PHYSICAL REVIEW B 91, 245150 (2015)

Spin-charge-separated quasiparticles in one-dimensional quantum fluids

F. H. L. Essler,1 R. G. Pereira,2 and I. Schneider3 1The Rudolf Peierls Centre for Theoretical Physics, Oxford University, Oxford OX1 3NP, United Kingdom 2Instituto de F´ısica de Sao˜ Carlos, Universidade de Sao˜ Paulo, C.P. 369, Sao˜ Carlos, SP, 13560-970, Brazil 3Department of Physics and Research Center OPTIMAS, University of Kaiserslautern, D-67663 Kaiserslautern, Germany (Received 30 April 2015; published 24 June 2015)

We revisit the problem of dynamical response in spin-charge separated one-dimensional quantum fluids. In the framework of Luttinger liquid theory, the dynamical response is formulated in terms of noninteracting bosonic collective excitations carrying either charge or spin. We argue that, as a result of spectral nonlinearity, long-lived excitations are best understood in terms of generally strongly interacting fermionic holons and spinons. This has far reaching ramifications for the construction of mobile impurity models used to determine threshold singularities in response functions. We formulate and solve the appropriate mobile impurity model describing the spinon threshold in the single-particle Green’s function. Our formulation further raises the question whether it is possible to realize a model of noninteracting fermionic holons and spinons in microscopic lattice models of interacting spinful fermions. We investigate this issue in some detail by means of density matrix renormalization group (DMRG) computations.

DOI: 10.1103/PhysRevB.91.245150 PACS number(s): 71.10.Pm, 71.10.Fd

I. INTRODUCTION algebra [19,20]: Understanding the essential features of a quantum many- = cd Za(θ1)Zb(θ2) Sab(θ1,θ2)Zd (θ2)Zc(θ1), body system usually entails finding a simple explanation of its † = − low-energy spectrum in terms of weakly interacting, long-lived Za(θ1)Zb(θ2) 2πδ(θ1 θ2)δa,b quasiparticles. For example, in Landau’s Fermi liquid theory † +Sda(θ ,θ )Z (θ )Z (θ ). (1) the quasiparticles are fermions carrying the same quantum bc 2 1 d 2 c 1 numbers as an electron, namely, charge e and spin 1/2. cd Here, Sab(θ1,θ2) is the purely elastic two-particle S matrix. The However, it is well established that the long-lived excitations corresponding elementary excitations are infinitely long lived, in strongly correlated systems may carry only a fraction of but generally strongly interacting, as can be seen from their the quantum numbers of the elementary constituents. In fact, scattering matrices [8–10]. Moreover, their quantum numbers, fractionalization is often invoked as a route towards exotic e.g., charge 0 and spin 1/2 for spinons in the Hubbard model, phases of matter such as spin liquids or high-temperature differ from those of the collective bosonic spin modes in superconductors [1]. the Luttinger liquid description. It is natural to assume that Perhaps the most prominent example of fractionalization breaking integrability slightly will not generically lead to a is spin-charge separation in one-dimensional (1D) quantum qualitative change in the nature of elementary holon and spinon fluids known as Luttinger liquids [2]. The hallmark of these excitations, but merely render the lifetimes finite. theories is a low-energy spectrum described by two decoupled How to reconcile this picture emerging from the exact free bosonic fields associated with collective spin and charge solution with Luttinger liquid theory? In the latter, the nature of degrees of freedom, respectively. On the experimental side, elementary excitations is obscured by the fact that, due to the the most direct evidence for spin-charge separation involves linear dispersion approximation, the spectrum is highly degen- the observation of multiple peaks associated with spin and erate and allows for many interpretations, which ultimately all charge collective modes in dynamical response functions at give the same results for physical observables. Examples are fairly high energies [3–6], beyond the regime where Luttinger chiral Luttinger liquid descriptions of quantum Hall edges [21] liquid theory is applicable. Spin-charge separation is known and the low-energy excitations of the Hubbard model. The to persist at high energies for integrable theories such as the latter can be understood both in terms of interacting, fermionic 2 1D Hubbard [7–16] and 1/r t-J [17,18] models. In these holons and spinons [7,22–24], and in terms of noninteracting cases, any eigenstate with a finite energy in the thermodynamic bosons associated with collective spin and charge degrees of limit can be classified in terms of elementary excitations freedom [25–27]. called holons (which carry charge e and spin 0) and spinons Going beyond the linear dispersion approximation is (which carry charge 0 and spin 1/2), along with their bound expected to remove ambiguities in the quasiparticle description states. A convenient way for describing such excitations of the spectrum: there will be a particular choice that maxi- as well as their scattering is to define the corresponding † mizes the lifetimes of elementary excitations. In integrable creation and annihilation operators Za(θ) and Za(θ), where models such as the Hubbard chain, these lifetimes are infinite. a labels the different types of elementary excitations and θ Over the last decade, it has been established in a series of is a variable that parameterizes the momentum pa(θ)(the works that going beyond the linear dispersion approximation form of this function depends on the particular model under is essential for correctly describing the dynamics of one- consideration). As a consequence of integrability, creation dimensional models with gapless excitations [28–56]. This has and annihilation operators fulfill the Faddeev-Zamolodchikov resulted in the so-called “nonlinear Luttinger liquid” (nLL)

1098-0121/2015/91(24)/245150(24) 245150-1 ©2015 American Physical Society F. H. L. ESSLER, R. G. PEREIRA, AND I. SCHNEIDER PHYSICAL REVIEW B 91, 245150 (2015) approach to gapless 1D quantum liquids; see Refs. [46,54] and V is the nearest-neighbor interaction strength. This model iα for recent reviews. The basic reason for the failure of linear has a U(1) symmetry, cj → e cj , α ∈ R, associated with Luttinger liquid theory is that at finite energies the running conservation of the total number of fermions N. Moreover, the coupling constants of irrelevant perturbations such as band model is integrable and the exact spectrum for arbitrary values curvature terms are in fact different from zero. Taking them of V can be calculated from the Bethe ansatz solution [57,58]. into account in perturbation theory leads to infrared singulari- There is a gapless phase extending between −2 fermion case was best understood renormalization group (RG) sense. In the standard Luttinger by considering weak interactions, we address the construction liquid approach, this term is dropped, which corresponds to ± of a microscopic model of interacting electrons giving rise to a linearizing the dispersion around the Fermi points kF . theory of noninteracting fermionic holons and spinons at low The bosonization formula for chiral spinless fermions reads energies, but in presence of spectral nonlinearities. We then η − √i ϕ(x) R(x) = √ e 2 , (5) derive mobile impurity models to analyze dynamical response 2π functions in our formulation. We demonstrate explicitly how to η¯ √i ϕ¯(x) recover results obtained previously by means of the approach L(x) = √ e 2 , (6) of Refs. [51,52]. Finally, we address the question how to 2π realize a lattice model of interacting electrons that gives rise where η,η¯ are Majorana fermions and ϕ(x),ϕ¯(x) are chiral to noninteracting holons and spinons at low energies. bosons that obey the commutation relations [ϕ(x),ϕ¯(y)] = 0, (7) II. SPINLESS FERMIONS [ϕ(x),ϕ(y)] = 2πisgn(x − y) =−[¯ϕ(x),ϕ¯(y)]. (8) Before we approach the problem of defining quasiparticles for spin-1/2 fermions, it is instructive to review the case of Throughout this paper we use “CFT normalizations” for spinless fermions. For concreteness, consider the simplest bosonic vertex operators: lattice model of interacting spinless fermions: 1 eiαϕ(x)e−iαϕ(y) = . (9) L (x − y)2α2 =− † + + H t (cj cj+1 H.c.) V nj nj+1. (2) j=1 j A consequence of employing these conventions is that vertex operators are dimensionful: Here, cj is the annihilation operator for a fermion at site j, = † iαϕ(x) = −α2 nj cj cj is the number operator, t is the hopping parameter, dim(e ) length . (10)

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We also define the canonical bosonic field (x) and its dual divergences [34,46]. The latter are associated with the huge (x)by degeneracy of states in the linear dispersion approximation: all states with an arbitrary number of bosons moving in = + (x) ϕ(x) ϕ¯(x), (11) the same direction carrying the same total momentum are (x) = ϕ(x) − ϕ¯(x), (12) degenerate [34,46]. A way to circumvent these difficulties in analyzing the non- which obey linear bosonic theory is suggested by reverting to the fermionic [ (x), (x)] = 4πisgn(x − x). (13) representation (18). At the free fermion point, the irrelevant interaction vanishes, g3 = 0, and one is left with the quadratic Bosonization of Eq. (4) then leads to the Hamiltonian dispersion term with effective mass m. Taking the nonlinear dispersion into account in the free fermion model removes H = H + H dx[ LL(x) irr(x)]. (14) the degeneracy of particle-hole pairs that carry the same total momentum. One can then approach the problem from free The first term, fermions with nonlinear dispersion and include interactions perturbatively [28]. This approach reveals that the most v 2 1 2 HLL(x) = K(∂x ) + (∂x ) , (15) pronounced effect of the interactions is to give rise to power- 16π K law singularities at the edges of the excitation spectrum. While is the Luttinger model. The velocity v and the Luttinger a complete analytical solution of model (4) taking into account  parameter K are given to first order in V by v ≈ v and K ≈ both band curvature and interaction is highly nontrivial, the − V 1 πt sin kF . The bosonic fields describe the collective low- edge singularities can be described by an effective impurity energy density mode of the quantum fluid. The uniform part model in analogy with the x-ray edge problem [28,60]. of the density operator is related to (x) by the bosonization Consider, for instance, the single-fermion spectral function relation =−1 † † 1 A(ω,k) Im Gret(ω,k), (20) Q(x) = R (x)R(x) + L (x)L(x) ∼− √ ∂x (x). (16) π π 8 where ∞ The total charge operator is given by the integral − † =− iωt ikla0  |{ }| +∞ Gret(ω,k) i dt e e ψ0 cj+l(t),cj ψ0 0 q = dx Q(x). (17) l −∞ (21) Thus, an elementary excitation√ with charge q = 1 corresponds is the retarded Green’s function, with |ψ0 the exact ground to a kink of amplitude π 8 in the bosonic field (x). As is state. We can separate the negative- and positive-frequency well known, the model in Eq. (15) correctly captures the long- parts of the spectral function: distance asymptotic decay of correlation functions for any 1D system in the Luttinger liquid universality class [25–27]. A(ω,k) = A<(ω,k) + A>(ω,k). (22) The limitations of Luttinger liquid theory appear when The Lehmann representation reads one considers dynamical response functions at small but finite frequency and momentum. At finite energy scales, it = | | | |2 + − A<(ω,k) 2π ψn ck ψ0 δ(ω En E0), (23) becomes necessary to take the irrelevant perturbations Hirr to n the Luttinger model into account. At weak coupling and in the = | | †| |2 − + absence of particle-hole symmetry (i.e., away from half-filling A>(ω,k) 2π ψn ck ψ0 δ(ω En E0), (24) in the lattice model), the leading corrections in Eq. (4)arethe n dimension-three operators where ck annihilates a fermion with momentum k and |ψn denotes an exact eigenstate of the Hamiltonian with energy 1 † 2 † 2 Hirr =− R ∂ R + L ∂ L En. Let us focus on the negative-frequency part for k

245150-3 F. H. L. ESSLER, R. G. PEREIRA, AND I. SCHNEIDER PHYSICAL REVIEW B 91, 245150 (2015) a narrower subband, while R(x) is split into two separate where ... stands for irrelevant operators, which are neglected subbands corresponding to the low-energy mode r(x) and the in the impurity model (since they only introduce subleading “high-energy” mode χ(x). Restricting the energy window to corrections to edge singularities). On the other hand, the the vicinity of the threshold with a single impurity, we now expression in Eq. (26) now becomes have to calculate the propagator of χ(x): +∞ ∞ † ∞ ∞ iωt Gret,<(ω,k) ≈−i dt e dxd (0,0)d (t,x) 0 iωt † −∞ Gret,<(ω,k) ≈−i dt e dxψ0|χ(0,0)χ (t,x)|ψ0 . 0 0 −∞ † ×F (0,0)F (t,x) 0, (35) (26) where  0 denotes the expectation value in the noninteracting At weak coupling, i.e., as long as the four-fermion in- ground state |ψ˜ 0 =U|ψ0 and F (x) is the string operator teraction strength V is small, we can substitute Eq. (25) ◦ + ◦ into Eq. (2) and bosonize the low-energy fields to derive an F (x) = ei[γϕ (x) γ¯ϕ¯ (x)]. (36) effective Hamiltonian for the single hole coupled to low-energy The correlation function in Eq. (35) can then be calculated collective modes. The result is the mobile impurity model [38] by standard methods [54]. The important point is that the v 1 scaling dimension of the operator F (x) changes continuously H = dx K(∂ )2 + (∂ )2 imp 16π x K x as a function of γ,γ¯. As a result, the effective impurity model predicts a power-law singularity in the spectral function, † † + χ (ε − iu∂x )χ + χ χ[f (q)∂x ϕ + f¯(q)∂x ϕ¯] , −1+2(γ 2+γ¯ 2) A<(ω,k) ∼ θ((k) − ω)|(k) − ω| . (37) (27) Importantly, the impurity mode χ(x)inEq.(27) carries = where ε ≡−(k) > 0 is the energy of the “deep hole” charge q 1 because it is defined from the original fermion = d cj at the noninteracting point. This is the particle that can be excitation, u dk is the velocity obtained by linearizing the dispersion around the center of the impurity subband, and f (q) identified with an elementary excitation in the Bethe ansatz and f¯(q) are momentum-dependent impurity-boson couplings solution for the integrable model. From the exact S matrix it is known that interactions between these elementary excitations of order V . The calculation of the Green’s function in Eq. (21) = is made possible by performing a unitary transformation that increase as V increases. Particularly at the SU(2) point, V 2, decouples the impurity from the low-energy modes: the elementary excitations are rather strongly interacting. By contrast, the transformed impurity operator d(x) = χ(x)F (x) − ∞ + † U = e i −∞ dx [γϕ(x) γ¯ϕ¯(x)]χ (x)χ(x). (28) carries a fractional charge that depends on the interaction strength, since the string F (x) in general does not commute The bosonic fields transform as with the charge operator in Eq. (17). ϕ◦(x) = Uϕ(x)U † = ϕ(x) − 2πγC(x), In the low-energy limit, an alternative approach to obtain (29) the edge singularity in the spectral function was put forward in ◦ = † = + ϕ¯ (x) Uϕ¯(x)U ϕ(x) 2πγC¯ (x), Ref. [44]. In this approach, one starts by introducing fermionic where quasiparticles that are asymptotically free at low energies. In ∞ our notation, the idea is to define chiral bosons φ,φ¯ by C(x) = dy sgn(x − y)χ †(y)χ(y). (30) √ −∞ (x) = K(φ + φ¯), (38) The transformed impurity field is 1 (x) = √ (φ − φ¯). (39) d(x) = Uχ(x)U † K

2 2 In terms of these, the Luttinger model (15) reads = χ(x)ei[γϕ(x)+γ¯ϕ¯(x)]e−iπ(γ −γ¯ )C(x). (31) v H (x) = [(∂ φ)2 + (∂ φ¯)2]. (40) Note that the impurity density is invariant under the unitary LL 8π x x transformation, i.e., χ †(x)χ(x) = d†(x)d(x). ˜ ˜ We choose the parameters γ,γ¯ as the solution of The quasiparticles R(x) and L(x) are defined by η − √i φ(x) ˜ = 2 f −v+ + u −v− γ R(x) √ e , (41) = , (32) 2π f¯ v− v− − u γ¯ η¯ √i φ¯(x) where L˜ (x) = √ e 2 . (42) 2π v 1 v± = K ± . (33) The commutator with the charge operator yields 2 K √ √ [q,R˜ †(x)] = KR˜ †(x), [q,L˜ †(x)] = KL˜ †(x). (43) With this choice, the Hamiltonian becomes noninteracting: √ Thus the quasiparticles carry charge K. On the other v ◦ 2 1 ◦ 2 Himp = dx K(∂x ) + (∂x ) hand, this refermionization procedure removes the marginal 16π K interaction between the quasiparticles since the chiral modes † are decoupled in Eq. (40)[29,44]. The leading interaction is + d (ε − iu∂x )d +··· , (34) then represented by the irrelevant operator g3 in Eq. (18), which

245150-4 SPIN-CHARGE-SEPARATED QUASIPARTICLES IN ONE- . . . PHYSICAL REVIEW B 91, 245150 (2015) can be neglected as a first approximation in the low-energy limit. The relation between the original right-moving fermion and the new quasiparticle is

R˜(x) = R(x)F0(x), (44) where F0(x) is the limit q = kF − k → 0 of the string operator in Eq. (36). At this point a noninteracting impurity mode χ˜ (x) can be introduced by projecting the free field R˜(x) into low-energy and high-energy subbands. This leads to a universal result for the exponent in the vicinity of the threshold, | − | 2 ω (k) q /m˜ , which corresponds√ to Eq. (37) with√ pa- rameters γ = √1 (1 − √1 − K ) andγ ¯ = √1 ( √1 − K ). 2 2 K 2 2 2 K 2 This result differs from the prediction of the linear theory [44]. In summary, there are two possible paths towards calculat- ing edge exponents in the nLL theory for spinless fermions: (i) starting with free fermions, one defines low-energy and impurity subbands, and then turns on interactions between FIG. 1. (Color online) Strategy map for calculating edge expo- the elementary excitations in the impurity model; after that, nents in nonlinear Luttinger liquid theory for spinful fermions in the the interaction with the impurity is removed by a unitary low-energy limit. See the main text for details. transformation, which introduces the string operator in the correlation function; or (ii) starting from the Luttinger model for interacting fermions, one refermionizes into weakly inter- energies. It is well known [7] that the low-energy degrees of acting quasiparticles, which differ from the original fermions freedom are collective spin and charge modes respectively, i.e., by a string operator, and then projects the quasiparticles into H −→ H + H +··· , (46) low-energy and impurity subbands. The projection onto the Hub charge spin impurity model is well controlled in both paths because the where the dots denote additional terms that are irrelevant model of interacting spinless fermions is smoothly connected in the renormalization group sense. Crucially, as H is with the free model, i.e., the parameters γ,γ¯, which quantify Hub spin rotationally symmetric, H must exhibit a spin SU(2) the scattering between high-energy and low-energy particles, spin symmetry. In order to parallel our analysis in the spinless vanish continuously as V → 0. However, it is important to case, we wish to express H in terms of fermionic fields emphasize the difference between the original fermions, which spin carrying spin quantum numbers ±1/2. This is possible in carry unit charge of the U(1) symmetry, and the quasiparticles an SU(2)-symmetric way only if the fermions are strongly with fractional charge. While the latter are always weakly interacting, i.e., the situation is similar to the V = 2 case interacting in the low-energy limit, the fermions that carry the for spinless fermions. In the charge sector the situation is correct quantum numbers become strongly interacting even at analogous unless we work at very low electron densities. In low energies as V increases. order to generalize the mobile impurity model construction Once the low-energy, weak-coupling regime is well un- reviewed in Sec. II to the spinful case, we therefore cannot derstood, the impurity model of nLL theory can be extended work with weakly interacting spinful fermions, but require a phenomenologically to high energies, strong interactions and model that gives rise to noninteracting fermions describing the thresholds with more than one impurity, as has indeed been collective spin and charge degrees of freedom. As such a model done successfully for spinless fermions [54]. is not known, we proceed along the lines sketched in Fig. 1. 1. Starting with weakly interacting spinful fermions at III. SPINFUL FERMIONS low energies, we derive the corresponding model of strongly As we have seen above, the spinless case is most easily interacting spin and charge fermions. understood by considering the vicinity of noninteracting 2. We then decrease the interactions in the spin and charge fermions. The situation is very different in the spinful case. fermion model, and derive a low-energy effective Hamiltonian In order to understand this point in some detail, let us consider in the vicinity of the “Luther-Emery point” [61] where the the particular example of the 1D Hubbard model spin/charge fermions become noninteracting. 3. Having completed this construction, we are in a position L to construct mobile impurity models by following the logic =− † + + HHub t (cj,σ cj+1,σ H.c.) U nj,↑nj,↓, employed in the spinless case. j=1 σ j 4. Having constructed a suitable mobile impurity model, (45) we may calculate threshold exponents by standard methods. =↑↓ where cj,σ annihilates a fermion with spin σ , at site 5. Through an appropriate tuning of the parameters defin- = †  j, nj,σ cj,σ cj,σ is the number operator, and U 0isthe ing our mobile impurity model, we may analyze the case strength of the on-site repulsion. We work at fixed density be- of weakly interacting SU(2)-invariant spinful fermions. This low half-filling with zero magnetization, N↑ = N↓

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A key ingredient in our approach is our “Luther-Emery Hirr(x) = λ1 cos s + λ2∂x c cos s model” of noninteracting holons and spinons. An obvious + ± 2 ± 2 question is what such a theory might look like in terms of λ3,α∂x c[(∂x α) (∂x α) ] interacting spinful fermions. We address this issue in Sec. VII. α=c,s

+ λ4∂x s ∂x s ∂x c +··· . (54) Here, v are the velocities of the collective charge and A. Bosonization at weak coupling α spin modes and Kα the corresponding Luttinger parameters. As our point of departure we choose a general extended The contributions Hirr are irrelevant in the renormalization Hubbard model below half-filling, where we allow fairly gen- group sense. A complete list of irrelevant operators with eral electron-electron interactions in addition to (45), provided scaling dimensions of at most four (for Ks = 1) is given in that they are invariant under the following symmetries: Appendix A. The velocities vα and Luttinger parameters Kα can be calculated exactly for the Hubbard model, but Eq. (52)is U(1)⊗ U(1) transformations in the charge and spin sectors generic for spinful Luttinger liquids if we regard vα and Kα as iασ cj,σ → e cj,σ , ασ ∈ R; phenomenological parameters. We note that, as a consequence

spin reflection cj,↑ ↔ cj,↓; of spin reflection symmetry, marginal interactions coupling spin and charge such as site parity cj,σ → c−j,σ ;

translations cj,σ → cj+1,σ . ∂x s (x)∂x c(x) (55) are not allowed. Hence the collective degrees of freedom at The kind of lattice model we have in mind is of the form low energies are described in terms of pure spin and pure charge modes, rather than linear combinations thereof (which H = HHub + Vr nj nj+r r1 j would be the case in presence of a magnetic field, see, e.g., Ref. [63]). + · + z z z Jr Sj Sj+r Jr Sj Sj+r , (47)  r 1 j B. Refermionizing in terms of spin and charge fields { z} where the coupling constants Vr ,Jr ,Jr must be such that the The next step is to refermionize (52) in terms of spin and model remains in a spin-charge-separated quantum critical charge fermion fields. In order to see how this should be phase. Lattice models of this type can be bosonized by done, we consider the limit of vanishing interactions. Here standard methods [26,27]. The generalization of Haldane’s the bosonization formulas simplify to (σ =↑,↓) bosonization formulas [62] to the spinful case is √ ikF x −ikF x cj,σ → a0[e Rσ (x) + e Lσ (x)], √ − − ∼ (σ ) ikF x(1 4n 2m) cj,σ a0 n,mAn,me ησ − i − i ∼ √ 2 ϕc(x) 2 sσ ϕs (x) ∈ Rσ (x) e , (56) n,m Z 2π − i ϕ (x)− i s ϕ (x) i 2n+m (x) −is m (x) × 2 c 2 σ s 2 c σ 2 s η¯ i i e e e . (48) σ ϕ¯c(x)+ sσ ϕ¯s (x) Lσ (x) ∼ √ e 2 2 , 2π Here, s↑ = 1 =−s↓, a0 is a short-distance cutoff, x = ja0, kF = πN↑/(La0) is the Fermi momentum, An,m are nonuni- where s↑ =−s↓ = 1. The idea is to decompose the right- (σ ) ≡ n m versal amplitudes and n,m ησ (ησ η¯σ ησ¯ η¯σ¯ ) (ησ¯ η¯σ¯ ) are moving spin-up electron into a right-moving holon field Rc Klein factors (and we use notations where, e.g., ↑=↓¯ ) that en- and a right-moving spinon field Rs in the form sure the correct anticommutation relations. The bosonic fields i(charge string) i(spin string) R↑(x) ∼ Rc(x)e Rs (x)e . (57) α(x) = ϕα(x) + ϕ¯α(x), (49) In a pure Luttinger liquid, there are infinitely many acceptable α(x) = ϕα(x) − ϕ¯α(x), (50) choices for Rα and string operators in Eq. (57). In Refs. [51,52], α = c,s with , obey the commutation relations the fermions Rα are chosen so as to have scaling dimension   1/2, in analogy with the procedure in the spinless case, see [ (x),  (x )] = 4πiδ  sgn(x − x ). (51) α α α,α Eqs. (41) and (42). This choice is such that the particles We note that in the CFT normalizations (9) the amplitudes are asymptotically free at low energies. However, they then An,m are dimensionful, i.e., they are proportional to carry fractional spin and charge quantum numbers. Such a appropriate powers of the lattice spacing a0. choice is not the most natural one for our purposes: it is In the spin-charge separated Luttinger liquid phase, the low- known that the scaling limit of the Hubbard model is given energy effective Hamiltonian for extended Hubbard models of by the U(1) Thirring model [SU(2) at half-filling], see, e.g., the type (47)is Ref. [26]. The U(1) Thirring model is integrable, and the elementary excitations are known to be strongly interacting fermionic spinless holons and neutral spinons (with a known H = dx[H (x) + H (x)], (52) LL irr S matrix) carrying charge ∓e and spin ±1/2, respectively. The principle guiding our construction is that charge and spin † † vα 2 1 2 H (x) = K (∂ ) + (∂ ) , (53) fermions created by Rc and Rs should carry the same quantum LL 16π α x α K x α α=c,s α numbers as the elementary holon and spin excitations. The

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U(1) charges corresponding to these quantum numbers are are O(1) at weak coupling Kα → 1. Moreover, gs,0 is always +∞ large as long as the spin SU(2) symmetry is unbroken, as in this = qα = dx Qα(x), (58) case the Luttinger parameter is fixed at Ks 1. This implies −∞ that the spin and charge fermions are strongly interacting. where This is consistent with known results for the exact S matrix of the Hubbard model [8–10]. Moreover, the spin sector of (65) = † + † Qc(x) [Rσ (x)Rσ (x) Lσ (x)Lσ (x)], describes a massive Thirring model perturbed by irrelevant σ =↑,↓ operators, which is precisely what one would expect on the (59) basis of the known S matrices for the Hubbard model [65]. = † + † Qs (x) sσ [Rσ (x)Rσ (x) Lσ (x)Lσ (x)]. =↑ ↓ σ , C. Bosonic representation of charge and spin fermions As usual, these expressions are to be understood in terms of a Our spin and charge fermion fields can be bosonized by standard point splitting and normal ordering prescription. We standard methods. Introducing chiral charge and spin (α = now require ∗ ∗ c,s) Bose fields ϕα,¯ϕα, and ignoring higher harmonics, we = † + † have Qc(x) Rc(x)Rc(x) Lc(x)Lc(x), † † (60) ηα − √i ϕ∗(x) η¯α √i ϕ¯∗(x) ∼ 2 α ∼ 2 α Qs (x) = R (x)Rs (x) + L (x)Ls (x), Rα(x) √ e ,Lα(x) √ e , (67) s s 2π 2π which ensure that the spin and charge fermions have the desired where η ,¯η are Klein factors fulfilling anticommutation re- U(1) charges: α α lations {ηα,ηβ }=2δα,β ={η¯α,η¯β }, {η¯α,ηβ }=0. Bosonizing † = † † = † the spin and charge fermions leads to the following expressions [qα,Rα(x)] Rα(x), [qα,Lα(x)] Lα(x). (61) for the original right- and left-moving spinful fermions: Our refermionization prescription then reads − √i ϕ∗(x)+ √i ∗ (x) R↑(x) ∝ e 2 α 4 2 α , R↑(x) ∼ η↑ O (x) O (x), c s α=c,s † (68) ↓ ∼ ↓ O O ∗ ∗ R (x) η c(x) s (x), (62) √i ϕ¯ (x)− √i (x) ↑ ∝ 2 α 4 2 α iπ x   L (x) e . − −∞ dx Qα (x ) Oα(x) ∼ Rα(x) e 2 . α=c,s ∗ ∗ Analogous relations hold for left-moving fermions. One issue The new Bose fields ϕα,¯ϕα are related to the usual spin and x   that arises here is that ns (x) = −∞ dx Qs (x ) is the number of charge bosons (52) by a canonical transformation −∞ spinons on the interval [ ,x], and therefore string operators ∗ √ = √α = ∗ of the form exp[iαns (x)] should be 2π-periodic functions of α , α 2 α. (69) α. When bosonizing string operators naively this periodicity 2 is lost. A simple way of dealing with this issue is via the Given (69), it is straightforward to rewrite (52) in terms of the replacement [64] new Bose fields −→ +  exp[iαns (x)] exp[i(2πm α)ns (x)]. (63) vα ∗ 2 ∗ 2 H = dx [(∂x ) + (∂x ) ] m 16π α α α The operators O (x) fulfill braiding relations for x = y: √ α + ∗ 2 − ∗ 2 + ∗ λα[(∂x α) (∂x α) ] λ1 cos( s / 2) − iπ − O O = 2 sgn(x y)O O α(x) α(y) e α(y) α(x). (64) α √ The low-energy effective Hamiltonian (52)isexpressedin λ2 ∗ ∗ +√ ∂x cos( / 2) +··· , (70) terms of our fermionic charge and spin fields as 2 c s

 1 vα 1 H = H + H + H where v = vα(Kα + ) and λα = (Kα − ). dx[ c(x) s (x) cs(x)], α 4Kα 16π 4Kα H = † −  − 2 + †  − 2 c Rc ivc∂x η∂x Rc Lc ivc∂x η∂x Lc + † † +··· IV. LUTHER-EMERY (LE) POINT gc,0RcRcLcLc , (65) FOR SPIN AND CHARGE H = R† −iv ∂ + iζ∂3 R + L† iv ∂ − iζ∂3 L s s s x x s s s x x s A particular case of the family of Hamiltonians (65) + † † + † + † +··· describes a free theory of noninteracting gapless fermionic gs,0Rs Rs Ls Ls gs,1(Rs Ls Ls Rs ) , spinons and holons. This LE point for both spin and charge H = † + † † + +··· cs g1(RcRc LcLc)(Rs Ls H.c.) . corresponds to A crucial feature of this expression is that the coupling constants of the marginal interactions, H = † −  − 2 +··· LE dx Rc ivc∂x η∂x Rc ∼ 1 − gα,0 2πvα Kα , (66) + †  − 2 +··· 4Kα Lc ivc∂x η∂x Lc

245150-7 F. H. L. ESSLER, R. G. PEREIRA, AND I. SCHNEIDER PHYSICAL REVIEW B 91, 245150 (2015) + † −  + 3 +··· Rs ivs ∂x iζ∂x Rs − (p) s †  + − 3 +··· ω s(p) Ls ivs ∂x iζ∂x Ls . (71) Here we have included the quadratic (cubic) term in the holon (spinon) dispersion to emphasize the nonlinearity. In order to 0 realize a Hamiltonian of this form fine-tuning a number of couplings is required, as can be seen by analyzing the stability of (71) to perturbations.

A. Stability of the LE point k − q −kF 0 F kF An obvious question is to what extent the LE point is stable. p The most important perturbations to (71)are FIG. 2. (Color online) Support in the energy/momentum plane H = † † of excitations with the quantum numbers of an electron or hole. For pert dx[gc,0RcRcLcLc commensurate band fillings, there is an absolute threshold that we take † † † † to follow the spinon/anti-spinon dispersions. Above the threshold, the + gs,1(R L + L R ) + gs,0R Rs L Ls s s s s s s single-particle spectral function is singular, and we aim to determine + † + † † + g1(RcRc LcLc)(Rs Ls H.c.)]. (72) the threshold exponent of the negative-frequency part at a momentum kF − q (green circle). In addition to (72) there are other, less relevant perturbations. A list of the ones with scaling dimension below four is given in B. Threshold singularities in the single Appendix B.Thegs,1 term in Eq. (72) is recognized as a mass term for spinons, and is the only strongly relevant perturbation. electron spectral function This implies that spinons are generically gapped, and in order Given the low-energy Hamiltonian at the LE point (71), we to reach a LE point with gapless spinons fine tuning gs,1 = 0 are now in a position to derive a mobile impurity model, valid a is necessary. Assuming that this is possible, we are left with priori at low energies. The usual continuity arguments suggest three perturbing operators of scaling dimension 2. While the that the restriction to low energies can be relaxed and the model gs,0 and gc,0 terms are scalar, the g1 term carries nonzero applied to energies of the order of the lattice scale t. Let us Lorentz spin. In order to assess the stability of the LE point focus on the mobile impurity model relevant for analyzing the to these perturbations, we have carried out a renormalization threshold behavior in the single-electron spectral function, group analysis. In principle, we need to work with different 1 cut-offs for the charge and spin degrees of freedom. However, A(ω,k) =− Im G (ω,k), at one-loop logarithmic divergences are encountered only in ret π the spin sector. We obtain RG equations of the form ∞ iωt −ikla0 Gret(ω,k) =−i dt e e (76) dg1 1 dυc 1 = g g , =− g2, (73) 0 l d 4πυ 1 s,0 d 2π 2υ 1 s s × |{ † }| ψ0 cj+l,σ (t),cj,σ ψ0 , dg 1 dg dυ c,0 =− 2 s,0 = s = g1, 0, (74) where |ψ is the ground state. d πυs d d 0 For commensurate band fillings the spectral function has where  = ln(L/a0) and a0 and L are short- and long-distance a threshold at low energies. To be specific, we will consider cutoffs, respectively. The RG equations are easily integrated,   the case vs ω>− (k), produced under the RG flow if the bare coupling is initially set A(ω,k) = s (77) A |ω +  (k)|μ if ω →− (k). to zero. We have checked that this remains true at two loops. 0 s s However, we cannot rule out that g may be generated at s,1 Here, s (k) denotes the spinon dispersion. higher orders and it is possible that setting it to zero requires fine tuning an infinite number of parameters in a lattice model. (2) The coupling gs,0 does not flow under the RG. This remains C. Threshold exponent at the LE point true at two-loop order. Hence, to this order, gs,0 needs to be Let us focus on momentum transfers kF − q, where we take fine-tuned to zero in order to reach the LE point. (3) If the initial 0

245150-8 SPIN-CHARGE-SEPARATED QUASIPARTICLES IN ONE- . . . PHYSICAL REVIEW B 91, 245150 (2015) we see that the relevant field theory correlator is this projection corresponds to  ∞ ∞  dp iωt+iqx ipx A(ω,k − q) ∼−i dt dx e rs (x) ∼ e Rs (p), F  0 −∞ − 2π (87) † q+ × |{ ↑ }| dp ψ0 R (t,x),R↑(0,0) ψ0 † ∼ i(p−q)x  | | χs (x) e Rs (p), q . −  = A<(ω,kF − q) + A>(ω,kF − q). (79) q  2π Substituting the decomposition (85) into our expression of the A A Here, > and < are, respectively, the positive and negative Hamiltonian density (71) and dropping oscillatory contribu- frequency parts of the spectral function. Using (62), we arrive tions under the integral, we arrive at the following mobile at the following expression for the latter: impurity model: ∞ ∞ − ∼− iωt+iqx H(0) = −  † − † A<(ω,kF q) i dt dx e MIM dx ivα(rα∂x rα lα∂x lα) −∞ 0 α=c,s × ψ |O† (0,0)O (t,x)|ψ . (80) 0 α α 0 + † − +··· α=c,s dx χs (εs ius ∂x )χs. (88) = − ≈  + At the LE point, we are dealing with a free fermion theory. Here, we have introduced notations εs s (kF q) vs q 3 ds  2 Hence correlation functions of the kind required in Eq. (80) ζq +··· and us =− | ≈ v + 3ζq . dk kF −q s can be expressed as Fredholm determinants [66],butwedonot Next, we need to work out the projection of the operator follow this route here. Instead, we construct a mobile impurity Os on low-energy (rs ) and impurity (χs ) degrees of freedom. model and use it to extract the threshold exponent. Using that rs(x), ls (x), and χs (x) are slowly varying fields, we At the LE point spin and charge degrees are perfectly can approximate the string operator as separated. As a consequence it is possible to construct a basis −iqx † † − † π [ e r (x)χ (x)−H.c.] O ∼ + iqx 2 q s s of energy eigenstates in the form s (x) [rs(x) e χs (x)]e iπ x  † † † − −∞ dx [rs (x)rs (x)+ls (x)ls (x)−χs (x)χs (x)] |nc ⊗|ns , (81) ×e 2 . (89) Here the second term is the contribution of the string arising where nc,s are appropriate charge and spin quantum numbers. from the upper boundary of integration x. By virtue of the The correlators required in Eq. (80) then have Lehmann iqx representations of the form presence of the strongly oscillatory factor e in the expres- sion (80) for the spectral function, the leading contribution to iE t−iP x 2 the spectral function arises from the part of O (x) proportional e nα nα |n |O (0,0)|ψ | ,α= c,s. (82) s α α 0 to e−iqx: nα O (x) = O(q)(x)e−iqx +··· . (90) The threshold singularity arises from excitations involving a s s single high-energy spinon with momentum q plus low-energy In terms of this component, we have excitations in the charge and spin sectors. This means that the ∞ ∞ − ∼− iωtO† O charge part of (80) can be calculated using bosonization. The A<(ω,kF q) i dt dx e c(0,0) c(t,x) bosonization identities (68) imply that 0 −∞ † × O(q) O − √i ∗ √i ∗ s (0,0) s (t,x) . (91) ϕc (x) c (x) Oc(x) ∝ e 2 e 4 2 . (83) In order to isolate the desired contribution, we expand the At the LE point the coupling constants λ , λ , λ in Eq. (70) second factor in Eq. (89): α 1 2 vanish, and a simple calculation gives π O(q) ∼ −iqx † † +··· s (x) e rs (x) rs (x)χs (x) †  − 1  2 − 1 2q O O ∝ − 2 2 − 2 16 c(0,0) c(t,x) (vct x) x vc t . (84) π 2 + χ †(x) 1 − {χ (x)r (x),r†(x)χ †(x)}+··· In order to work out the contribution from the spin part, we s 8q2 s s s s follow Ref. [67]. We decompose the spin fermions into low- † † † − iπ x dx[r (x)r (x)+l (x)l (x)−χ (x)χ (x)] energy and mobile impurity parts: × e 2 −∞ s s s s s s . (92) ∼ + −iqx † In order to proceed further, it is convenient to bosonize Rs (x) rs (x) e χs (x), (85) the low-energy degrees of freedom associated with rs , ls Ls (x) ∼ ls (x), using (68): O(q) ∼ −iqx † + ∗ +··· † s (x) e χs (x)[b0 b1∂x ϕs (x) ] where χs creates a hole in the spinon band. In terms of † √i ∗(x)+ iπ x dx χ (x)χ (x) momentum modes, × e 4 2 s 2 −∞ s s . (93) dp This can be simplified further using the appropriate operator R (x) ∼ eipx R (p), (86) s 2π s product expansions. In order to determine the threshold

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singularity, it is sufficient to retain only the term with the scope of our work. The functions fα(q), f¯α(q)aswellasthe lowest scaling dimension at low energies, which is parameters vα, Kα, εs , us depend on the microscopic details − † √i ∗(t,x) of the particular lattice realization of our field theory. We will O(q) ∼ iqx 4 2 s s (t,x) e χs (t,x)e . (94) show below how they can be fixed either numerically in the The two-point correlator of this operator is generic case or analytically when our theory is applied to the † Hubbard model. O(q) O(q) s (0,0) s (t,x) The mobile impurity model (98) can now be analyzed  by standard methods [54,60]. The interaction between the −  2 − 1 sin[ (x − us t)] ∼ e iqx x2 − v t2 16 eiεs t . impurity and the low-energy degrees of freedom can be s − (95) π(x us t) removed through a unitary transformation  Sending the cutoff  to infinity turns the last term into a ∞ ∗ ∗ † −i dx [γ ϕ (x)+γ¯ ϕ¯ (x)]χ (x)χ (x) U = e −∞ α α α α α s s . (99) delta function δ(x − us t). Substituting the resulting expression for (95) and the charge sector contribution (84) into the The transformed spin impurity field equals expression (80) for the hole spectral function and then carrying = † out the space and time integrals, we arrive at the following ds (x) Uχs (x)U ∗ ∗ 2 2 result for the threshold behavior: i [γα ϕ (x)+γ¯α ϕ¯ (x)] −iπ (γ −γ¯ )C(x) = χs (x)e α α α e α α α , (100) − ∝| + |−1/4 A<(ω,kF q) ω εs . (96) while the chiral spin and charge Bose fields transform as As expected there is a threshold singularity. The exponent is ◦ = ∗ † = ∗ − ϕα(x) Uϕα(x)U ϕα(x) 2πγαC(x), seen to be momentum independent. As we will see, this is (101) ◦ = ∗ † = ∗ + particular to the LE point. ϕ¯α(x) Uϕ¯α(x)U ϕ¯α(x) 2πγ¯αC(x), where V. MOBILE IMPURITY MODEL AWAY ∞ = − † FROMTHELEPOINT C(x) dy sgn(x y)χs (y)χs (y). (102) −∞ We now wish to generalize the above analysis to the Luttinger liquid phase surrounding the LE point. We will We note that assume that ◦ = ∗ − † ∂x ϕα(x) ∂x ϕα(x) 4πγαχs (x)χs (x), 1. the spinon mass term is fine-tuned to zero, i.e. g = 0; (103) s,1 ◦ = ∗ + † 2. the four fermion interactions in the spin and charge ∂x ϕ¯α(x) ∂x ϕ¯α(x) 4πγ¯αχs (x)χs (x).

sectors are attractive, i.e., gc,0,gs,0 < 0, and sizable. Adjusting the parameters γα,¯γα such that Under these assumptions holons and spinons remain gap- f u − v+ −v− γ less, and the g1 term in Eq. (72) is irrelevant so that we can α = α α α ¯ − + + , (104) drop it at low energies. Focussing again on the single-electron fα vα vα u γ¯α spectral function, using the decomposition (85), and finally, with bosonizing the low-energy spin and charge degrees of freedom, v 1 we arrive at a mobile impurity model of the form v± = α 2K ± , (105) α 2 α 2K α H = dx H + H + H , (97) the impurity decouples in the new variables: MIM α imp int α=c,s H = H + H v 1 MIM dx α imp , H = α ∗ 2 + ∗ 2 α (∂x α) 2Kα(∂x α) , α=c,s 16π 2Kα v 1 H = † − H = α ◦ 2 + ◦ 2 imp χs (εs ius ∂x )χs , (98) α (∂x α) 2Kα(∂x α) , (106) 16π 2Kα † ∗ ∗  † H = + ¯ H = d (ε˜s − ius ∂x )d . int χs χs fα(q)∂x ϕα fα(q)∂x ϕ¯α . imp s s α The interaction between the mobile impurity and the Here, we have dropped all terms that do not affect the threshold Luttinger liquid degrees of freedom is now encoded in the exponent and retained the same parametrization of the impurity ◦ ◦ boundary conditions of the transformed Bose fields α, α, part of the Hamiltonian, although the actual values of s and which are “twisted” by the presence of the impurity, see, us are of course not the same as the LE point. The Luttinger e.g., (103). The negative-frequency part of the single-electron = parameter in the spin sector varies from Ks 1/2atthe spectral function is again given by (91), where LE point to Ks = 1 in the SU(2)-invariant limit. The charge − † √i ∗ − √i ϕ∗ √i ∗ = O O(q) ∼ iqx 4 2 s 2 c 4 2 c Luttinger parameter equals Kc 1/2 at the LE point, and c(x) s (x) e χs e e e . (107) varies with doping and interaction strength otherwise. We In terms of the transformed fields, this reads note that close to the LE point (in the sense that gc,0, gs,0 are H − † i(γ + √1 )ϕ◦+i(¯γ + √1 )¯ϕ◦ small), there is√ an additional contribution to int of the form O O(q) ∼ iqx s 4 2 s s 4 2 s † ∗ c(x) s (x) e ds e χs χs cos( s / 2). The analysis of this case is very interesting ◦ ◦ i(γ − √3 )ϕ +i(¯γ + √1 )¯ϕ (see, e.g., Ref. [68] for a related problem), but beyond the ×e c 4 2 c c 4 2 c . (108)

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The threshold behavior of the hole spectral function can now transformed Hamiltonian (106), we obtain be calculated in the same way as at the LE point. The result is 2πv 1 q + q¯ 2 E = α α α − γ + γ¯ 1 LL L 4K 4π α α A (ω,k − q) ∼ , (109) α=c,s α < F | + |μ ω 0 − 2 + qα q¯α − − + + + − where the exponent μ is given by Kα γα γ¯α n[Mn,α Mn,α] . 4π = − 2 + 2 + 2 + 2 n>0 μ 1 2 νc,+ νc,− νs,+ νs,− , (116) Kc 1 1 1 ± νc,± = γc + γ¯c − √ ± √ γc − γ¯c − √ , Here, qα, q¯α, and Mn,α are “quantum numbers” characterizing a 2 2 2 8Kc 2 particular low-energy excitation. Their quantization conditions depend on the boundary conditions for the spin and charge Ks 1 1 νs,± = γs + γ¯s + √ ± √ (γs − γ¯s ). (110) Bose fields (112) in the presence of a high-energy mobile 2 2 2 8Ks impurity. These can be worked out by considering the “minimal” excitation that can be made in the sector where the As γα,¯γα are functions of q, the threshold exponent is now generally momentum dependent. However, it is shown in mobile impurity is present. This sector is reached by acting Appendix C that spin rotational SU(2) symmetry in the limit with the operator K → 1 enforces the particular values − † √i ◦ − √i ϕ◦ √i ◦ s O O(q) = iqx 4 2 s 2 c 4 2 c c(x) s (x) e χs e e e (117) 1 | γ = γ¯ =− √ , (111) on the ground state ψ0 . The latter is characterized by being s s ∗ ¯ ∗ 4 2 annihilated by aα,R,n, aα,L,n, Qα, and Qα. The quantum numbers of the “minimal” excitation are found by noting that for any value of q. ∗ O O(q) | = ∗ O O(q) | Qα c(x) s (x) ψ0 [Qα, c(x) s (x)] ψ0 γ γ = (0)O O(q) | A. Relation of α, ¯α to finite-size energy spectra qα c(x) s (x) ψ0 , (118) An obvious question is whether there is a way of directly where determining the parameters γ ,¯γ for a given microscopic α α 3π π π (0) = √ (0) = √ (0) =− (0) =−√ lattice model. To that end, let us consider the spectrum of our qc , q¯c ,qs q¯s . (119) mobile impurity model on a large, finite ring of circumference 2 2 2 ∗ ∗ L. The mode expansions of the Bose fields ϕα,¯ϕα are Let us denote the lowest-energy state with these quantum numbers by ∞ ∗ ∗ x ∗ 2 i 2πn x −i 2πn x † (0) (0) (0) (0) ϕ (x) = ϕ + Q + (e L a + e L a ), q ,q¯ ,q ,q¯ . (120) α α,0 L α n α,R,n α,R,n c c s s n=1 Higher excited states in the “impurity sector” can be obtained, ∞ ∗ ∗ x ∗ 2 −i 2πn x i 2πn x † for example, by making particle-hole excitations on top of the ϕ¯ (x) = ϕ¯ + Q¯ + (e L a + e L a ). α α,0 L α n α,L,n α,L,n state (120). The energies of such states are given by (116) n=1 ± by choosing (119) and in addition taking some of the Mn,α (112) different from zero. Crucially, the values of γα,¯γα are the ∗ ¯ ∗ ∗ ∗ same as for the state (120). For practical purposes, states with Here, Qα, Qα, ϕα,0, andϕ ¯α,0 are zero mode operators with the same momentum as (120) but different spin and charge commutation relations quantum numbers might be of particular interest as they are the [ϕ∗ ,Q∗ ] =−4πi =−[¯ϕ∗ ,Q¯ ∗ ]. (113) lowest energy states in certain sectors of quantum numbers and α,0 α α,0 α can therefore be more easily targeted in DMRG computations. ∗ ¯ ∗ The quantum numbers for such states are The eigenvalues qα, q¯α of the zero mode operators Qα, Qα ∗ ∗ depend on the boundary conditions on the fields ϕα,¯ϕα, which (0) π qα = q + √ (3mα + m¯ α), on general grounds will depend on whether or not a mobile α 2 impurity is present. In the presence of the impurity, the finite- π (121) = (0) + √ + size spectrum of (98) has the following structure: q¯α q¯α (3m¯ α mα), 2 −1 E = EGS + Eimp + ELL + o(L ). (114) where mα, m¯ α are integers. This follows from the mode expansion and the requirement that the bosonized expressions Here, Eimp is the contribution of the impurity to the energy. On for R↑(x), R↓(x), L↑(x), L↓(x) must be single valued. Using general grounds, it will have the following expansion in terms that at low energies, the charge, spin densities, and currents of the system size: are given by 1 = (0) + (1) + −1 =− √1 ◦ Eimp Eimp Eimp o(L ). (115) ρα(x) ∂x (x), L π 8 α

The other contribution to (114) arises from the Luttinger-liquid 1 ◦ jα(x) =− √ ∂x (x), (122) part of the theory. Applying the mode expansions to the π 8 α

245150-11 F. H. L. ESSLER, R. G. PEREIRA, AND I. SCHNEIDER PHYSICAL REVIEW B 91, 245150 (2015) we can identify that we obtain a set of equations in which the only unknown parameters are the γα,¯γα. This provides a numerical method (mc + m¯ c)/2 is the difference in charge (particle number) for determining them in a general lattice model. For integrable between the excitation and the state (120), theories like the Hubbard model, analytical techniques are available and we discuss this case next. (mc − m¯ c)/2 is the number of charge fermions transferred from the right-moving to the left-moving branch,

(ms + m¯ s )/2 is the difference in the number of down spins B. Hubbard model between the excitation and the state (120), The finite-size spectrum in presence of a high-energy spinon (ms − m¯ s )/2 is the number of spin fermions transferred excitation was calculated for the case of the Hubbard model from the right-moving to the left-moving branch. using the Bethe ansatz solution in Ref. [53]. The result for the lowest excited state above the spinon threshold is The values of γ ,¯γ can now in principle be extracted by α α 1 numerically computing finite-size energy levels by a method E = E −  (h) −  (h)δh GS s L s such as momentum-space DMRG [69,70]. The procedure is outlined in the following: 2πv (N − N imp)2 D 2 + c c c + K D − Dimp + s (1) The velocities vα and Luttinger parameters Kα are bulk 2 c c c L 8Kc 2 properties and can be determined by standard methods. We will assume them to be known quantities in the following. We − 1 − 1 2 2 2πv Ns Nc D − further denote the number of particles and down spins in the + s 2 2 + s + o(L 1), L 2 2 ground state by NGS and MGS, respectively. (2) We then numerically compute the lowest excitation (129) with momentum q and quantum numbers N = NGS − 1, M = MGS − 1. Its energy is where

(0) 1 (1) D = D = 0,N=−1,N= 0. (130) E− − = E + E + E s c c s 1, 1 GS imp L imp − h − 1  h h + (0) (0) (0) (0) + −1 The contribution s ( ) L s ( )δ is the finite-size ELL qc ,q¯c ,qs ,q¯s o(L ). (123) energy of the impurity. The velocities vα, Kα as well as imp imp (3) Next, we compute the lowest excitations with momen- the quantities Nc and Dc are expressed in terms of tum q but different values of N and M. For example, choosing solutions to coupled linear integral equations, and in practice are easily calculated numerically with very high precision. By N = N − 1,M= M − 2, (124) GS GS construction, the quantum numbers (130) correspond to our gives the excited state characterized by minimal excited state (120). By matching the Luttinger liquid part of the energy to (114) we then obtain the following results mc =−m¯ c = 1,ms =−2,m¯ s = 0. (125) for the parameters γα,¯γα: The zero-mode eigenvalues follow from (121) √ imp 1 imp Nc 5π π γc + γ¯c = √ − 2D ,γc − γ¯c =− √ , q(1) = √ , q¯(1) =−√ , 2 2 c 2 c 2 c 2 (131) (126) 1 7π π γs = γ¯s =− √ . q(1) =−√ , q¯(1) =−√ . 4 2 s 2 s 2 One subtlety to keep in mind when making contact between The corresponding energy is the Bethe ansatz calculation and (114) is that the former refers (0) 1 (1) only to highest weight states of the SU(2)⊗ SU(2) symmetry E− − = E + E + E 1, 2 GS imp imp algebra of the Hubbard model [71,72]. Descendant states need L + (1) (1) (1) (1) + −1 to be taken into account separately. Substituting (131)inthe ELL qc ,q¯c ,qs ,q¯s o(L ). (127) ± expressions for the quantities να (110), we find Here, we have asserted that the change in the impurity −1 1 + N imp contribution to the energy is of higher order in L . This has ± = ± =− imp ∓ √ c νs 0,νc KcDc . (132) been shown for the case of the Hubbard model using methods 4 Kc of integrability in Ref. [53]. We believe that this continues (1) Finally, the threshold exponent is obtained from (110): to hold true in general, because Eimp is sensitive only to the values of the parameters γα,¯γα, which are the same for all 1 + N imp 2 excitations we are considering. = − c − imp 2 μ 1 4Kc Dc . (133) The point is that by considering energy differences like 4Kc − = (1) (1) (1) (1) E−1,−2 E−1,−1 ELL qc ,q¯c ,qs ,q¯s This agrees with what was found in Ref. [53]usingthe approach of Schmidt, Imambekov, and Glazman [51,52], − E q(0),q¯(0),q(0),q¯(0) + o(L−1) LL c c s s as well as with the exponents reported previously in (128) Refs. [31–33].

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1. Other excited states right-moving spin quasiparticle in Eq. (138) is projected into The Bethe ansatz result (129) can be applied to other low-energy and impurity subbands: excited states as well. In particular, the holon plus three spinon R˜ (x) ∼ r˜ (x) + e−iqxχ˜ †(x). (140) excitation considered in Ref. [53] gives rise to an excitation s s s with the same momentum and quantum numbers: One then rewrites the electron operator in terms of the quasiparticles defined in Eq. (138). As this differs from ours, N =−1 = N ,D=−1,D= 1 . (134) c s s c 2 cf Eq. (67), the string-operator part in the expression for the The corresponding state in our mobile impurity model electron operator is also different: √ has quantum numbers (125), and the energies calculated √i i √1 −iqx † φs − ( Kα + )φα ∼ 2 4 Kα from (114) and (129) agree as they must. R↑(x) e χ˜s (x)e e α=c,s √ − i − √1 ¯ VI. RELATION TO THE APPROACH OF SCHMIDT, ( Kα )φα × e 4 Kα . (141) IMAMBEKOV AND GLAZMAN The next step is to write down an effective impurity model, The method of Refs. [51,52] is based on a different analogous to Eq. (98), but usingχ ˜ as the impurity. After prescription for defining fermionic quasiparticles in the charge s performing a unitary transformation that removes the coupling and spin sectors. We now summarize the main steps of this betweenχ ˜ and the low-energy modes, the electron operator approach and compare them to our framework. The starting s becomes point is the standard bosonized description (53) of the spinful √ √ i √1  −iqx † − ( Ks + −2 2−4γ )φs fermion model under consideration. One then introduces the ∼ ˜ 4 Ks s R↑(x) e ds (x)e chiral components φα,φ¯α by √ √ − i − √1 −  ¯ + + √1 −  [( Ks 4¯γs )φs ( Kc 4γc )φc] ×e 4 Ks Kc (x) = K (φ + φ¯ ), (135) √ α α α α i √1  − [( Kc− −4¯γ )φ¯c] ×e 4 Kc c , (142) = √1 − ¯ α(x) (φα φα). (136) ˜ = † Kα where ds Uχ˜s U is the free impurity field for the quasipar-   These chiral bosons diagonalize the spinful Luttinger ticle with fractional charge, and γα,γ¯α are the parameters of model (53) in the form the unitary transformation, which are not the same as γα,γ¯α discussed in Sec. V. vα 2 2   H (x) = [(∂ φ ) + (∂ φ¯ ) ]. (137) At the LE point, we set Kα = 1/2 and γ = γ¯ = 0 and LL 8π x α x α α α α=c,s Eq. (142) reduces to

√i √i Next, in analogy with the spinless case in Eqs. (41) and (42), −iqx † − φc (φs +φ¯s +φc+φ¯c) R↑(x) ∼ e d˜ (x)e 2 e 4 2 . (143) one defines the quasiparticle operators s

− √i φ (x) √i φ¯ (x) This result agrees with the refermionization in Sec. IV C, since ˜ ∼ 2 α ˜ ∼ 2 α Rα(x) e , Lα(x) e . (138) ˜ = = = ∗ ¯ = ∗ at the LE point dα dα χα, φα ϕα, and φα ϕ¯α; thus, As in the spinless case, this prescription removes the marginal the two approaches coincide. interactions between quasiparticles, rendering them asymptot- Moving away from the LE point, the expressions for ically free in the low-energy limit. However, these quasipar- physical operators in terms of impurity and low-energy fields ticles cannot be identified with the finite energy elementary will in general be different in the two approaches. However, excitations in integrable models, because they carry fractional the results for the edge exponents are still consistent because quantum numbers: the difference in string operators can be accommodated by the parameters of the unitary transformation and by imposing ˜ † = ˜ † ˜ † = ˜ † [qα,Rα(x)] 2KαRα, [qα,Lα(x)] 2KαLα. (139) proper boundary conditions on the bosonic fields. For instance, imposing SU(2) symmetry in the approach of Refs. [51,52] The only exception is the LE point Kα = 1/2, where the ˜ ˜ leads to the requirements operators Rα,Lα in fact carry the same quantum numbers as √ our spin and charge fermions. γ  = 2 − 1, γ¯  = 0. (144) Despite the lack of correspondence with long-lived ex- s s citations in (nearly) integrable models, the approach of This should be contrasted with Eq. (111). Refs. [51,52] allows one to compute threshold exponents, as long as the parameters in the effective impurity model are VII. REALIZING THE LE POINT adjusted appropriately. The situation is analogous to the two ways of determining threshold exponents discussed above in We have seen that a good starting point for understanding the spinless fermion case. threshold singularities in dynamical response functions is the In order to facilitate a direct comparison with our approach, LE point for both charge and spin. In Sec. IV, we considered we briefly review how to express the electron operator in order properties of the LE point in the field theory limit. An obvious to calculate the threshold exponent in the single-particle spec- question raised by these considerations is whether it is possible tral function within the approach of Refs. [51,52]. We assume to realize the LE point in practice in a lattice model of again that the lower threshold of the support corresponds interacting spinful fermions. We now investigate this issue to an excitation with a single high-energy spinon. First, the in some detail and present a number of preliminary results.

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A. Lattice model A SzSz ∼ O(2)(x) + 2O(2)(x) . (150) As discussed in Sec. IV A, realizing the LE point at j j+1 4 2 3 sufficiently low energies in an extended Hubbard model of the Here, x = ja0, A is a dimensionful amplitude, and kind (47) requires the fine-tuning of (at least) four parameters: O(2) = 1 cos s , K = K = 1 ,g = g = 0. (145) O(2) = 2 + 2 c s 2 s,1 1 2 (∂x ϕs ) (∂x ϕ¯s ) , O(2) = For the Hubbard model in zero magnetic field, spin rotational 3 ∂x ϕs ∂x ϕ¯s , symmetry fixes Ks = 1, while Kc varies with both band (2) 2 2 O = (∂x ϕc) + (∂x ϕ¯c) , filling and interaction strength U. In particular, it is well 4 = 1 →∞ O(2) = ∂ ϕ ∂ ϕ¯ . (151) known [23,24] that Kc 2 is obtained in the U limit of 5 x s x s the Hubbard model [14,73]. Values Ks < 1 can be realized by At weak coupling, the spin-charge separated Luttinger liquid adding spin-dependent interactions that break the spin SU(2) at low energies is therefore perturbed by symmetry, but retain spin inversion symmetry. The latter is crucial for avoiding marginal interactions between spin and 5 5 H = O(2) ≡ H charge sectors that lead to a more complicated conformal δ νj dx j (x) δ j , (152) spectrum involving a dressed charge matrix [7,23,24]. j=1 j=1 A minimal lattice model that may allow us to fulfill the where the νj are proportional to linear combinations of U, V1, conditions in Eq. (145)is z H H V2, and J1 . The effects of δ 2 and δ 4 are to renormalize the spin and charge velocities respectively, while δH3 and L−1 L † δH5 change the values of the Luttinger parameters Ks and H =−t (c c + + H.c.) + U nj,↑nj,↓ j,σ j 1,σ Kc. We must pay special attention to the perturbation δH1:as j=1 σ j=1 discussed in Appendix B, this operator is marginal at weak 2 L−r L−1 coupling, but gives rise to the relevant spinon mass term as we + + z z z Vr nj nj+r J1 Sj Sj+1. (146) approach the LE point. Our objective is to adjust U, V1, V2, and z r=1 j=1 j=1 J1 in such a way that Kc,s are reduced towards 1/2, while the coupling ν1 of the spinon mass term remains very small. The In anticipation of the DMRG computations of energy levels bosonization results (150) suggest the following prescription reported below we have imposed open boundary conditions. for achieving this at weak coupling: (1) increase V2 in order to = | | z In the following we set t 1, i.e., measure all energies in make ν1 very small. (2) Then adjust V1 and J1 in order to drive units of the hopping parameter. The idea is then to try to Kc,s towards 1/2. Importantly, the bosonization results (150) z adjust the four interaction strengths U, V1,2 and J1 in such indicate that at least at weak coupling this does not produce a way that (145) are achieved. In order to ascertain the sizable contributions to δH1. Our analysis below is guided by low-energy properties of (146), we compute the energies of these considerations, even though we are not operating in the the ground state and several low-lying excited states for a weak coupling regime, in which (150) are applicable. quarter-filled band, and compare the results to expectations based on Luttinger liquid theory. We choose to work at quarter B. Finite-size spectrum of unperturbed Luttinger liquid filling in order to simplify finite-size scaling analyses. As with open boundary conditions alluded to in Appendix A, working at commensurate fillings A standard procedure for determining the Luttinger pa- induces additional Umklapp interactions. In the case at hand, rameters K and K is to compare the finite-size spectrum this corresponds to the presence of an additional perturbation c s predicted by Luttinger liquid theory with the low-energy dx cos(2 ). However, this term has a scaling dimension c spectrum calculated numerically for a given lattice model. The 8K and is therefore highly irrelevant for K  1/2. We c c finite-size spectrum relative to the ground state of a Luttinger therefore discard it in the following analysis. liquid with open boundaries is given by Some insight into how the parameters Kc,Ks and gs,1 z z c s depend on U, V1,2 and J1 can be gained by bosonizing the E S ,Nc, m,m interactions at weak coupling. Using (48), we obtain in leading z 2 2 order = πvs (S ) + πvc(Nc) A Ks L 4KcL ∼ O(2) + O(2) − O(2) ∞ nj,↑nj,↓ 4 (x) 2 5 (x) 2 (x) πv dN πv 4 − c c + α α + −1 m o(L ). (153) − O(2) + O(2) 2KcL = = L 2 3 (x) 8η↑η¯↓η↓η¯↑ 1 (x) , (147) α c,s  1 z Here, S and Nc are quantum numbers in the spin and charge ∼ A O(2) + O(2) sectors, respectively, associated with the global U(1)⊗U(1) nj nj+1 4 (x) 2 5 (x) , (148) symmetry. The value of Sz measures the change in the total = − A magnetization and Nc N N0 measures the change in ∼ O(2) + O(2) + O(2) nj nj+2 3 4 (x) 6 5 (x) 2 (x) the total number of electrons with respect to a singlet ground 2 z state with N0 electrons. As usual, the values of S and Nc + O(2) − O(2) z 2 3 (x) 8η↑η¯↓η↓η¯↑ 1 (x) , (149) are constrained by the selection rule that 2S must be even

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(odd) if Nc is even (odd). The parameter d in Eq. (153) U>0, this can be done efficiently by increasing the next- is a dimensionless constant that depends on the definition nearest-neighbor interaction V2 > 0. In this process, we keep s = z = of the chemical potential to order 1/L. The parameters m V1 J1 0, so the SU(2) symmetry is preserved. As we keep c and m are non-negative integers that count the number of increasing V2, the marginal coupling constant will change sign c low-energy particle-hole pairs in each sector. We note that at some critical value V2 . Beyond this point the perturbation for open boundary conditions the total momentum is not becomes marginally relevant and the system undergoes a conserved and there are no quantum numbers associated with Berezinskii-Kosterlitz-Thouless (BKT) transition to a spin- current excitations (i.e. with transferring particles between gapped phase, analogous to the dimerization transition in the Fermi points). Thus (153) should not be confused with the J1 − J2 spin chain [74]. spectrum in Eq. (116), which is valid for periodic boundary Because of the exponentially small value of the gap in conditions. the vicinity of the critical point, it is difficult to pinpoint c The spin and charge velocities vs and vc can be extracted V2 by means of a gap scaling analysis. A better approach by matching the energies of the lowest excitations with is to determine the critical point by searching for a level z S = Nc = 0. Assuming vs

245150-15 F. H. L. ESSLER, R. G. PEREIRA, AND I. SCHNEIDER PHYSICAL REVIEW B 91, 245150 (2015) bare λ2 has not been fine tuned and is not guaranteed to be small. 0.15 We see that the size dependence becomes increasingly complex as Ks tends towards 1/2, which complicates the analysis of energy levels. A good way of achieving an accurate description of finite-size energies would be through (two-loop) E renormalization-group-improved perturbation theory around Δ 0.1 both the LE point and the weak-coupling limits. However, as this is quite involved, we content ourselves with the simpler form (159) (and stress again that our numerical analysis is to be considered as preliminary). 0.05 0.8 11.2 1. Determining the Luttinger parameter Ks V2 Equation (159) provides us with an alternative way of FIG. 3. (Color online) Energies of the four lowest excited states determining Ks by considering the system-size dependence in the neutral sector Sz = N = 0forU = 3, J z = V = 0, and of the quantity c 1 1 = c L 64 as a function of V2. The critical value V2 is identified through the crossing of the two nearly degenerate states in the (2s,0) multiplet. δ(L) ≡ 1 E (2,0) − E (1,0). (160) 4 0 0 Lines are guide to the eye. Using the expression (159), we obtain 4. Determining Kc ˜ ˜ = a˜1 + b1 + b2 +··· Finally, the energy difference related to the compressibil- δ(L) − − . (161) L˜ 2Ks 1 L˜ 2Ks L˜ 4Ks 1 ity (157) is found to have the following size dependence: E (0,2) + E (0,−2) πv bc By fitting δ(L) computed numerically for a range of system 0 0 = c + 2 +··· 4K −1 sizes to (161), we obtain a value for Ks . 16 8KcL˜ L˜ s ≡ f (L). (164)

2. Determining the spin velocity vs We can use (164) together with Ks and vc determined in Given Ks , we may determine vs from the size dependence Secs. VII D 1 and VII D 3, respectively, to fix the Luttinger in Eq. (159). Alternatively we can consider other excited parameter Kc through a finite-size scaling analysis of f (L), c states, whose finite-size energies can be analyzed similarly. taking b2 to be a fit parameter. For example, for vs

Using the results for Ks from the analysis described in The analysis presented below suggests that this corresponds Sec. VII D 1,Eq.(163) provides us with a means of deter- to Luttinger parameter values of Ks = 0.655 and Kc = 0.500. mining vc through a finite-size scaling analysis. This is probably as close to the LE point as one can get without

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0.001 0.04

0 0.03 -0.001 (L) f(L) δ

-0.002 0.02

-0.003

-0.004 0.01 50 100 150 200 50 100 150 L L

FIG. 4. (Color online) Crosses represent numerical results for FIG. 5. (Color online) Crosses represent numerical results for δ(L)asdefinedinEq.(160). The solid line is the best fit to f (L)asdefinedinEq.(164). The solid line is the best fit to (164) = = ˜ = Eq. (161) and is obtained for Ks 0.655,a˜1 0.009, b1 0.493, with Ks = 0.655. and b˜2 =−5.131.

˜c =− and b2 1.191. Combining (168) and (169), we conclude a more precise theoretical description of finite-size energies that based on a renormalization-group-improved perturbation the- ory analysis. K ≈ 0.500. (170) Figure 4 shows numerical results for the quantity δ(L) c defined in Eq. (160). The data have been fitted using (161). The resulting best-fit estimates are F. Friedel oscillations The analysis of finite-size energy levels presented above K = 0.655, (166) s is clearly rather involved, and an independent check on the results for Kc,s would clearly be very useful. Such a check a˜1 = 0.009, b˜1 = 0.493, and b˜2 =−5.131. The quality of the fit is visibly excellent (the residuals are ∼10−6). The numerical is provided by analyzing Friedel oscillations of the charge density on a chain with open boundary conditions, cf Ref. [78]. value for a˜1 is very small, confirming that we have almost succeeded with fine-tuning the spinon mass term to zero (on We summarize the main steps of how to calculate the charge the scale set by the system sizes we consider). density for open boundary conditions in (perturbed) Luttinger In the next step, we determine the spin velocity using (159) liquid theory in Appendix D. For a quarter filled band, we and by retaining the b and b terms. Setting K = 0.655, we obtain 1 2 s obtain π + p1 + 1 d sin + j p2 n ≈ + 1 + d 2 L 1 j 2K 2 (K +K )/2 v ≈ 1.707. (167) 2 (Dj ) s (Dj ) c s s + 2p1 + cos π + j 2p2 This value is consistent with the result obtained by considering + d L 1 +··· , (171) 3 2K { s = c = } (Dj ) c E(0,0, m δ,1,m 0 )inEq.(162). We have verified that the energy level corresponds to the first spin descendant state by tracking the tower of lowest-lying energies in the neutral sector along a path in parameter space connecting the Hubbard model to the point (165). 0.15

Having determined vs and Ks , we now turn to the charge }) l,1 sector. In Fig. 5, we present numerical results for the quantity δ = c f (L) defined in Eq. (164), together with a fit to the functional l

form posited in Eq. (164). The spin Luttinger parameter is =0,m s l fixed as Ks = 0.655. The best-fit estimates are 0.1

πvc = 2.000, (168) E(0,0,{m 8Kc Δ

c = O −5 0.05 and b2 0.271. The residuals are of order (10 ). 50 100 150 Finally, in Fig. 6, we present numerical results for L { s = c = } E(0,0, m 0,m δ,1 ). Fitting the data to (163) with Ks = 0.655 results in estimates FIG. 6. (Color online) Crosses represent numerical results for { s = c = } E(0,0, m 0,m δ,1 ). The solid line is the best fit to (163) vc ≈ 2.543, (169) for Ks = 0.655.

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0.9 Our construction differs in important aspects from pre- vious work by Schmidt, Imambekov, and Glazman [51,52]. 0.8 However, we demonstrated explicitly how and why results for threshold exponents obtained in the two approaches coincide. 0.7 Finally, we presented a preliminary analysis of the question 0.6 of how to realize, in the low-energy regime, the Luther-Emery j

n point for spin and charge in a lattice model of interacting 0.5 spinful fermions. We showed that the structure of allowed 0.4 perturbations to the Luther-Emery point is such that fine tunings of various interactions is required. Achieving these 0.3 fine tunings is very delicate, and we discussed in some detail what problems one encounters. 0.2 50 100 150 200 Our work raises a number of interesting questions that L deserve further attention. First and foremost, further numerical studies are required in order to identify a parameter regime in FIG. 7. (Color online) Crosses represent local density for a quar- = = = z = an appropriate extended Hubbard model that, at least approx- ter filled band and U 3, V1 0.85, V2 1.1, J1 5.2. The solid line is a fit to Eq. (171) with K = 0.65 and K = 0.5. imately, realizes the Luther-Emery point. Second, it would s c be very interesting to implement the numerical procedure we proposed for determining the parameters γα,¯γα characterizing where Dj denotes the threshold exponents. One first might want to reproduce the known exact results for the Hubbard model, before moving 2(L + 1) πj on to nonintegrable cases. Third, in close proximity to the D = sin . (172) j π L + 1 Luther-Emery point, the mobile impurity model involves an additional marginal interaction that cannot be removed by a In Fig. 7, we compare the prediction (171) to DMRG results unitary transformation. It would be interesting to analyze its for a quarter filled band and system size L = 208. We fix the effects on the threshold exponents. values of the Luttinger parameters to Ks = 0.65 and Kc = 0.5 and use the amplitudes dj and phase shifts pj as fit parameters. The agreement is very good except near the boundaries. The ACKNOWLEDGMENTS best fit is obtained for d1 =−0.05, d2 = 0.83, d3 = 0.01, p1 = This work was supported by the EPSRC under Grants −π/4, and p2 =−π/2. The good agreement between the expected behavior (171) EP/I032487/1 (FHLE) and EP/J014885/1 (FHLE), by the and the DMRG results provides a consistency check on the IRSES network QICFT (RGP and FHLE), the CNPq (RGP), and by the DFG under Grant SCHN 1169/2-1 (IS). We are values of Kc,s extracted from the analysis of finite-size energy levels. grateful to Eric Jeckelmann for providing the DMRG code used in the numerical part of our analysis.

VIII. SUMMARY AND CONCLUSIONS In this work, we have developed a new approach to deriving APPENDIX A: IRRELEVANT PERTURBATIONS TO THE mobile impurity models for studying dynamical correlations LUTTINGER LIQUID HAMILTONIAN in gapless models of spinful fermions. Our construction is One way of working out the allowed irrelevant perturbations based on the principle that our mobile impurity must carry to the Luttinger liquid Hamiltonian is by using symmetry the quantum numbers of a holon/antiholon (charge ±e,spin considerations. Our starting point are extended Hubbard 0) or a spinon (charge 0 and spin ±1/2). For the case of models of the kind (47). For the purposes of this appendix, integrable models like the Hubbard chain, it is known from we will assume all interactions to be small. the exact solution that elementary excitations at finite energies are (anti)holons and spinons with precisely these quantum numbers. In integrable models these excitations are stable (i.e. 1. Symmetries do not decay). Breaking integrability is expected to render Our lattice models of interest are invariant under various their lifetimes finite, but leaving their spin and charge quantum symmetry operations. These symmetries are inherited by the numbers intact. bosonic low-energy description (52) and we now discuss their To facilitate our construction, we first derived a rep- realizations. resentation of spinful nonlinear Luttinger liquids in terms of strongly interacting fermionic holons and spinons. At a particular Luther-Emery point for spin and charge, holons, a. Spin-flip symmetry and spinons become noninteracting. Using this as our point of reference, we derived a mobile impurity model appropriate for The lattice models of interest are invariant under exchange the description of threshold singularities in the single-particle of up and down spins, spectral function for a general class of extended Hubbard models in their Luttinger liquid phase. cj,↑ ↔ cj,↓. (A1)

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It the is realized at the level of the bosonic fields as 2. Dimension-two operators allowed by symmetry ⎛ ⎞ ⎛ ⎞ ϕs (x) −ϕs (x) We now list all the symmetry-allowed perturbations to ⎜ϕ¯ (x)⎟ ⎜−ϕ¯ (x)⎟ the Luttinger liquid Hamiltonian for noninteracting spinful ⎝ s ⎠ −→ ⎝ s ⎠. (A2) η↑ η↓ fermions with scaling dimensions 2, 3, and 4. Such perturba- η¯↑ η¯↓ tions will be of the form b. Translational invariance ˆ j λj dx O(x), (A11) The translation operator acts as j

† where λj are coupling constants and j are products of Klein T cj,σ T = cj+1,σ . (A3) factors. The only nontrivial combination of Klein factors We can implement this on the level of bosonic fields by allowed to appear is in fact imposing the transformation properties η↑η¯↑η↓η¯↓. (A12) ϕ (x) ϕ (x) − 2k a c −→ c F 0 , ϕ¯c(x) ϕ¯c(x) − 2kF a0 This is because all terms in our Hamiltonian (47)areofthe (A4) form ϕ (x) ϕ (x) s −→ s . † † † ϕ¯s (x) ϕ¯s (x) cj,σ ck,σ ,cj,σ ck,τ cl,τ cm,σ . (A13) We note that this transformation works for all higher harmonics Expressing the lattice fermion operators in terms of Bose fields in Eq. (48). Translational invariance of the Hamiltonian by (48), and then imposing that for a given interaction to appear then implies that it generically may not contain any vertex in the Hamiltonian it must not contain a rapidly oscillating operators of the charge boson. Exceptions to this rule occur at factor eij kF x , one finds that the Klein factors either cancel or commensurate fillings combine to η↑η¯↑η↓η¯↓. p Taking this into account, we find five symmetry-allowed k a = π ,p,q∈ N. (A5) dimension-two operators: F 0 q O(2) = Here operators of the form 1 cos s , O(2) = 2 + 2 q q 2 (∂x ϕs ) (∂x ϕ¯s ) , cos c , sin c (A6) 2 2 O(2) = 3 ∂x ϕs ∂x ϕ¯s , (A14) are allowed to occur. As long as q>4, such operators are (2) 2 2 O = (∂x ϕc) + (∂x ϕ¯c) , highly irrelevant and do not play a role in the following 4 O(2) = discussion. 5 ∂x ϕc∂x ϕ¯c.

c. U(1) ⊗ U(1) Invariance We note that none of these lead to a coupling between spin and charge sectors. In order to see that the operator cos s is By this we mean that the Hamiltonian commutes with allowed, but sin s is not, one needs to consider the structure particle number and the z component of the total spin: of Klein factors in Eq. (48). [H,Sˆz] = [H,Nˆ ] = 0. (A7) 3. Dimension-three operators allowed by symmetry This symmetry implies that no vertex operators involving the dual fields c, s are allowed to occur in the expression for H . The analogous analysis for dimension-three operators gives the three possible perturbations involving only the charge d. Site parity sector: The reflection symmetry acts on the lattice fermion opera- O(3) = 3 + 3 1 (∂x ϕc) (∂x ϕ¯c) , tors like O(3) = 2 + 2 2 (∂x ϕc) ∂x ϕ¯c (∂x ϕ¯c) ∂x ϕc, (A15) Pc P = c− . (A8) j,σ j,σ O(3) = 2 − 2 3 ∂x ϕc∂x ϕ¯c ∂x ϕ¯c∂x ϕc, We see that P can be realized in the field theory as and four possible perturbations that couple spin and charge Pϕa(x)P =−ϕ¯a(−x),Pϕ¯a(x)P =−ϕa(−x), sectors together: (A9) = Pησ P η¯σ . O(3) = 2 + 2 4 (∂x ϕs ) ∂x ϕc (∂x ϕ¯s ) ∂x ϕ¯c, Crucially, parity acts on the unphysical Klein degrees of (3) 2 2 O = (∂x ϕs ) ∂x ϕ¯c + (∂x ϕ¯s ) ∂x ϕc, freedom as well. Moreover, we obtain the following constraints 5 (A16) O(3) = + on the amplitudes An,m in Eq. (48): 6 ∂x ϕs ∂x ϕ¯s (∂x ϕ¯c ∂x ϕc), = O(3) = + An,m A1−n,−1−m. (A10) 7 (∂x ϕc ∂x ϕ¯c) cos( s ).

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4. Dimension-four operators allowed by symmetry LE point all symmetry-allowed operators in the Hamiltonian Finally, we consider all symmetry-allowed dimension four are local in terms of free holons and spinons, we shall operators. give the list of irrelevant operators directly in the fermionic representation. Among the operators listed in Appendix A, a. Charge sector only those that involve only derivatives of ϕα,ϕ¯α preserve the same scaling dimension at the LE point; their expressions in the We find six perturbations involving only the charge sector: fermionic basis can be obtained straightforwardly by using O(4) = 4 + 4 bosonization identities for spinless fermions. 1 (∂x ϕc) (∂x ϕ¯c) , On the other hand, operators that contain cos( s ) require a O(4) = 2 2 2 (∂x ϕc) (∂x ϕ¯c) , more careful analysis. First, we note that, taking into account O(4) = 3 + 3 the Klein factors, the operator is actually represented by 3 (∂x ϕc) ∂x ϕ¯c (∂xϕ¯c) ∂x ϕc, (A17) cos( ) → η↑η¯↓η↓η¯↑ cos( ). (B1) O(4) = 2 2 + 2 2 s s 4 ∂x ϕc (∂x ϕ¯c) , Recall that spin flip and parity act nontrivially on ησ ,η¯σ . O(4) = 2 2 5 ∂x ϕc∂x ϕ¯c, However, the product of four Majorana fermions in Eq. (B1) (4) is invariant under both transformations. As a result, the Klein O = ∂2ϕ (∂ ϕ¯ )2 − ∂2ϕ¯ (∂ ϕ )2. 6 x c x c x c x c factors can be safely omitted in the symmetry analysis of Here we have used that the bosonized perturbations to the Luttinger model. At the LE point, however, cos( ) must be refermionized into free ∂3ϕ ∂ ϕ¯ + ∂3ϕ¯ ∂ ϕ s x c x c x c x c spinons according to Eq. (67). In terms of spinon operators, =− 2 2 + 2∂x ϕc∂x ϕ¯c total derivative. (A18) spin-flip symmetry is equivalent to a particle-hole symmetry: R ↔ R†,L↔ L†, (B2) b. Spin sector only s s s s We find altogether eight symmetry allowed perturbations while leaving the spinon Klein factors ηs ,η¯s invariant. Parity involving only the spin sector: acts on spinon operators in the form

O(4) = 4 + 4 PRs (x)P = Ls (−x),PLs (x)P = Rs (−x), (B3) 7 (∂x ϕs ) (∂xϕ¯s ) , O(4) = 2 2 Pηs P = η¯s,Pη¯s P = ηs . (B4) 8 (∂x ϕs ) (∂x ϕ¯s ) , The refermionization of cos( ) at the LE point yields O(4) = 3 + 3 s 9 (∂x ϕs ) ∂x ϕ¯s (∂x ϕ¯s ) ∂x ϕs , √ cos( ) → cos( ∗/ 2) O(4) = 2 2 + 2 2 s s 10 ∂x ϕs ∂x ϕ¯s , (A19) ∼ π(η R†η¯ L + η R η¯ L†) O(4) = 2 2 s s s s s s s s 11 ∂x ϕs ∂x ϕ¯s , =−π(η η¯ R†L + η¯ η L†R ). (B5) (4) s s s s s s s s O = cos( s )∂x ϕs ∂x ϕ¯s , 12 Notice that the operator in Eq. (B5) is invariant under the O(4) = 2 + 2 13 cos( s )[(∂xϕs ) (∂x ϕ¯s ) ]. spin-flip transformation (B2) only if we take the spinon Klein factors into account explicitly. Nevertheless, in the following c. Terms coupling charge and spin list of irrelevant operators, we shall omit the Klein factors for Finally, there are eight symmetry-allowed perturbations short and write simply involving both spin and charge sectors: → † + † cos( s ) (Rs Ls Ls Rs ). (B6) O(4) = 15 cos( s )∂x ϕc∂x ϕ¯c, Since all the operators that stem from cos( s ) contain the same combination of Klein factors in Eq. (B5), we can adopt O(4) = cos( )[(∂ ϕ )2 + (∂ ϕ¯ )2], 16 s x c x c the prescription in Eq. (B6) supplemented by the ad hoc rule O(4) = 2 + 2 2 + 2 → † →− † 17 [(∂x ϕs ) (∂x ϕ¯s ) ][(∂xϕc) (∂x ϕ¯c) ], that spin-flip symmetry takes Rs Rs but Ls Ls . (4) 2 2 Importantly, the operator cos( s ) has scaling dimension 2 O = [(∂x ϕs ) + (∂x ϕ¯s ) ]∂x ϕc∂x ϕ¯c, 18 (A20) at the SU(2)-symmetric weak coupling regime, but dimension O(4) = 2 + 2 19 ∂x ϕs ∂x ϕ¯s [(∂xϕc) (∂x ϕ¯c) ], 1 at the LE point. Therefore the scaling dimension of perturba- tions that contain cos( ) is reduced by 1 as we go from weak O(4) = s 20 ∂x ϕs ∂x ϕ¯s ∂x ϕc∂x ϕ¯c, coupling to the LE point. For instance, the marginal operator (4) 2 2 2 2 O(2) = O = [(∂ ϕ ) + (∂ ϕ¯ ) ] ∂ ϕ − ∂ ϕ¯ , 1 cos( s ) at the SU(2) point becomes the relevant mass 21 x s x s x c x c √ † † term of the Thirring model, cos( ∗/ 2) ∼ R L + L R at O(4) = 2 − 2 s s s s s 22 ∂x ϕs ∂x ϕ¯s ∂x ϕc ∂x ϕ¯c . the LE point, while the irrelevant (dimension-three) operator O(3) = + 7 (∂x ϕ¯c ∂x ϕc) cos( s) at the SU(2) point gives rise APPENDIX B: LIST OF IRRELEVANT OPERATORS † † † to the marginal spin-charge coupling (Rs Ls + Ls Rs )(RcRc + ATTHELEPOINT † LcLc) at the LE point. This implies that, in order to have The symmetries of the Hamiltonian at the Luther-Emery the complete list of irrelevant operators up to dimension four point are the same as those listed in Appendix A. Since at the at the LE point, we have to consider the refermionization of

245150-20 SPIN-CHARGE-SEPARATED QUASIPARTICLES IN ONE- . . . PHYSICAL REVIEW B 91, 245150 (2015) operators that have dimension 5 at the SU(2) point and were (4) † † † † V = R R L L (iR ∂ R − iL ∂ L + H.c.), (B32) not included in the list in Appendix A. The latter correspond 15 c c c c s x s s x s V(4) V(4) V(4) = (iR†∂ R − iL†∂ L + H.c.)R†R L†L , (B33) to operators 21 through 27 in the list below. 16 c x c c x c s s s s V(4) = † † † † 17 RcRcLcLcRs Rs Ls Ls , (B34) 1. Dimension-three operators allowed by symmetry V(4) = † † + 18 ∂x (RcRc)(iRs ∂x Rs H.c.) † † V(3) = R ∂2R + L ∂2L + H.c., (B7) + † † + 1 c x c c x c ∂x (LcLc)(iLs ∂x Ls H.c.), (B35) V(3) = R†R L†i∂ L − L†L R†i∂ R + H.c., (B8) V(4) = † † + 2 c c c x c c c c x c 19 ∂x (RcRc)(iLs∂x Ls H.c.) V(3) = † † − † † + † † + 3 ∂x (RcRc)LcLc ∂x (LcLc)RcRc, (B9) ∂x (LcLc)(iRs ∂x Rs H.c.), (B36) V(3) = † + † V(4) = † − † † † 4 ∂x Rs ∂x Ls ∂x Ls ∂x Rs , (B10) 20 ∂x (RcRc LcLc)Rs Rs Ls Ls , (B37) V(3) = † † − † † + V(4) = † 2 + † 2 + † + † 5 iRs ∂x Rs Ls Rs iLs ∂x Ls Rs Ls H.c., (B11) 21 (Rc∂x Rc Lc∂x Lc H.c.)(Rs Ls Ls Rs ), (B38) V(3) = † † − † † + 6 RcRcRs i∂x Rs LcLcLs i∂x Ls H.c., (B12) (3) † † † † (4) † † † † V = R R L i∂x L − L L R i∂x R + H.c., (B13) V = − + 7 c c s s c c s s 22 (RcRcLci∂x Lc LcLcRci∂x Rc H.c.) V(3) = (R†R + L†L )R†R L†L , (B14) × † + † V(3) = 8† −c c† c +c s s †s s + † (Rs Ls Ls Rs ), (B39) 9 (iRc∂x Rc iLc∂x Lc H.c.)(Rs Ls Ls Rs ), V(4) = † † † − † † † + (B15) 23 RcRcRs i∂x Rs Ls Rs LcLcLs i∂x Ls Rs Ls H.c., V(3) = † † † + † (B40) 10 RcRcLcLc(Rs Ls Ls Rs ), (B16) V(4) = † † † − † † † + V(3) = † − † † + † 24 RcRcLs i∂x Ls Rs Ls LcLcRs i∂x Rs Ls Rs H.c., 11 ∂x (RcRc LcLc)(Rs Ls Ls Rs ). (B17) (B41) 2. Dimension-four operators allowed by symmetry V(4) = 2 † + † † + † 25 ∂x (RcRc LcLc)(Rs Ls Ls Rs ), (B42) a. Charge sector only V(4) = † + † † + † 26 ∂x (Rci∂x Rc Lci∂x Lc)(Rs Ls Ls Rs ), (B43) V(4) = iR†∂3R − iL†∂3L + H.c., (B18) V(4) = † † − † † † + † 1 c x c c x c 27 [∂x (RcRc)LcLc ∂x (LcLc)RcRc](Rs Ls Ls Rs ). V(4) = † † + 2 Rc∂x RcLc∂x Lc H.c., (B19) (B44) V(4) = † 2 † + † 2 † + 3 Rc∂x RcLcLc Lc∂x LcRcRc H.c., (B20) V(4) = † † + † † APPENDIX C: CONSTRAINING THE MOBILE IMPURITY 4 Rc∂x RcRc∂x Rc Lc∂x LcLc∂x Lc, (B21) MODEL IN THE SU(2)-SYMMETRIC CASE V(4) = † † 5 ∂x (RcRc)∂x (LcLc), (B22) We can impose SU(2) symmetry in the parameters of the V(4) = † † + † † + mobile impurity model of Sec. V by requiring that the edge 6 ∂x (RcRc)Lci∂x Lc ∂x (LcLc)Rci∂x Rc H.c. exponents of longitudinal and transverse spin-spin correlations (B23) coincide [43]. First, consider the longitudinal component of the spin density operator: b. Spin sector only z † † † S (x) ∼ R↑(x)R↑(x) − R↓(x)R↓(x) ∼ O (x)O (x), (C1) V(4) = † 3 − † 3 s s 7 iRs ∂x Rs iLs ∂x Ls , (B24) (4) † † where the charge strings are canceled in the sense of the lowest V = R ∂x R L ∂x L + H.c., (B25) 8 s s s s order in the OPE. We then project Os (x)soastocreatea V(4) = † † + † † high-energy spinon: 9 Rs ∂x Rs Ls ∂x Ls Ls ∂x Ls Rs ∂x Rs , (B26) † † † † ∗ V(4) = 2 + 2 + − † √i (x) R ∂ R L L L ∂ L R R H.c., (B27) O ∼ iqx 4 2 s 10 s x s s s s x s s s s (x) e χs (x)e , (C2) V(4) = † † + † † 11 Rs ∂x Rs Rs ∂x Rs Ls ∂x Ls Ls ∂x Ls , (B28) † while Os (x) acts only in the low-energy subband: V(4) = † † 12 ∂x (Rs Rs )∂x (Ls Ls ). (B29) † √i ϕ∗− √i ∗(x) O ∼ 2 s 4 2 s s (x) e . (C3) c. Terms coupling charge and spin Cancelling the neutral string, we obtain V(4) = † + † + (iR ∂x R H.c.)(iR ∂x R H.c.) ∗ 13 c c s s z −iqx † √i ϕ (x) S (x) ∼ e χ (x)e 2 s . (C4) + † + † + s (iLc∂x Lc H.c.)(iLs∂x Ls H.c.), (B30) V(4) = † + † + In terms of the transformed impurity field 14 (iRc∂x Rc H.c.)(iLs ∂x Ls H.c.) † † − † i( √1 +γ )ϕ∗+iγ¯ ϕ¯∗+iγ ϕ∗+iγ¯ ϕ¯∗ + + + z ∼ iqx 2 s s s s c c c c (iLc∂x Lc H.c.)(iRs ∂x Rs H.c.), (B31) S (x) e ds (x)e . (C5)

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Now, consider the transverse component: 1. Open boundary conditions † S−(x) ∼ R (x)R (x) For open boundary conditions we still have expansions ↓ ↑ like (48), but the chiral Bose fields no longer commute and ∼ O (x)O (x) σ s s hence the Klein factors n,m need to be adjusted accordingly. − † − √i ϕ∗(x)+ √i ϕ¯∗(x) The mode expansions for the chiral Bose fields are ∼ iqx 2 2 s 2 2 s e χs (x)e − √1 − ∗+ √1 + ∗ ∗ ∗ ∞ −iqx † i( γs )ϕs i( γ¯s )ϕs iγ ϕ +iγ¯ ϕ¯ = e d (x)e 2 2 2 2 e c c c c . (C6) a + π0 xϕ0 2 − s ϕ (x) = + + i (e iqnx α − H.c.), c 2 2(L + 1) n n Given the expressions (C5) and (C6) we can calculate threshold n=1 exponents in the Fourier transforms of the spin correlations ∞ + − a − π0 xϕ0 2 functions Sz(x,t)Sz(0,0) and S (x,t)S (0,0) and impose ϕ¯ (x) = + − i (eiqnx α − H.c.), c 2 2(L + 1) n n that they share the same exponents in the SU(2)-symmetric n=1 case. This has to be the case even for higher harmonics, taking (D4) into account backscattering processes [43]. A shortcut is to compare the scaling dimensions of the strings in Eqs. (C5) where [π0,ϕ0] = 8πi and qn = πn/(L + 1). The commutator and (C6). We must have between different chiralities is ⎧ √1 + = √1 − γs γs , (C7) ⎨0ifx = y = 0, 2 2 2 = + [ϕα(x),ϕ¯α(y)] ⎩2πi if 0

These values of γs ,γ¯s are of order 1, as expected since the spinons are strongly interacting at the SU(2) point. Moreover, Imposing the boundary conditions (D6) on the bosonized these values are consistent with the exact result for the Hubbard expression (48) [with Klein factors appropriate for the mode model at zero magnetic field (see Sec. VB). Using Eq. (C9) expansions (D4)] leads to conditions of the form and setting Ks = 1inEq.(110) yields c(0)|ψ0 =c1|ψ0 , c(L + 1)|ψ0 =c2|ψ0 , = = (D7) νs,+ νs,− 0. (C10) | = + | =  | s (0) ψ0 s (L 1) ψ0 c1 ψ0 . APPENDIX D: FRIEDEL OSCILLATIONS  Importantly, the actual values of the c1,2 and c1,2 depend on all In this Appendix, we summarize some useful facts regarding the amplitudes An,m in Eq. (48) and are thus nonuniversal. Friedel oscillations for open boundary conditions. At low energies, the charge density operator has an expansion of the 2. Friedel oscillations form For open boundary conditions, the 2k and 4k components ∼ F F nj ρ2nkF (x), (D1) of the total charge density are again given by expressions of the n=0 form (D2). Using the mode expansions, we then can determine the form of the Friedel oscillations where ρ2nkF (x) denotes the Fourier components with momenta ± 2nkF . For periodic boundary conditions it follows from (48)  sin 2k x − c1 and the site-parity symmetry (A9) that the leading contribu-  | | ∼ F 2 ψ0 ρ2kF (x) ψ0 B2kF (1+K )/2 , tions to the 2kF and 4kF components are 2 sin πx c  L+1 (D8) $ c s  ρ (x) = A sin − 2k x cos +··· , cos 4k x − c1 2kF 2kF F  | | ∼ F 2 2 ψ0 ρ4kF (x) ψ0 B4kF K , 2 sin πx 2 c = $ − +···  L+1 ρ4kF (x) A4kF cos ( c 4kF x) , (D2) where A$ and A$ are nonuniversal amplitudes that include where  is a short-distance cutoff, we have taken into account 2kF 4kF = Klein factors. The leading contributions to zero-momentum Kc 1 and defined component are  c1 − c2 k = kF + . (D9) 1 s F + ρ (x) = n − ∂ + A$ ∂ cos +··· , 4(L 1) 0 2π x c 0 x c 2 (D3) The upshot is that Friedel oscillations involve two nonuniver- sal interaction-dependent parameters: the phase shift c1 and where n is the band filling. the 1/L shift of kF .

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[1] C. Xu and S. Sachdev, Phys.Rev.Lett.105, 057201 (2010). [36] M. Khodas, M. Pustilnik, A. Kamenev, and L. I. Glazman, [2] V. V. Deshpande, M. Bockrath, L. I. Glazman, and A. Yacoby, Phys. Rev. Lett. 99, 110405 (2007). Nature (London) 464, 209 (2010). [37] M. Khodas, M. Pustilnik, A. Kamenev, and L. I. Glazman, Phys. [3] O. M. Auslaender et al., Science 295, 825 (2002). Rev. B 76, 155402 (2007). [4] B. J. Kim et al., Nat. Phys. 2, 397 (2006). [38] R. G. Pereira, S. R. White, and I. Affleck, Phys. Rev. Lett. 100, [5] Y. Jompol et al., Science 325, 597 (2009). 027206 (2008). [6] J. Schlappa et al., Nature (London) 485, 82 (2012). [39] A. Imambekov and L. I. Glazman, Phys.Rev.Lett.100, 206805 [7]F.H.L.Essler,H.Frahm,F.Gohmann,¨ A. Klumper,¨ and V. (2008). E. Korepin, The One Dimensional Hubbard Model (Cambridge [40] V. V. Cheianov and M. Pustilnik, Phys. Rev. Lett. 100, 126403 University Press, Cambridge, 2005). (2008). [8]F.H.L.EsslerandV.E.Korepin,Phys.Rev.Lett.72, 908 (1994). [41] E. Bettelheim, A. G. Abanov, and P. Wiegmann, J. Phys. A 41, [9]F.H.L.EsslerandV.E.Korepin,Nucl. Phys. B 426, 505 (1994). 392003 (2008). [10] N. Andrei, arXiv:cond-mat/9408101. [42] A. G. Abanov, E. Bettelheim, and P. Wiegmann, J. Phys. A 42, [11] T. Deguchi, F. H. L. Essler, F. Gohmann,¨ V. E. Korepin, A. 135201 (2009). Klumper,¨ and K. Kusakabe, Phys. Rep. 331, 197 (2000). [43] A. Imambekov and L. I. Glazman, Phys.Rev.Lett.102, 126405 [12] K. Penc, F. Mila, and H. Shiba, Phys. Rev. Lett. 75, 894 (1995). (2009). [13] J. Favand, S. Haas, K. Penc, F. Mila, and E. Dagotto, Phys. Rev. [44] A. Imambekov and L. I. Glazman, Science 323, 228 (2009). B 55, R4859 (1997). [45] R. G. Pereira, S. R. White, and I. Affleck, Phys.Rev.B79, [14] K. Penc, K. Hallberg, F. Mila, and H. Shiba, Phys. Rev. B 55, 165113 (2009). 15475 (1997). [46] R. G. Pereira, Int. J. Mod. Phys. B 26, 1244008 (2012). [15] H. Benthien, F. Gebhard, and E. Jeckelmann, Phys. Rev. Lett. [47] A. V. Rozhkov, Phys. Rev. Lett. 112, 106403 (2014). 92, 256401 (2004). [48] T. Price and A. Lamacraft, Phys.Rev.B90, 241415 (2014). [16] A. E. Feiguin and D. A. Huse, Phys.Rev.B79, 100507(R) [49] R. G. Pereira and E. Sela, Phys.Rev.B82, 115324 (2010). (2009). [50] R. G. Pereira, K. Penc, S. R. White, P. D. Sacramento, and J. M. [17] M. Arikawa, Y. Saiga, and Y. Kuramoto, Phys.Rev.Lett.86, P. Carmelo, Phys.Rev.B85, 165132 (2012). 3096 (2001). [51] T. L. Schmidt, A. Imambekov, and L. I. Glazman, Phys. Rev. [18] M. Arikawa, T. Yamamoto, Y. Saiga, and Y. Kuramoto, Lett. 104, 116403 (2010). Nucl. Phys. B 702, 380 (2004). [52] T. L. Schmidt, A. Imambekov, and L. I. Glazman, Phys. Rev. B [19] A. B. Zamolodchikov and Al. B. Zamolodchikov, Ann. Phys. 82, 245104 (2010). 120, 253 (1979). [53] F. H. L. Essler, Phys.Rev.B81, 205120 (2010). [20] L. D. Faddeev, Sov. Sci. Rev. Math. Phys. C 1, 107 (1980). [54] A. Imambekov, T. L. Schmidt, and L. I. Glazman, Rev. Mod. [21] R. A. J. van Elburg and K. Schoutens, Phys.Rev.B58, 15704 Phys 84, 1253 (2012). (1998). [55] O. Tsyplyatyev and A. J. Schofield, Phys. Rev. B 90, 014309 [22] G. I. Japaridze, A. A. Nersesyan, and P. B. Wiegmann, (2014). Nucl. Phys. B 230, 511 (1984). [56] L. Seabra, F. H. L. Essler, F. Pollmann, I. Schneider, and T. [23] F. Woynarovich, J. Phys. A 22, 4243 (1989). Veness, Phys.Rev.B90, 245127 (2014). [24] H. Frahm and V. E. Korepin, Phys.Rev.B42, 10553 (1990). [57] V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quan- [25] I. Affleck, in Fields, Strings and Critical Phenomena, edited by tum Inverse Scattering Method and Correlation Functions E. Brezin´ and J. Zinn-Justin, (Elsevier, Amsterdam, 1989). (Cambridge University Press, Cambridge, 1999). [26] A. O. Gogolin, A. A. Nersesyan, and A. M. Tsvelik, Bosoniza- [58] L. D. Faddeev and L. Takhtajan, Jour. Sov. Math. 24, 241 (1984). tion and Strongly Correlated Systems (Cambridge University [59] F. H. L. Essler and R. M. Konik, in From Fields to Strings: Press, Cambridge, 1999). Circumnavigating Theoretical Physics, ed. M. Shifman, A. [27] T. Giamarchi, Quantum Physics in One Dimension (Claredon Vainshtein, and J. Wheater (World Scientific, Singapore, 2005); Press, Oxford, 2004). and references therein; arXiv:cond-mat/0412421. [28] M. Pustilnik, M. Khodas, A. Kamenev, and L. I. Glazman, [60] K. D. Schotte and U. Schotte, Phys. Rev. 182, 479 (1969). Phys. Rev. Lett. 96, 196405 (2006). [61] A. Luther and V. J. Emery, Phys. Rev. Lett. 33, 589 (1974). [29] A. Rozhkov, Eur. Phys. J. B 47, 193 (2005). [62] F. D. M. Haldane, Phys. Rev. Lett. 47, 1840 (1981). [30] A. V. Rozhkov, Phys.Rev.B74, 245123 (2006). [63] H. Frahm and V. E. Korepin, Phys. Rev. B 43, 5653 (1991). [31] J. M. P. Carmelo, K. Penc, L. M. Martelo, P. D. Sacramento, [64] D. B. Abraham, F. H. L. Essler, and A. Maciolek, Phys. Rev. J. M. B. Lopes, Dos Santos, R. Claessen, M. Sing, and U. Lett. 98, 170602 (2007). Schwingenschlogl,¨ Europhys. Lett. 67, 233 (2004). [65] S. Coleman, Phys.Rev.D11, 2088 (1975). [32] J. M. P. Carmelo, K. Penc, P. D. Sacramento, M. Sing, and R. [66] M. B. Zvonarev, V. V. Cheianov, and T. Giamarchi, J. Stat. Claessen, J. Phys.: Condens. Matter 18, 5191 (2006). Mech. (2009) P07035. [33] J. M. P. Carmelo, D. Bozi, and K. Penc, J. Phys. Cond. Mat. 20, [67] H. Karimi and I. Affleck, Phys.Rev.B84, 174420 (2011). 415103 (2008). [68] C. G. Callan, I. R. Klebanov, A. W. W. Ludwig, and J. M. [34] R. G. Pereira, J. Sirker, J.-S. Caux, R. Hagemans, J. M. Maillet, Maldacena, Nucl. Phys. B422, 417 (1994). S. R. White, and I. Affleck, Phys.Rev.Lett.96, 257202 [69] T. Xiang, Phys. Rev. B 53, R10445(R) (1996). (2006). [70] U. Schollwock,¨ Rev. Mod. Phys. 77, 259 (2005). [35] E. Bettelheim, A. G. Abanov, and P. Wiegmann, Phys. Rev. Lett. [71] F. H. L. Essler, V. E. Korepin, and K. Schoutens, Phys. Rev. 97, 246401 (2006). Lett. 67, 3848 (1991).

245150-23 F. H. L. ESSLER, R. G. PEREIRA, AND I. SCHNEIDER PHYSICAL REVIEW B 91, 245150 (2015)

[72] F. H. L. Essler, V. E. Korepin, and K. Schoutens, Nucl. Phys. B [76] S. R. White, Phys.Rev.Lett.69, 2863 (1992). 372, 559 (1992). [77] S. R. White, Phys.Rev.B48, 10345 (1993). [73] M. Ogata and H. Shiba, Phys. Rev. B 41, 2326 (1990). [78] S. A. Soffing,¨ M. Bortz, I. Schneider, A. Struck, [74] S. Eggert, Phys.Rev.B54, R9612 (1996). M. Fleischhauer, and S. Eggert, Phys.Rev.B79, 195114 [75] K. Okamoto and K. Nomura, Phys. Lett. A 169, 433 (1992). (2009).

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