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From Luttinger liquids to Luttinger droplets via higher-order bosonization identities

We derive generalized Kronig identities expressing quadratic fermionic terms including momentum transfer to bosonic operators and use them to obtain the exact solution for one-dimensional fermionic models with linear dispersion in the presence of position-dependent interactions and scattering potential. In these Luttinger droplets, which correspond to Luttinger liquids with spatial variations or constraints, the position dependences of the couplings break the translational invariance of correlation functions and modify the Luttinger-liquid interrelations between excitation velocities.


I. INTRODUCTION
An important goal of condensed matter theory is a reliable description of the correlated behavior of electrons which is rooted in the Coulomb interaction between them. In one-dimensional geometries they exhibit a special coherence at low energies: 1,2 the dispersion can be approximately linearized in the vicinity of the Fermi points ±k F as k ±v F (k ± k F ), so that the energy δ = v F δk of a particle-hole excitation from k 1 to k 2 is a function only of the momentum transfer δk = k 2 −k 1 . By contrast, in higher dimensions the magnitude and relative orientation of the two momenta usually enter into δ , leading to a continuum of excitation energies for a given momentum transfer. This coherence in one dimension is prominently featured in the Tomonaga-Luttinger model, 3,4 which is based on the approximation that one can regard a physical electron field operator Ψ(x) for a wire of length L at low energies as a sum of two independent fields, where the lower summations limits −k F in (1a) were replaced by −∞ (1b). Then ψ R,L (x) = ψ 1,2 (∓x) and ψ η (x) = (2π/L) 1 2 k e −ikx c kη are defined in terms of canonical fermions c kη = C ±(kF+k) which correspond to the physical fermions C k near the two Fermi points for η = 1, 2. In the Tomonaga-Luttinger model the dispersion is linearized near the Fermi points and only forward-scattering density interactions between left-and right-moving fermions are kept. The Tomonaga-Luttinger model can be solved by bosonization, 3-13 which expresses the above-mentioned coherence of excitations into an exact mapping to bosonic degrees of freedom (at the operator level 9, [14][15][16][17][18] or in a path-integral formulation; [19][20][21][22] throughout we use Ref. 17's constructive finite-size bosonization approach, which is recapped below). Bosonization has led to such remarkable results and concepts as spin-charge separation of elementary excitations, 3,4 interactiondependent exponents of correlation functions, 9,[23][24][25] and the Luttinger-liquid paradigm 13,26 which states that the relations between excitation velocities and correlation exponents of the Tomonaga-Luttinger model remain valid even for weakly nonlinear dispersion. These topics are nowadays presented in many reviews 1, 14,15,17,[27][28][29][30] and textbooks. 2, [31][32][33][34][35][36] Characteristic signatures of onedimensional electron liquids have been observed in a variety of experiments. [37][38][39][40][41][42][43][44][45][46][47][48][49][50] The theory of nonlinear dispersion terms has been of particular further interest, 6,13,[51][52][53] including refermionization techniques which use bosonization identities in reverse to map diagonalized bosonic systems back to free fermions. 52,[54][55][56][57][58] arXiv:1911.03158v1 [cond-mat.str-el] 8 Nov 2019 For Luttinger liquids out of equilibrium [59][60][61][62][63][64][65][66][67][68][69][70][71][72] nonlinear dispersion effects are also essential. [73][74][75] The technical hallmarks of bosonization are the following. On the one hand, a two-body density interaction term for fermions c kη becomes quadratic in terms of canonical bosons, defined for q > 0 as b qη = −i k c + k−qη c kη / √ n q . Here the momentum sum runs over k = 2π L (n k − 1 2 δ b ) with integer n k , and the parameter 0 ≤ δ b < 2 fixes the boundary conditions, ψ η (x + L/2) = e iπδ b ψ η (x − L/2). On the other hand the fermionic kinetic energy also translates into free bosons by means of the so-called Kronig identity, 28,76,77 where the fermionic number operator is given by 78N η = k * * c + kη c kη * * , which commutes with b qη . The normal ordering * * · · · * * is defined with respect to the state |0 0 , where |N 0 is an eigenstate of all c + kη c kη (with eigenvalue 1 if n k ≤ N η and 0 otherwise). Furthermore, real-space fermionic and bosonic fields are related by the celebrated bosonization identity, [7][8][9]17 which allows the calculation of fermionic in terms of bosonic correlation functions. 9 Here the fermionic Klein factor F η decreases the fermionic particle numberN η by one, and ϕ η (x) = − q>0 b qη e −iqx−aq / √ n q . The regularization parameter a → 0 + is needed to obtain a finite commutator, 17 . A final ingredient to the solution of the Tomonaga-Luttinger model is a Boguljubov transformation, which absorbs the interaction between left-and right-moving fermions into the free bosonic theory. 3 In the present work we will study Luttinger liquids with additional spatial constraints, which we term Luttinger droplets. Namely, we consider a (spinless) fermionic Hamiltonian with linear dispersion, positiondependent interactions V (x) and U (x), and scattering potential W (x) (all assumed to be real symmetric functions of x), in terms of densities n R,L (x) = ψ + R,L (x)ψ R,L (x)/(2π). Without W (x) and with constant V (x) and U (x), H reduces to the usual translationally invariant Tomonaga-Luttinger model (with contact interactions). Below we will diagonalize (4a) exactly for the special case for otherwise arbitrary V (x) > −2πv F and a constant γ. This means that real Fourier components V q = V −q as well as U q=0 can be chosen freely; then γ = U 0 /[2πv F + V 0 ] and U q =0 = γV q . Thus γ characterizes the relative strength of interbranch interactions. Below we obtain the single-particle Green function for the ground state of this model, the exponents of which will reflect the spatial dependence of the couplings. We first derive generalized Kronig-type identities in Sec. II, which we then use to solve a single-flavor chiral version of (4) in Sec. III. We then proceed to the two-flavor case in Sec. IV, with a discussion of the similiarities and differences of the spectrum and Green function compared to the translationally invariant case. One representative choice of V (x) to be discussed below involves a central region with stronger repulsion than at the edges of the system, as shown in Fig. 1a. An explicit evaluation is provided for a piecewise constant V (x) as shown in Fig. 1b for integer exponents m ≥ 0 and momentum transfer q = 2π L n q with integer n q ; here and throughout real-space integrals without indicated endpoints extend over the interval [−L/2, L/2]. Arbitrary dispersion terms are included in (5a) for q = 0, such as (2a) for m = 1. Forming the product of (3) with its hermitian conjugate at different positions x and x + , canceling the Klein factors (F + η F η = 1), commuting the bosonic fields, taking a to zero, and combining exponentials, we obtain A generating function of the terms in (5a) then reads where we summed the Taylor series of the terms (5b), inserted relation (6), and performed the normal ordering. Taylor expanding the exponentials and taking coefficients of m on both sides of (7) now yields H (m) qη in terms of bosonic operators, as discussed below. Relation (7) thus provides explicit bosonic representations of general bilinear fermionic operators, including (2). 93 We also introduce operators which use the more convenient powers of the integer n k instead of momentum k, so that the terms (5a) are then given by H q η ] = −n q δ qq δ ηη . The operators K qη (λ), which are operator-valued formal power series in the (complex) indeterminate λ with coefficients K (m) qη , obey the intriguing operator algebra which is reminiscient of affine Lie algebras, 96 but not immediately recognizable. From (7), or alternatively from (9), the generating function (8a) becomes The coefficients K of λ m in these expression are given by Here −pη e ipx and B n (x) and B m (x 1 , . . . , x m ) are the Bernoulli and complete Bell polynomials, respectively, defined by 97 A detailed derivation of (10)-(11) will be presented elsewhere.

B. Bosonic representation of a fermionic scattering term
Generalized Kronig identities for arbitrary order m follow from the equivalence of (8b) and (11a), with the latter involving only fermionic number operators and normal-ordered bosonic operators. As a special case, we obtain for m = 1 and q = 0 the finite-q generalization of (2), which can also be expressed as so as to make the modification of the momentumdiagonal identity (2) more apparent.

A. Droplet model with only right movers
As a simple application of (14) and for later reference we first consider a single species of spinless fermions with density subjected to a single-particle potential w(x) = w(−x) and a position-dependent interaction g(x) = g(−x), with Fourier transforms w q = w(x) e −iqx dx/L = w −q and so on. For simplicity we choose antiperiodic boundary conditions (δ b = 1). For a linear dispersion the Hamiltonian of such a 'chiral Luttinger droplet' is given by

B. Diagonalization of the chiral model
On the one hand, we can now express the fermionic Hamiltonian H chiral in terms of bosonic operators. We define with symmetric parametersŵ q (that may containN ) and g q . Forg q = 2πv F δ q0 + g q andŵ q = w q L + g qN we find that On the other hand, the fermionic basis permits a full diagonalization as follows. Using (13) to eliminate the last term in (17) we arrive at a fermionic scattering Hamiltonian, We conclude that the four-fermion interaction terms in (16) cancel, as they do in the Kronig identity (2). In terms of field operators we obtain whereg(x) = 2πv F + g(x) as above, and s = −1/(2π).
Next we use the spectrum of the first-quantized Hamil- where the momentum k takes on the same discrete values k n as before. Here S j = s j (x 2 , x 1 ) with s j (x, x ) = x x dy (δ j0 + δ j1 w(y))/g(y). These eigenstates correspond to plane waves subject to a local scale transformation induced by the interaction potential, reminiscient of eikonal wave equations or semiclassical Schrödinger equations. We note the eigenstate expectation values w(X) = S 1 /S 0 = E k − s P . Setting x 1 = −x 2 = L/2 and δ b = 1 and requiring g q=0 > −2πv F , we thus diagonalize (16), (19), (20) in terms of new canonical fermions, {Ξ k , Ξ + k } = δ kk , as Note that the renormalized dressed Fermi velocityṽ is given by the spatial harmonic average of the renormalized 'local' Fermi velocity v F + g(x)/(2π) =g(x)/(2π).

C. Green function for the chiral model
From the above solution it is straightforward to obtain the time-ordered Green function for the Heisenberg operators of the chiral field, with θ(±t) = (1 ± sgn(t))/2. At zero temperature in a state with fixed particle number N we find where a → 0 + stems from a convergence factor that was included in the momentum sum. For constant g(x) and w(x) we recover the translationally invariant case, Position-dependent couplings, on the other hand, may lead to a substantial redistribution of spectral weight. The critical behavior however remains unaffected, in the sense that the exponent of the denominator involving R(x, x , t) remains unity for the chiral model.

A. Droplet model with with right and left movers
We now study a generalization of the two-flavor Tomonaga-Luttinger model to position-dependent interactions and scattering potentials. Such a 'Luttinger droplet' involves right-and left-moving fermions, ψ R (x) = ψ 1 (−x) and ψ L (x) = ψ 2 (x) (see introduction) with linear dispersion in opposite directions, subject to the one-particle potential W (x), as well as intrabranch and interbranch density interactions V (x) and U (x), respectively, as given in (4). In terms of fermions with flavor η = 1, 2 we have i.e., compared to (16) the couplings g(x) and w(x) were relabeled as V (x) and W (x), indices η were put on operators, and the interaction term with U (x) was included.
B. Diagonalization of the Luttinger droplet model

Rewritten with bosonic operators this becomes
H contains a standard (i.e., translationally invariant) Tomonaga-Luttinger model H TL involving only the zeromomentum (space-averaged) couplings, which by itself can be diagonalized by a Bogoljubov transformation. For position-dependent couplings, on the other hand, also H (linear in bosons) and H (quadratic in bosons with momentum transfer) are present.

Specialization to common spatial dependence
For simplicity we set from now on with constant prefactors V and U and f q = V q /V = U q /U = f −q for q = 0. We can then simplify the momentumoffdiagonal term H by a Bogoljubov transformation to K where E 0 is a constant energy shift, omitted from now on, which diverges due to the contact interactions in H.
Here and below we use the following abbreviations and relations, The Hamiltonian H has thus become diagonal in the new flavors σ except for the term H (2) in (29).

Specialization to interrelated interaction strengths
For simplicity we now assume thatV = 0, i.e., that the bare Fermi velocity v F and the strengths of the positionaveraged (V 0 and U 0 ) and position-dependent interactions (V and U ) combine so that H (2) is absent. This corresponds to the special case which together with (27) is equivalent to (4b). From now on we will thus consider v F , V q , γ to be chosen freely (with V 0 > −2πv F ), with the other parameters in H then being given by i.e.,Ū f q =γV q for q = 0. Then for σ = ±1 each decoupled Hamiltonian has precisely the form of the bosonic Hamiltonian (17) encountered in the chiral model, where we also introduced the renormalized Fermi velocityv and averaged one-particle potentialW which will emerge below.

Refermionization as separately diagonalizable chiral models
We thus refermionize each H σ , first in terms of new fermions ψ σ (x), with bosonic fields φ σ (x) = ϕ + σ (x) + ϕ σ (x) built from the K (0) qσ analogously to (3), Below we will fix the connection between the fermionic number operatorsN σ and their associated Klein factors F σ to the original fermions c kη , which is not determined by the purely bosonic Bogoljubov transformation (28).
Next each chiral-type Hamiltonian H σ is diagonalized with fermions Ξ kσ according to (21), with the two types of fermions ψ σ and Ξ σ related by in terms of the following functions and parameters

Rebosonization into canonical form with quadratic number operator terms
Due to the linear dispersion we can rebosonize the Ξ kσ in terms of new canonical bosons B qσ , which will also be needed for the calculation of Green functions below. The corresponding (re-)bosonization identity reads where F σ is another Klein factor which lowersN σ by one. We note that once we fix F σ , then F σ is determined by (36), (38a), (40), although its explicit form is not needed in the following. The transformation (40) yields We observe that even for position-dependent interactions, collective bosonic excitations with linear dispersion emerge.
To complete the diagonalization of H in (41), we must still define the new number operatorsN σ (with integer eigenvalues) and Klein factors F σ in terms of the original N η and F η (which also appear in H N ). We set which ensures that the ground state (without bosonic excitations B + qσ ) remains in a sector with finiteN 1 =N 2 , because then only the density terms (N 2 1 +N 2 2 ) andN 1N2 appear in the Hamiltonian. We note that no other form ofN σ that is linear inN 1 andN 2 has this feature The corresponding Klein factors are then given by Collecting terms, the diagonalization of the Luttinger droplet Hamiltonian (4) is then finally complete, in which the following parameters appear, andv andW were defined in (35b). Here the total and relative fermionic number operators,N andĴ , take on integer values and commute with the two flavors of bosonic operators. We note the ground-state value ofN may shift due to the one-particle potential W (x) according to the value , which also depends on the interaction viaW . We consider (44) to be the canonical form of the diagonalized Luttinger droplet Hamiltonian, as it is essentially the same as that of the bosonized translationally invariant Tomonaga-Luttinger model. Namely, both are characterized by the renormalized Fermi velocityv for collective bosonic particle-hole excitations with linear dispersion, as well as v N ,J for total and relative particle number changes. For the Luttinger droplet, however, spatial dependencies enter into the diagonalization and lead to qualitatively different behavior for the fermionic degrees of freedom, as discussed below.
C. Spectrum of the Luttinger droplet model

Recovery of the translationally invariant case
For position-independent potentials, the translationally invariant case is fully recovered by setting f q =0 = 0, so thatv =γṽ F and ∆v = 0. We thus find that i.e., the parameter γ of (32) only relates v F , V 0 , U 0 to one another, as the interactions V and U are absent for the translationally invariant case. As before, γ characterizes the relative strength of (translationally invariant) interbranch interactions. It is one of the characteristic properties of a Luttinger liquid 13 that the relations remain valid even if the dispersion in H TL is weakly nonlinear. This connects the excitation velocitiesv, v N , v J as well as the power-law exponents in the single-particle Green function, which contain the parameter γ, as discussed below.

Excitation velocities for position-dependent interactions
By contrast, for the Luttinger droplet (4) with position-dependent interactions, the renormalized Fermi velocityv depends on V (x) according to (35b), so thatv can be varied independently from the average interaction potential V 0 . Namely ifv =γṽ F in (45) the three velocitiesv, v N , v J are independent of each other (but together determine γ).
In the following, however, we will adopt a different perspective. We regard γ as given by the interactions as in (4b), Then it follows from (45) that the velocities are related bȳ which replaces (47). Hence we may already conclude that the Luttinger droplet (4) is strictly speaking not a Luttinger liquid, in the sense thatv = (48) holds, so that the Luttinger liquid relation (47) is violated and the linear relations (50) between the velocitiesv, v N , v J ,ṽ F hold instead. Note also that while the canonical form of the Hamiltonian (44) and its eigenvalues are very similar to the translationally invariant case, their relation to the original fermions is more complex since it was obtained from a position-dependent canonical transformation. As a result, the position dependence of the interaction appears in the Green function, which we calculate next.

Rebosonization route to the Green function
As in the translationally invariant case, the Green function is obtained from the bosonization identity (3) and the Bogoljubov transformation (28), but also makes use of the refermionization (36) and the rebosonization (40).
To evaluate correlation functions of this field, we need to express it in the diagonalizing fermionic basis (38a). We define the auxiliary functions in terms of which we can express the bosonic fields as Here further auxiliary functions were introduced, x 0 (r) = r + 2 q>0R q sin qr q , where x 0 (r) is the unique inverse function of r 0 (x), which was substituted in the integral in (55) and expressed in terms ofR q via Fourier transform in (56) for later reference. The rebosonization relation (40) then yields finally expressing the fermionic field (51) in the diagonal bosonic basis (44). For the Green function we also need the time dependence of the Klein factors, which originates from H N + H in (26) and (29). This leads to a sum over K (0) qση which we calculate from the inversion K Using the hyperbolic relation e ∓θ (1 ± γ/2) =γ(u 3 ∓ v 3 ) and eliminating U 0 with (32), the time-dependent Klein factor then becomes We evaluate the Green function in the ground state witĥ N η = N /2 =N σ and B + qσ B qσ = 0 for all q > 0, where N is the integer closest to − /(2v N ), with θ(±t) = (1±sgn(t))/2. The greater and lesser Green functions, are then flavor-diagonal. They are evaluated by first clearing the Klein factors, inserting the Bogoljubovtransformed bosonic fields, separate them according to the index σ, and then express them withN σ , B qσ , B + qσ . This leads to with a phase factor and σ-diagonal exponential bosonic expectation values with x > = x, x < = x , t > = t, x < = 0, and a velocity parameter given by v ≷ = (ṽ F −γκ 0 v 3 /(2π))(N + 1 ± 1) + (ṽ F γ +γκ 0 u 3 /(2π))N + LW 0 /π −ṽ F . To evaluate the remaining expectation value, we use the identity 17 valid for linear bosonic operators A 1 , A 2 , A 3 and eigenstates of the bosonic particle numbers. We obtain where the index σ was omitted because M σ is independent of it, and we used the abbreviations Using the explicit wave functions and the definition (55), they evaluate tō which for n = 0 depend on the position dependence of V (x) throughR q of (55). Putting (65), (67), (69) into (64), the calculation of the Green function is complete, and can be summarized as with the factors given by (65) and (67). We now discuss this result for different settings, referring for simplicity only to G > η (x, x ; t).

Recovery of the translationally invariant case
In the translationally invariant case (46) we have r 0 (x) = x, due to the constant function r 0 (x) =v/(γṽ F ) = 1, cf. (39). AlsoR q = δ q0 , so that all sums overR q (with q > 0) vanish. InS survives, so that the contributions to the Green function for L → ∞ become The Green function then takes the familiar power-law form , depends only on the velocity ratio of v N /v J , which is a characteristic feature of the Luttinger liquid that remains valid even for weakly nonlinear dispersions. 13 Furthermore, in the translationally invariant case without interaction we have γ = 0 and hence v = 0, so that only the first factor with unit exponent correctly remains in (74).

Weak quadratic position dependence of the interactions
Next we consider position-dependent potentials that are regular at the origin, i.e., V (x) = V (0) + V (0)x 2 /2 + O(x 4 ), which is sketched in Fig. 1a for the repulsive case. From (39) we find for the function r 0 (x) that We will be interested in the asymptotic behavior of Green functions (rather than their periodicity in L) and thus will eventually take the limit L → ∞. We therefore consider a weak correction to the linear behavior r (0), i.e., x 0 (r) =ᾱr −βr 3 + O(β 2 r 5 ) ,ᾱ = 1 α ,β = β α 4 . For the potential this means The following choice of coefficientsR q turn out to produce this behavior,R where c is positive dimensionless parameter, because from (56) we find which for small |x/L| corresponds to (76) with The functions (70) are evaluated from (78) as For large L, the last logarithmic term in the exponent of (67) then dominates, containinḡ To leading order in x/L, x /L, the Green function then becomes i.e., translational invariance is only broken in finite-size corrections.
Note that according to (83) a fermionic single-particle perturbation near x = 0, as measured by the Green function, propagates with velocityv/α =v/r 0 (0) = γ(v F + V (0)/(2π). This which differs from the translationally invariant case (47) with corresponding velocitȳ γ(v F +V 0 /(2π) for which only the position-averaged interaction V 0 matters. For the Luttinger droplet, the position dependence of V (x) is thus observable in the propagation velocity described by the Green function. This can be observed in more detail for a stronger position dependence of V (x), as discussed in the next subsection.
We also note that the exponent v 2 (expressed in terms of γ in (31)) is no longer related only to the velocity ratio of v N /v J , hence this feature of the Luttinger liquid is also no longer present.

Piecewise constant interaction potential
As a minimal example which explicitly breaks the translational invariance of the Green function, we consider an interaction potential that is piecewise constant, i.e., the particles interact differently inside a central region and outside of it, as depicted in Fig. 1b for the repulsive case. The average of this function is given by Here r is the fraction of the central region with interaction V (0), which tends to zero if we consider a fixed finite central interval of width 2R but let L tend to infinity, see below. For the potential (84) we find v =γ 2π with the abbreviations α = r 0 (0) = s s + (s −s)r ,α = r 0 ( L 2 ) =s s + (s −s)r .
From now on we consider only fixed finite R and let L → ∞, i.e., r → 0. The second fraction in (86c) involving the sine function can then be replaced by unity. In this limit the summations (70) evaluate tō The logarithmic term inS [s,a] 2 (x, y) then again provides the leading term in (67) for L → ∞, The Green function then takes a power-law form with piecewise linear argument with the exponent v 2 given in terms of γ in (31). As listed in (86b), in the present case r 0 (x) is piecewise linear in x with a change in slope at |x| = R. Hence if x and x lie both inside or both outside the central region, the Green function is essentially the same as in the case of weak position dependence (83) or the translationally invariant case (74), respectively. However, if only one of x and x is inside the central region, the two coordinates enter with different prefactors into the Green function, breaking its translational invariance. The Green function (90) and velocity relation (50) indicate that for the interaction potential (84) the Luttinger droplet (4) is distinguishable from the Luttinger liquid.
Moreover, the Green function (90) shows that a fermionic single-particle perturbation created at position x will initially propagate with velocityv/r 0 (x) = γ(v F + V (x)/(2π), which is piecewise constant in the present case. As might have been expected, the position dependence of V (x) thus translates into a positiondependent 'local' propagation velocity. Its relation to the other excitation velocitues of the Luttinger droplet model will be discussed the next section.

V. TOWARDS A LUTTINGER DROPLET PARADIGM
The translationally invariant Tomonaga-Luttinger model obeys the relations (46) between excitation velocities and Green function exponents, i.e., in our notation betweenv, v N , v J , and γ. In particular, the dressed Fermi velocityv appears in the Green function (74) as the velocity with which a fermion ψ + η (x) propagates when added to the Luttinger liquid ground state. For the Luttinger droplet model (4) (with linear dispersion) we found different relations between the excitation velocities and γ, as given in (50). Furthermore, the Green functions of Sec. IV D show that a fermion ψ + η (x), inserted into the Luttinger droplet ground state at position x, initially propagates with velocityv/r 0 (x). This behavior was observed explicitly for a weak and piecewise constant position dependence of the interaction potential V (x) in (83) and (90), respectively. It can be traced to (39), where a phase r 0 (x)k appears in the exponent of the eigenfunctions ξ k (x) of the refermionized model (36). We can therefore expect that a 'local' propagation velocity of fermionic perturbations, will appear in the Green function also for more general V (x). Compared to the translationally invariant case this is a new range of velocities, which we will now relate to the other excitation velocities of the Luttinger droplet. For this purpose we first seek to characterize the scales of v loc (x). One way to do this uses its arithmetic and harmonic averages over the entire system. For these we find where, as above,ṽ F = v F + V 0 /(2π). For general V (x) these two averages are different, but coincide in the translationally invariant case. With the excitations of the Luttinger droplet characterized by the velocitiesv, v loc , v N , v J , we then obtain their interrelation from (50), where the prefactors are given by Furthermore γ, which characterizes the relative strength of interbranch interactions, determines the Green function exponent v 2 according to (31). The dependence of the coefficients (94) on γ is shown in Fig. 2. We note that for γ = 0, the two branches in the Hamiltonian do not mix; in this case v N and v J contribute equally tov and v loc equals v N . On the other hand, for only interbranch interactions (γ → ±1),γ vanishes and hence so do v N ,J . A preliminary physical interpretation of the velocities (92) might be thatv plays the role of group velocity, asvq is the energy of a bosonic excitation in (44) which involves a nonlocal and mixed-flavor superposition of original fermions. On the other hand, since v loc (x) plays the role of a local phase velocity, its scale is presumably captured by the arithmetic average v loc . Note that for the translationally invariant casev =γṽ F , and indeed the group velocity and (position-independent) phase velocity are both given byv, cf. (44), (46), (74). We conclude that for the Luttinger droplet model (4) the quantitiesv, v loc , v N , v J , and γ are related, extending the Luttinger liquid relations betweenv, v N , v J , γ to the position-dependent case. However, it remains to clarify how the relations (93) evolve away from the special case (4b). Furthermore, in order to be regarded as a paradigm for one-dimensional electronic systems with position-dependent interactions, these relations would have to remain valid also for weak nonlinearites in the dispersion. Both of these questions would therefore be worthwhile to address, e.g., by perturbative methods.

VI. CONCLUSION
Using higher-order bosonization identities, i.e., Kronigtype relations with finite momentum transfer, we solved the Luttinger droplet model (4) for a large class of position-dependent interactions and arbitrary oneparticle potentials. While the diagonalized Hamiltonian has the same operator expression as for the Luttinger liquid, the relation between its velocity parameters is not fulfilled in general, as the bosonic excitations and particle number changes involve different averages of the interaction potential over all positions. Similarly the Green functions retain their power-law form for weak positiondependence of the interaction potential, but their exponents also no longer depend only on the ratio of excitation velocities for particle-number changes. For weak positiondependent interactions the Luttinger-liquid characteristics are rather robust regarding their functional form, although the interrelation of the dressed scales and exponents is somewhat different. On the other hand, for an interaction potential with different (e.g., constant) values inside or outside a central region of finite width, not only are the Luttinger-liquid velocity relations modified, but also the Green function is no longer translationally invariant and exhibits a position-dependent propagation velocity of single-particle excitations. This may mean that the group velocity of such an excitation differs from its (position-dependent) phase velocity, in contrast to the Luttinger liquid. We conclude that the Luttinger droplet model has a ground state with different characteristics than the Luttinger liquid. It remains to be seen how the velocity relations obtained for (4) evolve for more general one-dimensional models with position-dependent interactions, and whether a Luttinger droplet paradigm emerges for them.

ACKNOWLEDGMENTS
The authors would like to thank Matthias Punk and Jan von Delft for valuable discussions. M.K. would also like to thank Sebastian Diehl, Erik Koch, Volker Meden, Lisa Markhof, Aditi Mitra, Herbert Schoeller, and Eva Pavarini for useful discussions. S.H. gratefully acknowledges support by the German Excellence Cluster Nanosystems Initiative Munich (NIM) and by the Deutsche Forschungsgemeinschaft under Germany's Excellence Strategy EXC-2111-390814868. M.K. was supported in part by Deutsche Forschungsgemeinschaft under Projektnummer 107745057 (TRR 80) and performed part of this work at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.