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Effects of finite-range interactions on the one-electron spectral properties of one-dimensional metals: Application to Bi/InSb(001)

We study the one-electron spectral properties of one-dimensional interacting electron systems in which the interactions have finite range. We employ a mobile quantum impurity scheme that describes the interactions of the fractionalized excitations at energies above the standard Tomonga-Luttinger liquid limit and show that the phase shifts induced by the impurity describe universal properties of the one-particle spectral function. We find the explicit forms in terms of these phase shifts for the momentum dependent exponents that control the behavior of the spectral function near and at the (k,omega)-plane singularities where most of the spectral weight is located. The universality arises because the line shape near the singularities is independent of the short-distance part of the interaction potentials. For the class of potentials considered here, the charge fractionalized particles have screened Coulomb interactions that decay with a power-law exponent l>5. We apply the theory to the angle-resolved photo-electron spectroscopy (ARPES) in the highly one-dimensional bismuth-induced anisotropic structure on indium antimonide Bi/InSb(001). Our theoretical predictions agree quantitatively with both (i) the experimental value found in Bi/InSb(001) for the exponent alpha that controls the suppression of the density of states at very small excitation energy omega and (ii) the location in the (k,omega) plane of the experimentally observed high-energy peaks in the ARPES momentum and energy distributions. We conclude with a discussion of experimental properties beyond the range of our present theoretical framework and further open questions regarding the one-electron spectral properties of Bi/InSb(001).


I. INTRODUCTION
One-dimensional (1D) interacting systems are characterized by a breakdown of the basic Fermi liquid quasiparticle picture. Indeed, no quasiparticles with the same quantum numbers as the electrons exist when the motion is restricted to a single spatial dimension. Rather, in a 1D lattice, correlated electrons split into basic fractionalized charge-only and spin-only particles 1,2 . Hence the removal or addition of electrons generates an energy continuum of excitations described by these exotic fractionalized particles which are not adiabatically connected to free electrons. Hence they must be described using a different language.
These models share common low-energy properties associated with the universal class of the Tomonaga-Luttinger liquid (TLL) 1-4 . To access their high-energy dynamical correlation functions beyond the low-energy TLL limit, approaches such as the pseudofermion dynamical theory (PDT) 5 or the mobile quantum impurity model (MQIM) 6,7 must be used. Those approaches incorporate nonlinearities in the dispersion relations of the fractionalized particles.
An important low-energy TLL property of 1D correlated electronic metallic systems is the universal powerlaw scaling of the spectral intensity I(ω, T ) such that I(0, T ) ∝ T α and I(ω, 0) ∝ |ω| α . Here the exponent α controls the suppression of the density of states (SDS) and ω is a small excitation energy near the ground-state level. The value SDS exponent α = (1 − K c ) 2 /(4K c ) is determined by that of the TLL charge parameter K c 1,2,8 . Importantly, this exponent provides useful information about the range of the underlying electron interactions.
In the case of integrable 1D models solvable by the Bethe ansatz 9 (such as the 1D Hubbard model 10,11 ), the PDT and MQIM describe the same mechanisms and lead to the same expressions for the dynamical correlation functions 12 . The advantage of the MQIM is that it also applies to non-integrable systems 7 . The exponents characterizing the singularities in these systems differ significantly from the predictions of the linear TLL theory, except in the low-energy limit where the latter is valid.
For integrable 1D lattice electronic models with only onsite repulsion (such as the Hubbard model), the TLL charge parameter K c is larger than 1/2 and thus the SDS exponent α = (1 − K c ) 2 /(4K c ) is smaller than 1/8. In non-integrable systems a SDS exponent α larger than 1/8 stems from finite-range interactions 8 .
In fact, as shown in Table I, for the metallic states of both 1D and quasi-1D electronic systems, the SDS exponent α frequently has experimental values in the range 0.5 − 0.8 1, 2,8,[13][14][15][16][17][18] . In actual materials, a finite effective range interaction 19  long-range Coulomb interactions with potentials vanishing as an inverse power of the separation with an exponent larger than one. In general, such finite-range interactions in 1D lattice systems represent a complex and unsolved quantum problem involving non-perturbative microscopic electronic processes. Indeed, as originally formulated, the MQIM does not apply to lattice electronic systems with finite-range interactions whose screened Coulomb potentials vanish as an inverse power of the electron distance.
Recently, the MQIM has been extended to a class of electronic systems with effective interaction ranges of about one lattice spacing, compatible with the highenergy one-electron spectral properties observed in twin grain boundaries of molybdenum diselenide MoSe 2 16,24 . This has been achieved by suitable renormalization of the phase shifts of the charge fractionalized particles. That theoretical scheme, called here "MQIM-LO", accounts for the effects of only the leading order (LO) in the effective range expansion 19,20 of such phase shifts.
In this paper we consider a bismuth-induced anisotropic structure on indium antimonide which we henceforth call Bi/InSb(001) 17 . Experimentally, strong evidence has been found that Bi/InSb(001) exhibits 1D physics 17,18 . However, a detailed understanding of the exotic one-electron spectral properties revealed by its angle resolved photo-emission spectroscopy (ARPES) 17,18 at energy scales beyond the TLL has remained elusive. In particular, the predictions of the MQIM-LO for the lo-cation in the (k, ω) plane of the experimentally observed high-energy peaks in the ARPES momentum distribution curves (MDC) and energy distribution curves (EDC) of Bi/InSb(001) do not lead to the same quantitative agreement as for the ARPES in the MoSe 2 line defects 16,24 . This raises the important question of what additional effects must be included to obtain agreement with the experimental data.
In this paper, we answer this question by extending the MQIM-LO to a larger class of 1D lattice electronic systems with finite-range interactions by accounting for higher-order terms in the effective range expansion 19,20,[25][26][27] of the phase shifts of the fractionalized charged particles. While the corresponding higher order "MQIM-HO" corresponds in general to a complicated, non-perturbative many-electron problem, we find, unexpectedly, that the interactions of the fractionalized charged particles with the charge mobile quantum impurity occur in the unitary limit of (minus) infinite scattering length [28][29][30] . In that limit, the separation between the interacting charged particles (the inverse density) is much greater than the range of the interactions, and the calculations simplify considerably.
The unitary limit plays an important role in the physics of many physical systems, including the dilute neutron matter in shells of neutron stars 31 and in atomic scattering in systems of trapped cold atoms 29,30 . Our discovery of its relevance in a condensed matter system is new and reveals new physics.
The results of the MQIM-HO are consistent with the expectation that the microscopic mechanisms behind the one-electron spectral properties of Bi/InSb(001) include finite-range interactions. Indeed, accounting for the effective range of the corresponding interactions [21][22][23] leads to theoretical predictions that quantitatively agree with both (i) the experimental value of the SDS exponent (α ∈ [0.6−0.7]) in Bi/InSb(001) observed in I(ω, 0) ∝ |ω| α and (ii) the location in the (k, ω) plane of the experimentally observed high-energy peaks in the ARPES MDC and EDC.
Since Bi/InSb(001) is a complex system and the MQIM-HO predictions are limited to the properties (i) and (ii), in the discussion section of this paper we consider other possible effects beyond the present theoretical framework that might contribute to the microscopic mechanisms determining spectral properties of Bi/InSb(001).
In this paper we employ units of = 1 and k B = 1. In Sec. II we introduce the theoretical scheme used in our studies. The effective-range expansion of the phase shift associated with the interactions of the charge fractionalized particles and charge hole mobile impurity, the corresponding unitary limit, and the scattering lengths are all issues we address in Sec. III. In Sec. IV the effective range expression is derived and expressed in terms of the ratio of the renormalized and bare scattering lengths. In Sec. V we show how our approach predicts the location in the (k, ω) plane of the experimentally observed highenergy Bi/InSb(001) ARPES MDC and EDC peaks. In Sec. VI, we discuss our results and experimental properties outside the present theoretical framework, mention open questions on the Bi/InSb(001) spectral properties, and offer concluding remarks.

II. THE MODEL
The 1D model Hamiltonian associated with the MQIM-HO for electronic density n e ∈]0, 1[ is given by, Hereρ j,σ = c † j,σ c j,σ − 1 2 , V e (0) = U/2, V e (r) = U F e (r)/r for r > 0, and F e (r) is a continuous screening function such that F e (r) ≤ 1/4, which at large r vanishes as some inverse power of r whose exponent is larger than one, so that lim r→∞ F e (r) = 0.
We use a representation of the fractionalized c (charge) and s (spin) particles that also naturally emerges in the MQIM-LO 16 . For simplicity, in general in this paper they are called c particles and s particles, respectively. They occupy a c band and an s band whose momentum values q j and q j , respectively, are such that q j+1 − q j = 2π/L and q j+1 − q j = 2π/L. In the thermodynamic limit one often uses a continuum representation in terms of corresponding c band momentum variables q and s band momentum variables q with ground-state occupancies q ∈ [−2k F , 2k F ] and q ∈ [−k F , k F ], respectively, where 2k F = πn e . The energy dispersions for c and s particles,ε c (q) andε s (q ), respectively, are defined for these momentum intervals in Eqs. (A2) and (A4)-(A10) of Appendix A.
Most of the weight of the one-electron spectral function is generated by transitions to excited states involving creation of one hole in the c band, one hole in the s band, plus low-energy particle-hole processes in such bands. Processes where both holes are created away from the c band and s band Fermi points ±2k F and ±k F , respectively, contribute to the spectral-function continuum. Processes where the c band hole is created at momentum values spanning its band interval q ∈] − 2k F , 2k F [ and the s hole (spinon) is created near one of its Fermi points ±k F contribute to the c and c branch lines whose spectra run from k ∈] − k F , k F [ and k ∈] − 3k F , 3k F [, respectively. Since in such processes the c band hole is 1: Sketch of the s (spin) and c and c (charge) branch lines in the one-electron removal spectral function of the lattice electronic correlated models discussed in this paper. The soft grey region refers to the small spectral-weight distribution continuum whereas the darker grey regions below the branch lines typically display more weight. In the actual spectral function, Eq. (2), this applies to k subdomains for which the exponents that control the line shape near those lines are negative. The lack of spectral weight in some of the figure (k, ω)-plane regions is imposed by kinematical constraints.
created away from the c band Fermi points, we call it a c (charge) hole mobile impurity. Finally, processes where the s band hole is created at momentum values in the interval q ∈] − k F , k F [ and the c hole (holon) is created near one of its Fermi points ±2k F contribute to the s branch line whose spectrum runs from k ∈] − k F , k F [. In the case of these processes it is the s band hole that is created away from the corresponding s band Fermi points. Hence we call it s (spin) hole mobile impurity. See a sketch of such spectra in Fig. 1. In the remainder of this paper the charge (and spin) hole mobile impurity is merely called c (and s) impurity. The one-electron operators matrix elements between energy eigenstates in the expressions for the spectral function involve phase shifts and the charge parameter ξ c = √ 2K c whose value is determined by them. Its range for the present lattice systems isξ c = √ 2K c ∈ ]1/2, ξ c ], where the bare parameter ξ c ∈]1, √ 2[ defined by Eq. (A16) of Appendix A refers to the 1D Hubbard model. Note that the model in Eq. (1), becomes the 1D Hubbard model at the bare charge parameter value,ξ c = ξ c . In this limit, the SDS exponent reads α 0 = (2 − ξ 2 c ) 2 /(8ξ 2 c ) ∈ [0, 1/8] with α 0 = 0 and α 0 = 1/8 for u → 0 and u → ∞, respectively. For n e ∈]0, 1[ there is a ξ c →ξ c transformation 16 for each fixed value of ξ c andξ c < ξ c such that ξ c ∈]1, √ 2[ andξ c ∈]1/2, 1[ ; ]1, ξ c ]. This maps the 1D Hubbard model onto the model, Eq.
(1), upon gently turning on F e (r). Consistent with this result, limξ c →ξc F e (r) → 0 for r ∈ [0, ∞]. For ξ c < ξ c the corresponding SDS exponent intervals are The phase shifts in the one-electron matrix elements play a major role in our study by appearing explicitly in the expressions of the momentum-dependent exponents of the oneelectron removal spectral function. These phases shifts are 2πΦ c,s (±2k F , q ) and 2πΦ c,c (±2k F , q). Specifically, −2πΦ c,s (±2k F , q ) and −2πΦ c,c (±2k F , q) are the phase shifts, respectively, imposed on a c particle of c band momentum ±2k F by a s and c impurity created at momentum q ∈ [−k F , k F ] and q ∈ [−2k F , 2k F ]. (Their explicit expressions are given below.) The charge parameterξ c is given by a superposition of charge-charge phase shifts, The expressions for the exponents of spectral functions also involve the phase shifts 2πΦ s,c (±k F , q) = ∓ π √ 2 and 2πΦ s, induced on a s particle of s band momentum ±k F by a c and s impurity created at momentum q ∈ [−2k F , 2k F ] and q ∈ [−k F , k F ], respectively. Their simple expressions are invariant under the ξ c →ξ c transformation and, due to the spin SU (2) symmetry, are interaction, density, and momentum independent. (Except for (−1) δ q ,±k F in the 2πΦ s,s (±k F , q ) expression at q = ±k F .) For small energy deviations (ω β (k) − ω) > 0 and (ω s (k) − ω) > 0 near the β = c, c branch lines and s branch line, the spectral function behaves as, respectively. Here C β,ι and C s are n e , u = U/4t, andξ c dependent constants for energy and momentum values corresponding to the small energy deviations (ω β (k) − ω) > 0 and (ω s (k) − ω) > 0, respectively, and ω < 0 are high energies beyond those of the TLL. The upper bounds of the constants C c,ι , C c ,ι , and C s in Eq. (2) are known from matrix elements and sum rules for spectral weights, but their precise values remain in general an unsolved problem. The expressions for the γ = c, c , s spectraω γ (k) and exponentsζ γ (k) are given in Eqs. (A1) and (A3) of Appendix A, respectively. As discussed in Appendix B, the MQIM-HO also applies to the low-energy TLL limit in which such exponents have different expressions. For the present high-energy regime, they have the same expressions as for the MQIM-LO except that the phase shift 2πΦ c,c (±2k F , q) in that of the spectral function exponentsζ c (k) andζ c (k) has MQIM-HO additional terms.
That the s branch line coincides with the edge of the support for the spectral function ensures that near it the line shape is power-law like, as given in Eq. (2). For the c, c branch likes, which run within the spectral weight continuum, the β = c, c lifetime τ β (k) in Eq. (2) is very large for the intervalξ c ∈]1, ξ c [, so that the expression given in that equation is nearly power-law like,B(k, ω) ∝ (ω β (k) − ω)ζ β (k) . The finite-range interaction effects increase upon decreasingξ c in the interval ξ c ∈ [ξ c , 1[ whereξ c = 1/ξ c . In it the corresponding c impurity relaxation processes associated with large lifetimes τ c (k) and τ c (k) in Eq. (2) for the k intervals for whichζ c (k) < 0 andζ c (k) < 0, respectively, start transforming the power-law singularities into broadened peaks with small widths. Such effects become more pronounced upon further decreasingξ c in the intervalξ c ∈]ξ c , 1]. As discussed in more detail below in Sec. V D, for k ranges for which the exponentsζ c (k) andζ c (k) become positive upon decreasingξ c , the relaxation processes wash out the peaks entirely.

III. THE EFFECTIVE-RANGE EXPANSION AND THE UNITARY LIMIT
A. The effective-range expansion As we shall establish in detail below, the finite-range electron interactions have their strongest effects in the charge-charge interaction channel. In contrast, for the charge-spin channel, the renormalization factor of the phase shift, remains that of the MQIM-LO. For small relative momentum k r = q ∓ 2k F of the c impurity of momentum q and c particle of momentum ±2k F the phase shiftΦ c (k r ) = −2πΦ c,c (±2k F , ±2k F + k r ) associated with the charge-charge channel obeys an effective range expansion, This equation is the same as for three-dimensional (3D) s-wave scattering problems if k r is replaced by |k r | 19,20 . The first and second terms involve the scattering length a and effective range R eff , respectively. The third and higher terms are negligible and involve the shape parameters 19,20,[25][26][27] . One finds that in the bare charge parameter limit, ξ c = ξ c , the effective range expansion reads cot(Φ c (k r )) = −1/(a k r ) where Φ c (k r ) = −2πΦ c,c (±2k F , ±2k F + k r ), 2πΦ c,c (±2k F , q) is the bare phase shift defined in Eqs. (A11)-(A15) of Appendix A, and a = limξ c →ξcã is the bare scattering length.
Due to the 1D charge-spin separation at all MQIM energy scales, the repulsive electronic potential V e (r) gives rise to an attractive potential V c (x) associated with the interaction of the c particle and c impurity at a distance x. To go beyond the MQIM-LO, we must explicitly account for the general properties of V c (x) whose form is determined by that of V e (r). The corresponding relation between the electron and c particle representations is discussed see Appendix C. The attractive potential V c (x) is negative for x > x 0 where x 0 is a non-universal distance that either vanishes or is much smaller than the lattice spacing a 0 . Moreover, for the present class of systems V c (x) vanishes for large x as, Here µ is a non-universal reduced mass, l is an integer determined by the large-r behavior of V e (r), and 2r l is a length scale whose l dependence forξ c < 1 is given below in Sec. IV. (And is twice the van der Waals length at l = 6).
Since V c (x) has asymptotic behavior 1/x l , the scattering length, effective range, and shape parameter terms in Eq. (4) only converge if l > 5, l > 7, and l > 9, respectively 27 . We shall find that agreement with the experimental results is achieved provided that the effective range studied in Sec. IV is finite and this requires that l > 5 in Eq. (5).
Similarly to the potentials considered in Refs. 23 and 32, the class of potentials with large-distance behavior, Eq. (5) and whose depth is larger that the scattering energy of the corresponding interactions considered here are such that the positive "momentum" 2µ(−V c (x)) obeys a sum rule of general form, Here ∆a/ã where ∆a = a−ã is a relative fluctuation that involves two uniquely defined yet non-universal scattering lengths, a andã. As justified in Sec. IV, in the present unitary-limit case discussed in Sec. III B, they are the bare and renormalized scattering lengths, respectively, defined in that section. The physically important renormalized charge parameter range isξ c ∈]1/2, 1[ for which α > 1/8. The term θ c π/[2(l − 2)] in Eq. (6) refers to a potential boundary condition 23,32 with θ c = 1 for that interval. (In that regime, the expressions in Eq. (6) are similar to those in Eqs. Our choice of potentials with large-x behavior given in Eq. (5) is such that the sum rule, Eq. (6), is obeyed yet for small x the form of V c (x) is not universal and is determined by the specific small-r form of V e (r) itself. The zero-energy phase Φ in Eq. (6) whose physics is further clarified below can be expressed as, . Here x 2 ≈ 2r l for ξ c ∈]1/2, 1[ with the ratio x 2 /2r l decreasing from 1.342 at l = 6 to 1 at l = ∞. Forξ c ∈]1, ξ c ] it is an increasing function ofξ c such that limξ c →ξc x 2 /2r l = ∞ for l finite. The universal form of the spectral function near the singularities, Eq. (2), is determined by the large x behavior of V c (x), Eq. (5), and sum rules, Eqs. (6) and (7). In the spectral-weight continuum, its form is not universal, as it depends on the specific small x form of V c (x) determined by V e (r).
The length scale 2r l in Eq. (5) is found below in Sec. IV to obey the inequality √ 2 (2r l ) 23 in units of Bohr radius a 0 = 0.529177Å with µ and α corresponding to µ and γ c , respectively.) This inequality justifies why V c (x) = V asy c (x) for x > x 2 and implies that x2 x0 dx 2µ(−V c (x)) in Eq. (7) has forξ c ∈]1/2, 1[ large values. This is consistent with the above mentioned requirement of the scattering energy of the residual interactions of the c particles and c impurity being smaller than the depth −V c (x 1 ) of the potential V c (x) well, which since ∞ x0 dx 2µ(−V c (x))/π 1 must be large. Here x 1 is a small non-universal potential-dependent x value such that x 0 < x 1 < a 0 at which ∂V c (x)/∂x = 0 and −V c (x) reaches its maximum value.

B. The unitary limit and the scattering lengths
As confirmed below in Sec. IV, the expression for the phase shift in the thermodynamic limit, for lim kr→0Φc (k r ) remains the same as for the MQIM-LO. Its use along with that of ∓(ξ c −1) 2 π/ξ c for the bare phase shift lim kr→0 cot(Φ c (k r )) in the leading term of the corresponding effective-range expansions gives the scattering lengths. In the thermodynamic limit they read, respectively. This is known as the unitary limit 29,30 . The validity of the MQIM-HO refers to this limit, which occurs provided that ξ = 1,ξ c = 1, and as confirmed below thatξ c > 1/2. The dependence of the bare charge parameter ξ c = ξ c (n e , u) on the density n e and u = U/4t is defined by Eq. (A16) of Appendix A. It is such that ξ c = √ 2 for u → 0 and ξ c = 1 for u → ∞ for n e ∈]0, 1[ and ξ c = 1 for u > 0 and ξ c = √ 2 at u = 0 for both n e → 0 and n e → 1. This implies that a = −∞ provided that the relative momentum obeys the inequality |k r | tan(π ne) 4u . This excludes electronic densities very near n e = 0 and n e = 1 for all u values and excludes large u values for the remaining electronic densities.
The phase shiftsΦ c = −2πΦ c,c (±2k F , q) incurred by the c particles from their interactions with the c impurity near the c band Fermi points −2k F and 2k F , respectively. As discussed in Appendix B, the creation of an impurity in the c band intervals q ∈ 2k F ] refers to the low-energy TLL regime. Its velocity becomes that of the low-energy particle-hole excitations near −2k F and 2k F , respectively. In this regime, the physics is different, as the impurity loses its identity, since it cannot be distinguished from the c band holes (TLL holons) in such excitations.
The small momentum k 0 F c can be written as k 0 F c = πn 0 F c . The unitary limit refers to the corresponding low-density n 0 F c of c particle scatterers with phase shift Φ c = −2πΦ c,c (±2k F , q) and c band momentum values [−2k F , −2k F + k 0 F c ] and [2k F − k 0 F c , 2k F ] near −2k F and 2k F , respectively. They, plus the single c impurity constitute the usual dilute quantum liquid of the unitary limit whose density is thus n 0 Here N 0 F c is the number of c particle scatterers in n 0 F c = N 0 F c /L. In the thermodynamic limit one has that n 0 F c is very small or even such that lim L→∞ n 0 F c → 0. Consistent with this result, the following relations of the usual unitary limit of the dilute quantum liquid unitary limit 29 , R eff 1/k 0 F c |ã| and 0 1/k 0 F c |a| hold. The effective range R eff derived in Sec. IV is such that R eff → ∞ asξ c → 1/2. The unitary limit requirement that R eff 1/k 0 F c in the thermodynamic limit is the reason that the valueξ c = 1/2 is excluded from the regime in which the MQIM-HO is valid.

IV. THE EFFECTIVE RANGE
A. The effective-range general problem and cancellation of its unwanted terms The MQIM-HO accounts for the higher terms in the effective range expansion, Eq. (4), so that as anticipated the phase shift 2πΦ c,c (±2k F , q) acquires an additional term, 2πΦ R eff c,c (k r ), relative to the MQIM-LO, The second term in the expression for the phase shift 2πΦã c,c (±2k F , q) reveals that its renormalization is controlled by the scattering lengths associated with the leading term in the effective range expansion. Theξ c = ξ c bare phase shift 2πΦ c,c (±2k F , q) in that expression is defined in Eqs. (A11)-(A15) of Appendix A. Furthermore, the function P c (k r ) in the expression of 2πΦ R eff c,c (k r ) vanishes for l < 8 and is such that its use in the term on the left-hand side of Eq. (4), cot(Φ c (k r )) = cot(−2πΦã c,c (±2k F , q) − 2πΦ R eff c,c (k r )), gives rise to all the shape parameter terms in the expansion, Eq. (4), beyond the two leading terms, −1 a kr + 1 2 R eff k r . Fortunately, in the unitary limit all properties that are characterized by these higher-order terms become irrelevant also for l > 7.
Hence 2πΦ R eff c,c (k r ) is given by arctan 1 2 R eff k r sin 2 ([(ξ c − 1) 2 /ξ c ]π) , which gives cot(Φ c (k r )) = −1 a kr + 1 2 R eff k r at small k r . (That 2πΦ R eff c,c (∓2π/L) vanishes in the thermodynamic limit confirms that at k r = ∓2π/L the phase shift 2πΦ c,c (±2k F , ±2k F + k r ) has the same value ±(ξ c − 1) 2 π/ξ c , Eq. (8), as for the MQIM-LO.) Both the unitary limit and the fact that forξ c ∈]1/2, 1[ the scattering energy of the residual interactions of the c particles and c impurity are much smaller than the depth −V c (x 1 ) of the potential V c (x) will play important roles in the following derivations of the effective range R eff in the expression of 2πΦ R eff c,c (k r ), Eq. (14).
in the effective range expansion, Eq. (4), contributes only to the leading term in that expansion , −1 a kr . Thus it does not contribute to the effective range R eff . Indeed, that phase shift term reads ∓(ξ c − 1) 2 π/ξ c , Eq. (8), at k r = ∓2π/L whereas it vanishes at k r = 0, so that in the thermodynamic limit the derivative (14) follows from standard scatteringtheory methods, and becomes 21-23 This integral converges provided that l > 5. The bare limitξ c = ξ c boundary condition V c (x) = 0 for all x corresponds to the wave function ψ 0 c (x) in Eq. (15). It is the zero-energy solution of the Schrödinger equation for the free motion, Here µ is the reduced mass of the c particle and c impurity. The function ψ 0 c (x) then has the form ψ 0 In contrast, the wave function ψ c (x) in Eq. (15) is associated with the potential V c (x) induced by the potential V e (r) in Eq. (1). The former is associated with the interaction of the c particle and c impurity. That wave function is thus the solution of a corresponding Schrödinger equation at zero energy, with the boundary condition ψ c (0) = 0. It is normalized The charge parameter intervalξ c ∈]1, ξ c ] for which α < 1/8 that corresponds to the secondξ c expression in Eq. (12) is of little interest for our studies, as similar α values are reachable by the 1D Hubbard model. Two boundary conditions that must be obeyed by R eff in that parameter interval are limξ c →ξc R eff = 0 and limξ c →1 R eff = a 0 . They are satisfied by the following phenomenological effective range expression, In the case of the intervalξ c ∈]1/2, 1[ for which α > 1/8 that corresponds to the firstξ c expression in Eq. (12), the explicit derivation of the integral in the effective range expression, Eq. (15), simplifies because the inequalities √ 2µγ c 1 in units of a 0 = 1 and Φ/π 1 are found to apply, as reported in Sec. III. This ensures that for ξ c < 1 the scattering energy of the residual interactions of the c particles and c impurity is much smaller than the depth −V c (x 1 ) of the potential V c (x).
The following analysis applies to general scattering lengths a, finite or infinite, and potentials with these general properties. They imply that the large-x function ψ c (x) obeying a Schrödinger equation, whose attractive potential is given by its large-distance asymptotic behavior V asy c (x) = −C c /(x/2r l ) l , Eq. (5), which has the general form, This expression can be used for all x ∈ [0, ∞] provided that V c (x) at small distances x ≈ x 1 where it is deep is replaced by a suitable energy-independent boundary condition. This is also valid for 3D s-wave scattering problems whose potentials have the above general properties and whose scattering lengths are parametrically large 23 .
From the use in Eq. (18) of the asymptotic behavior, J ν (y) ≈ y ν /(2 ν Γ(1 + ν)), of the Bessel functions for x 1 and thus y 1 one finds that the normalization at x → ∞ as ψ c (x) = ψ 0 c (x) = 1 − x/a requires that, and It is convenient to write the integrand in Eq. (15) as and the functions f c (x) and g c (x) are given by, (22) and respectively. The separation in Eq. (21) is convenient because the divergences all appear in the functions (ψ 0 c (x)) 2 and f c (x). That B 1 and B 2 have the expressions given in Eqs. (19) and (20), respectively, ensures that both 2 and thus the divergences from (ψ 0 c (r)) 2 and f c (x) exactly cancel each other under the integration of Eq. (15). Hence R eff can be expressed as,

B. The energy-independent boundary condition
The only role of f c (x), Eq. (22), is to cancel (ψ 0 c (x)) 2 within g virtual c (x), Eq. (21), under the integration in Eq. (15). In the general expression for the effective range given in that equation, ψ c (x) is the solution of Eq. (16) with the actual potential V c (x) defined in its full domain, x ∈ [0, ∞]. The alternative use of Eq. (24), which was derived by using the function ψ c (x) large-x expression, Eq. (18), for the whole domain x ∈ [0, ∞], also leads to the effective range, Eq. (15). This applies provided that V c (x) is replaced at small distances near x = x 1 , where it is deep, by the energy-independent boundary condition defined below. It accounts for the effects from V c (x) for small distances.
In the unitary limit the inverse scattering length, a −1 = 0, which appears in the B 2 expression, Eq. (20), is at the middle of negative a −1 < 0 and positive a −1 > 0 values and could refer to a = −∞ or a = ∞. Hence in that limit there is not much difference between the repulsive and attractive scattering cases. As discussed in Ref. 41 for the case of two particles with a s-wave interaction, the scattering lengths in the attractive a = −∞ and repulsive a = ∞ cases of the unitary limit merely refer to different states of the same a −1 = 0 scattering problem.
For a potential V (r) with a finite scattering length a and having the general properties reported above, at small distances r where the potential is deep it can be replaced by an energyindependent boundary condition such that the ratio HereΦ π is a potential dependent zero-energy phase, V (r 0 ) = 0, and where ∆a = a −ā, ∆a/ā is a relative fluctuation, and a given in Eq. (20) is a mean scattering length determined by the asymptotic behavior ∝ 1/r l of the potential V (r) through the integer l > 5 and the length scale 2r l . For instance, in terms of the constants A = B 1 − B 2 and B = −B 2 tan π l−2 , of the scattering problem studied in Ref. 23, the ratioā/a = B 2 /B 1 on the left hand side of the above boundary condition reads For the present rangeξ c ∈]1/2, 1[ the length scale 2r l whose expression is given below is finite in the unitary limit and thus the related length scaleā in Eq. (20) is also finite. It follows both thatā/a = 0 and the constant B 2 = B 0 2 , Eq. (20), vanishes. This result is clearly incorrect. The reason is that we have have not yet accounted for the behavior of V c (x) at small distances through a suitable energy-independent boundary condition. In the case of the unitary limit, this boundary condition renders both a/a and B 2 = B 0 2 in Eq. (20) finite. Specifically, the scattering length a is suitably mapped under it into a finite scattering length a f = aā a , so that B 0 2 is mapped onto the following corresponding finite constant B 2 , Here tan(Φ f ) = tan(Φ) yetΦ f π may be different from Φ π in Eq. (10). Indeed, the relation tan(Φ f ) = tan(Φ) is insensitive to such phase differences. In the unitary limit the boundary condition is thus equivalent to a transformation a → a f such that tan(Φ f ) = tan(Φ).
The energy-independent boundary condition in Eq. (25) is in terms of the finite scattering length a f such that , similarly to scattering problems of the same universality class whose scattering lengths are parametrically large 23,32 . The positivity of a f = aā a often occurs for potentials that for large distances are attractive 23 . If a f were negative,ā/a f = −ã/a, then tan(Φ f ) would be given by 2 cot π l−2 −tan(Φ), which would violate both the requirements that tan(Φ f ) = tan(Φ) and that tan(Φ f ) = 0 in the bare limit,ξ c = ξ c .
Importantly, the cancellation (ψ 0 c (x)) 2 − f c (x) = 0 is independent of the value of the scattering length in the expressions for ψ 0 c (x) and f c (x). Hence all results associated with Eqs. (15)-(24) remain the same, with a replaced by a f . This includes the effective range R eff , Eq. (24), remaining determined only by g c (x).
The main property of the transformation a → a f is the corresponding exact equality of the ratios,ā/a f =ã/a. This actually justifies why the scattering lengths a and a, Eqs. (9), can be used in tan(Φ) in Eq. (6). That transformation is also the mechanism through which the renormalized scattering lengthã emerges in ψ c (x).
Hence similarly to finite-a scattering problems of the same universality class, as for instance those studied in Refs. 23 and 37, the relations of general form, Eq. (6), apply. In the present unitary limit the scattering length ratioã/a in them equals the ratioā/a f also given by Eq. (10). The sum rule Φ = x2 x0 dx 2µ(−V c (x)), Eq. (7), encodes the effects from V c (x) for small distance near C. The effective range dependence on the scattering length finite ratioã/a The use of the function g c (x), Eq. (23), with the constants B 1 and B 2 given in Eqs. (19) and (25), respectively, in Eq. (24) leads forξ c ∈]1/2, 1[ to, Description 2r l length scale in the large-x potential decay with exponent l > 5, V asy c (x) ∝ (x/2r l ) −l , Eqs. (5) and (29)  a0 lattice spacing related to 2r l (twice the van der Waals length at l = 6) as given in Eq. (27) a andã scattering lengths at ξc andξc, respectively, within the ξc →ξc transformation, Eqs. (9) and (10) R eff effective range R eff = a0 1 − c1 ã a + c2 ã a 2 for the intervalξc < 1 of physical interest, Eqs. (26) and (28)  After performing the integrations, one finally reaches the following expression valid for the present charge parameter intervalξ c ∈]1/2, 1[ and α > 1/8, That here the coefficient, is identified with the lattice spacing results from the imposition of R eff having the same value forξ c → 1 − and ξ c → 1 + . The coefficients c 1 and c 2 in Eq. (26) can be expressed in terms the usual Γ function and are given by, respectively. They decrease from c 1 = c 2 = 2 at l = 6 to c 1 = 1 and c 2 = 1/3 for l → ∞. The effective range R eff , Eq. (26), appears in the expression of the spectral function exponentsζ c (k) and ζ c (k), Eq. (A3) of Appendix A, through the expression for the phase shift 2πΦ c,c (±2k F , q), Eq. (14). R eff = ∞ forξ c → 1/2 is excluded, as it is outside the range of validity of the unitary limit. The R eff values found below in Sec. V for Bi/InSb(001) are given in Table V. They obey the unitary limit inequality, R eff 1/k 0 F c . The effective range, Eq. (26), can alternatively be expressed in terms of the ratioā/a f involving the finite scattering lengthsā ∝ 2r l and a f defined by Eqs. (20) and (25), respectively.
The expression for the lattice spacing a 0 , Eq. (27), contains important physical information: Its inverse gives the following expression valid forξ c ∈]1/2, 1[ for the length scale 2r l in the potential V asy c (x) expression, Eq. (5), and the related length scale x 2 , Eq. (7), (29) Here 2r l is given by 5.95047 a 0 at l = 6, reaches a maximum 6.48960 a 0 at l = 10, and decreases to 4.93480 a 0 as l → ∞, so that √ 2 (2r l ) l−2 2 = √ 2µγ c 1 in units of a 0 = 1 as given in Table V. Thus Φ/π 1 for l = 6 − 12. As in the case of 3D s-wave atomic scattering problems 23,32 , this shows that forξ c ∈]1/2, 1[ the scattering energy of the interactions of the c particles and c impurity is indeed much smaller than the depth −V c (x 1 ) of the potential V c (x) well. This confirms the consistency of the derivation of the effective range forξ c ∈]1/2, 1[ that assumed the validity of such properties.
The length scales involved in the MQIM-HO description are explicitly defined in Table II.

Agreement
Description First type overall (k, ω)-plane shapes of the theoretical branch-line spectra, Eq. (A1), versus ARPES experimental spectra Second type (k, ω)-plane location of the singularities corresponding to negative exponents, Eq. (A3), versus ARPES peaks Third type SDS exponent α from the dependence of the exponents, Eq. (A3), onξc versus its low-ω experimental value ARPES measurements were performed at ω = 15 eV and taken at 8 K in the CASSIOPÉE beamline of SOLEIL synchrotron. The photoelectron kinetic energy at E F and the overall energy resolution of the ARPES setup were calibrated by the Fermi edge of the photoelectron spectra from Mo foils attached to the sample. The energy resolution was ∼20 meV. The ARPES taken at 8 K is shown in Fig. 3.
The theoretical predictions reported in this paper refer to (i) the (k, ω)-plane location of the high-energy Bi/InSb(001) MDC and EDC ARPES peaks and (ii) the value of the power-law SDS exponent α associated with the angle integration to detect the low-energy suppression of the photoelectron intensity that were performed at k y = 0.2Å −1 , near the boundary of the (1 × 3) surface Brillouin zone (0.23Å −1 ).

B. Criteria for agreement between ARPES and the present theory
Refs. 17 and 18 found strong experimental evidence that Bi/InSb(001) at y momentum component k y = 0.2Å −1 and temperature 8 K displays 1D physics with an SDS exponent that for small |ω| < 0.10 eV has values in the interval α ∈ [0.6, 0.7]. As discussed and justified below in Sec. VI A, the oneelectron spectral properties of Bi/InSb(001) are expected to be controlled mainly by the interplay of one dimensionality and finite-range electron interactions, despite a likely small level of disorder. Consistent with an SDS exponent α larger than 1/8 stemming from finite-range interactions 8 , here we use the MQIM-HO to predict oneelectron spectral properties of Bi/InSb(001).
As discussed in Sec. VI A, Bi/InSb(001) is a complex system and some of its experimental properties beyond those studied here may involve microscopic processes other than those described by the MQIM-HO and the Hamiltonian, Eq. (1). This includes coupling to twodimensional (2D) physics if k y = 0.2Å −1 is smoothly changed to k y = 0.
As reported in Sec. III A, the MQIM-HO can describe both the low-energy TLL regime and the spectral function, Eq. (2), at high energies near the (k, ω)-plane singularities. At and in the vicinity of those singularities, the renormalization from its bareξ c = ξ c form is determined by the large x behavior of V c (x), Eq. (5), and its sum rules, Eqs. (6) and (7), which refer to a high energy regime that goes well beyond the TLL limit.
Hence we can predict two properties of the one-electron spectral function : (i) the location in the (k, ω) plane of the experimentally observed high-energy peaks in the ARPES MDC and EDC and (ii) the value of the lowenergy SDS exponent α. Our T = 0 theoretical results describe the former high-energy experimental data taken at 8 K for which the smearing of the spectral function singularities by thermal fluctuations is negligible. The quantitative agreement with the corresponding experimental data taken at fixed momentum k y = 0.2Å −1 reached below provides further evidence of 1D physics and electron finite-range interactions in Bi/InSb(001).
A first type of agreement of the theoretical branch-line energy spectra with the (k, ω)-plane shape of the ARPES image spectra must be reached for well-defined fixed values of electronic density n e and interaction u = U/4t. Through Eq. (A16) of Appendix (A), these uniquely determine the value of the bare charge parameter ξ c = ξ c (u, n e ) to be used in the ξ c →ξ c transformations suited to Bi/InSb(001). In addition, that first type of agreement also determines the value of the transfer integral t.
The experimental values of the lattice spacing a 0 and of the momentum width of the spectra at ω = 0 provide the Fermi momentum k F = (π/2a 0 ) n e and thus the electronic density n e . At the density n e , the ratioW s /W c of the experimental energy bandwidthsW s ≡ |ω s (0)| of the s branch line spectrum andW c ≡ |ω c (0)| = |ω c (0)| of the c and c branch line spectra at momentum k = 0 uniquely determine u = U/4t. (See such energy bandwidths in the sketch of the theoretical spin s and charge c and c branch lines in Fig. 1.) Finally, the experimental values ofW s andW c determine the value of the transfer integral t.
As discussed below in Sec. V C, from the available experimental data it is not possible to trace the energy dispersion of the s branch line. However, combining the experimental data on the EDC with kinematic constraints of the MDC provides information about the most prob-able value of the energy at which its bottom is located, which equals the branch line energy bandwidthW s . A second type of agreement is between the momentum interval and corresponding energy interval for which the exponentsζ c (k),ζ c (k), andζ s (k), Eq. (A3) of Appendix A, are negative and the (k, ω)-plane location of the experimentally observed high-energy ARPES MDC and EDC peaks. That agreement must be reached at the fixed u and n e values and corresponding bare charge parameter ξ c = ξ c (u, n e ) value obtained from the first type of agreement. This second type of agreement is reached at some values of the integer quantum number l > 5 in the large-x potential V c (x) expression, Eq. (5), and of the renormalized charge parameterξ c (and thus of tan(Φ), see Eqs. (12) and (13)).
For the theoretically predicted high-energy ARPES peaks located on the s branch line, there is only limited experimental information. Hence we start by finding theξ c and l > 5 values at which the second type of agreement is reached concerning the momentum intervals where the exponentsζ c (k) andζ c (k) are negative and the corresponding (k, ω)-plane location of the experimentally observed high-energy ARPES MDC and EDC peaks. Fortunately, it turns out that theξ c values lead to a prediction of location in the (k, ω)-plane of the highenergy ARPES peaks associated with the s branch line that is consistent with the available experimental EDC data.
This second type of agreement is reached for specific ξ c values. This then provides a prediction for the SDS exponent α = (2 −ξ 2 c ) 2 /(8ξ 2 c ) obtained from a different low-energy experiment that detects the suppression of the photoelectron intensity. That the SDS exponent α determined by theξ c values for which the second type of agreement is reached is also that measured within the low-energy angle integrated photoemission intensity then becomes the required third type of agreement.
In the Lehmann representation of the spectral function, the first and second types of agreement correspond to the energy spectra and the overlaps of the oneelectron matrix elements, respectively. The exponents in Eq. (A3) of Appendix A involved in the second type of agreement depend both onξ c and momentum-dependent phase shiftsΦ c,c (±2k F , q) andΦ c,s (±2k F , q ). There is no apparent direct relation between the high-energy ARPES MDC peaks and the low-energy SDS. That the MQIM-HO describes the main microscopic mechanisms behind the specific one-electron spectral properties of Bi/InSb(001) then requires that the third type of agreement is fulfilled.
The three types of agreement between theory and experiment are explicitly described in Table III. C. Searching for agreement between theory and experiments

First type of agreement
The MDC spectral shape plotted in Fig. 3(c) displays two peaks centered at well defined Fermi momentum values −k F = −0.06Å −1 and k F = 0.06Å −1 , re-spectively. Furthermore, the experimental circles (with error bars) in Fig. 3(e) clearly indicate that the MDC peaks are located on two lines that in the limit of zero energy start at such two Fermi momenta. Since the experimental data lead to π/a 0 ≈ 0.68Å −1 , one finds from k F = (π/2a 0 ) n e ≈ 0.06Å −1 a small electronic density, n e ≈ 0.176. The experimental value of the c branch line energy bandwidthW c is directly extracted from analysis of the experimental MDC data provided in Fig. 3(e). From analysis of the EDCs in Fig. 3(d) alone one finds that there is a uncertainty 0.05±0.05 eV concerning the energy at which the bottom of the s branch line is located. It is clear that in this energy region there is a hump that cannot be explained by assuming the single peak at 0.25 eV, which refers to the bottom of the c branch line.
The zero-energy level of the theoretically predicted downward-convex parabolic-like dispersion of the s branch line plotted in Fig. 3(e) (see also sketch depicted in Fig. 1) refers to the Fermi level. Hence the s branch line energy bandwidthW s equals that of its bottom. While the energy range uncertainty of that bottom energy is experimentally rather wide, one can lessen it by combining the experimental ARPES MDC intensity distribution shown in Fig. 3(c) with its kinematical constraints, which follow from the finite-energy bandwidth of the theoretical s branch line. One then finds that the most probable value of the s branch line bottom energy and thus ofW s is between 0.05 and 0.10 eV.
The maximum momentum width of the ARPES MDC intensity distribution shown in Fig. 3(c) for energy |ω| = 0.05 eV allowed by such kinematic constraints involves the superposition of two maximum momentum widths ∆k, centered at −k F and k F , respectively. Within the MQIM-HO, these kinematical constraints explain the lack of spectral weight in well-defined (k, ω)-plane regions shown in Fig. 1. Fortunately, the lines that limit such regions without spectral weight only involve the s band dispersion spectrum.
In the case of the spectral weight centered at −k F and k F , respectively, such kinematical constraints imply that for each energy value |ω| = −ω the corresponding maximum momentum width reads, whereW s = |ω s (0)| and the s band dispersion spectrum ω s (k) is given in Eq. (A1) of Appendix A. For |ω| W s the kinematical constraints, Eq. (30), are those of a TLL, ∆k = 2|ω|/v s (k F ), consistent with v s (k F ) = min(v s (k F ), v c (2k F )) 33 . However, for energy |ω| = −ω larger than the s branch line energy bandwidth W s = |ω s (0)|, which is that at which the s branch line bottom is located in the experimental data, there are no kinematical constraints.
The absolute value of the derivative with respect to k of the ARPES MDC intensity plotted in Fig. 3(c) increases in a |k| interval |k| ∈ [k F , k F + k MDC ] and decreases for |k| > k MDC . Theoretically, the ARPES MDC intensity should be symmetrical around k = 0. Its actual experimental shape then introduces a small uncertainty in the value of k MDC . The relatively large intensity in the tails located at the momentum region |k| > k MDC is explained by the larger uncertainty in the s branch line bottom energyW s . Indeed, the ARPES MDC under consideration refers to an energy |ω| = 0.05 eV within that uncertainty. And, as given in Eq. (30), there are no kinematic constraints for |ω| >W s .
One can then identify the most probable value ofW s within its uncertainty interval as that for which at the energy |ω| = 0.05 eV the kinematic constraints would limit the ARPES MDC intensity to momentum values within the interval |k| ≤ k MDC . The corresponding momenta k = ±k MDC are the inflection points at which the derivative of the variation of the ARPES MDC intensity with respect to k changes sign in Fig. 3(c). The momentum width associated with |k| ≤ k MDC is thus that of the ARPES MDC shown in that figure if one excludes the tails.
The corresponding maximum momentum width ∆k, Eq. (30), of the two overlapping spectral weights centered at k F and −k F , respectively, that at |ω| = 0.05 eV would lead to the kinematic constraint ∆k = 2(k MDC − k F ), so that ±(k F + ∆k/2) = ±k MDC . According to the kinematic constraints in Eq. (30), this is fulfilled when at k = ±(k F −∆k/2) = ±(2k F −k MDC ) so that the s branch line energy spectrum reads |ω s (k)| = −ω s (k) = 0.05 eV. Accounting for the combined k MDC andW s uncertainties, the most probable value of the energy bandwidth W s is larger than 0.05 eV and smaller than 0.10 eV, as that of the theoretical s branch line plotted in Fig. 3(e).
At electronic density n e = 0.176 the best second type of agreement between theory and experiments discussed in the following is reached within that combined uncertainty by the u = U/4t and t values that are associated with the energy bandwidthW s of such a theoretical s branch line. They read u = 0.30 and t = 1.22 eV, as determined from the corresponding ratioW s /W c and experimentalW c value in Fig. 3(e). Hence within the MQIM-HO the first type of agreement with the ARPES spectra is reached by choosing these parameter values for the electronic density n e = 0.176.

Second type of agreement
The second type of agreement involves the theoretical γ = c, c , s exponentsζ γ (k), Eq. (A3) of Appendix A. They are plotted for u = 0.30 and n e = 0.176 as a function of the momentum k in Fig. 4(a) for l = 6 and in Fig. 4(b) for l = 12. In Appendix A, they are plotted as a function of k for several additional values of l. The different curves in each figure are associated with different values of the charge parameterξ c and thus of the SDS exponent α = (2−ξ 2 c ) 2 /(8ξ 2 c ) and effective range R eff . The black solid lines refer to the bare charge pa-rameter limit,ξ c = ξ c = 1.242. The valuesξ c = 0.805, 0.857, and 1.166 correspond toξ c = 1/ξ c ,ξ − c , andξ + c , respectively.
As justified in Sec. V B, we start by finding theξ c and l > 5 values at which the second type of agreement is reached. It refers to the momentum intervals (and corresponding energy ranges) at which the exponentsζ c (k) andζ c (k) are negative. Those are required to agree with the corresponding (k, ω)-plane location of the experimentally observed high-energy ARPES MDC and EDC peaks in Figs , respectively. In the case of the exponentζ c (k), the momentum k ex F c /k F is vanishing or very small in the thermodynamic limit. It is the experimental value of the small theoretical c band momentum k 0 F c = πn 0 F c associated with the low density n 0 F c of c particle scatterers near the c band Fermi points −2k F and 2k F considered in Sec. III B.
Concerning the momentum interval k ∈ [−k ex c , k ex c ] for which the exponentζ c (k) must be negative, there is a small uncertainty in the value of k ex c . It is such that Fig. 3(e) is the momentum width of the ARPES image crossed by the c branch line. This small uncertainty, which in the units used in the figures corresponds to δk 0 ∈ [0, 0.07π], implies corresponding small uncertainties in theξ c and α values at which for each l agreement with the experimentally observed high-energy ARPES MDC and EDC peaks is reached. The corresponding two limiting values of such ξ c and α uncertainties at which the exponentζ c (k) in Fig. 4 and in Figs. 5 and 6 of Appendix A crosses zero at k ≈ 0 and k ≈ 0.07π, respectively, are given in Table  IV for each integer l = 6 − 12.
Following the direct relation between the c and c branch lines spectra, that δk 0 ∈ [0, 0.07π] ensures that the exponentζ c (k) is indeed negative for k intervals where k ex F c /k F 1, as also required for the second type of agreement to be reached.
Hence regarding the c and c branch lines, agreement between theory and experiments is reached by theξ c and l > 5 values that in Fig. 4   The two values of the charge parameterξc and corresponding SDS exponent α = (2 −ξ 2 c ) 2 /(8ξ 2 c ) that at each integer l = 6 − 12 are those at which the exponentζ c (k) plotted as a function of k in Fig. 4(a) for l = 6 and in Fig. 4(b) for l = 12 crosses zero at k ≈ 0 and k ≈ 0.07π, respectively. The same applies to the exponentζ c (k) plotted for l = 7 − 11 in Figs. 5(a) and (b) and 6(a)-(c) of Appendix A.
number l > 5. For theξ c values for which the c branch line exponent curves cross zero between k ≈ 0 and k ≈ δk 0 ≈ 0.07π in Fig. 4

and in Figs. 5 and 6 of Appendix A, the exponentζ s (k) is negative in corresponding intervals
Here k * F s is a function of n e , u, andξ c and k = ±(k F − k * F s ) are the two momentum values at whichζ s (k) vanishes.
The predicted location at k ∈] − k F + k * F s , k F − k * F s [ of the ARPES MDC peaks associated with the s branch line cannot be confirmed from the available experimental data. Indeed and as mentioned in Sec. V B, it is not possible to extract from such data the dispersion of that line. However, the corresponding energy intervals |ω| ∈ [|ω s (k F − k * F s )|,W s ] are consistent with the available experimental data from the EDCs in Fig. 3(d). Here |ω| =W s = |ω s (0)| is the bottom of the s branch line energy, as estimated in Sec. V C 1 from the interplay of the kinematical constraints, Eq. (30), and the ARPES MDC shown in Fig. 3(c) for |ω| = 0.05 eV.

Third type of agreement
From the above results we see that for l = 6 − 12 agreement with the experimentally observed high-energy ARPES MDC and EDC peaks in Figs. 3(e) and (f) is reached by the exponents curves referring toξ c and α values belonging to the small intervals reported in Table  V. The overlap of the subintervals obtained for each l = 6 − 12 given in that table then leads to the theoretical predictionsξ c ∈ [0.66, 0.69] and α ∈ [0.610 − 0.700].
Table V also provides the corresponding intervals of the effective range R eff in units of the lattice spacing that refer to first and second types of agreements. The effective range dependence on the bare charge parameter ξ c = ξ c (n e , u), renormalized charge parameterξ c , and integer quantum number l > 5 values at which such agreements have been reached is defined by combining Eqs. are also provided.
(10) and (26). That table also provides the values of the length scale 2r l in the same units whose dependence on l is given in Eq. (29). Upon increasing l from l = 6 to l = 12, the effective range R eff c values for which there is agreement with the experiments change from R eff c ≈ 5r l to R eff c ≈ r l , respectively. According to the analysis of Sec. V C 2, agreement with the experimentally observed high-energy (k, ω)plane ARPES MDC and EDC peaks distribution has been reached for the SDS exponent range α ∈ [0.610 − 0.700]. The third type of agreement between theory and experiments defined in Sec. V C 2 is reached provided that such a predicted SDS exponent range agrees with the α values measured within the low-energy angle integrated photoemission intensity. An experimental uncertainty α = 0.65 ± 0.05 of the SDS exponent was found for −ω < 0.1 eV in Ref. 17.
The remarkable quantitative agreement of the MQIM-HO predictions within the third criterion reported in Sec. V C 2 provides evidence of finite-range interactions playing an active role in the Bi/InSb(001) spectral properties and confirms the 1D character of its metallic states also found in Ref. 17.

D. Interplay of relaxation processes with the momentum dependence of the exponents
Here we discuss the physical mechanisms within the MQIM-HO that underlie the dependence of the exponentsζ c (k),ζ c (k), andζ s (k) on the charge parameterξ c . These exponents are plotted in Fig. 4 and in Figs. 5 and 6 of Appendix A.
In the bare charge parameter limit,ξ c = ξ c , the exponents being negative or positive just refers to a dif-ferent type of power-law behavior near the corresponding charge and spin branch lines. Forξ c < ξ c , this applies only to the spin s branch line. It coincides with the edge of support of the one-electron removal spectral function that separates (k, ω)-plane regions without and with finite spectral weight. Hence conservation laws impose that, near that line, the spectral function remains of power-law form, Eq. (2), for both intervalsξ c ∈]1/2, 1[ andξ c ∈]1, ξ c ]. As confirmed from an analysis of the s branch-line exponents plotted in Fig. 4 and in Figs. 5 and 6 of Appendix A, the effect of decreasing the charge parameterξ c from its initial bare value ξ c (and thus in- is merely to increase the spin branch line exponentζ s (k). Except for two regions near −k F and k F corresponding to |k| ∈ [(k F − k * F s ), k F ], that exponent remains negative, so that the singularities prevail. In the complementarily small momentum regions near ±k F defined by |k| ∈ [(k F − k * F s ), k F ] where the exponent is positive, the line shape remains of power-law type.
Analysis of the c and c branch-line exponents curves plotted in the same figures reveals that the situation is different for the one-electron removal spectral function in the vicinity of the charge c and c branch lines, Eq. (2). These are located in the continuum of the one-electron spectral function. The physics is though different for the subintervalsξ c ∈]1, ξ c ] andξ c ∈]1/2, 1[, respectively.
Smoothly decreasingξ c from its initial bare value ξ c toξ c → 1, produces effects quite similar to those of increasing U within the 1D Hubbard model to U → ∞ 12 . Indeed, these changes renderζ c (k) andζ c (k) more negative and lead to an increase of the width of the k intervals in which they are negative. Within theξ c ∈]1, ξ c ] interval, a large number ofξ c = ξ c conservation laws that are behind the factorization of the scattering S matrix into two-particle scattering processes survive, which tend to prevent the c impurity from undergoing relaxation processes. Hence the lifetimes τ c (k) and τ c (k) in Eq. (2) are very large for the k intervals for which the corresponding branch line exponents are negative, so that the expression given in the equation for the spectral function near the β = c, c branch lines is nearly power-law like, The effects of the finite-range interactions increase upon decreasingξ c within the intervalξ c ∈ [ξ c , 1[ wherẽ ξ c = 1/ξ c = 0.805 for n e = 0.176 and u = 0.30. Indeed, smoothly decreasingξ c within that interval tends to remove an increasing number of conservation laws, which strengthens the effects of the impurity relaxation processes. Such effects become more pronounced when ∆a/ã ∈] − 1, 0[ and tan(Φ) > 0, upon further decreasing ξ c within the intervalξ c ∈]1/2,ξ c ].
In the k intervals for which the β = c, c branch line exponentsζ β (k) remain negative, the lifetimes τ β (k) in Eq.
(2) remain large and the c impurity relaxation processes only slightly broaden the spectral-function power-law sin-gularities, as given in Eq. (2). For the complementary k ranges for which such exponents become positive upon decreasingξ c and thus increasing α, the high-energy singularities are rather washed out by the relaxation processes.
As confirmed by analysis of the curves plotted in Fig.  4 and in Figs. 5 and 6 of Appendix A, starting at |k| = 3k F − k 0 F c and downwards, smoothly decreasingξ c fromξ c first gradually enhances the k domains wherẽ ζ c (k) is positive. Further decreasingξ c after the c branch line singularities are fully washed out leads to the emergence of a c branch line k domain starting at |k| = 0 and upwards in which that line singularities are finally fully washed out up to |k| = k F − k 0 F c below a smallerξ c value.

VI. DISCUSSION AND CONCLUDING REMARKS
A. Discussion of other effects and properties outside the range of the MQIM-HO As reported in Sec. V A, the ARPES data were taken at 8 K and the angle integrations to detect the suppression of the photoelectron intensity were performed at k y = 0.2Å −1 , near the boundary of the (1 × 3) surface Brillouin zone (0.23Å −1 ).
As shown in Fig. 2(b) of Ref. 17, at k y = 0.2Å −1 there is an energy gap between the spectral features studied in this paper within a 1D theoretical framework and a bulk valence band. Due to that energy gap, the coupling between the two problems is negligible, which justifies that the system studied here corresponds to 1D physics.
Smoothly changing k y from k y = 0.2Å −1 to k y = 0 corresponds to smoothly turning on the coupling to the 2D physics. As shown in Fig. 2(a) of Ref. 17, at k y = 0 the energy gap between the spectral features studied in this paper and that bulk valence band has been closed. The study of the microscopic mechanisms involved in the physics associated with turning on the coupling to the 2D physics by smoothly changing k y from k y = 0.2Å −1 to k y = 0 is an interesting problem that deserves further investigation.
Another interesting open problem refers to theoretical prediction of the MDC for extended momentum intervals and of the EDC for corresponding energy ranges. The universal form of the spectral function near the singularities, Eq. (2), is determined by the large x behavior of the potential V c (x), Eq. (5), which follows from that of the potential V e (r) in Eq. (1), and potential sum rules, Eqs. (6) and (7). As reported in Eqs. (12) and (13), the value of the renormalized charge parameterξ c behind the renormalization of the phase shifts in the exponents of that spectral function expression, Eq. (2), is indeed controlled by the value of the initial bare charge parameter ξ c = ξ c (n e , u), the integer quantum number l > 5 associated with the potential V c (x) large-x behavior, Eq. (5), and the zero-energy phase Φ determined by that potential sum rules, Eqs. (6) and (7). Plotting a MDC for extended momentum intervals and an EDC for corresponding energy ranges is a problem that involves non-universal properties of the one-electron removal spectral function. This would require additional information of that function in (k, ω)-plane regions where it is determined by the detailed non-universal dependence on r of the specific electronic potential V e (r) suitable to Bi/InSb(001).
Another interesting issue refers to the validity of the MQIM-HO to describe the Bi/InSb(001) one-electron spectral properties. The question is whether the interplay of one dimensionality and electron finite-range interactions is indeed the main microscopic mechanism behind such properties. As in all lattice electronic condensed matter systems, it is to be expected that there are both some degree of disorder effects and electronelectron effects in the Bi/InSb(001) physics. However, we can provide evidence that the interplay of latter effects with the Bi/InSb(001) metallic states one dimensionality is the dominant contribution to the one-electron removal spectral properties.
The first strong evidence that this is so is the experimentally observed universal power-law scaling of the spectral intensity I(ω, T ). (Here ω = 0 refers to the Fermi-level energy.) For instance, at ω = 0 and finite T and at T = 0 and low ω it was found in Ref. 17 to have the following TLL behaviors for Bi/InSb(001), respectively, where α is the SDS exponent. If there were important effects from disorder, its interplay with electron-electron interactions would rather give rise in the case of 1D and quasi-1D systems to a spectral intensity I(ω, T ) with the following behaviors [34][35][36] , for ω C 2 0 /(32πD 0 ). Here D 0 ∝ l is the bare diffusion coefficient and C 0 is a constant that depends on the effective electron-electron interaction and electronic density.
The behaviors in Eq. (32) are qualitatively different from those reported in Eq. (31), which are those experimentally observed in Bi/InSb(001). This holds specially for I(0, T ), in which case disorder effects cannot generate such a temperature power-law scaling. Also the experimentally found behavior I(ω, 0) ∝ |ω| α disagrees with that implied by Eq. (32).
Further, in the limit of low ω and T , the MQIM-HO describes the corresponding TLL limit in which the universal power-law scaling of the spectral intensity I(ω, T ) has the behaviors reported in Eq. (31). Theoretically, the value of the SDS exponent α depends on those of the electronic density n e , the interaction u = U/4t, and the renormalized charge parameterξ c . Within the MQIM-HO phase shifts constraints, its values span the intervals α ∈ [α 0 , 1/8[ ; ]1/8, 49/32[. The theoretically predicted α value has been determined from agreement of the T = 0 one-electron spectral function, Eq. (2), with the ARPES peaks location in the (k, ω)-plane. The quantitative agreement then reached refers to the experimental value α = 0.65 ± 0.05 obtained for I(ω, 0) ∝ |ω| α at |ω| < 0.1 eV in Ref. 17. For low temperatures and ω = 0 the MQIM-HO also leads to the I(0, T ) behavior given in Eq. (31).
Finally, despite bismuth Bi, indium In, and antimony Sb being heavy elements, the present 1D surface metallic states do not show any detectable spin-orbit coupling effects and nor any related Rashba-split bands. In this regard, it is very important to distinguish the system studied in this paper with 1-2 monatomic layers of Bi thickness whose ARPES data were first reported in Ref. 17 from the system with a similar chemical name which was studied in Ref. 37, which refers to 5-20 monatomic layers of Bi thickness. One expects, and indeed observes, significant qualitative differences in these two systems.

B. Concluding remarks
In this paper we have discussed an extension of the MQIM-LO used in the theoretical studies of the ARPES in the line defects of MoSe 2 16 . This MQIMtype approach 6,7 accounts only for the renormalization of the leading term in the effective range expansion of the charge-charge phase shift, Eq. (4). As shown in Ref. 24, this is a good approximation if the effective range of the interactions of the c particles and the c impurity is of about one lattice spacing.
The MQIM-HO developed in this paper accounts for the renormalization of the higher terms in the effective range expansion of the charge-charge phase shift, Eq. (4). It applies to a class of 1D lattice electronic systems described by the Hamiltonian, Eq. (1), which has longer range interactions. The quantum problem described by that Hamiltonian is very involved in terms of many-electron interactions. However, we found that a key simplification is the unitary limit associated with the scattering of the fractionalized charged particles by the c impurity. We have shown a theory based on the MQIM-HO with finite-range interactions, Eq. (1), applies to the study of some of the one-electron spectral properties of Bi/InSb(001) measured at y momentum component k y = 0.2Å −1 and temperature 8 K.
Consistent with the relation of the electron and c particle representations discussed in Appendix C, the form of the attractive potential V c (x) associated with the interaction of the c particles and the c impurity at a distance x is determined by that of the electronic potential V e (r) in Eq. (1). The universal behavior of the spectral function near the singularities given in Eq. (2) whose (k, ω)-plane location refers to that of the experimentally observed high-energy ARPES peaks, is determined by the large x behavior of V c (x) shown in Eq. (5) and sum rules, Eqs. (6) and (7). Otherwise the spectral function expression in the continuum is not universal.
Despite the limited available experimental information about the ARPES peaks located on the spin branch line, we have shown that all the three criteria associated with the different types of agreement between theory and experiments considered in Sec. V B are satisfied. This provides further evidence to that given in Ref. 17 for the interplay of one dimensionality and finite-range interactions playing an important role in the one-electron spectral properties of the metallic states in Bi/InSb(001).  The phase shifts 2πΦ c,s (±2k F , q ) and 2πΦ c,c (±2k F , q) that in Eq. (A3) appear in units of 2π are defined in Eqs. (3) and (14), respectively. The bare phase shifts 2πΦ c,s (±2k F , q ) and 2πΦ c,c (±2k F , q) in the latter equations are defined below. The MQIM-HO phase shift term 2πΦ R eff c,c (k r ) in Eq. (14) accounts for effects of finite-range interactions beyond the MQIM-LO through the spectral function exponentsζ c (k) andζ c (k) in Eq. (A3).
The small momentum k 0 F s in Eq. (A3) such that k 0 F s /k F 1 is in general smaller than the momentum k * F s considered in the discussions of Sec. V C 2. It has the same role for the s band as k 0 F c for the c band, concerning the crossover between the low-energy TLL and high-energy regimes. As discussed below in Appendix B, for physical momenta k associated with creation of the c impurity at q in the small c band momentum in the absolute value interval |q| ∈ [2k F − k 0 F c , 2k F ] and creation of the s impurity at q in the small s band momentum absolute value interval |q | ∈ [k F − k 0 F s , k F ] corresponding to the low-energy TLL regime the expressions for the exponents, Eq. (A3), are not valid.
The bare energy dispersions ε c (q) and ε s (q ) in Eq. (A2) are defined as follows, ε c (q) =ε c (k(q)) and ε s (q ) =ε s (Λ(q )) wherē The distributions 2t η c (k) and 2t η s (Λ) appearing here are solutions of the coupled integral equations, The rapidity distribution functions k(q) and Λ(q ) for the c and s impurity occupancies q ∈ [−2k F , 2k F ] and q ∈ [−k F , k F ], respectively, in the arguments of the auxiliary dispersionsε c andε s in Eq. (A4) are defined in terms of their inverse functions q = q(k) where k ∈ [−Q, Q] and q = q (Λ) where Λ ∈ [−∞, ∞], respectively. The latter are defined by the equations, The parameter Q in Eqs. (A4), (A6), and (A7) is defined by the relations, Furthermore, the distributions 2πρ(k) and 2πσ(Λ) in Eq. (A7) are the solutions of the coupled integral equations, and In the u → 0 and u 1 limits the solution and the use of Eqs. (A4)-(A10) leads to the following analytical expressions for the dispersions ε c (q) and ε s (q ), respectively.
The bare phase shifts 2πΦ c,s (±2k F , q ) and 2πΦ c,c (±2k F , q) in the expressions of the phase shifts 2πΦ c,s (±2k F , q ) and 2πΦ c,c (±2k F , q) provided in Eqs. (3) and (14), respectively, are given by, where the quantities 2πΦ c,c (±r Q , r ) and 2πΦ c,s (±r Q , r ) where r Q = sin Q u are particular cases of the rapidity dependent auxiliary phase shifts 2πΦ c,c (r, r ) and 2πΦ c,s (r, r ). Those are the solution of the following integral equations, 2πΦ c,c (r, r ) = D 0 (r − r ) and 2πΦ c,s (r, r ) = − arctan sinh π 2 (r − r ) respectively, where, The choice of theξc intervals corresponding to the lines whose negative exponents ranges agree with the experimentally observed high-energy ARPES (k, ω)-plane MDC and EDC peaks in Fig. 3(e) obeys the same criterion as in Fig. 4. For l = 7, l = 9, and l = 11 such intervals whose limiting values are given in Tables IV  and V and Γ(z) is the usual Γ function.
In the u → 0 and u 1 limits the solution and the use of Eqs. (A11)-(A15) leads to the following analytical expressions for the bare phase shifts 2πΦ c,c (±2k F , q) and 2πΦ c,s (±2k F , q ), The dependence on the electronic density n e ∈]0, 1[ and interaction u = U/4t of the bare charge parameter ξ c is defined by the following relation and equation, where D(r) is given in Eq. (A15). Its limiting behaviors are, where lim u→0 Q = k F for n e ∈]0, 1[. Finally, the extended domain of relative fluctuation ∆a/ã, Eq. (11), is briefly discussed. For the physical processes of interest for the problem studied in this paper, ξ c in Eq. (A16) varies in the domain ξ c ∈]1, √ 2[. The relative fluctuation ∆a/ã in Eq. (11) where ∆a = a −ã applies though to all finite negative values ]−∞, 0[ of a/L andã/L. This refers to an extended domain ξ c ∈]1, 2[. Its new subinterval ξ c ∈ [ √ 2, 2[ corresponds to electronic potentials of the same general form, V e (0) = U/2 and V e (r) = U F e (r)/r for r > 0, as those in Eq. (1) but for which U ∈] − ∞, 0[. In the general case there are ξ c →ξ c transformations within which the function F e (r) is smoothly turned on from F e (r) = 0 atξ c = ξ c to (i) positive F e (r) > 0 and (ii) negative F e (r) < 0 values, respectively. Considering all such processes and U ∈] − ∞, ∞[ values leads to charge parameters that vary in the intervals ξ c ∈]1, 2[ andξ c ∈]1/2, 2[, respectively, for which the relations given in Eqs. (12) and (13) remain valid. The scattering length symmetry relations, a(ξ c ) = a(1/ξ c ) andã(ξ c ) =ã(1/ξ c ), then confirm that both a/L ∈] − ∞, 0[ andã/L ∈] − ∞, 0[ in ∆a/ã.

Appendix B: Relation to the TLL regime and crossover to it
Both the MQIM-LO and the MQIM-HO also apply to the low-energy TLL regime whose spectral-function exponents near the c, c , s branch lines are different from those given in Eq. (A3). In the high energy regime whose spectral function expression, Eq. (2), was used in this paper to predict the (k, ω)-plane location of the highenergy Bi/InSb(001) ARPES peaks, the velocity of the c or s impurity is different from the velocity at the c or s band Fermi points, respectively.
In contrast, in the TLL regime the (i) c or (ii) s impurity is created in its band at a momentum in one of the intervals (i) q ∈ [−2k F , −2k F + k 0 F c ] and q ∈ [2k F − k 0 F c , 2k F ] or (ii) q ∈ [−k F , −k F + k 0 F s ] and q ∈ [k F − k 0 F s , k F ]. (Here both k 0 F c /2k F 1 and k 0 F s /k F 1.) The group velocity of that impurity thus becomes that of the low-energy particle-hole excitations near the corresponding Fermi point (i) −2k F and 2k F or (ii) −k F and k F , respectively. Hence they loses their identity, as they cannot be distinguished from the c or s holes (usual holons and spinons) in such excitations.
It turns out that in the TLL regime the expressions for the γ = c, c , s exponents in the above equation lose one of the four 2∆ ι γ s. It is the 2∆ ι γ whose sign of ι = ±1 is that of the Fermi point whose velocity is the same as the γ = c or γ = s impurity velocity. The expressions of the exponents, Eq. (A3), in the high-energy spectral function expressions Eq. (2) used in the theoretical study of the Bi/InSb(001) high-energy ARPES peaks are thus different from those of the TLL regime.
In the case of a large finite system, there is a crossover regime between the high energy regime and the low-energy TLL regime within which the above quantity 2Φ ι c or 2∆ ι s is gradually removed as the energy decreases. This cross-over regime refers to (k, ω)-plane regions whose momentum and energy widths are very small or vanish in the thermodynamic limit. It is an interesting theoretical problem, but the details of its physics have no impact on the specific problems discussed in this paper.
Appendix C: Electron and c particle representations In the bare limit,ξ c = ξ c , the c particles are directly related to rotated electrons for which double occupancy is a good quantum number for u > 0. Their operators, c † j,σ =Û † c † j,σÛ ,c j,σ =Û † c j,σÛ ,ñ j,σ =c † j,σc j,σ , (C1) andñ j = σñ j,σ are generated from those of the electrons by the unitary operatorÛ = eŜ. It is uniquely defined in Ref. 40 in terms of the 4 L × 4 L matrix elements between all the model energy eigenstates.
The present quantum problem is defined in a subspace without rotated-electron double occupancy. Hencê U = eŜ merely removes the corresponding electron double occupancy from all sites around that of index j at whichc † j,σ orc j,σ acts. In that subspace, the electron double occupancy reads D = N e ne 4 f (n e , u) where f (n e , u) = ln 2 u 2 1 − sin(2πne) 2πne for u 1 and lim u→0 f (n e , u) = 1, c † j,↑ = ( 1 2 −S z j,s )f † j,c , andc † j,↓ =S + j,s f † j,c . Here f † j,c is the c particle creation operator andS z j,s andS + j,s are usual spin operators written in terms of rotated-electron operators.
The same rotated-electron basis and corresponding c particle representation can be used for theξ c < ξ c renormalized Hamiltonian, Eq. (1). The difference relative to the bare limit is that the states generated from the fractionalized particles configurations are not in general energy eigenstates but in the relevant subspaces for which the c impurity has a large lifetime, they have overlap with single energy eigenstates.
Importantly, despite the infinite number of terms in the Hamiltonian, Eq. (1), when expressed in terms of the rotated-electron operators, Eq. (C2), its relevant term for our study is that in Eq. (C4) with V e (r) replaced by rotated-electron potential V re (r) renormalized by the higher kinetic terms in the expansion, Eq. (C2). The latter are in turn renormalized by V e (r). ±1,j,ι × (ñ j+r +ñ j−r +ñ j+r+ι +ñ j−r+ι ) .
Higher kinetic terms also only involve the operatorsT 0,j,ι andT ±1,j,ι , Eq. (C3), andñ j at different relative sites. The interaction between a c particle at site j and the c impurity at site j + r refers to a Hamiltoninan term of the form −V re (r) f † j,c f j,c f j+r,c f † j+r,c . In terms of V c (x) ∝ −V re (r), it refers to suitably transformed operatorsf † j,c and corresponds to V c (x)f † j,cf j,cfj+x,cf † j+x,c 38 . The part of the V re (r) renormalization by the infinite