conserved densities and currents (Section 5), second to computethe density-density correlation function in the ground state (Section 6). We finish with theconclusions. Appendices are devoted to complementary computations. In Appendix A werecall the computation of the 1ph contribution to the density-density correlation function. InAppendix B, based on the TBP and 2ph form-factors of the density in the Lieb-Liniger model,we conjecture an expression for 3ph form-factors. In Appendix C we fill in a small gap in thederivation of the Hadamard regularization scheme for the spectral sum.
The Lieb-Liniger model is defined by the following Hamiltonian [66, 67] H = − N (cid:88) j =1 ∂ ∂x j + 2 c N (cid:88) j>k δ ( x j − x k ) , (5)in units where (cid:126) = 1, 2 m = 1 and with c being the interaction parameter. We consideronly repulsive interactions ( c > N [62, 66, 67]. Here, we are concerned with theThermodynamic Bethe Ansatz [61,63] solution valid in thermodynamic limit when L → ∞ withthe density of particles N/L fixed. In that case, the eigenstates of the system are described bythe filling function ϑ ( λ ) and related density of particles ρ p ( λ ), with Lρ p ( λ )d λ the number ofparticles in range [ λ, λ + d λ ]. The density of particles is given by ρ p ( λ ) = ϑ ( λ ) ρ t ( λ ) , (6)where ρ t ( λ ) is a solution to the linear integral equation, ρ t ( λ ) = 12 π + ˆ d λ (cid:48) ϑ ( λ (cid:48) ) T ( λ, λ (cid:48) ) ρ t ( λ (cid:48) ) , (7)with the differential scattering kernel T ( λ, λ (cid:48) ) = 1 π c ( λ − λ (cid:48) ) + c . (8) The 1ph form-factors do not have singularities when p → h and moreover the presence of the δ -functions fixesuniquely p and h so there is no summation. ρ h ( λ ) = (1 − ϑ ( λ )) ρ t ( λ ) = ρ t ( λ ) − ρ p ( λ ) . (9)The density of particles ρ p ( λ ) determines macroscopic variables of the state like the totaldensity NL = ˆ d λ ρ p ( λ ) , (10)and its energy and momentum E [ ϑ ] = L ˆ d λ ρ p ( λ ) λ , P [ ϑ ] = L ˆ ∞−∞ d λ ρ p ( λ ) λ. (11)We assign to a filling function ϑ ( λ ) a state | ϑ (cid:105) . It can be constructed from (normalized) sum ofits microscopic realizations [24, 26, 27, 58]. Their number is counted by the (log of) entropy [61] S [ ϑ ] = L ˆ d λ ( ρ t ( λ ) ln ρ t ( λ ) − ρ p ( λ ) ln ρ p ( λ ) − ρ h ( λ ) ln ρ h ( λ )) . (12)We are then interested in the dynamic correlation functions in such state (cid:104) ϑ |O ( x, t ) O (0) | ϑ (cid:105) , (13)where O ( x ) is a local operator and O ( x, t ) = e iHt − iP x O (0) e − iHt + iP x . (14)In this work our main focus lies on correlation functions of local particle density ˆ q ( x )( ≡ ˆ ρ ( x )), the associated particle current ˆ j ( x ) and their higher ”spin” generalizations ˆ q j ( x, t ) andˆ j j ( x, t ) present in the integrable theories. We introduce the following notation C ij ( x, t ) = (cid:104) ϑ | ˆ q i ( x, t )ˆ q j (0) | ϑ (cid:105) , Γ ij ( x, t ) = (cid:104) ϑ | ˆ j i ( x, t )ˆ j j (0 , | ϑ (cid:105) , (15)and reserve S ( x, t )(= C ( x, t )) for the special case of the density-density correlation function.The Fourier transform of S ( x, t ), the dynamic structure factor (DSF), is important for the Braggspectroscopy experiments with ultra-cold atoms [29–32]. The correlations of higher conservedcharges and currents are important for the construction of the Generalized Hydrodynamics [54,55]. The charge and current operators obey the continuity equation ∂ t ˆ q j ( x, t ) + ∂ x ˆ j j ( x, t ) = 0 , (16)which implies that the associated charge Q j = ˆ d x ˆ q j ( x, t ) , (17)is conserved. The expectation value of the charge ˆ q j on the thermodynamic state | ϑ (cid:105) is (cid:104) ϑ | ˆ q j | ϑ (cid:105) = ˆ d λ ϑ ( λ ) h j ( λ ) , (18)where h j ( λ ) is the single particle eigenvalue with h ( λ ) = 1 for the density, h ( λ ) = λ for themomentum and h ( λ ) = λ for energy of particles.Local operator, like ˆ q j ( x ), is not capable of modifying the thermodynamic state macroscopi-cally, its action connects different microscopic realizations of the same state. In this work we areconcerned with operators conserving the number of particles. Such operators connect states with4he same number of particles. The different realizations are then parametrized as m particlesand holes modifying the filling function ϑ ( λ ; p , h ) = ϑ ( λ ) + 1 L m (cid:88) j =1 ( δ ( λ − p j ) − δ ( λ − h j )) , (19)with respect to a chosen reference state | ϑ (cid:105) . We denote the corresponding state | ϑ, h , p (cid:105) . Theenergy and momentum of such excited states, with respect to the energy and momentum of | ϑ (cid:105) are ω ( p , h ) = m (cid:88) j =1 ( ω ( p j ) − ω ( h j )) , k ( p , h ) = m (cid:88) j =1 ( k ( p j ) − k ( h j )) , (20)where ω ( λ ) = λ + 2 ˆ d µ ϑ ( µ ) µF ( µ | λ ) , k ( λ ) = λ + ˆ d µ ϑ ( µ ) F ( µ | λ ) , (21)with the back-flow function F ( µ, λ ) satisfying F ( µ | λ ) = θ ( µ − λ )2 π + ˆ d µ (cid:48) ϑ ( µ (cid:48) ) ϑ ( µ (cid:48) ) T ( µ, µ (cid:48) ) F ( µ (cid:48) | λ ) . (22)The phase shift θ ( λ ) = 2 atan( λ/c ) and is related to the differential phase shift T ( λ, λ (cid:48) ) = 12 π ∂θ ( λ − λ (cid:48) ) ∂λ . (23)After characterizing the relevant excited states, we turn our attention to the correlationfunctions and write the resolution of the identity, in the subspace of the Hilbert space with afixed number of particles, as = ∞ (cid:88) m =0 m !) d p m d h m | ϑ, h , p (cid:105)(cid:104) ϑ, h , p (cid:105) , (24)with the integration measure defined asd p m d h m = m (cid:89) j =1 d p j d h j ρ h ( p j ) ρ ( h j ) . (25)and Hadamard regularization which excludes contributions when any particle p j coincides withany hole h k . For example, the connected density-density correlation function in the spectralrepresentation is S ( x, t ) = ∞ (cid:88) m =1 m !) d p m d h m e ik ( p , h ) x − iω ( p , h ) t |(cid:104) ϑ | ˆ ρ (0) | ϑ, p , h (cid:105)| . (26)In this work, we focus on their Fourier transform S ( k, ω ) = ˆ d t d x e − ikx + iωt S ( x, t ) , (27)Contributions to both expressions (26) and (27) can be readily classified by the number ofparticle-hole pairs involved. For S ( k, ω ) we write S ( k, ω ) = ∞ (cid:88) m =1 S m ph ( k, ω ) , (28)5here, as advertised in the introduction, S m ph ( k, ω ) = (2 π ) ( m !) d p m d h m |(cid:104) ϑ | ˆ ρ (0) | ϑ, p , h (cid:105)| δ ( k − k ( p , h )) δ ( ω − ω ( p , h )) . (29)The main aim of this work is to understand the low momentum and energy contributionsto S ( k, ω ).The thermodynamic form factors (cid:104) ϑ |O (0) | ϑ, p , h (cid:105) can be computed either directly from amicroscopic theory in the case where the microscopic form-factors are known, for example dueto the Algebraic Bethe Ansatz [62, 68, 69] or the vacuum Form-Factors bootstrap program [57].An alternative and direct route is provided by Thermodynamic Bootstrap Program [27, 28].The full thermodynamic form-factor of the density operator in the Lieb-Liniger model wascomputed in [24]. Its small momentum limit was explored in [25, 26]. Independently, smallmomentum limits of thermodynamic form-factors for higher conserved charges and currents andfor 1 particle-hole excited state were investigated in [6, 7]. The universal form of the 2 particle-hole thermodynamic form factors were conjectured in [54, 55]. The same structure arises inthe Thermodynamic Bootstrap Program [27, 28] as the consequence of the annihilation poleaxiom. From this point of view it is also possible to conjecture analogous expressions for higherparticle-hole form-factors. In the following we summarize these findings, focusing on the leadingcontributions to the form-factors.The one particle-hole form-factor of the conserved charge density is (cid:104) ϑ | ˆ q j | ϑ, h, p (cid:105) = h dr j ( h ) + ( . . . ) . (30)The two particle-hole form-factors is (cid:104) ϑ | ˆ q j (0) | ϑ, h , p (cid:105) = 2 πk ( p , h ) × (cid:32) T dr ( h , h ) h dr j ( h ) k (cid:48) ( h ) k (cid:48) ( h )( p − h ) + T dr ( h , h ) h dr j ( h ) k (cid:48) ( h ) k (cid:48) ( h )( p − h )+ T dr ( h , h ) h dr j ( h ) k (cid:48) ( h ) k (cid:48) ( h )( p − h ) + T dr ( h , h ) h dr j ( h ) k (cid:48) ( h ) k (cid:48) ( h )( p − h ) + ( . . . ) (cid:33) . (31)Combining conjectures from [55] and [27] we can guess the structure of higher form-factors. Forexample, the next one is (cid:104) ϑ | ˆ q j (0) | ϑ, h , p (cid:105) = (2 πk ( p , h )) k (cid:48) ( h ) k (cid:48) ( h ) k (cid:48) ( h ) × (cid:32) T dr ( h , h ) T dr ( h , h ) h dr j ( h )( p − h )( p − h ) + (cycl. perm.) + ( . . . ) (cid:33) , (32)where (cycl. perm.) refers to independent cyclic permutation of indices of particles and holes.The reasoning leading to this last formula is presented in Appendix B. In all these expressions( . . . ) stands for the subleading contributions in the particle-hole differences p i − h j .The index dr appearing in the formulas above describes the dressing procedure: f ( λ ) dressing −→ f dr ( λ ), defined as f dr ( θ ) = f ( θ ) + ˆ d λ (cid:48) ϑ ( λ (cid:48) ) T ( λ, λ (cid:48) ) f dr ( θ ) . (33)The dressing is relative to the filling function ϑ ( λ ). The (left-)dressing of the differential scat-tering kernel T ( λ, λ (cid:48) ) is T dr ( λ, λ (cid:48) ) = T ( λ, λ (cid:48) ) + ˆ d λ (cid:48)(cid:48) ϑ ( λ (cid:48)(cid:48) ) T ( λ, λ (cid:48)(cid:48) ) T dr ( λ (cid:48)(cid:48) , λ (cid:48) ) , (34)6ith resulting function inheriting the symmetry in exchanging the arguments.The filling function ϑ ( λ ) characterizes the state of the system. At (generalized) thermalequilibrium ϑ ( λ ) is given by ϑ ( λ ) = 11 + e (cid:15) ( λ ) , (35)with (cid:15) ( λ ) solving the generalized TBA equation [61, 64, 65] (cid:15) ( λ ) = (cid:88) j µ j h j ( λ ) − µ − π ˆ d λ (cid:48) K ( λ − λ (cid:48) ) log (cid:16) e − (cid:15) ( λ (cid:48) ) (cid:17) . (36)The case of a thermal gas at temperature T is given by µ = 1 /T (we set k B = 1) with otherchemical potentials µ j equal to zero.The thermodynamic form-factors are known for smooth filling function ϑ ( θ ), the thermalone (35) being an important example. However, once the correlation function is computed it ispossible to take the limit to discontinuous filling function, for example the one describing theground state [66], ϑ ( λ ) = 1 − Θ( | λ | − λ F ) , (37)with λ F the Fermi rapidity set by the density of particles N/L . In eqs. (30), (31) and (32), the most singular part of the form-factors in p i − h j is shown. Itforms the leading part of the form-factor whenever particles and holes can be paired such thatfor each pair ( p i , h i ) the difference p i − h i is small. We call such excitations small particle-holeexcitations . In this section we will formalize this concept by i) introducing a criteria sayingwhen p i − h i is small and ii) exploring a relation between states formed with small excitationsand states of small momentum and energy.Let us start with a single particle-hole excited state | ϑ, h, p (cid:105) specified by pair ( p, h ). Itskinematics, according to (20), is k = k ( p ) − k ( h ) , ω = ω ( p ) − ω ( h ) . (38)When p − h is small, given the analyticity of k ( λ ) and ω ( λ ), we can expand both formulas to find k = k (cid:48) ( h )( p − h ) + 12 k (cid:48)(cid:48) ( h )( p − h ) + O (cid:0) ( p − h ) (cid:1) , (39) ω = ω (cid:48) ( h )( p − h ) + 12 ω (cid:48)(cid:48) ( h )( p − h ) + O (cid:0) ( p − h ) (cid:1) . (40)The small excitations assumption breaks certainly when the first and second order terms inthese expansions become similar. These leads to the constraints | p − h | (cid:46) (cid:12)(cid:12)(cid:12)(cid:12) k (cid:48) ( h ) k (cid:48)(cid:48) ( h ) (cid:12)(cid:12)(cid:12)(cid:12) , | p − h | (cid:46) (cid:12)(cid:12)(cid:12)(cid:12) ω (cid:48) ( h ) ω (cid:48)(cid:48) ( h ) (cid:12)(cid:12)(cid:12)(cid:12) . (41)From numerical solutions we observe that the second condition is tighter and implies the first .This leads to the following condition on when the excitation is small. | p − h | < (cid:12)(cid:12)(cid:12)(cid:12) ω (cid:48) ( h ) ω (cid:48)(cid:48) ( h ) (cid:12)(cid:12)(cid:12)(cid:12) . (42) This is partially because functions k (cid:48) ( λ ) and ω (cid:48)(cid:48) ( λ ) are bounded from below, whereas k (cid:48)(cid:48) ( λ ) and ω (cid:48) ( λ ) areodd functions of the rapiditiy. This makes, at least at small h , the momentum bound to diverge and the energybound to fall to zero and therefore making it tighter. left panel: ) The phase space for small 1ph excitations. The color codes for the phasespace factor ϑ ( h )(1 − ϑ ( h )), the black line is the bound (42) and dotted red lines show thedispersion relation for the excitations with velocity ¯ v eff . ( right panel: ) The same but now for2ph excitations. The plots are for the thermal state with T = 1, c = 4 and in units of Fermimomentum k F = πN/L and Fermi energy (cid:15) F = k F .Such excitation has then a linear spectrum ω = v eff ( h ) k , with the velocity dependent on itsposition in the rapidity space. Alternatively, we can think of h as labelling different particlestypes each with a different velocity of propagation. The effective velocity is defined as v eff ( λ ) = ω (cid:48) ( λ ) k (cid:48) ( λ ) . (43)We can now check for what values of k and ω small excitations exist. That is, we fix k and ω , find corresponding p and h at the linear order and then check if their difference satisfies thebounds. Additionally, to get an impression which excitations are important we look at the valueof ϑ ( h )(1 − ϑ ( p )) which controls how much space there is to create an excitation. The resultsare shown in fig. 1. We see that the possible excitations are localized along two rays determinedby the average velocity ¯ v eff = ´ d h ϑ ( h )(1 − ϑ ( h )) v eff ( h ) ´ d h ϑ ( h )(1 − ϑ ( h )) . (44)The excluded region is a region with energy ω relatively small compared to k ¯ v eff and corre-sponds to excitations localized in the center of the distribution ϑ ( λ ), that is with h and p closeto 0. The energy ω ( λ ) has a global minimum in this region making braking the bound easy. Onthe other hand for the thermal states of not too high temperature, the center of distribution ϑ ( λ ) is largely filled which effectively diminishes their contribution to the correlation functions.This shows that the relevant low momentum 1ph states can be understood as small excitationsstates.The situation gets only slightly more complicated for two particle-hole excitations pairs. Wewant both pairs to constitute small excitations, that is | p i − h i | (cid:46) (cid:12)(cid:12)(cid:12)(cid:12) ω (cid:48) ( h i ) ω (cid:48)(cid:48) ( h i ) (cid:12)(cid:12)(cid:12)(cid:12) , i = 1 , . (45)8he phase space is then constructed in the standard way: each point on the single ph phasediagram becomes now a point from which rays of the second ph excitation originate, see fig. 1.In the consequence, the 2ph excitations are not focused only 1ph along the two rays, but rathercover the whole plane of small k − ω including the 1ph excluded region.Are there small momentum-energy excited states that escape this structure? For the energyand momentum of the excited states to be small, in the leading order, we have relations k ( p ) + k ( p ) ≈ k ( h ) + k ( h ) , ω ( p ) + ω ( p ) ≈ ω ( h ) + ω ( h ) . (46)One can interpret this condition from the point of view of elastic scattering process in (1 + 1) d with initial rapidities p and p and final rapidities h and h . The set of initial and finalrapidities must then coincide up to a permutation (cid:40) p ≈ h ,p ≈ h , (cid:40) p ≈ h p ≈ h , (47)with h and h free parameters. Whether the energy bound (42) is fulfilled depends then ontheir values. In any case, there are choices of h and h such that both pairs constitute smallexcitations. Such choices are again dominating by the phase space argument. Therefore therelevant 2ph excited states of small momentum and energy are formed by the small particle-holeexcitations.Considering 3ph and higher excited states the momentum and energy constraints, togetherwith the phase space argument, are not enough to restrict the structure of the excited state tobe of small excitations type.In summary, we have argued that small excitations reign the contributions to the DSFin the small momentum-energy limit of 1ph and 2ph types. Each small ph excitation has alinear dispersion relation with possibly different velocity, however for thermal states focusedaround ¯ v eff . In this section, we evaluate the contribution to the DSF from small 2ph excitations. For thecompleteness in Appendix A we have included the computations of the 1ph case. We start byspecializing eq. (31) to the density form factor, using h dr0 ( λ ) = 2 πρ t ( λ ), (cid:104) ϑ | ˆ q j (0) | ϑ, h , p (cid:105) = (2 π ) k ( p , h ) × (cid:18) T dr ( h , h ) ρ t ( h ) k (cid:48) ( h ) k (cid:48) ( h )( p − h ) + T dr ( h , h ) ρ t ( h ) k (cid:48) ( h ) k (cid:48) ( h )( p − h )+ T dr ( h , h ) ρ t ( h ) k (cid:48) ( h ) k (cid:48) ( h )( p − h ) + T dr ( h , h ) ρ t ( h ) k (cid:48) ( h ) k (cid:48) ( h )( p − h ) + ( . . . ) (cid:19) . (48)The kinematics (20), for two small particle-hole excitations, is k ( p , h ) = k (cid:48) ( h )( p − h ) + k (cid:48) ( h )( p − h ) ,ω ( p , h ) = v eff ( h ) k (cid:48) ( h )( p − h ) + v eff ( h ) k (cid:48) ( h )( p − h ) . (49)Here we have implicitly assumed a pairing ( p with h and p with h ) of p with h . Giventhe symmetry of the form-factor, to account for the other choice it suffices to multiply thecontribution to the DSF by 2. It is convenient to change the variables from ( p , h ) to ( α , h )where α = p − h . The contribution (29) to the DSF is then S ( k, ω ) = (2 π ) ˆ d h d h d α d α F ( h , α ) δ ( k − k ( α + h , h )) δ ( ω − ω ( α + h , h )) , (50)9here F ( h , α ) = ρ p ( h ) ρ p ( h ) ρ h ( h + α ) ρ h ( h + α ) |(cid:104) ϑ | ˆ ρ (0) | ϑ, h , α + h (cid:105)| . (51)We will now perform integrations over α with the help of the δ -functions. Certain care is requireddue to the Hadamard regularization. At first imagine that there is no regularization and thatthese are regular integrals. Presence of the δ -function puts the form-factor on the ( k, ω )-shell.That is, given the momentum and energy constraints: k = k ( α + h , h ) and ω = ω ( α + h , h ), wecan solve for the differences α i to find¯ α = ω − v eff ( h ) kk (cid:48) ( h )( v eff ( h ) − v eff ( h )) , ¯ α = ω − v eff ( h ) kk (cid:48) ( h )( v eff ( h ) − v eff ( h )) . (52)The product of the δ -functions can be now disentangled giving δ ( k − k ( α + h , h )) δ ( ω − (cid:15) ( α + h , h )) = δ ( α − ¯ α ) δ ( α − ¯ α ) k (cid:48) ( h ) k (cid:48) ( h ) | v eff ( h ) − v eff ( h ) | . (53)This yields F ( h , α ) δ ( k − k ( α + h , h )) δ ( ω − (cid:15) ( α + h , h )) = F h ( k, ω ) δ ( α − ¯ α ) δ ( α − ¯ α ) , (54)where F h ( k, ω ) is the ( k, ω )-shell form-factor together with the density factors and extra contri-butions from the integrals over δ -functions F h ( k, ω ) = F ( h , ¯ α ) k (cid:48) ( h ) k (cid:48) ( h ) | v eff ( h ) − v eff ( h ) | = ϑ ( h ) ϑ ( h ) ρ h ( h + α ) ρ h ( h + α ) |(cid:104) ϑ | ˆ ρ (0) | ϑ, h , ¯ α + h (cid:105)| (2 π ) | v eff ( h ) − v eff ( h ) | . (55)Ignoring the Hadamard regularization in (50) we then find S ( k, ω ) = (2 π ) ˆ d h d h F h ( k, ω ) . (56)Let us look into the structure of F h ( k, ω ) and start with evaluating the form-factor (cid:104) ϑ | ˆ ρ (0) | ϑ, h , p (cid:105) with positions of the particles p = h + ¯ α fixed by the momentum and energy ( k, ω ) and positionsof holes h . Recall from (48), that the form factor consists of 4 terms potentially diverging like p i − h j and some regular parts. Additionally, the form-factor is multiplied by the total momen-tum. In the small momentum limit only the diverging parts of the form-factor contribute asthey are also of order of k . For the choice of the excited state, the terms containing p − h and p − h are small and therefore constitute the leading contribution. The form-factor can besimplified to (cid:104) ϑ | ˆ ρ (0) | ϑ, h , p (cid:105) = (2 π ) k ( h , p ) (cid:18) T dr ( h , h ) ρ dr t ( h ) k (cid:48) ( h ) k (cid:48) ( h )( p − h ) + T dr ( h , h ) ρ dr t ( h ) k (cid:48) ( h ) k (cid:48) ( h )( p − h ) (cid:19) . (57)Using that T dr ( h , h ) = T dr ( h , h ) and k (cid:48) ( h ) = 2 πρ t ( h ) together with the definition of α , theexpression simplifies further to (cid:104) ϑ | ˆ ρ (0) | ϑ, h , p (cid:105) = 2 πk ( h , p ) T dr ( h , h ) k (cid:48) ( h ) α + k (cid:48) ( h ) α k (cid:48) ( h ) k (cid:48) ( h ) α α . (58)Putting the form-factor on ( k, ω )-shell means setting α i = ¯ α i . In that case we find (cid:104) ϑ | ˆ ρ (0) | ϑ, h , ¯ p (cid:105) = − πT dr ( h , h ) k ( v eff ( h ) − v eff ( h )) ( ω − v eff ( h ) k )( ω − v eff ( h ) k ) . (59)10e see now the problem with ignoring the Hadamard regularization. The Hadamard regular-ization takes care of an unphysical divergence when particle and hole are placed on top of eachother. This corresponds to the pole in the form-factor when α i ≈
0. The ( k, ω )-shell form-factorstill has the same pole, it appears now as a pole when ω − v eff ( h ) k ≈
0. This happens, whenone of the particle-hole excitations carries the whole energy and momentum of the excited state,meaning, that for the other excitation, α i ≈
0. By properly executing the Hadamard regular-ized integrals we should then find the integrations over h i also properly regularized. The correctanswer turns out to be simply S ( k, ω ) = (2 π ) h ∗ d h d h F h ( k, ω ) , (60)with the double Hadamard integral. Parameter h ∗ is defined as the unique solution to ω = v eff ( h ∗ ) k . The uniqueness comes from the fact that v eff ( h ∗ ) is a strictly monotonic function. Inthe remaining part of this section we present the derivation of this result. The Hadamard regularization assigns a finite value to an integral of a function with a double pole.We need to evaluate two Hadamard integrals of a function with the pole structure 1 / | α α | .Recall the prescription for Hadamard regularization for a generic function f ( x ) having a doublepole at x = 0, d x f ( x ) = lim (cid:15) → + ˆ d x (cid:18) f ( x )Θ( | x | − (cid:15) ) − (cid:15) δ ( x ) f (1) ( x ) (cid:19) , (61)where f (1) ( x ) ≡ x f ( x ), which has now a finite value for x = 0 . We introduce the followingnotation for the form-factors F ( i ) ( h , α ) = α i F ( h , α ) , F (1 , ( h , α ) = α α F ( h , α ) , (62)and for their ( k, ω )-shell counterparts F ( i ) h ( k, ω ) = ¯ α i F h ( k, ω ) , F (1 , h ( k, ω ) = ¯ α ¯ α F h ( k, ω ) , (63)and consider F h ( k, ω ) = d α d α F ( h , α ) δ ( k − k ( α + h , h )) δ ( ω − (cid:15) ( α + h , h )) . (64)Evaluating the two integrals, according to the prescription (61), leads to 3 different contributions,classified by the number of (cid:15) i ’s multiplying the contribution: zero, one or two. The first one(without (cid:15) i prefactor) is ˆ d α d α F ( h , α ) δ ( k − k ( α + h , h )) δ ( ω − (cid:15) ( α + h , h ))Θ( | α | − (cid:15) )Θ( | α | − (cid:15) )= F h ( k, ω )Θ( | ¯ α | − (cid:15) )Θ( | ¯ α | − (cid:15) ) . (65)There are two contributions of the second type. One of the form − (cid:15) ˆ d α d α F (1) ( h , α ) δ ( k − k ( α + h , h )) δ ( ω − (cid:15) ( α + h , h )) δ ( α )Θ( | α | − (cid:15) )= − (cid:15) F (1) h ( k, ω ) δ ( ¯ α )Θ( | ¯ α | − (cid:15) ) , (66) In practice to evaluate the Hadamard regularized integral we need only f (1) ( x ) at x = 0. We keep the x -dependence as it simplifies the notation in the 2 d case. (cid:15) i ’s inthe prefactor, is4 (cid:15) (cid:15) ˆ d α d α F (1 , ( h , α ) δ ( k − k ( α + h , h )) δ ( ω − (cid:15) ( α + h , h )) δ ( α ) δ ( α )= 4 (cid:15) (cid:15) F (1 , h ( k, ω ) δ ( ¯ α ) δ ( ¯ α ) , (67)Summing all the contributions we find F h ( k, ω ) = lim (cid:15) i → (cid:18) F h ( k, ω )Θ( | ¯ α | − (cid:15) )Θ( | ¯ α | − (cid:15) ) − (cid:15) F (1) h ( k, ω ) δ ( ¯ α )Θ( | ¯ α | − (cid:15) ) − (cid:15) F (2) h ( k, ω )Θ( | ¯ α | − (cid:15) ) δ ( ¯ α ) + 4 (cid:15) (cid:15) F (1 , h ( k, ω ) δ ( ¯ α ) δ ( ¯ α ) (cid:19) . (68)For a given value of k and ω this expression should be integrated over h and h and then thelimits (cid:15) i → α i depends on both h and h . We would like to rewrite this expression in such a way that conditions on h are explicit. Thisis easy to achieve for the first term of (68), where we can use two properties of the HeavisideΘ-function, Θ( x/a ) = Θ( x ) , Θ( f ( x )) = Θ( x ) , (69)which hold for a > f ( x ) with positive values for x >
0. We have thena chain of relationsΘ( | ¯ α | − (cid:15) ) = Θ( | v eff ( h ∗ ) − v eff ( h ) | − (cid:15) (cid:48) ) = Θ( | h ∗ − h | − (cid:15) (cid:48)(cid:48) ) , (70)with new (cid:15) (cid:48) = | k (cid:48) ( h )( v eff ( h ) − v eff ( h )) | (cid:15) and (cid:15) (cid:48)(cid:48) = (cid:15) (cid:48) / | kκ ( h ∗ ) | . Here κ ( λ ) = ∂ λ v eff ( λ )and we used that v eff ( λ ) is a monotonically increasing function. We can now perform similartransformations with other expressions appearing in (68). For example δ ( ¯ α ) = (cid:15) (cid:48)(cid:48) (cid:15) δ ( h ∗ − h ) . (71)Transformation of F ( i ) h ( k, ω ) is more subtle. Recall that its definition is the following: havingfunction f ( x, y ) with a double pole at x we define f (1) ( x ) = x f ( x, y ). F (1) h ( k, ω ) is a functionhaving a pole for α = 0. What we want is however to restate this in terms of the h variable,with the pole now for h − h ∗ = 0. In the vicinity of the pole we can then write F (1) h ( k, ω ) = (cid:18) (cid:15) (cid:15) (cid:48)(cid:48) (cid:19) ¯ F (2) h ( k, ω ) . (72)with ¯ F (2) h ( k, ω ) defined as ¯ F (2) h ( k, ω ) = ( h − h ∗ ) F h ( k, ω ) . (73)Collecting together the rescaling of Dirac δ -function and of F (1) h ( k, ω ) we find that the secondterm of (68) transforms as − (cid:15) F (1) h ( k, ω ) δ ( ¯ α )Θ( | ¯ α | − (cid:15) ) = − (cid:15) (cid:48)(cid:48) ¯ F (2) h δ ( h − h ∗ )Θ( | h − h ∗ | − (cid:15) (cid:48)(cid:48) ) . (74)Transforming in the similar way the remaining terms, dropping the bar from ¯ F ( i ) h ( k, ω ) andexchanging indices of (cid:15) j ’s so they match indices of h j ’s, we find F h ( k, ω ) = 12 lim (cid:15) i → (cid:18) F h ( k, ω )Θ( | ∆ h ∗ | − (cid:15) )Θ( | ∆ h ∗ | − (cid:15) ) − (cid:15) F (1) h ( k, ω ) δ (∆ h ∗ )Θ( | ∆ h ∗ | − (cid:15) ) − (cid:15) F (2) h ( k, ω )Θ( | ∆ h ∗ | − (cid:15) ) δ (∆ h ∗ ) + 4 (cid:15) (cid:15) F (1 , h ( k, ω ) δ (∆ h ∗ ) δ (∆ h ∗ ) (cid:19) . (75)12ntegration over F h ( k, ω ) takes now a form of a double Hadamard integral of a function of twovariables with a double pole in each variable at h i = h ∗ . The 2ph contribution to DSF thenequals S ( k, ω ) = (2 π ) h ∗ d h d h F h ( k, ω ) . (76)with the following ingredients entering this formula F h ( k, ω ) = G h ( k, ω ) × k | v eff ( h ) − v eff ( h ) | ( ω − v eff ( h ) k ) ( ω − v eff ( h ) k ) , (77) G h ( k, ω ) = ϑ ( h ) ϑ ( h ) ρ h ( h + ¯ α ) ρ h ( h + ¯ α ) × (cid:18) T dr ( h , h )2 π (cid:19) . (78)The limiting expressions are F (1) h ∗ ,h ( k, ω ) = lim h → h ∗ ( h − h ∗ ) F h ( k, ω ) = G h ∗ ,h ( k, ω ) × | v eff ( h ) − v eff ( h ∗ ) | κ ( h ∗ ) ,F (1) h ,h ∗ ( k, ω ) = lim h → h ∗ ( h − h ∗ ) F h ( k, ω ) = G h ,h ∗ ( k, ω ) × | v eff ( h ) − v eff ( h ∗ ) | κ ( h ∗ ) , (79) F (1 , h ∗ ,h ∗ ( k, ω ) = lim h → h ∗ lim h → h ∗ ( h − h ∗ ) ( h − h ∗ ) F h ( k, ω ) = 0 . For smooth distributions of rapidities and in the leading order in k , G h ( k, ω ) is further simplifiedby setting ¯ α i = 0 in the arguments of ρ h ( h ). In that case G h ( k, ω ) = G h is a function of positionof two holes only.Eq. (76) is the main result of this work. We will now turn to two applications of thisformula. First, it should be clear that the derivation does not depend on the details of the form-factor considered, rather on its general analytic structure and therefore holds also for correlationfunctions of higher conserved charges and also of conserved currents. The computations arepresented in Section 5. Second, similarly as in the 1ph case (see Appendix A), we can considerthe ground state correlation function. This is presented in Section 6. Before that, we discusscertain properties of the correlator. We close this section with a discussion of properties of S ( k, ω ). To this end, we consider analternative representation of F h ( k, ω ) which follows from parametrizing ω and k with h ∗ , thatis writing ω = v eff ( h ∗ ) k . This leads to F h ( h ∗ ) = G h × | v eff ( h ) − v eff ( h ) | ( v eff ( h ∗ ) − v eff ( h )) ( v eff ( h ∗ ) − v eff ( h )) . (80)This representation is convenient for considering various features of the correlator. For ex-ample, it is easy to see that S ( k, ω ) = S ( k (cid:48) , ω (cid:48) ) if ω/k = ω (cid:48) /k (cid:48) and as a consequence S ( k, ω ) = S ( − k, − ω ). This is the low energy limit of the detailed balance relation, whichin full generality reads S ( k, ω ) = e − ω/T S ( − k, − ω ) . (81)In the leading order in energy, the exponential term can be neglected leading to the aforemen-tioned equality.It is equally straightforward to show that S ( k, ω ) = S ( − k, ω ). To this end we usethat changing the sign of k leads to a change in the sign of h ∗ since v eff ( h ) is an odd func-tion. The change of sign of h ∗ in the form-factor can be compensated by changing signs ofthe integration variables h and h . This leads back to the original expression for the cor-relator upon assumption that the thermodynamic functions ϑ ( h ), ρ t ( h ) are even functions13nd T dr ( h , h ) = T dr ( − h , − h ). This is the case at the thermal equilibrium. Therefore S ( k, ω ) = S ( − k, ω ) and similarly S ( k, ω ) = S ( k, − ω ). The first of these equalitiesreflects the spatial parity invariance of the thermal state. The second relation can be interpretedfrom the point of view of the f-sum rule, which reads ˆ d ω π ωS ( k, ω ) = NL k . (82)Given the aforementioned symmetry, the 2ph contribution to this expression is zero.Also of interest are the limiting expressions of zero momentum and energy. The resultdepends on the order of the limits. Taking first the zero momentum and then zero energycorresponds to taking h ∗ to ±∞ which gives zero contribution to the DSF. The opposite limitcorresponds to h ∗ → k → lim ω → S ( k, ω ) = (2 π ) d h d h ϑ p ( h ) ϑ p ( h ) ρ h ( h ) ρ h ( h ) × (cid:16) T dr ( h , h ) (cid:17) | v eff ( h ) − v eff ( h ) | ( v eff ( h ) v eff ( h )) . (83)We note that the dependence on the order of the limits is caused by the form-factor itself andnot by the density functions and therefore we expect the same feature also in more general,beyond the thermal equilibrium, settings. The computation presented in the previous section was performed for the specific density-density correlation function. However similar computations hold also for correlation functionsof higher conserved densities and conserved currents given their similar analytic structure. Fol-lowing [55], we recall that (cid:104) ϑ | ˆ q i | ϑ, p , h (cid:105) = k ( p , h ) f i ( p , h ) , (84) (cid:104) ϑ | ˆ j i | ϑ, p , h (cid:105) = (cid:15) ( p , h ) f i ( p , h ) , (85)where in the case of 2ph f i ( p , p , h , h ) = 2 πT dr ( h , h ) h dr i ( h ) k (cid:48) ( h ) k (cid:48) ( h )( p − h ) + 2 πT dr ( h , h ) h dr i ( h ) k (cid:48) ( h ) k (cid:48) ( h )( p − h )+ 2 πT dr ( h , h ) h dr i ( h ) k (cid:48) ( h ) k (cid:48) ( h )( p − h ) + 2 πT dr ( h , h ) h dr i ( h ) k (cid:48) ( h ) k (cid:48) ( h )( p − h ) + ( . . . ) . (86)Applying the procedure presented in the previous section we start by computing the ( k, ω )-shell form-factors by setting ¯ p = h + ¯ α with ¯ α given in (52). Function f i ( p , p , h , h ) becomes f i ( p , p , h , h ) = T dr ( h , h ) k ( v eff ( h ) − v eff ( h )) (cid:18) h dr i ( h ) ρ t ( h )∆ v eff1 − h dr i ( h ) ρ t ( h )∆ v eff2 (cid:19) , (87)where we introduced ∆ v eff i = v eff ( h i ) − v eff ( h ∗ ). The ( k, ω )-shell form-factors are thus (cid:104) ϑ | ˆ q i | ϑ, h , ¯ p (cid:105) = T dr ( h , h )( v eff ( h ) − v eff ( h )) (cid:18) h dr i ( h ) ρ t ( h )∆ v eff1 − h dr i ( h ) ρ t ( h )∆ v eff2 (cid:19) , (88) (cid:104) ϑ | ˆ j i | ϑ, h , ¯ p (cid:105) = v eff ( h ∗ ) T dr ( h , h )( v eff ( h ) − v eff ( h )) (cid:18) h dr i ( h ) ρ t ( h )∆ v eff1 − h dr i ( h ) ρ t ( h )∆ v eff2 (cid:19) . (89)14hese two classes of form-factors display different behaviour in the small k and ω limits whichtranslates into a different behaviour of the corresponding correlation functions. Recall, thattaking first k to zero and then ω to zero corresponds to taking h ∗ to ±∞ . In this limit thecharge density form-factor vanishes whereas the current density form-factor approaches a finitevalue, lim h ∗ →±∞ (cid:104) ϑ | ˆ j i | ϑ, h , ¯ p (cid:105) = − T dr ( h , h ) (cid:16) v eff ( h ) − v eff ( h ) (cid:17) (cid:18) h dr i ( h ) ρ t ( h ) − h dr i ( h ) ρ t ( h ) (cid:19) . (90)In the opposite limit (first ω → k → h ∗ → (cid:104) ϑ | ˆ q i | ϑ, h , ¯ p (cid:105) = T dr ( h , h )( v eff ( h ) − v eff ( h )) (cid:18) h dr i ( h ) ρ t ( h ) v eff ( h ) − h dr i ( h ) ρ t ( h ) v eff ( h ) (cid:19) , (91)whereas the conserved current form-factor vanishes. We will apply now these observations tothe correlation functions.Consider the connected two-point correlation functions of conserved charges and currents, C ij ( x, t ) = (cid:104) ϑ | ˆ q i ( x, t )ˆ q j (0) | ϑ (cid:105) , Γ ij ( x, t ) = (cid:104) ϑ | ˆ j i ( x, t )ˆ j j (0 , | ϑ (cid:105) , (92)and their Fourier transforms C ij ( k, ω ) and Γ ij ( k, ω ). The 2ph contributions to these correlationfunctions in the zero momentum and energy limit arelim ω → lim k → C ij ( k, ω ) = 0 , (93)lim k → lim ω → C ij ( k, ω ) = 12 d h d h ϑ ( h ) ϑ ( h ) ρ h ( h ) ρ h ( h )( T dr ( h , h )) | v eff ( h ) − v eff ( h ) |× (cid:18) h dr i ( h ) ρ t ( h ) v eff ( h ) − h dr i ( h ) ρ t ( h ) v eff ( h ) (cid:19) (cid:32) h dr j ( h ) ρ t ( h ) v eff ( h ) − h dr j ( h ) ρ t ( h ) v eff ( h ) (cid:33) , (94)lim ω → lim k → Γ ij ( k, ω ) = 12 ˆ d h d h ϑ ( h ) ϑ ( h ) ρ h ( h ) ρ h ( h )( T dr ( h , h )) | v eff ( h ) − v eff ( h ) |× (cid:18) h dr i ( h ) ρ t ( h ) − h dr i ( h ) ρ t ( h ) (cid:19) (cid:32) h dr j ( h ) ρ t ( h ) − h dr j ( h ) ρ t ( h ) (cid:33) , (95)lim k → lim ω → Γ ij ( k, ω ) = 0 (96)To obtain these expressions we simply use our main result, eq. (76), and adopt the formula (55)for F h ( k, ω ) to the form-factors of higher conserved densities and charges. Finally, we take theappropriate limit h ∗ → h ∗ → ∞ .We observe, as was already noted in [55], that the integrand in Γ ij ( k, ω ) is a regular function(without the double poles) which turns the Hadamard integral into an ordinary one. This is notthe case for the C ij ( k, ω ) which still requires the regularization. We specialize now back to the density-density correlation function and consider the T = 0limit of the 2ph contribution. In this case, the product of density factors ϑ ( h i ) ρ h ( h i + ¯ α i ),appearing in (77) for F h ( k, ω ), localize the integrals to the vicinity of the edge in the groundstate distribution (37), ϑ ( h i ) ρ h ( h i + ¯ α i ) = ρ t ( q F ) | ¯ α i | ( δ ( h i − q F )Θ( α i ) + δ ( h i + q F )Θ( − α i )) + . . . , (97)15here ( . . . ) are contributions of zero measure of the form Θ( x )(1 − Θ( x )). This leads to 2classes of contributions: one with both excitations on different edges, and the second with bothexcitations on the same edge. The form-factor F h ( k, ω ) vanishes when positions of holes coincideand therefore only the first class of excitations contribute. Given the symmetry in labelling theexcitations we can assume that h = q F and h = − q F . We will multiply the contribution tothe correlation function by 2 to account for the other choice. With this choice for the positionsof the holes we find ¯ α = ω + v F k v F k (cid:48) ( q F ) , ¯ α = − ω − v F k v F k (cid:48) ( q F ) , (98)where we used eq. (52) for ¯ α i and Fermi velocity v F ≡ v eff ( q F ). Factor Θ( α )Θ( − α ) implementscondition ω > v F | k | with the form-factor equal zero otherwise. We find G h ( k, ω ) = ρ t ( q F ) | ¯ α || ¯ α | (cid:18) T dr ( q F , − q F )2 π (cid:19) × δ ( h − q F ) δ ( h + q F )Θ( ω − v F | k | ) . (99)This expression is valid when both ¯ α i are small compared to q F which implies that ω + v F | k | (cid:28) q F v F k (cid:48) ( q F ) , (100)where we used that non-zero contribution is possible only for positive energy ω (in fact theenergy has to be at least v F | k | ). Recall that, according to (77), the full form-factor is F h ( k, ω ) = G h ( k, ω ) × k | v eff ( h ) − v eff ( h ) | ( ω − v eff ( h ) k ) ( ω − v eff ( h ) k ) , (101)which, including the mentioned above factor 2, leads to F h ( k, ω ) = 1 π (cid:18) T dr ( q F , − q F )2 π (cid:19) v F k | ω − v F k | × δ ( h − q F ) δ ( h + q F )Θ( ω − v F | k | ) , (102)where we used that k (cid:48) ( h ) = 2 πρ t ( h ). To implement the Hadamard integral we need also thelimiting expressions F (1) h ∗ ,h ( k, ω ), etc. Setting h = h ∗ yields ¯ α = 0 which leads to factor ϑ ( h ) ρ h ( h ) which is zero in the ground state. In turn F (1) h ∗ ,h ( k, ω ) = 0. Therefore, in theHadamard integral only the double integral term remains with h and h dependence onlythrough the δ -functions in (102). The remaining integration is a product oflim (cid:15) → ˆ d h δ ( h − q F )Θ( | h − h ∗ | − (cid:15) ) = lim (cid:15) → Θ( | q F − h ∗ | − (cid:15) ) = Θ( | q F − h ∗ | ) , (103)with an analogous expression for the second hole. The Θ-functions encode the condition thatthere is no contribution when one of the excitations carries the whole energy and momentumwhich is equivalent to the condition already present in (102). For the 2ph contribution to theDSF we have then, S ( k, ω ) = 2 v F k | ω − v F k | (cid:18) T dr ( q F , − q F )2 π (cid:19) Θ( ω − v F | k | ) . (104)This expression is valid for ω and k obeying (100). The contribution exhibits a one-sidedsingularity of the form ( ω − v F | k | ) − for ω approaching v F | k | . Presence of such edge singularityis expected from the general theory of non-linear Luttinger liquids [70–76]. Here, we see it in asimple setting as an effect of interactions between the linearly dispersing excitations at oppositeedges of the Fermi sea. 16 Conclusions and outlook
In this work, we have considered dynamic two-point correlation functions in the Lieb-Linigermodel. For local operators conserving the number of particles, such correlation functions canbe expanded in thermodynamic form-factors with a number of particle-hole excitations. Ourfocus lied in analysing the contribution from two particle-hole pairs, given by a Hadamardintegral constrained by the energy and momentum conserving δ -function. We have shown how toevaluate such integrals for the form-factors of the density operator ˆ ρ ( x ) and for higher conserveddensities and currents present in integrable theories. In our analysis, we have focused on theleading contribution to those form-factors which come from small particle-hole excitations. Ourresults extend the findings of [55] where the zero momentum and energy limit of the current-current correlation functions was derived. Here we have extended them to finite, albeit smallmomenta and energies, and considered two-point functions also of conserved densities. In bothcases, the resulting expressions involve double Hadamard integrals. We have also considered thezero-temperature limit of the dynamic density factor showing that it exhibits, as expected, edgesingularities.In our consideration we have focused only on the case of small momentum and energy withthe diverging part of the 2ph form-factor controlling the leading contribution. This is also thepart of the form-factor that requires Hadamard regularization when performing the spectralsum. The subleading parts contain simple poles and regular terms. Hadamard integral oversuch terms reduces to the Cauchy principal value and regular integrals respectively. With thisin mind, we can readily evaluate the 2ph contribution of the full form-factor and not only itsleading, diverging part. Given (cid:104) ϑ |O (0) | ϑ, p , h (cid:105) with simple poles whenever p i ∼ h j we have that S ( k, ω ) = (2 π ) ˆ d h p i = h j d p |(cid:104) ϑ |O (0) | ϑ, p , h (cid:105)| δ ( ω − ω ( p , h )) δ ( k − k ( p , h ))= (2 π ) h ∗ d h ρ p ( h ) ρ p ( h ) ρ h ( h + α ) ρ h ( h + α ) k (cid:48) ( h ) k (cid:48) ( h ) | v eff ( h ) − v eff ( h ) | |(cid:104) ϑ |O (0) | ϑ, ¯ α + h , h (cid:105)| , (105)where ¯ α follows from solving the energy-momentum constraint k = k ( ¯ α + h , h ) , ω = ω ( ¯ α + h , h ) . (106)This makes it possible to implement a numerical approach to the evaluation of such contributionswhen the full thermodynamic form-factor is known, e.g for the density operator in the Lieb-Liniger model. We also observe that the approach can be extended to higher particle-holecontributions given that the singular part of the higher form-factors follows the regular pattern.While finishing this work, a new results, based on the perturbative expansion of the density-density correlation function in 1 /c , appeared [77]. The authors computed the correlation functionon any finite entropy density state up to 1 /c order. The 1 /c terms involve contributions from2ph excitations which should allow in a future to compare the results of the two approaches. Acknowledgments
The author acknowledges the support from the National Science Centre, Poland, under theSONATA grant 2018/31/D/ST3/03588.
A 1ph contribution
For the completeness, we present in this Appendix the computation of the 1ph contribution. Forthe case of the density-density correlation function these computations were performed in [25].Here we present their straightforward generalization to a two-point function of any two conserved17ensities. The relevant form-factor is given by eq. (30). The form-factor has no singularitiesand the Hadamard regularization of the spectal sum is not needed. The 1ph contribution to thecorrelation function of two conserved densities is C ij ( k, ω ) = (2 π ) ˆ d p d h ρ h ( p ) ρ p ( h ) h dr i ( h ) h dr j ( h ) δ ( k − k ( p, h )) δ ( ω − ω ( p, h )) . (107)We define α = p − h and rewrite the integration as C ij ( k, ω ) = (2 π ) ˆ d h d αρ h ( h + α ) ρ p ( h ) h dr i ( h ) h dr j ( h ) δ ( k − k (cid:48) ( h ) α ) δ ( ω − v eff ( h ) k ) . (108)Executing the integrals with the help of the δ -functions yields, C ij ( k, ω ) = 2 πϑ ( h ∗ ) ρ h ( h ∗ + ¯ α ) h dr j ( h ∗ ) h dr j ( h ∗ ) | kκ ( h ∗ ) | , (109)where h ∗ is a unique solution to ω = v eff ( h ) k , ¯ α = k/k (cid:48) ( h ∗ ) and κ ( h ) = ∂ h v eff ( h ). We have alsoused that k (cid:48) ( h ) = 2 πρ t ( h ). For a smooth distribution of rapidities the shift ¯ α can be neglected,which results in C ij ( k, ω ) = 2 πϑ ( h ∗ ) ρ h ( h ∗ ) h dr i ( h ∗ ) h dr j ( h ∗ ) | kκ ( h ∗ ) | , (110)with the dependence on k and ω given implicitly by h ∗ such that ω = v eff ( h ∗ ) k . This represen-tation makes it easy to compute the real space-time correlation function (strictly speaking the1ph contribution to it) with the result C ij ( x, t ) = ϑ (˜ h ) ρ h (˜ h ) h dr i (˜ h ) h dr j (˜ h ) | tκ (˜ h ) | , (111)with ˜ h the unique solution to x = v eff (˜ h ) t .At the ground state, the rapidities distribution have a sharp edge at Fermi rapidity q F , ρ p ( λ ) = ρ t ( λ ) (1 − Θ( | λ | − q F )) ,ρ h ( λ ) = ρ t ( λ )Θ( | λ | − q F ) , (112)Therefore the product ϑ ( h ∗ ) ρ h ( h ∗ ) = 0 and the non-zero contribution comes from the shift ¯ α .Consider first the case of positive momentum k >
0. Then the particle-hole excitation must belocalized at the right Fermi edge: h ∗ ≤ q F and h ∗ + ¯ α ≥ q F . We have then ϑ ( h ∗ ) ρ h ( h ∗ + ¯ α ) = ¯ αρ t ( q F ) δ ( h ∗ − q F ) + ( . . . ) = k π δ ( h ∗ − q F ) + ( . . . ) , (113)where ( . . . ) stands for terms of measure zero and terms of higher order in ¯ α . This yields thefollowing contribution to the correlation function h dr i ( q F ) h dr j ( q F ) | κ ( h ∗ ) | δ ( h ∗ − q F ) = | k | h dr i ( q F ) h dr j ( q F ) δ ( ω − v F k ) , (114)where we used that 1 | κ ( q F ) | δ ( h ∗ − q F ) = δ ( v eff ( h ∗ ) − v F ) = | k | δ ( ω − v F k ) , (115)where v F = v eff ( q F ) is the Fermi velocity. 18n a similar fashion, for k <
0, the excitation is localized at the left Fermi edge: h ∗ ≥ − q F and h ∗ + α ≤ − q F and then ρ p ( h ∗ ) ρ h ( h ∗ + ¯ α ) = − ¯ αρ t ( q F ) δ ( h ∗ + q F ) + ( . . . ) = | k | π δ ( h ∗ + q F ) , (116)with the contribution to the correlation function | k | h dr i ( q F ) h dr j ( q F ) δ ( ω + v F k ) . (117)Adding the two contributions we find C ij ( k, ω ) = | k | h dr i ( q F ) h dr j ( q F ) δ ( ω − v F | k | ) . (118)Finally, specializing to the DSF, h dr i ( λ ) = 2 πρ t ( λ ), and using, 2 πρ t ( q F ) = ( N/L ) v − F we find S ( k, ω ) = 2 π NL | k | v F δ ( ω − v F | k | ) . (119)The static correlator S ( k ) = ˆ d ω π S ( k, ω ) = NL | k | v F , (120)in agreement with the results of [25]. B From the TBP to form-factors in the Lieb-Liniger model
In this Appendix, we conjecture the singular structure of the density operator form-factor in theLieb-Liniger model with higher number of particle-hole excitations from the ThermodynamicBootstrap Program [27, 28]. We adopt a shorthand notation for the form-factors f O ϑ ( θ , . . . , θ n ) = (cid:104) ϑ |O (0) | ϑ, θ , . . . , θ n (cid:105) . (121)The thermodynamic bootstrap program for Integrable Quantum Field Theories predicts thesingular structure of the form-factors when rapidities of the particles and holes coincide. Inthe relativistic notation, with θ being the rapidity, a form-factor of a local operator O with 4particles has the following singular structure f O ϑ ( θ , θ , θ , θ ) = − π (cid:88) σ ∈ P S σ ( θ , . . . , θ ) θ σ − θ σ − iπθ σ − θ σ − iπ T dr ( θ σ , θ σ ) f O ϑ ( θ σ θ σ ) . (122)We recall that shifting a rapidity by iπ , in a relativistic theory, corresponds to transformingit into a hole. For the correspondence with the Lieb-Liniger model, we choose θ = h + iπ , θ = p , θ = h + iπ and θ = p . This reduces the diverging part from 24 terms to 4 terms.We can moreover assume that p i ∼ h i so there are only 2 diverging terms. With these choicesonly two permutations matter: σ = (1234) and σ = (3412) = (23)(12)(34)(23). The latter leadsto the following combination of the S -matrices S (3412) ( θ , . . . , θ ) = S ( p − h − iπ ) S ( p − p ) S ( h − h ) S ( h + iπ − p )= S ( p − p ) S ( h − h ) S ( p − h ) S ( h − p ) . (123)This expression, in the leading order in p i ∼ h i , is equal to 1. Therefore f O ϑ ( p , h , p , h ) = − πT dr ( p , p ) (cid:18) h − p h − p f O ϑ ( h , p ) + h − p h − p f O ϑ ( h , p ) (cid:19) . (124)19o simplify the notation we assumed here that T dr ( p , p ) is a symmetric function of its argu-ments.Formally, the expression (124) is valid in the relativistic theory and to make a connection withthe Lieb-Liniger model one would need to consider an appropriate non-relativistic limit [78, 79].Instead, we will assume that in all the terms we can simply substitute the corresponding Lieb-Liniger quantities. We will see that this leads to the correct result for the 2ph form-factor andconjecturally also for higher ones. The only caveat, is the difference in the normalization of theform factors which in the Lieb-Liniger model are additionally divided by (2 πρ s ( λ )) / for eachparticle or hole rapidity λ . Therefore, in the limit p i ∼ h i each particle-hole pair leads to theadditional factor 2 πρ s ( h i ) = k (cid:48) ( h i ). Denoting Lieb-Liniger form-factors with a bar we find¯ f O ϑ ( p , h , p , h ) = − πT dr ( p , p ) (cid:18) h − p k (cid:48) ( h )( h − p ) ¯ f O ϑ ( h , p ) − h − p k (cid:48) ( h )( h − p ) ¯ f O ϑ ( h , p ) (cid:19) . (125)Recall that for p i ∼ h i , k = k (cid:48) ( h )( p − h ) + k (cid:48) ( h )( p − h ) . (126)The terms h i − p i in the numerators, in the leading order of the diverging part can be thenreplaced by k/k (cid:48) ( h i ). This gives¯ f O ϑ ( p , h ) = 2 πk ( p , h ) T dr ( p , p ) (cid:18) ¯ f O ϑ ( h , p ) k (cid:48) ( h ) k (cid:48) ( h )( h − p ) + ¯ f O ϑ ( h , p ) k (cid:48) ( h ) k (cid:48) ( h )( h − p ) (cid:19) . (127)Finally, specializing to the conserved charge density, in the limit of small excitation ¯ f q j ϑ ( h, p ) = h dr j ( h ), we find the singular part of the form-factor reported in (31) for p i ∼ h i . The fullexpression (31) can be recovered by separately symmetrizing the labelling of holes and particles.For 3 particle-hole form factors, the TBP predicts the following singular structure in thelimit of small ph excitations [28] f O ϑ ( θ + iπ + κ , θ , θ + iπ + κ , θ , θ + iπ + κ , θ ) == (2 π ) (cid:18) κ κ + κ κ κ (cid:19) T dr ( θ , θ ) f O ϑ ( θ + iπκ , θ ) + (cycl. perm.) , (128)where (cycl. perm.) refers to independent cyclic permutation of indices { , , } for particles andholes. Isolating the most diverging part and following the same steps as for the 2ph form-factorwe conjecture the following expression for the Lieb-Liniger model¯ f O ϑ ( p , h ) = (2 πk ( p , h ) k (cid:48) ( h ) k (cid:48) ( h ) k (cid:48) ( h ) (cid:18) T dr ( p , p ) T dr ( p , p ) ¯ f O ϑ ( h , p )( p − h )( p − h ) + (cycl. perm.) (cid:19) . (129)Specializing to the conserved charge density, in the limit of small excitation ¯ f q j ϑ ( h, p ) = h dr j ( h ),we find the form-factor reported in (32). C Hadamard representation of the spectral sum
In [55] it was argued that the spectral sum over the 2ph excitations can be brought to theHadamard form by an appropriate redefinition of the thermodynamic form-factor. In this ap-pendix, we show that such redefinition is not necessary and the Hadamard regularization appearsautomatically when considering the thermodynamic limit of the spectral sum. We study thethermodynamic limit of the spectral sum following the approach introduced in [15].We consider the 2ph contribution to the finite-size correlation function, in which we droppedthe subleading contributions in the system size, S L ( x, t ) = 14 1 L (cid:88) p ,p ,h ,h e ik ( p , h ) x − iω ( p , h ) t |(cid:104) ϑ | ˆ ρ (0) | ϑ, p , h (cid:105)| . (130)20he summations extends over possible particle-hole excitations over a chosen discretization ofthe thermodynamic state | ϑ (cid:105) with N particles. The thermodynamic limit of S L ( x, t ) does notdepend on this choice of discretization. To rewrite the sum over holes as an integration weintroduce the counting function Q p ( λ ) defined as Q p ( λ ) = L ˆ λ −∞ d λ (cid:48) ρ p ( λ (cid:48) ) , (131)with the defining property that Q p ( λ j ) = Lj for λ j the j -th rapidity in the chosen discretizationof | ϑ (cid:105) . With the help of the counting function we rewrite the sum over h as1 L (cid:88) h F ( h ) = 1 L N (cid:88) j =1 ˛ γ j d z F ( z ) Q (cid:48) p ( z ) e πiQ (cid:48) ( z ) −
1= 1 L ˛ Γ F ( z ) Q (cid:48) p ( z ) e πiQ p ( z ) − − πiL (cid:88) k ∈{ , } res z → p k (cid:18) F ( z ) Q (cid:48) p ( z ) e πiQ p ( z ) − (cid:19) , (132)where we used a shorthand notation F ( h ) ≡ e ik ( p , h ) x − iω ( p , h ) t |(cid:104) ϑ | ˆ ρ (0) | ϑ, p , h (cid:105)| . The positive-oriented contour γ j encircles λ j for each j , while avoiding p j . In the second step, we combinedall contours γ j into a single contour Γ going around the real axis. In the process, we crossedthe annihilation poles at z = p j and thus had to subtract their contributions. We are now inthe position to take the thermodynamic limit. To this end observe that Q (cid:48) p ( z ) = Lρ p ( z ) and wedefine g L ( z ) = (cid:16) e πiQ p ( z ) − (cid:17) − . (133)From Q p ( x + i(cid:15) ) ≈ Q (cid:48) p ( x ) + i(cid:15)Lρ p ( x ) + O ( (cid:15) ) it follows thatlim th g L ( x + i(cid:15) ) = (cid:40) − , if (cid:15) > , , if (cid:15) < , (134)and the first term in (132) giveslim th (cid:18) L ˛ Γ F ( z ) Q (cid:48) p ( z ) g L ( z ) (cid:19) = ˆ R + i(cid:15) d z F ( z ) ρ p ( z ) . (135)For the computation of the second term we first interchange the thermodynamic limit withtaking the residue lim th res z → p k ( F ( z ) ρ p ( z ) g L ( z )) = res z → p k (cid:18) F ( z ) ρ p ( z ) lim th g L ( z ) (cid:19) . (136)To compute the residue we need to know the thermodynamic limit of g L ( z ) for real z = p k ,which eq. (134) does not provide. We can work it out around by representing g L ( z ) through thecontour integral g L ( z ) = 12 πi ˛ γ d z (cid:48) g L ( z (cid:48) ) z (cid:48) − z . (137)This amounts to assigning to g L ( z ) the average value of g L ( z (cid:48) ) for z (cid:48) in the neighbourhood of z . For z in the vicinity of the real number p k the contour γ has to encircle p k and thereforeextends above and below the real line. For large L , only the part of the contour above the realline, denoted γ + , contributes and we findlim th g L ( z ) = − πi ˆ γ + d z (cid:48) z (cid:48) − z = − , (138)21herefore, lim th (cid:18) res z → p k ( F ( z ) ρ p ( z ) g L ( z )) (cid:19) = −
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