Correlations of zero-entropy critical states in the XXZ model: integrability and Luttinger theory far from the ground state
SSciPost Physics Submission
Correlations of zero-entropy critical states in the XXZ model:integrability and Luttinger theory far from the ground state
R. Vlijm, I.S. Eli¨ens, J.-S. CauxInstitute for Theoretical Physics, University of Amsterdam,Science Park 904, 1098 XH Amsterdam, The NetherlandsNovember 13, 2018
Abstract
Pumping a finite energy density into a quantum system typically leads to ‘melted’ states char-acterized by exponentially-decaying correlations, as is the case for finite-temperature equilib-rium situations. An important exception to this rule are states which, while being at highenergy, maintain a low entropy. Such states can interestingly still display features of quantumcriticality, especially in one dimension. Here, we consider high-energy states in anisotropicHeisenberg quantum spin chains obtained by splitting the ground state’s magnon Fermi seainto separate pieces. Using methods based on integrability, we provide a detailed study ofstatic and dynamical spin-spin correlations. These carry distinctive signatures of the Fermisea splittings, which would be observable in eventual experimental realizations. Going further,we employ a multi-component Tomonaga-Luttinger model in order to predict the asymptoticsof static correlations. For this effective field theory, we fix all universal exponents from ener-getics, and all non-universal correlation prefactors using finite-size scaling of matrix elements.The correlations obtained directly from integrability and those emerging from the Luttingerfield theory description are shown to be in extremely good correspondence, as expected, forthe large distance asymptotics, but surprisingly also for the short distance behavior. Finally,we discuss the description of dynamical correlations from a mobile impurity model, and clarifythe relation of the effective field theory parameters to the Bethe Ansatz solution.
Contents a r X i v : . [ c ond - m a t . s t r- e l ] S e p ciPost Physics SubmissionA Simplification for symmetric seas 23B Moses states and generalized TBA 24C Perturbative expressions for effective-field-theory parameters 24References 25 The ground states and low-lying excitations of one-dimensional many-body quantum systemsoften display interesting features associated to the Tomonaga-Luttinger liquid universalityclass [1,2], notable examples being one-dimensional quantum gases and spin chains. Bosoniza-tion techniques [3, 4] then provide a route for the description of the long-range asymptoticsof basic correlations. One-dimensional quantum physics is also famous for possessing a classof integrable theories for which all eigenstates can be exactly obtained by Bethe Ansatz [5].Prototypical examples of Bethe Ansatz solvable models are the Lieb-Liniger model [6, 7] ofdelta-interacting bosons and the Heisenberg spin chain [8, 9]. For these and other integrablemodels, the algebraic Bethe Ansatz [10] and more specifically Slavnov’s theorem [11, 12] pro-vides a method to compute matrix elements of local operators between two Bethe states.Summations over relevant matrix elements of particle-hole like excitations can then be imple-mented, leading to efficient quantitative evaluations of dynamical correlation functions andexpectation values of local operators.The correspondence between results from integrability on the one hand, and Tomonaga-Luttinger theory on the other, are well-known for ground states of the above-mentioned mod-els. The bosonized Tomonaga-Luttinger liquid descriptions are characterized by universalLuttinger parameters related to the compressibility and sound velocity, which can be fittedfrom the energy levels of the model at finite system size computed by alternative computationalmethods [13]. For example, the compressibility is readily fitted by the energy levels calcu-lated from integrability upon addition or removal of particles from the Fermi sea. Moreover,analysis of the system size scaling of matrix elements of Umklapp states [14–17] calculatedby algebraic Bethe Ansatz methods allows for the determination of non-universal exponentsand prefactors appearing in correlation functions of the bosonized theory.It is interesting to ask whether these correspondences between exact results and field-theory predictions are strictly limited to the vicinity of the ground state, or whether otherregions of Hilbert space can similarly be ‘captured’ both by integrability and by an appropriatefield theory. From integrability, things look promising: since the Bethe Ansatz does not carewhether the wave functions one writes are near or far from the ground state (these remainexact, irrespective of which energy they have), correlations on such high-energy states can inprinciple be obtained by extensions of existing ground state-based summation methods. Onthe other hand, the applicability of Tomonaga-Luttinger theory relies on the linearity of thedispersion relations of the effective excitations around the given state, which cannot be genericfor high-energy states. Here, we concentrate on a class of states for which this bosonization2 ciPost Physics Submission procedure can be applied. The setup is easy to visualize: starting from the ground state’sFermi sea configuration, we give a finite opposite momentum to two groups of particles, infact splitting the Fermi sea in two seas [18], yielding what we coin a ‘Moses’ state which,while having a macroscopic energy above the ground state, still possesses a zero entropydensity (and thus a potential for displaying critical power-law behavior in its correlations.The corresponding multi-component Tomonaga-Luttinger liquid theory and its correlationswere recently obtained for the Lieb-Liniger model [19, 20]. The entanglement entropy of thesestates is nontrivial [21] and consistent with predictions from the effective conformally invariantfield theory.The physical motivation to study such states with a double Fermi sea with opposite mo-mentum kicks originates from experiments where a Bragg pulse is applied to a gas of interact-ing bosons to create an initial state with counterpropagating particles, leading to a quantumrealization of the famous Newton’s cradle [22]. It has recently been shown that late-time cor-relations in the quantum Newton’s cradle experiment are better described by a finite entropyversion of these states, which can be constructed in the Lieb-Liniger model by theoreticallymimicking the application of a Bragg pulse on the ground state and for which the peaks aresmoothened and show the characteristic ghost-shaped momentum distribution function at latetimes [23]. The zero-entropy nature of the states we consider here translates into a quasi-condensate momentum distribution with sharp peaks at finite momenta similar to the onesobserved in cold atoms after domain-wall melting of a one-dimensional Mott insulator [24]. Atpresent it is not clear whether a connection exists between the steady state for this quantumquench and the split-Fermi-sea states we consider. Finally, another motivation is related tospin ladders, where states with a Fermi sea consisting of distinct pockets can appear as theground state [25, 26]. The comparison of their work with ours puts into sharp focus whichcharacteristics should be attributed to the out-of-equilibrium nature of the state and whichshould not.The aim of the present article is to extend the approach for the Bose gas with doubleFermi seas elaborated in Ref. [20] to the anisotropic Heisenberg spin chain. We compute thedynamical structure factor for this class of excited zero-entropy states from integrability. Thestatic spin-spin correlations are subsequently studied from both the matrix element summa-tion approach from algebraic Bethe Ansatz and the multi-component Tomonaga-Luttingerpredictions supported by parameter fitting from integrability, which show surprisingly goodcorrespondence for both the long-range asymptotics and short distances. Going further, wealso extend recent advances in the computation of time-dependent correlations by means ofmobile impurity models and apply these extensions here. The correspondence with the effec-tive field theory becomes more delicate in this case, but if the separation between seas is largeenough the method still gives adequate results.This article is structured in the following way. Section 2 describes the setup from BetheAnsatz of the zero-entropy critical states consisting of a state with two Fermi seas, while thedynamical structure factor of these states are evaluated using algebraic Bethe Ansatz matrixelement summations in section 3. Section 4 gives the multi-component Tomonaga-Luttingerliquid approach to the real space correlations, which are compared to the algebraic BetheAnsatz results in section 5. Finally, a description of dynamical correlations by a mobileimpurity model is given in section 6. 3 ciPost Physics Submission
The Hamiltonian of the anisotropic Heisenberg spin chain (XXZ model) in a longitudinalmagnetic field is given as [8, 9] H = J N (cid:88) j =1 (cid:20) S xj S xj +1 + S yj S yj +1 + ∆ (cid:18) S zj S zj +1 − (cid:19) − hS zj (cid:21) , (1)with periodic boundary conditions S N +1 = S . We fix J ∆ > | ∆ | <
1. The Tomonaga-Luttinger theory is applicable to these regimes,allowing for a comparison of both approaches.The commutation of the total spin operator along the z -axis S z tot = (cid:80) Nj =1 S zj with Hamil-tonian (1) assures a splitting of the Hilbert space in sectors of fixed magnetization M . Insuch a fixed- M subsector, the Bethe Ansatz wave functions [5, 9] are constructed from thefully polarized vacuum state | (cid:105) = ⊗ Nj =1 | ↑ j (cid:105) as plane waves of magnons |{ λ }(cid:105) = (cid:88) j <... 1. The momentum in Eq. (2) is expressible as p ( λ ) = π − θ ( λ ).The Bethe quantum numbers J j are (half-odd) integers for N + M (even) odd, and formthe starting point of the construction of Bethe states at finite system size. The set of rapidities { λ } can be obtained by solving the logarithmic Bethe equations numerically by an iterativeprocedure for a given set of non-coinciding quantum numbers { J } . With the set of rapiditiesof a Bethe state at hand, properties of the state such as its energy can be calculated directly, E = − J ϕ ∆ M (cid:88) j =1 θ (cid:48) ( λ j ) − h (cid:18) N − M (cid:19) , (4)with ϕ ∆ = 1 for ∆ = 1 and ϕ ∆ = sin( ζ ) for | ∆ | < 1, while the momentum of a Bethe state isexpressed directly in terms of its quantum numbers as q = πM − πN M (cid:88) j =1 J j . (5)In general, rapidities can take on complex values, while the full set { λ } solving Betheequations must remain self-conjugate, leading to an arrangement of the rapidities in terms of4 ciPost Physics Submission string solutions. String solutions will not be considered for the initial state consisting of twoseparate Fermi seas of real rapidities. However, string solutions can occur in the intermediatestates of the matrix element summations described in section 3.The sets of quantum numbers specifying the Bethe states are bounded by limiting quantumnumbers J ∞ , which are derived by taking the limit of one of the rapidities to infinity andcomputing the associated quantum number from the Bethe equations (3). The maximumallowed quantum number to still give a finite valued rapidity is then J max = J ∞ − 1. Fora Bethe state consisting of M finite real rapidities, one obtains J ∞ = ( N − M + 1) and J max = ( N − M − J max + 1 = N − M available possibilities todistribute M quantum numbers.The ground state in both zero and finite magnetic field consists of real rapidities [27],where the magnetization sector M (and therefore also the number of rapidities) is fixed bythe magnetic field, with the field strength h acting as a Lagrange multiplier. The quantumnumbers of the ground state are J GS j = − M + 12 + j for 1 ≤ j ≤ M. (6)At zero field ( M = N/ | Φ s (cid:105) with separated Fermi seas at magnetization M by applying ashift s to the ground state quantum numbers as (we take M to be even) J Φ s j = J GS j − s if 1 ≤ j ≤ M ,J GS j + s if M < j ≤ M. (7)The effect of shifting the quantum numbers, separating the Fermi sea, is illustrated in Fig. 1.We like to call such a state a Moses state. { J GS j } ◦ ◦ ◦ ◦ • • • • | • • • • ◦ ◦ ◦ ◦{ J Φ s j } ◦ ◦ • • • • ◦ ◦ | ◦ ◦ • • • • ◦ ◦ Figure 1: Illustration of the distribution of quantum numbers for the ground state (top) andthe state with split Fermi sea (bottom) at finite magnetic field. The size of the gap betweenthe two Fermi seas is 2 s holes.Four Fermi points are identified, labeled by i, j, k = 1 , a, b, c = 1 , k ia = πN J ia , while signs for each left or right Fermi point in a sea are definedas s = − s = 1. With these notations, generalization to n > ciPost Physics Submission The dynamical structure factor constitutes an important connection between inelastic neutronscattering experiments and theory for quantum spin systems. It is directly related to thedifferential cross section from scattering experiments on the one hand, while it is computabletheoretically from both analytical and (exact) numerical methods. The dynamical structurefactor is defined as the Fourier transform of the spin-spin correlation S a ¯ a ( q, ω ) = 1 N N (cid:88) j,j (cid:48) e − iq ( j − j (cid:48) ) (cid:90) ∞−∞ d t e iωt (cid:104) S aj ( t ) S ¯ aj (cid:48) (0) (cid:105) c , (8)where the label a = z, ± distinguishes the longitudinal and transversal structure factorsrespectively. Transverse structure factors have been computed using field theoretical methodsin Ref. [28].For spin chains at zero temperature, the expectation value in the expression for the spin-spin correlation is evaluated with respect to the ground state of Hamiltonian (1). In thecontext of computations for the state with split Fermi sea, the reference state is taken to bethe state | Φ s (cid:105) defined from the quantum numbers given by Eq. (7).The dynamical structure factor can be evaluated by inserting a resolution of the identityin Eq. (8), such that the correlator turns into a sum over matrix elements of the split Fermisea state and its excitations, S a ¯ a ( q, ω ) = 2 π (cid:88) α |(cid:104) Φ s | S aq | α (cid:105)| δ ( ω + ω Φ s − ω α ) . (9)The intermediate states | α (cid:105) are composed of excitations on the state with double Fermiseas. An adaptive scanning procedure through the most relevant intermediate states is ap-plied using the ABACUS algorithm [29] in order to evaluate the dynamical structure factor.The rapidities of the intermediate states can be obtained by solving Bethe equations (3) withcorresponding quantum numbers by an iterative numerical procedure. Subsequently, deter-minant expressions in terms of the rapidities [30–33] based on Slavnov’s formula [11, 12] areemployed to evaluate the matrix elements (cid:104) Φ s | S aq | α (cid:105) . Besides intermediate states consisting ofreal rapidities, the states | α (cid:105) can also contain string solutions. In this case, the Bethe-Gaudin-Takahashi equations [34] in terms of string centers are solved, while their matrix elements canbe computed by using reduced determinant expressions [33].By integrating over energy and summing over all momenta for Eq. (8), sum rules for thetotal intensities of the structure factors can be derived, yielding a quantitative measure onthe completeness of the truncated matrix element summations. The sum rules are given by (cid:90) ∞−∞ dω π N (cid:88) q S zz ( q, ω ) = 14 − (cid:104) S z (cid:105) , (10) (cid:90) ∞−∞ dω π N (cid:88) q S ±∓ ( q, ω ) = 12 ± (cid:104) S z (cid:105) , (11)where the magnetization is (cid:104) S z (cid:105) = − MN . Sum rule saturations for all ABACUS computationresults of dynamical structure factors used throughout this article are listed in Tab. 1. Thetable displays significantly lower sum rule saturations for the S + − structure factor compu-tations as opposed to the S zz and S − + structure factors. An intuitive explanation for the6 ciPost Physics Submission computational difficulties of S + − is provided in the Bethe Ansatz language, where the op-erator S + q removes a rapidity from the state on which it acts, while the operator S − q mustadd a rapidity to the state. For the latter, it might however be the case that all availablequantum number slots around momentum q are already filled, such that much more extensivereorganizations of the state are necessary. Thus, computing S + − requires summing over amuch more extensive set of intermediate states than for S − + . Given limited computationalresources, the saturation of the former will thus be markedly lower than those of the latter. s S zz S − + S + − ∆ = . 50% 99 . 50% 98 . . 50% 98 . 52% 91 . . 50% 97 . 73% 89 . . 16% 95 . 27% 85 . . 50% 99 . 50% 98 . . 73% 99 . 49% 94 . . 05% 99 . 90% 90 . . 00% 98 . 16% 88 . . 38% 99 . 50% 94 . . 04% 99 . 50% 89 . . 00% 99 . 12% 87 . . 80% 98 . 80% 86 . l , r . 67% 98 . N = 200and M = 50 for various values of the anisotropy and the momentum shift in the Fermi seas.The bottom row shows the saturations for an asymmetric shift of the quantum numbers.The longitudinal dynamical structure factors (Eq. (8) with a = z ) are shown in Fig. 2 forvarious values of the anisotropy, containing plots for the ground state, as well as for states witha double Fermi sea with varying momentum shifts in the quantum numbers. The transversedynamical structure factors (Eq. (8) with a = − and a = +) have been plotted in Figs. 3and 4, respectively.In order to be able to visualize the delta functions in energy appearing in Eq. (9), Gaussiansmoothening has been applied to the data as δ ( ω ) → e − ω /(cid:15) / ( √ π(cid:15) ), the width (cid:15) being of theorder of 1 /N .The boundaries of the spectra shown in Fig. 2 can be explained by tracking the energiesof particle-hole excitations on top of the Moses sea for which either the particle or hole iscreated at one of the Fermi points. The boundaries of the spectra are equivalent to Fig. 2 inRef. [20], for the particle-hole excitations for the double Fermi sea in the Lieb-Liniger model.In particular, for increasing momentum split s , the energy of the reference state increaseswith respect to the ground state, opening up the possibility to populate branches of the spec-trum at negative energy. Moreover, the incommensurate points (at zero energy) start movingin momentum. Broad continua of the spectrum remain with sharply defined thresholds, suchthat the Tomonaga-Luttinger liquid paradigm retains its validity.7 ciPost Physics Submission − / , s = 0 ∆ = 1 / , s = 0 ∆ = 1 , s = 0 − / , s = 2 ∆ = 1 / , s = 2 ∆ = 1 , s = 2 − / , s = 6 ∆ = 1 / , s = 6 ∆ = 1 , s = 6 − π/ π π/ 2∆ = 1 / , s = 12 0 π/ π π/ 2∆ = 1 / , s = 12 0 π/ π π/ π ∆ = 1 , s = 12 ω S zz ( q , ω ) ω S zz ( q , ω ) ω S zz ( q , ω ) ω q q q S zz ( q , ω ) Figure 2: Longitudinal dynamical structure factor S zz ( q, ω ) at N = 200, M = 50 computedfrom summations of matrix elements obtained from algebraic Bethe Ansatz. From left toright, the anisotropies are ∆ = , , 1. From top to bottom, the momentum shifts in thequantum numbers are s = 0 , , , 12. The corresponding sum rule saturations of the data arelisted in Tab. 1. 8 ciPost Physics Submission − 101 ∆ = 1 / , s = 0 ∆ = 1 / , s = 0 ∆ = 1 , s = 0 − 101 ∆ = 1 / , s = 2 ∆ = 1 / , s = 2 ∆ = 1 , s = 2 − 101 ∆ = 1 / , s = 6 ∆ = 1 / , s = 6 ∆ = 1 , s = 6 − π/ π π/ 2∆ = 1 / , s = 12 π/ π π/ 2∆ = 1 / , s = 12 π/ π π/ 2∆ = 1 , s = 12 ω S − + ( q , ω ) ω S − + ( q , ω ) ω S − + ( q , ω ) ω q q q S − + ( q , ω ) Figure 3: Transverse dynamical structure factor S − + ( q, ω ) at N = 200, M = 50 computedfrom summations of matrix elements obtained from algebraic Bethe Ansatz. From left toright, the anisotropies are ∆ = , , 1. From top to bottom, the momentum shifts in thequantum numbers are s = 0 , , , 12. The corresponding sum rule saturations of the data arelisted in Tab. 1. 9 ciPost Physics Submission / , s = 0 = 1 / , s = 0 = 1 , s = 0 / , s = 2 = 1 / , s = 2 = 1 , s = 2 / , s = 6 = 1 / , s = 6 = 1 , s = 6 ⇡/ ⇡ ⇡/ / , s = 12 0 ⇡/ ⇡ ⇡/ / , s = 12 0 ⇡/ ⇡ ⇡/ ⇡ = 1 , s = 12 ! S + ( q , ! ) ! S + ( q , ! ) ! S + ( q , ! ) ! q q q S + ( q , ! ) Figure 4: Transverse dynamical structure factor S + − ( q, ω ) at N = 200, M = 50 computedfrom summations of matrix elements obtained from algebraic Bethe Ansatz, which is further-more symmetrized around momentum q = π in order to obtain higher sum rule saturationswith lower computational time. From left to right, the anisotropies are ∆ = , , 1. Fromtop to bottom, the momentum shifts in the quantum numbers are s = 0 , , , 12. The corre-sponding sum rule saturations of the data are listed in Tab. 1, which are substantially lowerthan the other structure factors. 10 ciPost Physics Submission The Hamiltonian of the XXZ model can be mapped exactly to spinless fermions on a latticeby the Jordan-Wigner transformation S − j → ( − j cos π (cid:88) l 2. We will adhereto the equilibrium terminology and call these marginal while terms of higher scaling dimensionare called irrelevant. 11 ciPost Physics Submission Note that by considering the momentum of particle or hole excitations constructed bycreating a hole or particle at the Fermi points k ia in quantum number space, it is clear thatthe k ia do not change when we vary the interaction parameter ∆, as the total momentum iscompletely determined by the quantum number configuration. We thus observe that a kindof generalized Luttinger’s theorem fixes the k ia independent of interactions.We bosonize the chiral fermions according to ψ ia ∼ √ π e − i √ πφ ia , ρ ia = − s a √ π ∂ x φ ia (16)(where s R,L = ± φ ia = (cid:88) ia,jb U ia,jb ϕ jb . (17)This results in the diagonal form of the effective Hamiltonian H TL = (cid:88) ia s a v ia π (cid:90) dx ( ∂ x ϕ ia ) (18)where the effective velocities v ia are now related to the dressed dispersion of the XXZ modelwith ∆ (cid:54) = 0. While the interaction parameters g ia,jb cannot be reliably obtained, the Bogoli-ubov parameters U ia,jb —which also determine the exponents of physical correlation functions—are related to finite-size energy contributions when we extend the filled quantum numberblocks by N ia particles at Fermi point k ia . The correction δE = E [ { N ia } ] − E [ { } ] is then toorder 1 /N given by δE = (cid:88) ia (cid:15) ia N ia + (cid:88) ia,jb,kc πN s c v kc U ia,kc U jb,kc N ia N jb . (19)Here N ia is the number of added (or removed when N ia < 0) particles corresponding to chiralspecies ψ ia and (cid:15) ia is the energy associated to Fermi point k ia . Eq. (19) gives a relationbetween the U ia,jb and the finite-size energy differences upon addition or removal of a particleat the Fermi points k ia , k jb . Thanks to the properties of the matrix U ia,jb , this relationcan in fact be inverted and leads to a way to determine U ia,jb and v ia directly from thesefinite-size corrections. The U ia,jb generalizes the universal compressibility parameter K usedin equilibrium situations. In the case of a symmetric combination the derivation may besimplified as detailed in appendix A.The Bogoliubov parameters U ia,jb have a beautiful interpretation in terms of the phaseshifts of the modes at Fermi point k ia upon addition of a particle at Fermi point k jb . Thiscan be argued upon refermionization of the effective Tomonaga-Luttinger theory and can bemade precise in terms of the shift function F ( λ | λ (cid:48) ) describing the change of the rapiditieswhen the system is excited. In the thermodynamic limit the shift function is determined bythe integral equation F ( λ | λ (cid:48) ) + (cid:88) j (cid:90) λ j λ j dµ a ( λ − µ ) F ( µ | λ (cid:48) ) = θ ( λ − λ (cid:48) )2 π (20)12 ciPost Physics Submission where a j ( λ ) = (2 π ) − ddλ θ j ( λ ). The relation to the Bogoliubov parameters is then U ia,jb = δ ia,jb + s b F ( λ jb | λ ia ) , (21)which can be shown by comparing the finite size corrections to the energy. A derivationof this relation will be presented elsewhere [38] (also see appendix B). In equilibrium it isknown [39] and the shift function plays an important role in going beyond the Luttingerliquid approximation in computing dynamic correlation functions [40, 41].Physical correlations generally translate into (products of) two-point functions of vertexoperators in bosonized language, which in our conventions are evaluated according to (cid:104) e iα √ πϕ ia ( x ) e − iα √ πϕ ia (0) (cid:105) = ( s a i/x ) α . (22)Here, x is measured in units of the lattice spacing.Asymptotes of spin correlation functions are now obtained by applying the Jordan-Wignertransformation, taking the continuum limit and using the bosonization identities in order toobtain a correlator of the bosonic fields φ ia . The Bogoliubov transformation then expressesthis in terms of the free fields ϕ ia for which correlation functions are easily evaluated, leadingto an expression involving the U ia,jb .For example, the prediction for the real space longitudinal correlation from the multi-component Tomonaga-Luttinger model is (cid:104) S z ( x ) S z (0) (cid:105) TL = s z − (cid:80) ia,jb,kc s a s b U ia,kc U jb,kc π x (23)+ (cid:88) ia (cid:54) = jb A ia,jb π ( − δ ab (1 − δ ij ) cos[( k ia − k jb ) x ] (cid:18) x (cid:19) µ ia,jb (24)with µ ia,jb = (cid:88) kc ( U ia,kc − U jb,kc ) . (25)Here, non-universal prefactors A ia,jb are included for the fluctuating terms and behave as1+ O (∆). These do not follow from the Tomonaga-Luttinger construction but can be obtainedfrom the finite-size scaling of matrix element detailed in the next section.The real space transverse correlation is given from the multi-component Luttinger theoryas (cid:104) S − ( x ) S + (0) (cid:105) TL = (cid:88) ia (cid:88) (cid:15) = ± B ia,(cid:15) π ( − δ sa,(cid:15) e − ik ia,(cid:15) x (cid:18) x (cid:19) µ ia,(cid:15) (26)where k ia,(cid:15) = k ia + (cid:15)πM/N and again non-universal prefactors, now denoted B ia,(cid:15) , are included.The exponents are µ ia,(cid:15) = (cid:88) kc (cid:88) jb ( (cid:15)/ s a δ ia,kc ) U jb,kc . (27)In the next section we discuss how all parameters are obtained from numerical evaluation of thespectrum and matrix elements and how these predictions compare to real-space correlationsobtained from the ABACUS data. 13 ciPost Physics Submission The real space spin-spin correlations of the double Fermi sea state are directly obtained fromthe ABACUS dynamical structure factor data from section 3 by applying an inverse Fouriertransform, (cid:104) S aj ( t ) S ¯ a (0) (cid:105) = 1 N (cid:88) α |(cid:104) Φ s | S aq α − q Φ s | α (cid:105)| e i ( q α − q Φ s ) j − i ( ω α − ω Φ s ) t . (28)All computations have been carried out at system size N = 200 at half-magnetization S z = N − M , M = 50, for various values of anisotropy and momentum shifts in the Fermi seas. Thesum rule saturations of all ABACUS data used in this section are listed in Tab. 1. The resultsfor the static real space correlations ( t = 0) are plotted as data points in Figs. 7-8 (longitudinal, a = z ) and Figs. 9-10 (transverse, a = − ) respectively. The multi-component Tomonaga-Luttinger model predictions for the correlations with fitted parameters from integrabilityare incorporated in the figures as well. The predictions for real-space correlations from thetransverse structure factor with a = + differ only at x = 0 from a = − . Although the sum-rule saturation for this correlation (see Tab. 1) is considerably less, we have checked that thefit of the real-space correlation is comparable to the case a = − .The longitudinal real space correlation from Eq. (24) requires the determination of threeclasses of parameters. First, the parameters U ia,jb can be deduced from the behavior of theenergy upon removal or addition of particles to all four Fermi points. We therefore considerthe second derivative of the finite size corrections to the Tomonaga-Luttinger model (Eq. (19))and define a matrix G ia,jb as G ia,jb = Nπ ∂ E ∂N ia ∂N jb = (cid:88) kc s c v kc U ia,kc U jb,kc . (29)By considering all possible combinations of adding and removing particles to the four Fermipoints, the second derivative with respect to energy can be calculated from the energy levelsobtained from Bethe Ansatz (Eq. (4)). Subsequently, the eigenvectors of the matrix G ia,jb yield all the parameters U ia,jb .The remaining non-universal prefactors A ia,jb and exponents µ ia,jb can be obtained usingthe system size scaling behavior of the Umklapp matrix elements, from the relation |(cid:104) Φ s | S zj | ia, jb (cid:105)| = A ia,jb π (cid:18) πN (cid:19) µ ia,jb , (30)where | ia, jb (cid:105) is defined as the Umklapp state by removing a particle at the Fermi pointlabeled by ia and placing it back at the Fermi point labeled by jb . By scaling the system size N , the parameters A ia,jb and µ ia,jb are directly obtained by a linear fit to the logarithm ofEq. (30) for all 6 combinations of Umklapp states. This procedure is repeated for all valuesof anisotropy and momentum shifts in the quantum numbers considered here. The values ofthe prefactors and exponents are plotted in Fig. 5 as function of anisotropy for a fixed valueof the momentum shift to the Fermi seas.In order to compare the multi-component Tomonaga-Luttinger model predictions to finitesize results, a conformal transformation x → Nπ sin( πx/N ) (31)14 ciPost Physics Submission . . . . . 820 0 . . . . . . 530 0 . . . . . . . . . . 40 0 . . . . . . . . A i a j b ∆ A , A A , A A A µ i a j b ∆ µ , µ µ , µ µ µ B i a (cid:15) ∆ B , B − B , B − B , B − B , B − µ i a (cid:15) ∆ B , B − B , B − B , B − B , B − Figure 5: Non-universal prefactors (left column) and exponents (right column) as a functionof anisotropy for the static real space (cid:104) S z ( x ) S z (0) (cid:105) (top row) and (cid:104) S − ( x ) S + (0) (cid:105) (bottom row)correlations of the multi-component Tomonaga-Luttinger model, computed from integrabilityfor a state with a double Fermi sea at N = 200, M = 50 and momentum shift s = 12. . . . . . . . 14 0 0 . 05 0 . . 15 0 . . 25 0 . . 35 0 . . 45 0 . U i a , j b ∆ U U U U U U Figure 6: Bogoliubov parameters as afunction of anisotropy for the static realspace (cid:104) S z ( x ) S z (0) (cid:105) correlations of themulti-component Tomonaga-Luttingermodel. The figure shows a compar-ison between the expansion in smallanisotropy (dashed lines) and the pa-rameters computed from integrability(solid lines) for a state with a doubleFermi sea at N = 200, M = 50 andmomentum shift s = 12.15 ciPost Physics Submission . . . . . 066 0 5 10 15 20 25 30 h S z ( x ) S z ( ) i x ∆ = ∆ = ∆ = 1 Figure 7: Longitudinal static real spacecorrelation (cid:104) S z ( x ) S z (0) (cid:105) , at N = 200, M = 50 for several values of theanisotropy, for a state with momen-tum split in the Fermi sea by s = 12.The figure compares the ABACUS re-sults (points) to the multi-componentTomonaga-Luttinger model (lines). Theagreement holds not only at large dis-tances, but also down to very short ones. . . . 06 0 2 4 6 s = 2 s = 6 s = 120 2 4 6 s = 2 s = 6 s = 120 2 4 6 s = 2 s = 6 s = 12 h S z ( x ) S z ( ) i x x x ∆ = ∆ = ∆ = 1 Figure 8: Longitudinal static real spacecorrelation (cid:104) S z ( x ) S z (0) (cid:105) for very shortdistances, at N = 200, M = 50 for sev-eral values of the anisotropy, for stateswith a momentum split in the Fermi seaby s = 2 , , 12 (from left to right panelsrespectively). The figure compares theABACUS results (points) to the multi-component Tomonaga-Luttinger model(lines).is applied to the scaling behavior of Eq. (24). The resulting expressions for the longitudinal realspace correlations, along with the parameters obtained by the procedure described above, areplotted in Figs. 7 and 8. Fig. 7 shows the correspondence of the multi-component Tomonaga-Luttinger model to the matrix element summations obtained from ABACUS at N = 200, M = 50, for a momentum shift in the Fermi seas of s = 12. For all distances but the verysmallest, both approaches show good agreement. Fig. 8 displays a comparison at very shortdistances for different momentum shifts and anisotropy, still showing a large agreement inboth approaches, in a regime where the Tomonaga-Luttinger model is not a priori expectedto give bonafide predictions.Similar to the determination of the non-universal prefactors and exponents for the longi-tudinal case, the scaling relation |(cid:104) Φ s | S − j | ia, (cid:15) (cid:105)| = B ia,(cid:15) π (cid:18) πN (cid:19) µ ia,(cid:15) (32)allows for the determination of the parameters B ia,(cid:15) and µ ia,(cid:15) for the purpose of Eq. (26).The Umklapp state | ia, (cid:15) (cid:105) is defined by the removal of a particle at the Fermi point labeledby ia , while (cid:15) = ± dictates the direction of the shift in the quantum numbers due to changein the parity after changing the number of particles. Again, the prefactors and exponents areobtained by fitting the finite size scaling behavior of Eq. (32) and their values are plotted inFig. 5 as function of anisotropy for a fixed value of the momentum shift to the Fermi seas.The transverse real space correlations from Eq. (26) are plotted for several values of16 ciPost Physics Submission − . − . . . 04 0 5 10 15 20 25 30 h S − ( x ) S + ( ) i x ∆ = ∆ = ∆ = 1 Figure 9: Transverse static real spacecorrelation (cid:104) S − ( x ) S + (0) (cid:105) , at N = 200, M = 50 for several values of theanisotropy, for a state with momen-tum split in the Fermi sea by s = 12.The figure compares the ABACUS re-sults (points) to the multi-componentTomonaga-Luttinger model (lines). Thecorrespondence works well for distanceslarger than a handful of sites. − . . . . . . 25 0 5 10 s = 2 s = 6 s = 120 2 4 6 s = 2 s = 6 s = 120 2 4 6 s = 2 s = 6 s = 12 h S − ( x ) S + ( ) i x x x ∆ = ∆ = ∆ = 1 Figure 10: Transverse static real spacecorrelation (cid:104) S − ( x ) S + (0) (cid:105) for very shortdistances, at N = 200, M = 50 for sev-eral values of the anisotropy, for stateswith a momentum split in the Fermi seaby s = 2 , , 12 (from left to right panelsrespectively). The figure compares theABACUS results (points) to the multi-component Tomonaga-Luttinger model(lines).17 ciPost Physics Submission the anisotropy in Fig. 9 for a fixed value of the momentum shift and in Fig. 10 for shortdistances and three separate values of the momentum shift, respectively. The non-universalprefactors and exponents are obtained by the aforementioned method, while the conformaltransformation to finite size from Eq. (31) has been applied as well. Both figures show againperfect agreement for large distances, while the agreement is also good for short distanceswith respect to system size. The smallest momentum shift ( s = 2 in Fig. 10) shows the worstagreement at very short distances.Finally, all previous procedures have been applied to a state where an asymmetric momen-tum shift is employed to separate the Fermi seas. Fig. 11 shows the corresponding longitudinaland transverse dynamical structure factors and real space correlations obtained from ABACUSat N = 200, M = 50, and the multi-component Tomonaga-Luttinger model. The parametersfor the latter have been obtained from the fitting procedure described in this section, appliedto system size scaling behavior of Umklapp matrix elements on the asymmetric Fermi points.Once again, the real space correlations display agreement for both the asymptotics as well asfor short distances for both approaches. − π/ π π/ π . . . . . − π/ π π/ π − . − . . . 040 5 10 15 20 ω q S zz ( q , ω ) h S z ( x ) S z ( ) i x ω q S − + ( q , ω ) h S − ( x ) S + ( ) i x Figure 11: Dynamical (left column) and static (right column) correlations for a state withan asymmetric momentum shift for the two Fermi seas by s l = 6 , s r = 18, at N = 200, M = 50, ∆ = 1. The panels in the right column show the resemblance between the real spacecorrelations obtained from ABACUS (points) and the multi-component Tomonaga-Luttingermodel (lines). The saturation of the ABACUS data is given in Tab. 1.18 ciPost Physics Submission The fact that the zero-entropy states we are considering are far from equilibrium is not visiblewhen we restrict attention to the static correlations, which would be the same if this statewas obtained as the ground state of a different Hamiltonian. In order to make the out-of-equilibrium nature apparent we have to probe the energies of ‘excitations’, i.e. modificationsof the Moses sea by creating additional particles and holes, which may now have both positiveand negative energy differences with respect to the reference state. A physically meaningfulway to probe the energy landscape is by computing time-dependent correlations which canin principle be related to observable quantities. These are already encoded in the dynamicalstructure factors computed with ABACUS and can be obtained by Fourier transformation.Recent years have witnessed a revolutionary increase in understanding of dynamical cor-relations in critical one-dimensional systems from the perspective of both effective field theorymethods and integrability [17, 40, 42–67]. The threshold behavior of many dynamical correla-tions in energy and momentum space can be understood in terms of specific configurations ofparticle and hole excitations. These lead to a scattering phase shift of the modes close to theFermi energy which is identified as the cause of the characteristic power-law singularities bymeans of Anderson’s orthogonality catastrophe. This threshold behavior, which also deter-mines the asymptotic behavior of the correlations in real space and time, is then described byan effective model in which the high energy particle or hole excitation is treated as a mobileimpurity interacting with the low-energy modes.We generalize this mobile impurity approach to the present out-of-equilibrium context byextending our multi-component Tomonaga-Luttinger model to include the appropriate impu-rity configurations and interactions. To be specific, we will compute the spin autocorrelation C ( t ) = (cid:104) S zj ( t ) S zj (0) (cid:105) = (cid:104) Ψ † ( t )Ψ( t )Ψ † (0)Ψ(0) (cid:105) , (33)where Ψ( t ) = Ψ( x = 0 , t ) denotes the Jordan-Wigner fermion and we used translationalinvariance. By imagining to obtain C ( t ) as a Fourier transform in ( k, ω )-space taking the k -integral first, one can argue that as a function of ω singular behavior stems from the ‘Fermipoints’ and points where the edge of support has a tangent with vanishing velocity. Thisidentifies the important impurity configurations for this function as corresponding to a particleor hole with vanishing velocity, i.e. either at the bottom or the top of the band. Let us assumethat the Moses state leaves the corresponding quantum numbers unoccupied, which is validfor a symmetric configuration with an even number of seas. This means that there are onlyhigh-energy particle impurities. We will use an index γ = 0 , H MIM = (cid:90) dx (cid:88) ia s a v ia ∂ x ϕ ia ) + (cid:88) γ d † γ (cid:18) (cid:15) γ − ∂ x m γ (cid:19) d γ − (cid:88) ia,γ s a κ ia,γ √ π d † γ d γ ∂ x ϕ ia . (34)Note that the last term is just a density-density interaction between the impurity modes andthe chiral fermions parametrized by the coupling constants κ ia,γ , while the first two termscorrespond to the impurity dispersions and the TL modes. In the small ∆ limit all parametersin H MIM can be obtained from H in Eq. (13) as in appendix C. In general they can be obtainedfrom integrability. 19 ciPost Physics Submission The impurity modes are decoupled from the Tomonaga-Luttinger model up to irrelevantoperators by the unitary transformation U = exp i (cid:90) dx (cid:88) ia,γ κ ia,γ s a v ia √ π d † γ d γ ϕ ia . (35)The effect is that correlators of the TL model are still computed in the same way, but theimpurity operator obtains an extra vertex operator in terms of the bosonic modes d → d exp (cid:40) − i (cid:88) ia κ γ,ia s a v ia √ π ϕ ia (cid:41) . (36)The logic is identical to the ground state case, and this also suggests that we can identifythe parameter κ ia,γ /v ia as the phase shift at the Fermi points upon creating the impurityaccording to [40, 41] κ ia,γ v ia = 2 πF ( λ ia | λ γ ) . (37)For the computation of the autocorrelation C ( t ), the crucial observation is now that theasymptotic behavior, determined by the behavior around a few singular points of the longitu-dinal structure factor, is well captured by certain correlations computable using the Hamilto-nian H MIM . In marked contrast to the equilibrium case, the TL model does not account forzero-energy states only, and therefore even the contributions to Eq. (33) that do not involvethe impurity will display the energy difference of the Fermi points leading to fluctuating terms e i ( (cid:15) ia − (cid:15) jb ) t (cid:104) ψ † ia ( t ) ψ jb ( t ) ψ † jb (0) ψ ia (0) (cid:105) (38)where (cid:15) ia is the energy associated to Fermi point k ia . The TL contributions sum up to anexpression similar to the static correlation in Eq. (23). The impurity contributions are of theform e i ( (cid:15) ia − (cid:15) γ ) t (cid:104) ψ † ia ( t ) d γ ( t ) d † γ (0) ψ ia (0) (cid:105) . (39)Using the impurity correlator (cid:104) d γ ( t ) d † γ (0) (cid:105) = (cid:90) dk π e − i k mγ = (cid:114) m γ πit (40)we find the result C ( t ) ∼ s − (cid:88) ia,jb,kc s a s b U ia,kc U jb,kc πv kc t + (cid:88) ia,jb A ia,jb cos( (cid:15) ia − (cid:15) jb )4 π (cid:89) kc (cid:18) is c v kc t (cid:19) [ U ia,kc − U jb,kc ] + (cid:88) ia,γ A (cid:48) ia,γ e i ( (cid:15) ia − (cid:15) γ ) t π (cid:114) m γ πit (cid:89) kc (cid:18) is c v kc t (cid:19) [ U ia,kc + κia,γ πscvkc ] . (41)The prefactor A (cid:48) ia,γ = 1 + O (∆) can in principle be obtained from finite-size scaling of matrixelements similar to A ia,jb but with the Umklapp state replaced by the appropriate impuritystate [68]. We have checked this expression for the autocorrelation against Fourier transformedABACUS data for small values of ∆, and find that it converges to the exact result on quite20 ciPost Physics Submission − . . . . . . . . . 08 5 10 15 20 25 30 35Real Imag5 10 15 20 25 30 35Real Imag h S z ( t ) S z ( ) i t t ABACUSFree fermionsNLL Figure 12: Comparison of computationsfor the autocorrelations (cid:104) S z ( t ) S z (0) (cid:105) ,from ABACUS at N = 200, M = 50,∆ = , s = 12, from free fermions(∆ = 0), and from non-linear Luttinger(NLL) theory with effective field param-eters taken for free fermions.short time scales for a configuration when the Fermi points k ia are well away from the bandtop and bottom at k γ = 0 , π (Fig. 12), but the correspondence for short to moderate timesbecomes noticeably worse when we decrease the separation between the two seas. This couldbe a finite size effect since the number of states in between the impurity mode and the Fermiedges becomes small, but rather we believe that the correlation is not well-captured by theimpurity model in that case as a clear separation in sub-bands becomes questionable.When the current mobile-impurity approach works well, this tells us that the time-dependent correlation is determined by the modes very close to the Fermi points k ia whichcorrespond to particle-hole excitations involving only the quantum numbers close to the edgesof the two Fermi seas. The role of the spectrum at the Fermi points and of the impurity istwo-fold: (i) The energy differences determine the frequencies of fluctuations. (ii) The Fermivelocities v ia and impurity mass m γ change the prefactor of the separate terms. Note that thedecay of the correlation on the other hand is determined by energy independent data, namelyby the appropriate phase shifts and Anderson’s orthogonality catastrophe, very similar to theequilibrium case. We have considered high energy zero-entropy states for the anisotropic Heisenberg spin chaindefined by a double-Fermi sea quantum-number configuration. Our focus was on dynamicalcorrelations computed by summing over relevant matrix elements of particle-hole excitationsat finite system size, the matrix elements being given by algebraic Bethe Ansatz. Correlationsin real space and time are obtained from these by numerical Fourier transformation.Zero-entropy Bethe states provide an interesting class of eigenstates of integrable modelswhich, while far from equilibrium, share many features with the ground state. In particularwe have shown that, when in the critical regime of the anisotropic Heisenberg spin chain,these states display critical correlations characterized by fluctuations with power-law decay.Starting from the Bethe Ansatz solution, we have constructed an effective field theory interms of a multi-component Tomonaga-Luttinger model capturing these correlations withgreat accuracy for large, but also surprisingly short, distances in real space. Similar to theground state case, fluctuations correspond to the differences of the generalized Fermi momenta k ia , the power-law exponents are related to a Bogoliubov transformation diagonalizing the21 ciPost Physics Submission effective Hamiltonian, while non-universal prefactors determine the relative amplitudes of thefluctuating terms.The Bogoliubov parameters U ia,jb can be related to finite-size energy differences uponadding or removing particles at the Fermi points, and the non-universal correlation prefactorsto the finite-size scaling of matrix elements with Umklapp excitations. We have used thisto obtain all field theory parameters from integrability yielding parameter-free fits for thestatic correlations. A surprising fact is that, while static correlations do not know about theHamiltonian for time evolution, the energies computed to obtain the U ia,jb do feature theFermi velocities from the spectrum of the actual XXZ Hamiltonian, which does not determinethe statistical ensemble in the present case. This implies that the relation of the spectrumand the Bogoliubov parameters is universally valid irrespective of the energy function oneuses.The parameters U ia,jb turn out to correspond to the phase shifts of the modes at oneof the generalized Fermi points k jb upon creating an excitation at Fermi point k ia . At zerotemperature, the Luttinger parameter K can similarly be regarded as a parametrization ofthe phase shift at the left and right Fermi points upon creating single-particle excitationswith momentum ± k F , and it has recently been realized that this implies a universal descrip-tion of dynamical correlations that go beyond the linear-spectrum approximation made inthe Tomonaga-Luttinger model [48]. The applicability of the multi-component Tomonaga-Luttinger model suggests that, similar to the ground state case, a universal description ofdynamical correlations in a domain close to the Fermi points may be obtained. The descrip-tion of domains near the edge of support is likely captured by a mobile impurity model. Asa first step of this generalized use of what is known as non-linear Luttinger liquid theory,we have compared the longitudinal autocorrelation function to predictions obtained from amobile impurity model at small ∆. The correlation seems to converge to the mobile impurityresult in quite reasonable times at least when the two Fermi seas are well-separated.In conclusion, we have presented results on dynamical correlations of zero-entropy states inthe anisotropic Heisenberg chain. The distinctive features, which may serve for identificationin experiment, can be understood by adapting the familiar ground state reasoning. By makingthe appropriate adjustments to equilibrium techniques based on the Tomonaga-Luttingermodel many aspects can furthermore be understood in great detail and with quantitativeagreement once a handful of parameters is fixed from integrability. Although we have focusedon states which have vanishing entropy density, we may consider thermal-like dressings tothe split seas. In the Tomonaga-Luttinger description, finite temperatures are treated by asimple functional replacement for the fundamental correlators. On the side of integrability,recent work has shown that finite temperature correlators are also numerically accessible, atleast for the Lieb-Liniger model [69]. How the correspondence between integrability resultsand the field theory works out in split-sea configurations at finite temperature remains to beinvestigated. Acknowledgements We thank Yuri van Nieuwkerk for useful comments. The authors acknowledge support fromthe Foundation for Fundamental Research on Matter (FOM) and from the Netherlands Or-ganization for Scientific Research (NWO). This work forms part of the activities of the Delta-22 ciPost Physics Submission Institute for Theoretical Physics (D-ITP). Funding information The work of R. P. V. and J.-S. C. was supported by NWO VICIgrant number 680-47-605. The work of I. S. E. and J.-S. C. was supported by FOM program128, ‘The singular physics of 1D electrons’. A Simplification for symmetric seas In the case of a symmetric configuration of n seas, we have − k iL = k n +1 − iR and − v iL = v n +1 − iR and in general that the system is symmetric under simultaneously exchanging i ↔ n + 1 − i and L ↔ R . It is then convenient to combine the fields at Fermi points at oppositemomenta and define φ i = 1 √ φ iL − φ n +1 − iR ) , θ i = 1 √ φ iL + φ n +1 − iR ) . (42)with inverse transformation φ iL = 1 √ θ i + φ i ) , φ n +1 − iR = 1 √ θ i − φ i ) . (43)The Hamiltonian for the multi-component TL model can then be written as H TL = (cid:88) i v i (cid:90) dx [( ∂ x φ i ) + ( ∂ x θ i ) ] + (cid:88) ij (cid:90) dx π [ g + ij ∂ x φ i ∂ x φ j + g − ij ∂ x θ i ∂ x θ j ] (44)with v i = v n +1 − iR and g ± ij = g iLjL ± g n +1 − iRjL .In order to respect the canonical commutation relations and diagonalize H TL we introducenew fields according to φ i = (cid:88) j U ij ϕ j , θ i = (cid:88) j [ U − ] ji ϑ j (45)where the U ij are related to the Bogoliubov parameters U ia,jb as U ij = U iLjL − U iLn +1 − jR (46)by virtue of the symmetry U ia,jb = U n +1 − i ¯ a,n +1 − j ¯ b . The effective Hamiltonian takes thefamiliar diagonal form H TL = (cid:88) i v i (cid:90) dx (cid:2) ( ∂ x ϕ i ) + ( ∂ x ϑ i ) (cid:3) . (47)The matrix U ij is straightforwardly obtained from calculations in finite size and finite particlenumber from corrections to the energy upon creating particle-number or current excitations δE = (cid:88) i πv i L (cid:88) j [ U − ] ij ∆ N j + (cid:88) j U ji ∆ J j . (48)Here ∆ N i = N n +1 − iR + N iL , ∆ J i = N n +1 − iR − N iL (49)in terms of the numbers N ia of particles added at Fermi point k ia .23 ciPost Physics Submission B Moses states and generalized TBA Finite-size corrections to the spectrum for the ground state are usually discussed by takingthe thermodynamic limit of the Bethe equations and the energy, keeping finite size correctionsusing the Euler-Maclaurin formula [10]. The class of Moses states can be considered as the zerotemperature limit of a generalized Gibbs ensemble for which a generalized thermodynamicBethe Ansatz and a treatment of finite-size corrections similar to the ground state exists[70, 71]. However, in our case we still use the energy dispersion obtained from the true XXZHamiltonian determining the time-evolution rather than the statistical ensemble (which wedefine by hand in the micro-canonical sense). Since we also obtain the critical exponents usingthe true XXZ Hamiltonian and not a GGE Hamiltonian, the fact that the results coinciderequires clarification. The standard derivation of the finite size corrections of the energy makesuse of the property that the corresponding dressed energy vanishes at the Fermi points, butin our case (cid:15) ( λ ia ) (cid:54) = 0 if we measure energies by using the true XXZ Hamiltonian.Upon changing the particle number N ia at one of the Fermi points k ia the extremalrapidities λ ia experience a shift of order 1 /L . The change in the Fermi points and rapiditiescan be shown to satisfy δk ia = − s a N ia πL , (50) δλ ia = (cid:88) jb Ls a ρ ( λ ia ) [ δ ia,jb + s a F ( λ ia | λ jb )] N jb . (51)The finite-size corrections to the energy due to adding or removing particles from the extrem-ities of the Fermi-sea-like blocks is then δE = (cid:88) ia (cid:15) ( λ ia ) N ia + 1 N (cid:88) ia,jb,kc s c (cid:15) (cid:48) ( λ kc )2 ρ ( λ kc ) [ δ ia,kc + s c F ( λ kc | λ ia )][ δ jb,kc + s c F ( λ kc | λ jb )] N ia N jb (52)where ± (cid:15) ( λ ) is the energy of a particle (hole) with rapidity λ on top of the Moses state. Incontrast to the equilibrium case where energy is measured or a GGE dressed energy satisfying (cid:15) GGE ( λ ia ) = 0, the funtion (cid:15) ( λ ) cannot be defined through an integral equation but has anadditional contribution stemming from the finite energy at the Fermi points. Details will bepresented elsewhere. C Perturbative expressions for effective-field-theory parame-ters A convenient way to obtain perturbative expressions for the exponents of asymptotes ofcorrelation functions is to derive the effective Hamiltonian to first order in the interaction.Starting from the Hamiltonian of spinless fermions on the lattice we use the mode expansionΨ( x ) ∼ (cid:88) ia e ik ia x ψ ia ( x ) + (cid:88) γ e ik γ x d γ ( x ) (53)and bosonize the chiral fermions. The kinetic term leads to the velocities for the chiralfermions v ia = J sin( k ia ) and the impurity parameters in the noninteracting limit (cid:15) γ = ∓ J ciPost Physics Submission and m γ = ± J . The interaction term H int = (cid:88) x J ∆ n ( x ) n ( x + 1) (54)renormalizes these values. By plugging in the bosonization identities, normal ordering andneglecting irrelevant terms we get the first order in ∆ expressions g ia,jb = J ∆ (cid:40)(cid:80) ld cos( k ia − k ld ) , ( ia = jb )1 − cos( k ia − k jb ) , ( ia (cid:54) = jb ) . (55)These give U ia,jb = δ ia,jb − [1 − δ ia,jb ] J ∆ π s b [1 − cos( k ia − k jb )] v ia − v jb . (56)Next, we focus on the terms from the interaction involving the impurity. This leads to adensity-density interaction of the impurity modes with the chiral fermions with parameters κ ia,γ = 2 J ∆[1 − cos( k ia − k γ )] . (57)There is also a first order correction to the impurity energy appearing from Eq. (54) from theterms proportional to d † d (after normal ordering), which is (cid:15) γ =0 , = ∓ J (cid:32) ∓ n ∆ + (cid:88) ia ∆ π s a sin( k ia ) (cid:33) . (58)This corresponds to the Hartree-Fock correction δ(cid:15) γ = (cid:88) i (cid:90) k iR k iL dk π [ V (0) − V ( k γ − k )] (59)with V ( q ) = 2 J ∆ cos( q ), which corresponds to the interaction potential in Eq. (54): H int = L (cid:80) q V ( q ) n q n − q .The non-universal prefactors can also be obtained perturbatively using the methods dis-cussed in Ref. [16], but we have not done this calculation. References [1] F. D. M. 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