Universal theory of one-dimensional quantum liquids with finite-range ...

Universal theory of one-dimensional quantum liquids with finite-range interactions J. M. P. Carmelo∗ ,1, 2, 3 T. Čadež,3, 2 Y. Ohtsubo,4, 5 and S.-i. Kimura4, 5 1 Department of Physics, University of Minho, Campus Gualtar, P-4710-057 Braga, Portugal Center of Physics of University of Minho and University of Porto, P-4169-007 Oporto, Portugal 3 Beijing Computational Science Research Center, Beijing 100193, China 4 Graduate School of Frontier Biosciences, Osaka University, Suita 565-0871, Japan 5 Department of Physics, Graduate School of Science, Osaka University, Toyonaka 560-0043, Japan

arXiv:1801.07732v1 [cond-mat.str-el] 23 Jan 2018

2

The universal theory for the high-energy spectral properties of a wide class of onedimensional correlated systems based on the mobile-quantum-impurity model has not been extended to finite-range interactions. Here that problem is addressed by adding to the Hubbard model screened-Coulomb potentials of general form. A representation in terms of neutral fractionalized particles whose scattering corresponds to the unitary limit as that in shells of neutron stars reveals unexpected universality. It concerns the independence from the short-distance part of the potentials of the high-energy oneelectron spectral functions near and at the (k, ω)-plane singular features where most of the weight is located. Such systems are found to share universal properties with three-dimensional atomic scattering problems for which the potential at small distances can be replaced by a universal energy-independent boundary condition. The universal theory is successfully applied to the angle-resolved-photoemission spectroscopy in onedimensional states on a InSb(001) surface covered with Bi.

2 ∗

e-mail: [email protected]

and [email protected] The large neutron-neutron scattering length (SL) of the dilute neutron matter in shells of neutron stars1 stimulated great interest in the so called unitary limit (UL) of infinite (negative) SL, a = −∞2 . It can be realized directly in systems of trapped neutral alkali atoms3,4 , which are tuned through a Feshbach resonance5 . At low densities, the two-body s-wave interaction dominates and can be described by the effective range (ER) re expansion6 , cot δkr = − kr1 a + 12 re kr + O(kr3 ). Here δkr is the phase shift at relative wave-vector kr . In the UL all properties of the potential that are characterized by higher terms in the ER expansion become irrelevant. One-dimensional (1D) non-perturbative many-electron systems often produce surprising results7 . Some such systems are solvable by the Bethe ansatz8−11 whose S-matrix naturally factorizes into two-particle scattering problems at all densities and energy scales12,13 . The corresponding dressed phase shifts refer to fractionalized particles or their composite particles rather than to the underlying electrons. The mobile quantum impurity model (MQIM)14,15 and the pseudofermion dynamical theory (PDT)16−18 rely on representations in terms of such fractionalized particles. (For integrable models, the MQIM and the PDT describe exactly the same processes19,20 ). They have allowed to access high-energy one-particle spectral functions of 1D correlated quantum problems beyond the low-energy Tomonaga-Luttinger liquid (TLL) limit21−30 . (By “high energy” it is meant higher than for the TLL). They apply to the Hubbard model with onsite repulsion U and transfer integral t31,32 , which on a 1D lattice is often used as a toy model for the description of 1D and quasi-1D conductors26−30 . A decisive low-energy property of such systems is the suppression of the density of states (SDS) at the Fermi level24−34 . The experimental value of the SDS power-law exponent α is typically larger than 1/224−34 . In turn, α < 1/8 for the integrable 1D Hubbard model (1DHM), consistently with an α > 1/8 stemming from finite-range interactions29 . Addition to the 1DHM of finite-range interaction potentials renders it non-integrable. The MQIM applies both to integrable and non-integrable 1D correlated models14,15 . However, its extension to 1D models with finite-range interactions remains a unsolved interesting and complex problem. In this article we develop a universal renormalized theory (URT) that relies on a transformation that maps the 1DHM into 1D non-integrable lattice models with finite-range screened Coulomb potentials Ve (r). They do not refer to a simple nearest-neighbour repulsion V and are of general form Ve (r) = U/2 at electron distance r = 0 and Ve (r) = U F (r)/r for r > 0. F (r) is here a continuous decreasing function of r such that limr→0 F (r) = 1/4 and limr→∞ F (r) = 0. Otherwise, it can have different forms, specific to the material or system under consideration. (For example, F (r) = e−r/b /4 with a finite screening length b). We use a suitable representation in terms of neutral fractionalized particles. A unexpected universal form of the one-electron spectral function near and at the (k, ω)-plane singular features where most of the spectral weight is located stems from such particles scattering referring to the UL2 . The corresponding scattering problem is found to share universal properties with three-dimensional atomic scattering problems for which the finite-range attractive potential at small distances can be replaced by a universal energyindependent boundary condition35 . The URT is successfully applied to the angle-resolved-photoemission spectroscopy (ARPES) in an anisotropic InSb(001) surface covered with Bi33,34 . In contrast to integrable 1D models, for non-integrable correlated models there is no pseudofermion representation for which there is only zero-momentum forward-scattering at all energy scales. This is due to the lack of an infinite number of conservation laws12,13,36 . The universality found in the framework of the MQIM for the spectral functions of non-integrable models14,15 refers to specific (k, ω)-plane regions. They correspond both to the low-energy TLL and energy windows near the high-energy threshold and branch-line spectra singularities. In the vicinity of them, the spectral functions of non-integrable 1D correlated models are of power-law form, with negative momentumdependent exponents. This universality means that at both these (k, ω)-plane regions there is for such models a suitable representation in terms of pseudofermions that undergo only zero-momentum forward-scattering events. Their phase shifts control the spectral functions at the corresponding energy scales. The URT extends this universality to non-integrable correlated models with finite-range interactions. This includes extension of approximations for finite-range interactions that though ignore their ER, which is taken to be zero37 . This approximation is only justifiable in 1D metallic states for which the ER is around one lattice spacing, as found by the present URT for the ARPES in line defects of MoSe2 37–39 . Indeed, only in the TLL limit the fractionalized particles infinite SL alone controls all scattering properties. In the vicinity of the high-energy singular features where most of the spectral weight is located, their (k, ω)-plane distribution is found to be rather controlled by the interplay of the infinite SL and the ER of such particles interaction. There is a universal √ transformation (UT) for each chosen initial √ fixed 1DHM finite values u = U/4t ∈]0, ∞[ and ξc = ξc (u, ne ) ∈]1, 2[ where ne is the electronic density and ξc = 2Kc . Here Kc is the TLL charge parameter24−29 . Turning on adiabatically the finite-r part of the above class of potentials Ve (r), such a UT maps ξc onto a renormalized

3

0.0

Intensity 0.2 (arb. units)

k (1/Å) 0.0

-0.2

a

Intensity (arb. units)

ω (eV)

-0.1 -0.2 -0.3

Intensity (arb. units)

-0.4

0.0

c -0.4 Min.

Max

Intensity

e

0.0

s

-0.2

c c’

-0.3 -0.4 -0.2

-0.2 0.0 ω (eV)

f

-0.1

0.0 k (1/Å) Min.

ω (eV)

-0.1 ω (eV)

d

b

-0.2 -0.3

0.2

-0.4 -0.2

0.0 k (1/Å)

0.2

Max

Figure 1. Bi/InSb(001) ARPES and theoretical branch lines. a Raw Bi/InSb(001) ARPES data for hν = 15 eV. b An ARPES energy distribution curve at k = 0 (1/Å). c ARPES MDC at ω = -0.05 eV. d ARPES energy distribution curves from k = -0.16 (bottom) to +0.16 (top) (1/Å). The thick line is the normal-emission spectrum (k = 0 (1/Å)). e,f Second-derivative ARPES images. Derivation was made along momentum in e and energy in f. Circles and error bars in e indicates the MDC peaks positions. Solid and dashed lines overlaid in e are the theoretical s (red), c (light blue), and c’ (black) branch lines for uef f = 0.3, t = 1.40 eV, and electronic density ne = 0.176. Only for the solid-line k ranges in e for which such features exponents are negative in Figs. 2 and 3 can they be seen in the ARPES image. It is from the experimental data also discussed in refs. 33 and 34.

parameter ξ˜c =

p ˜ c, 2K √ ξc ∈]1, 2[ → ξ˜c ∈]1/2, 1[ and ∈]1, ξc ] .

(1)

Here α0 = (2 − ξc2 )2 /(8ξc2 ) ∈]0, 1/8[ is the 1DHM SDS exponent that is mapped onto α = (2 − ξ˜c2 )2 /(8ξ˜c2 ) ∈ [α0 , 1/8[ and ∈]1/8, 49/32[, ξ˜c = 1 and α = 1/8 are found below not to correspond to the UL, and the inequality ξ˜c > 1/2 is imposed by the phase shifts to which ξ˜c is related (see Methods equation (11)). The UT universality stems from the existence of a whole subclass of different potentials Ve (r) = U F (r)/r, all corresponding to the same fixed values of u, ξc , and ξ˜c . The regime of more physical interest is ξ˜c ∈]1/2, 1[ for which α > 1/8. For the (k, ω)-plane regions where the URT applies the spectral functions have the MQIM universal form14,15 .

4 1.0

a

1.0

l=6

ζc

ζ c0

0.5

0.0

0.0

-1.0

α = 0.050, α = 0.100, α = 0.125, α = 0.580,

0

0.05

k/π

0

0.05

c Reff c Reff c Reff c Reff

= 0.67 = 0.97 = 1.00 = 11.53

0.1

α = 0.605, α = 0.622, α = 0.650, α = 0.680,

0.15

k/π

0.2

c Reff c Reff c Reff c Reff

0.25 0

ζ c0

-0.5

= 13.90 = 15.70 = 19.01 = 23.12

0.05

k/π

l=8

ζc

0.5

-0.5

b

ζs

-1.0

α = 0.050, α = 0.100, α = 0.125, α = 0.630,

0

0.05

k/π

0

0.05

c Reff c Reff c Reff c Reff

ζs

= 0.67 = 0.97 = 1.00 = 3.97

0.1

α = 0.660, α = 0.680, α = 0.710, α = 0.740,

0.15

k/π

0.2

c Reff c Reff c Reff c Reff

= 5.01 = 5.80 = 7.15 = 8.75

0.25 0

0.05

k/π

Figure 2. Spectral function momentum dependent exponents for l = 6, 8. The exponents that control the line shape near the Bi/InSb(001) ARPES MDC peaks33,34 and corresponding theoretical c, c0 , and s branch lines, respectively, in Fig. 1e. They are here plotted as a function of the wave vector k within the URT for ueff = 0.3, ne = 0.176, and a l = 6 and b l = 8 and corresponding α values. (See Fig. 3 and Methods 4 for l = 10, 12 and odd integer l values l = 9, 11, 13, respectively). The √ black solid lines correspond to the conventional 1DHM (α0 = 0.017 and ξc = 2Kc = 1.242) and the black dashed and the c dashed-dotted lines refer to α < 1/8 and α > 1/8 values, respectively. The corresponding Reff values are also provided. The lines whose negative exponents ranges agree with the ARPES (k, ω)-plane MDC peaks in Fig. 1e whose c0 branch-line exponent crosses zero between k/π = 0 and k/π ≈ 0.07 are for l = 6 and l = 8 those referring to α = 0.605 − 0.622 and α = 0.660 − 0.680, respectively.

Only their spectra and momentum dependent exponents are renormalized under the UT relative to those of the PDT. The PDT relies on an exact representation of the 1DHM in terms of charge c and spin s pseudofermions16−18 . It is a generalization to u > 0 of a u  1 scheme40−42 that profits from the wave functions factorization43−45 . The 1DHM pseudofermions neutrality (see Methods), t, ne , and the β = c, s momentum bands occupancy configurations that generate the energy eigenstates associated with the (k, ω)-plane regions where the URT applies remain invariant under the UT. The c band is occupied in the ground state for q ∈ [−2kF , 2kF ] and unoccupied for |q| ∈]2kF , π] where 2kF = π ne . The spin s band is exotic, as its q bandwidth is q ∈ [−kF , kF ]. It is full for the ground state. The 1DHM one-electron spectral function consists of a (k, ω)-plane continuum within which well-defined singular branch lines emerge16−18 . Most of the spectral weight is located at and near such singularities whose distribution is controlled by phase shifts Φβ,β 0 (±qF β , q) of β pseudofermions at and near the β = c, s bands Fermi points ±qF c = ±2kF and ±qF s = ±kF . More generally, ±Φβ,β 0 (q, q 0 ) is a phase shift in units of 2π defined by integral equations16−18 . It is acquired by a β pseudofermion of momentum q upon scattering off one β 0 pseudofermion (+) or β 0 band hole (-) created at momentum q 0 under a one-electron excitation. The phase shifts Φβ,β 0 (q, q 0 ) for which q 6= ±qF β are among those that control the (k, ω)-plane spectral weight over the continuum, away from the singularities. They transform under the UT into mere auxiliary mathematical functions without direct physical meaning. In turn, the phase shifts ˜ β,β 0 (±qF β , q). Rather than all 1DHM β = c, s Φβ,β 0 (±qF β , q) are mapped onto renormalized physical phase shifts Φ pseudofermion scatterers whose ground state numbers are Nc = Ne and Ns = Ne /2, the corresponding URT scattering problem thus only involves a finite number NβF of them, near the ±qF β Fermi points. They constitute a low-density F nF β = Nβ /L dilute quantum liquid. For system length L  1 their momenta ±qF β ∓ δq belong to two small ranges 18 associated with the interval δq ∈ [0, πnF β ] that corresponds to the PDT energy range ω ∈ [−Ω, 0] of processes (C) . F F The precise Nβ value is not needed for the following analysis for which only that nβ → 0 in the thermodynamic limit (TL) L → ∞ yet NβF  1 matters. ˜ β,β 0 (±qF β , q) of such N F pseudofermions control the one-electron removal spectral function near The phase shifts Φ β ˜ ˜ ω) ∝ (˜ three main τ = s, c, c0 branch lines. There it has the universal power-law form B(k, ωτ (k) − ω)ζτ (k) for small (˜ ωτ (k) − ω) > 0 where ω < 0 and ω ˜ τ (k) < 0. The spectrum ω ˜ τ (k) is expressed in terms of the β = c, s energy dispersions ε˜β (q) that are uniquely defined by integral equations. The dispersion ε˜s (q) = εs (q) is not and ε˜c (q) is slightly renormalized under the UT, respectively (see Methods equation (26)). The exponent ζ˜τ (k) is expressed below in terms of pseudofermions phase shifts. The τ = s, c, c0 branch lines refer to singular features for the k ranges for which it is negative. Importantly, such k ranges continuously vary upon changing under the UT the parameter ξ˜c and related SDS exponent α values. The 1DHM U becomes under it an effective interaction Ueff = U such that ε˜s (q) = εs (q) has the same dependence on ueff = Ueff /4t for ξ˜c < ξc as on u = U/4t at ξ˜c = ξc . For ξ˜c < ξc , U 6= Ueff in Ve (r) = U F (r)/r and Ueff has

5 contributions from both onsite and finite-range interactions. The s pseudofermion phase shifts Φs,β (±kF , q) remain in˜ c,c (ι2kF , ι2kF ), Φ ˜ c,c (ι2kF , −ι2kF ), variant under the UT. The expressions of the ι = ±1 c pseudofermion phase shifts Φ ˜ ˜ and Φc,s (ι2kF , ±ιkF ) remain those of the 1DHM with ξc replaced by ξc (see Methods equation (12)). (Here and in F ˜ c,c (ι2kF , ι2kF − ιδq) for δq ∈ [ 2π , πnF ˜ the following they stand for Φ c ], Φc,c (ι2kF , −ι2kF + ιδq) for δq ∈ [0, πnc ], and L F F ˜ c,s (ι2kF , ±ιkF ∓ ιδq) for δq ∈ [0, πn ], respectively). Moreover, Φc,β (±2kF , q) for q ∈ [−qF β + πn , qF β − πnF ] Φ s β β ˜ c,β (±2kF , q) that for β = c has the form where β = c, s, and thus q ∈] − qF β , qF β [ in the TL, is mapped onto Φ c ˜ eff ˜ c,c (±2kF , q) = Φ ˜ ac,c ˜R ˜ ˜ a˜ Φ (±2kF , q) + Φ c,c (kr ). The renormalization of Φc,s (±2kF , q) and Φc,c (±2kF , q) is again found  ˜   c c 2 k R 1 eff ˜R arctan r eff sin2 (ξc −1) π (see Methods equations (19) to be controlled by that of ξ˜c whereas Φ c,c (kr ) reads π

ξ˜c

4

and (30)). ˜ c,c (±2kF , q) associated with the interaction of The ER expansion2 of the one-electron removal phase shifts −2π Φ the c pseudofermion of momentum ±2kF and the c band hole of momentum q emerging under one-electron removal (MQIM mobile quantum impurity) reads, ˜ c,c (±2kF , q)) = −1 + 1 kr Rc , lim cot(−2π Φ eff kr →0 a ˜ kr 2

(2)

F whereas limkr →0 cot(−2π Φc,c (±2kF , q)) = a−1 kr for the 1DHM. Here kr = (q ± 2kF ) such that |kr | ∈ [0, 4kF − πnc ] = ]0, 4kF [ is their relative momentum, a ˜ and a are negative SLs associated with their attractive interaction, and the ER c Reff is induced by Ve (r). (The c - c pseudofermion interaction is repulsive). One finds in the TL,

˜ c,c (±2kF , ±2kF ∓ 2π )) tan(−2π Φ L ∓ 2π L ! 2 ˜ L (ξc − 1) =− tan π = −∞ for ξ˜c 6= 1 . 2π ξ˜c

a ˜=−

(3)

Similarly, a = −∞, for the 1DHM. Hence the ξ˜ 6= 1 scattering problem refers to the UL. That NcF  1 in both 1 F 2 a| = 21 NcF tan((ξ˜c − 1)2 /ξ˜c π) and πnF πnF c |a| = 2 Nc tan((ξc − 1) /ξc π), implies that the usual dilute quantum liquid c |˜ c F c F is derived below. a| hold where Reff UL relations 0  1/(πnc )  |a| and Reff  1/(πnc )  |˜ P P ˜ι ˜ The exponents ζτ (k) = −1 + β ι 2∆β of the spectral function near the τ = s, c, c0 branch lines involve four  2 ˜ ι = ιδN F + Φ ˜ F (ιqF β ) − Φ ˜ β β (ιqF β , q) . Here Φ ˜ F (ιqF β ) = important β = c, s and ι = ±1 functionals, 2∆ τ β β,ι β β P P ˜ 0 F F F is a phase shift where δN 0 0 = δN 0 are deviations under one-electron re0 Φβ β 0 (ιqF β , ι ιqF β 0 ) δN 0 0 0 β

β ,ι ι

ι

β ,ι ι

β ,±1

moval processes of the right (ι = +1) and left (ι = −1) β 0 = c, s pseudofermion numbers that for the ground state read NβF0 ,±1 = NβF0 /2. (For the δNβF0 ,±1 values specific to the τ = s, c, c0 branch lines, see Methods equation (20)). ˜ β β (ιqF β , q) reads βτ = c for the c and c0 branch lines and βτ = s for the s branch line. Moreover, the index βτ in −Φ τ ˜ ιs = 2∆ιs = 0 for the s branch line The spin SU (2) symmetry and the invariance under the UT imply that 2∆ 2 0 ι ι ˜ ˜F and 2∆s = 2∆s = (1 + ι) /8 for the c and c branch lines, as for the 1DHM. The phase shift Φ c (ιqF c ) = P P ˜ 0 F ι ˜ 0 0 Φ (ιq , ι ιq ) δN in the two ι = ±1 nontrivial charge functionals 2 ∆ results from low-energy 0 0 0 0 cβ Fc Fβ c β ,ι ι β ι P ˜ 0 F processes within the dilute quantum liquid. For its term, Φ (ιq , ι ιq ) δN , all properties of the po0 0 c c F c F c c,ι ι ι tential that are characterized by higher-order terms in the ER expansion, equation (2), become irrelevant under ˜ c,c (ι2kF , ι2kF ) and such processes and everything is characterized by the SLs. Consistently, the expressions of Φ c ˜ c,c (ι2kF , −ι2kF ) do not involve R (see Methods equation (12)). Importantly, in the TLL limit the four functionals Φ  2 eff ι F F ˜ ˜ read 2∆ = ιδN + Φ (ιqF β ) and only that type of processes occur. In contrast, near the high-energy features β

β,ι

β

˜ c c (ι2kF , q) where most of the weight is located, additional high-energy processes associated with the phase shift −Φ ι ˜ where q ∈] − 2kF , 2kF [ also emerge in the general 2∆c expression. Under such high-energy processes, both the SL a ˜ c and the ER Reff terms in the expansion, equation (2), must be accounted for. Furthermore, the only potentially non-universal renormalization that may not follow directly from that of ξc → ξ˜c c and may depend on the specific form of the electronic potential Ve (r) refers to Reff . The form of the renormalized finite-range potential Vc (x) felt by the c pseudofermion and c band hole due to Ve (r) is for small values of their spatial distance x non-universal, as it is determined by the specific form of Ve (r) itself. The UL is behind a unexpected c universality, as Reff is found not to depend on such non-universal potential part and to involve the UL finite ratio a ˜ a

=

πnF ˜ c a πnF c a

=

˜ c,c (2kF ,2kF −2π/L)) tan(2π Φ tan(2π Φc,c (2kF ,2kF −2π/L)) .

The c pseudofermion and c band hole neutrality implies the following large-x

6 1.0

a

1.0

l = 10

ζc

ζ c0

0.5

0.0

0.0

-1.0

α = 0.050, α = 0.100, α = 0.125, α = 0.640,

0

0.05

k/π

0

0.05

c Reff c Reff c Reff c Reff

= 0.67 = 0.97 = 1.00 = 2.71

0.1

α = 0.670, α = 0.695, α = 0.730, α = 0.760,

0.15

0.2

c Reff c Reff c Reff c Reff

ζ c0

-0.5

= 3.44 = 4.16 = 5.34 = 6.55

0.25 0

k/π

l = 12

ζc

0.5

-0.5

b

ζs

0.05

-1.0

α = 0.050, α = 0.100, α = 0.125, α = 0.640,

0

k/π

0.05

0

0.05

k/π

c Reff c Reff c Reff c Reff

ζs

= 0.67 = 0.97 = 1.00 = 2.19

0.1

α = 0.674, α = 0.700, α = 0.730, α = 0.760,

0.15

k/π

0.2

c Reff c Reff c Reff c Reff

= 2.89 = 3.53 = 4.40 = 5.42

0.25 0

0.05

k/π

Figure 3. Spectral function momentum dependent exponents for l = 10, 12. The same exponents as in Fig. 2 for a l = 10 and b l = 12. (See Methods 4 for odd integer l values l = 9, 11, 13). The lines whose negative exponents ranges agree with the ARPES (k, ω)-plane MDC peaks in Fig. 1e whose c0 branch-line exponent crosses zero between k/π = 0 and k/π ≈ 0.07 are for a l = 10 and b l = 12 those referring to α = 0.670 − 0.695 and α = 0.674 − 0.700, respectively.

l 6 7 8 9 10 11 12

α 0.605 − 0.622 0.640 − 0.662 0.660 − 0.680 0.665 − 0.691 0.670 − 0.695 0.672 − 0.699 0.674 − 0.700

Rceff /a0 13.90 − 15.70 6.91 − 8.11 5.01 − 5.80 3.95 − 4.79 3.44 − 4.16 3.10 − 3.81 2.89 − 3.53

Rceff /rl 4.66 − 5.27 2.19 − 2.57 1.56 − 1.80 1.22 − 1.48 1.06 − 1.28 0.96 − 1.18 0.90 − 1.10

rl /a0 2.98 3.15 3.22 3.24 3.24 3.23 3.22

Table I. The SDS exponent α intervals and corresponding ER Rceff intervals in units of lattice spacing a0 for which there is agreement between the high-energy theoretical branch lines and the ARPES MDC peaks in Figs. 1e,f for electronic density ne = 0.176, ueff = 0.3, and l integer values l = 6 − 12. The values of the length scale rl , equation (5), in units of lattice spacing for each l = 6 − 12 and the corresponding ratio Rceff /rl are also provided. Upon increasing l from l = 6 to l = 12, the ER for which there is agreement with the experiments changes from Rceff ≈ 5rl to Rceff ≈ rl , respectively.

universal behaviour for Vc (x), Vc (x) = −

(2rl )l−2 Cc where C = . c xl µl

(4)

Here rl is an important length scale (van der Waals length for l = 6) found in Methods to have the universal form,        1 2 2 Γ 3 2 l−2 Γ l−2 l−2 3π π (l − 2)     , rl = (5) sin 1 4 l−2 2 Γ Γ 4 l−2

l−2

c in units of lattice spacing. It is closely related to Reff (see Table I), which is found to depend on Cc µl = (2rl )l−2 and not on the non-universal reduced mass µl in equation (4). Moreover, there l is an integer l > 5. The ξ˜c and l values are determined by the large-r behaviour of Ve (r).
R∞ c c For neutral particles, Reff can be found from the integral46 , Reff = 2 0 dr (ψc0 (x))2 − (ψc (x))2 . Here ψc0 (x) and ψc (x) are the solution of the Schrödinger equations at zero energy for Vc (x) = 0 and Vc (x) 6= 0, respectively (see Methods equations (32) and (34)). The URT records the a value within the boundary condition ψc (x) = ψc0 (x) = c 1 − x/a for x → ∞ whereas ψc (0) = 0 and ψc0 (x) = 1 − x/a for x ∈ [0, ∞]. The Reff integral converges for the present potentials such that limr→∞ rVe (r) = U limr→∞ F (r) = 0. This occurs provided that l > 5. As justified in Methods, c the contribution of ψc (x) at small distances to the Reff integral on the whole is negligible and vanishes in the TL. Importantly, in scattering problems of neutral particles with that property the potential at small distances can be replaced by a universal energy-independent boundary condition35 involving the ratio a ˜/a, which is found in Methods

7 to lead to the universal ER expression, c Reff

= a0

 2 !   a ˜ a ˜ , + c2 1 − c1 a a

(6)

valid for ξ˜c ∈]1/2, 1[ and α > 1/8. Here a0 is the lattice spacing whose relation to rl leads to equation (5). ξ˜c = 1/2 c c gives Reff = ∞, which is excluded for the present class of potentials whereas Reff ≈ a0 (1 − a ˜/a) for ξ˜c ∈]1, ξc ]. Since c c1 and c2 are functions of l (see Methods equation (58)), Reff only depends on ξc , ξ˜c , and l > 5. The use of the URT provides strong evidence of 1D physics and corresponding spin-charge separation47−50 at both low and high energy beyond the TLL in a 1D Fermi contour of an anisotropic InSb(001) surface covered with Bi whose −1 −1 ARPES is shown in Fig. 1. The experimentally found aπ0 ≈ 0.68 Å and kF = 2aπ0 ne ≈ 0.06 Å 33,34 give ne ≈ 0.176. The ueff value that both matches the s and c branch lines in Fig. 1e with the ARPES momentum distribution curves (MDC) peaks and leads to agreement with their (k, ω)-plane distribution is ueff = 0.3 for t = 1.40 eV and l = 6 − 12. At finite temperature the T = 0 theoretically predicted singular branch lines survive as features displaying peaks. The high-energy results do not apply at very small energy where the TLL exponents are different, as for the PDT18 . Accounting for that, one finds from analysis of the ARPES images in Figs. 1e,f that the parameter ξ˜c values for which there is agreement with the high-energy branch lines are those for which each τ = s, c, c0 exponent ζ˜τ (k) ˜ ˜ ω) ∝ (˜ in B(k, ωτ (k) − ω)ζτ (k) is negative for a specific k range. For the s, c, and c0 branch lines it refers to k ∈ [−kF + δks , kF − δks ], k ∈ [−2kF + δkc , 2kF − δkc ], and k ∈ [−δkc0 , δkc0 ]. Here δks /kF ≈ 0.15, δkc /kF is −1 vanishing or very small, and there is a δkc0 uncertainty corresponding to δkc0 ∈ [0, δk0 ]. In Fig. 1e, 2δk0 ≈ 0.10 Å is the wave-vector width of the ARPES image crossed by the c0 branch line. The corresponding high-energy singular features have no direct relation to the low-energy SDS exponent α. That the URT contains the main microscopic mechanisms behind the 1D Bi/InSb(001) metallic state would require that α values corresponding to ξ˜c values for which the high-energy singular branch lines agrees with the ARPES images are also those observed experimentally at low energy. The theoretical τ = c, c0 , s exponents ζ˜τ (k) are plotted as a function of the wave vector k for ueff = 0.3 and ne = 0.176 in Fig. 2a,b for l = 6 and l = 8 and in Fig. 3a,b for l = 10 and l = 12. (See Methods Fig. 4 for l = 7, 9, 11). The different curves are associated with different ξ˜c values and thus α values. The black solid lines refer −1 to the 1DHM for which α0 = 0.017. Note that δk0 ≈ 0.05 Å corresponds in such figures units to δk0 ≈ 0.07π. Hence concerning the α values in such figures for which there is agreement with the ARPES image MDC peaks, there is for each l value an α uncertainty whose minimum and maximum values correspond to the c0 branch line exponent curves crossing zero at k ≈ δk0 and at k ≈ 0, respectively. For l integer values l = 6 − 12 the agreement with the ARPES image MDC peaks in Fig. 1e is reached by the exponents curves referring to α ∈ [0.605 − 0.700], as given in Table I c c for each l. The corresponding Reff and Reff /rl intervals and rl values are also provided. An experimental uncertainty α = 0.65 ± 0.05 was found in ref. 33 for −ω < 0.1 eV. The astonishing agreement of the above URT predictions with such α experimental values provides strong evidence for the assignment of the spin s and charge c, c0 branch lines observed in the experiments to spin-charge separation in a 1D metal. It will be shown elsewhere that the URT leads to similar agreement with 1D metallic states ARPES of other materials and physical systems associated with quite different ne and ueff values. The URT power follows from it not referring to a mere first-neighbouring V term and rather applying to screened Coulomb potentials of arbitrary form. Its universality is revealed in terms of the scattering of fractionalized particles emerging in 1D non-perturbative systems rather than in terms of their more complex underlying many-electron interactions.

Methods Sample growth and experimental methods. The Bi/InSb(001) surface was prepared on the clean InSb(001) substrate by repeated cycles of Ar sputtering and annealing up to 680 K. On the clean InSb(001) surface, Bi was evaporated up to nominally 3 monolayers (ML): 1 ML is defined as the atom density of bulk-truncated substrate. Then, the substrate was flash-annealed up to 680 K for ∼ 10 seconds. The resulted surface showed p (1×3) low-energy electron diffraction pattern. Detailed characterization of the Bi/InSb(001) surface sample was shown in refs. 33 and 34. ARPES measurements were performed at the CASSIOPÉE beamline of synchrotron SOLEIL at hν=15 eV. The photoelectron kinetic energy at EF and the overall energy resolution of the ARPES setup was calibrated by the Fermi edge of the photoelectron spectra from Mo foils attached to the sample. The energy resolution was ∼20 meV in this work. The URT physical parameters. Both the ξ˜c value and the specific integer l > 5 in the large-distance expression of the potential Vc (x), equation (4), are determined by the corresponding large-distance behaviour of the

8 electronic potential Ve (r). Interestingly, within the new found universality the specific relation between the latter potential form and the ξ˜c value and l > 5 integer in the potential Vc (x) associated with the corresponding c pseudofermion - c band hole attractive interaction is not needed for applications of the URT to actual 1D and quasi-1D systems. For given ueff and ξc values, the result of changing l is merely gently changing the ξ˜c value and thus the α value at which agreement with the ARPES image of actual 1D systems is reached. In Table I, such α values are given concerning the agreement reached between the high-energy branch lines and the Bi/InSb(001) ARPES MDC peaks for l = 6, 7, 8, 9, 10, 11, and 12, respectively. This refers to the ueff = 0.3 and ξc = 1.242 values found to be suitable to the Bi/InSb(001) 1D metallic states. The coefficients c1 and c2 in the ER expression, equation (6), are found below to decrease from c1 = c2 = 2 at l = 6 to c1 = 1 and c2 = 1/3 for l → ∞. Since they are found to read c1 = 1.1055728 and c2 = 0.4222912 at l = 12, the ER varies very little upon further increasing l from l = 12 at constant ξ˜c , ueff and ne values. This justifies why our c study focuses on the l = 1 − 12 range. The ER is found to vary in interval Reff ∈ [0, ∞], its value being rather mostly c c controlled by that of ξ˜c . Specifically, Reff = 0 for ξ˜c → ξc and Reff = ∞ for ξ˜c → 1/2. The ξ˜c = 1/2 value is excluded from the theory because the present class of potentials does not allow an infinite ER. Analysis of the data in Table I, reveals that changing the integer l in the l = 1 − 12 range leads to small deviations in the SDS exponent α values that exactly correspond to its experimental uncertainty, α = 0.65 ± 0.5. Hence l cannot be considered a fitting parameter. The parameters whose values are chosen following the matching of the theoretical predictions and experimental data are ueff and the transfer integral t concerning the spectra and ξ˜c concerning the high-energy (k, ω)-plane peaks distribution controlled by the momentum dependent exponents. (The initial ξc value is uniquely determined by ueff and ne ). The ξ˜c = 1 value (and thus α = 1/8) refers to a singular point for which the scattering theory does not correspond to the UL. Therefore, ξ˜c = 1 and α = 1/8 are excluded from the URT. The UL occurs for both the ranges ξ˜c ∈]1/2, 1[ and ξ˜c ∈]1, ξc ]. This includes the ξ˜c = ξc starting 1DHM for u ∈]0, ∞[. The pseudofermions and their neutrality. There are two directly related representations of the 1DHM. They refer to pseudoparticles and pseudofermions, respectively18 . The pseudofermions are related to the pseudoparticles by a unitary transformation that merely shifts the latter discrete momentum values by well-defined amounts smaller than 2π/L. It preserves the discrete momentum values separation 2π/L up to O(1/L) and does not affect the β = c, s bands momentum widths. Hence if within the TL one replaces the discrete momentum values qj by continuum momentum variables q, pseudoparticles and pseudofermions carry similar momenta. In addition, the 1DHM pseudofermion dynamical theory (PDT)16,17,18 used as starting point of the URT refers to the pseudofermion representation. Hence for simplicity and except for the present note, in this article only the pseudofermion representation is used and mentioned. For the 1DHM the c and s pseudofermions are neutral particles (see Supplementary Sections I and II). This means they do not couple to charge and spin probes. Their occupancy configurations rather describe the translational degrees of freedom of the charge and spin fractionalized particles, respectively, that couple to such probes. That such a neutrality persists under the UT is an attribute of their scattering properties used below to derive the ER universal expression in equation (6). Renormalization of the 1DHM quantities. The URT is an extension to 1D non-integrable correlated lattice models with finite-range interactions of the universal theory considered in refs. 14 and 15. As such a theory, the URT is valid for energy windows near the high-energy threshold and branch-line spectra singularities in the vicinity of which the spectral function is of power-law form, with negative momentum-dependent exponents. Its dependence on the excitation spectra and exponents is the same as for that universal theory and the 1DHM PDT. The exponents have the same expression in terms of renormalized pseudofermion phase shifts as those of the PDT in terms of 1DHM pseudofermion phase shifts. The 1D non-integrable correlated lattice models under consideration have Hamiltonians with the same operator terms as the 1DHM except for the interaction term, L/2−1

VˆR =

X r=0

Ve (r)

L XXX σ

 where ρˆj,σ =

ρˆj,σ ρˆj+r,σ0

σ 0 j=1

c†j,σ cj,σ −

1 2

 .

(7)

Here both the σ and σ 0 sums are over the spin projections ↑ and ↓ and the general properties of Ve (r) have been reported above in this article.

9 We use in our analysis parameters ξc for the 1DHM and ξ˜p c for the URT that are directly related to the usual charge √ TLL parameters Kc and K˜c 21−29 as ξc = 2Kc and ξ˜c = 2K˜c , respectively. The criterion behind using ξc and ξ˜c rather than Kc and K˜c is that the former parameters are directly related to charge c pseudofermion phase shifts. The UT ξc → ξ˜c that results from adiabatically turning on the finite-distance part of Ve (r) of any of the nonintegrable models belonging to the universal class under consideration uniquely transforms the 1DHM SDS exponent α0 as follows, α0 =

(2 − ξc2 )2 (2 − ξ˜c2 )2 → α = , 8 ξc2 8 ξ˜c2

(8)

where α0 ∈]0, 1/8[ and α ∈ [α0 , 1/8[ and ∈]1/8, 49/32[, respectively. Both the electronic potential Ve (r) and the attractive potential Vc (x) felt by the c pseudofermions and c band holes ˜ c,c (±2kF , q), refer to the charge degrees of freedom. Due to the spin-charge separation involved in the phase shifts Φ that occurs at the (k, ω)-plane regions where the URT applies, the s pseudofermion phase shifts and corresponding spin s-band energy dispersions remain invariant under the UT. Within the URT such phase shifts thus read, ˜ s,s (±kF , q) = Φs,s (±kF , q) Φ ±(ξs − 1)(ξs + (−1)δq,±kF ) 2ξs ˜ s,c (±kF , q) = Φs,c (±kF , q) = − ±ξs , Φ 4 =

(9)

√ where ξs = 2. The s band energy dispersion remains that of the 1DHM (see Supplementary Section III.B). The URT parameter ξ˜c can be expressed as ξ˜c = ξ˜c (rQeff ). Here the corresponding function ξ˜c (r) obeys the modified integral equation, Z rQ eff ξ˜c ξ˜c (r) = + dr0 D(r − r0 ) ξ˜c (r0 ) ξc −rQ eff

sin Q and where rQeff = ueff Z 1 ∞ cos(ω r) D(r) = . dω π 0 1 + e2ω

(10)

Relative to the corresponding PDT integral equation (see Supplementary Section III.D), under the replacement of u by ueff only the free term has been changed from 1 to ξ˜c /ξc . Importantly, the kernel D(r) in equation (10) is the same as for the 1DHM (see Supplementary Section III.D) and thus remains invariant under the UT. It also preserves the following relations, whose form is similar to that of the 1DHM with ξ˜c replaced by ξc , ˜ c,c (2kF , 2kF ) − Φ ˜ c,c (2kF , −2kF ) ξ˜c = 1 + Φ ξ˜c ˜ c,s (2kF , kF ) − Φ ˜ c,s (2kF , −kF ) =Φ 2 1 ˜ c,c (2kF , 2kF ) + Φ ˜ c,c (2kF , −2kF ) =1+Φ ˜ ξc ˜ c,s (2kF , kF ) + Φ ˜ c,s (2kF , −kF ) . 0=Φ

(11)

The notation used here and in the following within the TL is again that for a corresponding large finite system for which ˜ c,c (2kF , 2kF ), Φ ˜ c,c (2kF , −2kF ), and Φ ˜ c,s (2kF , ±kF ) stand for Φ ˜ c,c (2kF , 2kF − δq) for δq ∈ [ 2π , πnF ], L  1 and Φ c L ˜ c,c (2kF , −2kF + δq) for δq ∈ [0, πnF ], and Φ ˜ c,s (2kF , ±kF ∓ δq) for δq ∈ [0, πnF ], respectively, Φ c s ˜ β,β 0 (q, q 0 ) = −Φ ˜ β,β 0 (−q, −q 0 ), leads Inversion of the relations, Eq. (11), combined with the use of the symmetry, Φ to the following phase shifts expressions, 2 ˜ ˜ c,c (±2kF , ±2kF ) = ± (ξ − 1) Φ 2ξ˜c

˜2 ˜ c,c (±2kF , ∓2kF ) = ∓ (ξ − 1) Φ 2ξ˜c ˜ ˜ c,s (±2kF , ±kF ) = −Φ ˜ c,s (±2kF , ∓kF ) = ± ξc . Φ 4

(12)

10 Their form is again similar to that of the 1DHM with ξc replaced by ξ˜c . Analysis of these relations reveals that such phase shifts are transformed under the UT as, Φc,c (ι2kF , ±ι2kF ) → Cc∓

ξc (ξ˜c − 1) Φc,c (ι2kF , ±ι2kF ) ξ˜c (ξc − 1)

˜ c,c (ι2kF , ±ι2kF ) =Φ ξ˜c Φc,s (ι2kF , ±ιkF ) → Φc,s (ι2kF , ±ιkF ) ξc ˜ c,s (ι2kF , ±ιkF ) =Φ Cc± =

where

ξ˜c ± 1 and ι = ±1 . ξc ± 1

(13)

˜ c,β (±2kF , q) where β = c, s and q ∈ Next we consider the renormalized c pseudofermion physical phase shifts Φ F F [−qF β + πnβ , qF β − πnβ ] for a large finite system and thus q ∈] − qF β , qF β [ in the TL. (Here qF c = 2kF and qF s = kF .) c c ˜ c eff ˜ c,c (±2kF , q) = Φ ˜ ac,c ˜R ˜ Reff For β = c, it has the form Φ (±2kF , q) + Φ c,c (kr ) where the Φc,c (kr ) expression involves Reff , c a ˜ ˜ c,c (±2kF , q) of Φ ˜ c,c (±2kF , q) is related whereas Reff = 0 for the 1DHM. Consistently, only the first phase shift term Φ to the invariant kernel D(x) appearing in equation (10). ˜ a˜ (±2kF , q) and the renormalized c pseudAs for the 1DHM, the renormalized c pseudofermion phase shift term Φ c,c ˜ ofermion phase shift Φc,s (±2kF , q) can be written as,   sin kc (q) ˜ ˜¯ a˜ ±r ˜ ac,c and Φ (±2kF , q) = Φ , Q c,c eff ueff   Λs (q) ˜¯ ˜ c,s (±2kF , q) = Φ Φ , (14) c,s ±rQeff , ueff a ˜ ˜ ˜ ¯ c,c (±rQ , r) and Φ ¯ c,s (±rQ , r) obey modified integral equations given below. respectively. Here Φ eff eff ˜ The function ξc (r), which obeys the integral equation, equation (10), can be expressed in terms of the general a ˜ ˜ ˜ ¯ c,c (r, r0 ) and Φ ¯ c,s (r, r0 ). The corresponding expressions are a generalization of those provided in auxiliary functions Φ Eq. (11) for ξ˜c = ξ˜c (rQeff ). From analysis of such general expressions and accounting for the invariance under the UT of the kernel D(r) in the integral equation, equation (10), one finds that the 1DHM free term −Dc,s (r) is transformed as,

˜ ˜ c,s (r) = − ξc Dc,s (r) where −Dc,s (r) → −D ξc   π  1 Dc,s (r) = arctan sinh r . 2π 2

(15)

Such ξ˜c (r) general expressions and D(r) invariance are in addition consistent with the following transformation property of the 1DHM free term −Dc,c (r) (see Supplementary Section III.D), d − dr

dDc,c (r) dr

D(r)

!

d =0→− dr

˜ c,c (r) dD dr

D(r)

! = 0.

(16)

In the case of the integrable 1DHM, the vanishing derivative in equation (16) results from the relation D(r) =   d dr

˜ c,c (r) dD /D(r) dr

˜ c,c (r) 1 dD A dr

dDc,c (r) . dr

= 0 implies that D(r) = where A is for |r| < 2rQeff some r independent coefficient A = A(ξ˜c ). It is a unknown function of ξ˜c that must obey the boundary condition, limξ˜c →ξc A = 1. Out of the infinite mathematical solutions for A, only one is physically acceptable. This follows from the requirement ˜ c,c (±2kF , q) must be such that for kr = q ∓ 2kF → 0 the URT one-electron removal c pseudofermion phase shift −2π Φ ˜ c,c (±2kF , q)) = −1/(˜ that cot(−2π Φ a kr ), as given in Eq. (2) for kr → 0. This reveals that A is for |r| < 2rQeff the coefficient that renormalizes Φc,c (±2kF , ±2kF ) in equation (13). The renormalization at |r| = 2rQeff is also uniquely determined by that of Φc,c (±2kF , ∓2kF ) in that equation. One then finds that the 1DHM free term −Dc,c (r) is For the URT, the vanishing derivative

11 transformed as, ˜ c,c (r) where −Dc,c (r) → −D     1 r r Γ + i Γ 1 − i 2 4 4 i   , ln  Dc,c (r) = 2π Γ 1 − i r Γ 1 + i r 2

4

4

˜ ˜ c,c (r) = Cc− ξc (ξc − 1) Dc,c (r) D ξ˜c (ξc − 1) ξc (ξ˜c − 1)2 Dc,c (r) for |r| < 2rQeff = ξ˜c (ξc − 1)2

ξc (ξ˜c − 1) Dc,c (r) ξ˜c (ξc − 1) ξc (ξ˜c2 − 1) = Dc,c (r) for |r| = 2rQeff , ξ˜c (ξc2 − 1) = Cc+

(17)

and Cc± is given in equation (13). a ˜ ˜ ˜¯ (±r ¯ c,c (±rQ , r) and phase shift Φ The phase shift term Φ c,s Qeff , r) then obey the modified integral equations, eff a ˜

˜ ¯ c,c (±rQ , r) = −D ˜ c,c (±rQ − r) Φ eff eff Z rQ eff ˜¯ a˜ (r0 , r) and + dr0 D(±rQeff − r0 ) Φ c,c −rQeff

˜ ¯ c,s (±rQ , r) = −D ˜ c,s (±rQ − r) Φ eff eff Z rQ eff 0 ˜¯ (r0 , r) , + dr D(±rQeff − r0 ) Φ c,s

(18)

−rQeff

a ˜

˜¯ ˜¯ (±r respectively. The solution of such integral equations reveals that the Φ Qeff , r) and Φc,s (±rQeff , r) renormalization c,c relative to the corresponding PDT phase shifts is, ξc (ξ˜c − 1)2 ˜ ˜ ac,c Φc,c (±2kF , q) Φ (±2kF , q) = ξ˜c (ξc − 1)2 for q ∈] − 2kF , 2kF [ ˜c ξ ˜ c,s (±2kF , q) = Φc,s (±2kF , q) Φ ξc for q ∈] − kF , kF [ . c Reff

(19) c Reff

˜ ˜ c,c (kr ) in Φ ˜ c,c (±2kF , q) = Φ ˜ ac,c ˜ c,c (kr ) considered below vanishes The additional phase-shift term Φ (±2kF , q) + Φ at kr = 0. In the case of a large finite system for which L  1, that term is absent from the expression of the ˜ c,c (±2kF , q) for q = ∓(2kF − δq) where δq ∈ [0, πnF phase shift Φ c ]. For such a system there is in a small q region for c c Reff eff ˜ c,c ˜R |q| < (2kF − πnF ) a crossover between the two regimes for which Φ (kr ) = 0 and Φ c,c (kr ) 6= 0, respectively. In c the TL it gives rise to a singular behaviour associated with its values at q = ∓2kF and for q → ∓2kF , respectively. It is of the same type as that of the 1DHM phase shift Φs,s (±kF , q) at q = ±kF and for q → ±kF , equation (9). ˜ ι and corresponding MQIM shift functions δ˜ιββτ (see Supplementary Section The β = c, s and ι = ±1 functionals 2∆ β III.C) remain having the same form under the replacement of the 1DHM parameters and phase shifts by those of the ˜ ι general expression for one-electron removal excitations, URT. In the functionals 2∆ β  2 F ˜ ιβ = ιδNβ,ι ˜F ˜ 2∆ +Φ with β (ιqF β ) − Φβ βτ (ιqF β , q) XX ˜F ˜ β β 0 (ιqF β , ι0 ιqF β 0 ) δNβF0 ,ι0 ι where Φ Φ β (ιqF β ) = β0

ι0

F F δNc,±1 = −1/2 and δNs,±1 = 0 for τ = s ,

F F F δNc,±1 = ±1/2 , δNs,+1 = −1 and δNs,−1 =0

for τ = c, c0 and q ∈] − qF βτ , qF βτ [ ,

(20)

12 the index βτ reads βτ = s for the s branch line and βτ = c for the c and c0 branch lines. (As given below, the c F and c0 branch lines differ in the corresponding subranges of q ∈] − qF c , qF c [). That the numbers deviations δNc,±1 are half-integer numbers ±1/2 is justified by c band overall q → q ± L/π momentum shifts that occur under the transitions from the ground state to the one-electron removal excited energy eigenstates. F F The use in the expression, Eq. (20), of the number deviations δNc,±1 = −1/2 and δNs,±1 = 0 suitable for the s branch line leads to,

˜ ιc = 2∆



ι ˜ c s (ι2kF , q) −Φ − 2ξ˜c

2

˜ ιs = 2∆ιs = 0 and ∆

ss ˜ss δ˜ιcs ι ˜ c s (ι2kF , q) and δι = δι = 0 . =− −Φ 2π 2π 2π 2ξ˜c

(21)

F F F For the c and c0 branch lines the use of the number deviations δNc,±1 = ±1/2, δNs,+1 = −1, and δNs,−1 = 0 in that equation gives,

˜ ιc = 2∆

!2 ξ˜c ˜ c c (ι2kF , q) and ∆ ˜ ιs = 2∆ιs = 1 (1 + ι)2 −Φ 4 8

sc ˜sc δ˜ιcc ι ˜ c c (ι2kF , q) and δι = δι = − √ = −Φ . 2π 2π 2π 2 2

(22)

The renormalized one-electron removal spectral function in the (k, ω)-plane vicinity of the τ = s, c, c0 branch lines has the universal form, ˜ ˜ ω) ∝ (˜ B(k, ωτ (k) − ω)ζτ (q) ,

(23)

P P ˜ι where ζ˜τ (k) = −1 + β ι 2∆ ωτ (k) − ω) > 0 where ω < 0. β . This expression is valid for small (˜ The excitation wave vector k and the s pseudofermion q are here related as k = −q ∈ [−kF , kF ] for the s branch line. Moreover, the wave vector k and the c pseudofermion q are related as k = −sgn{k}kF − q ∈ [−kF , kF ] and k = sgn{k}kF − q ∈ [−3kF , 3kF ] for the c and c0 branch lines, respectively.

The corresponding s branch line spectrum ω ˜ s (k) = ωs (k) involves the s band energy dispersion ε˜s (q) = εs (q) that remains invariant under the UT (see Supplementary Section III-B). The τ = c, c0 branch-line spectra ω ˜ τ (k) have the same form as for the 1DHM PDT when expressed in terms of the slight renormalized c band energy dispersion ε˜c (q) given below. Hence, ω ˜ s (k) = ε˜s (k) = εs (k) ≤ 0 k = −q ∈ [−kF , kF ] ω ˜ c (k) = ε˜c (|k| + kF ) ≤ 0 k = −sgn{k}kF − q ∈ [−kF , kF ] ω ˜ c0 (k) = ε˜c (|k| − kF ) ≤ 0 k = sgn{k}kF − q ∈ [−3kF , 3kF ] .

(24)

These spectra are plotted in Fig. 1e as a function of the wave vector k for ueff = 0.3, t = 1.40 eV, and ne = 0.176. P P ˜ι ˜ι Upon the use in ζ˜τ (k) = −1 + β ι 2∆ β of the specific values of the four β = c, s and ι = ±1 functionals 2∆β 0 given in equations (21) and (22), which are those suitable to the τ = s and τ = c, c branch lines, respectively, the exponents in the spectral-function expression, equation (23), also remain having the same form as for the 1DHM PDT

13 1.0

a

1.0

l=7

ζc

ζ c0

0.5

0.0

0.0

-1.0

α = 0.050, α = 0.100, α = 0.125, α = 0.610,

0

0.05

k/π

0

0.05

c Reff c Reff c Reff c Reff

= 0.67 = 0.97 = 1.00 = 5.50

0.1

α = 0.640, α = 0.662, α = 0.690, α = 0.720,

0.15

0.2

c Reff c Reff c Reff c Reff

0.25 0

-1.0

0.05

α = 0.050, α = 0.100, α = 0.125, α = 0.640,

0

k/π

1.0

ζ c0

-0.5

= 6.91 = 8.11 = 9.83 = 11.99

k/π

l=9

ζc

0.5

-0.5

b

ζs

0.05

0

0.05

c Reff c Reff c Reff c Reff

= 0.67 = 0.97 = 1.00 = 3.25

0.1

k/π

c

ζs

α = 0.665, α = 0.691, α = 0.720, α = 0.750,

0.15

k/π

0.2

c Reff c Reff c Reff c Reff

= 3.95 = 4.79 = 5.89 = 7.22

0.25 0

0.05

k/π

l = 11

ζc

ζ c0

ζs

0.5

0.0

-0.5

-1.0

α = 0.050, α = 0.100, α = 0.125, α = 0.640,

0

0.05

k/π

0

0.05

c Reff c Reff c Reff c Reff

= 0.67 = 0.97 = 1.00 = 2.40

0.1

α = 0.672, α = 0.699, α = 0.730, α = 0.760,

0.15

k/π

0.2

c Reff c Reff c Reff c Reff

= 3.10 = 3.81 = 4.77 = 5.86

0.25 0

0.05

k/π

Figure 4. Spectral function momentum dependent exponents for l = 7, 9, 11. The exponents in equation (25) that control the line shape near the Bi/InSb(001) ARPES MDC peaks33,34 and corresponding theoretical c, c0 , and s branch lines, respectively, in 1e. They are here plotted as a function of the wave vector k within the URT for ueff = 0.3, ne = 0.176, and a l = 7, b l = 9, and c l = 11 and corresponding α values. (See √ Figs. 2 and 3 for even integer l values). The black solid lines correspond to the conventional 1DHM (α0 = 0.017 and ξc = 2Kc = 1.242) and the black dashed and the dashed-dotted c are provided. The lines whose negative lines refer to α < 1/8 and α > 1/8 values, respectively. The corresponding ERs Reff exponents ranges agree with the ARPES (k, ω)-plane MDC peaks in 1e whose c0 branch-line exponent crosses zero between k/π = 0 and k/π ≈ 0.07 are for l = 7, 9, 11 those referring to α = 0.640 − 0.662, α = 0.665 − 0.691, α = 0.672 − 0.699, respectively.

when expressed in terms of the renormalized parameters and phase shifts. They read, !2 X ξ˜c 1 ˜ ˜ ζc (k) = − + − Φc,c (ι2kF , q) 2 ι=±1 4 k = ∈ [−kF , kF ] q = −sgn{k}kF − k ∈ [−2kF , −kF ] ; [kF , 2kF ] !2 X ξ˜c 1 ˜ c,c (ι2kF , q) ζ˜c0 (k) = − + −Φ 2 ι=±1 4 k = ∈ [−3kF , 3kF ] q = sgn{k}kF − k ∈ [−2kF , kF ] ; [−kF , 2kF ] 2 X  ι 0 ˜ ˜ + Φc,s (ι2kF , q ) ζs (k) = −1 + ˜ ι=±1 2ξc k ∈ [−kF , kF ] and q 0 = −k ∈ [−kF , kF ] .

(25)

Such exponents are plotted as a function of the wave vector k for electronic density ne = 0.176 and ueff = 0.3 in Figs. 2, 3, and 4 for l = 6, 8, l = 10, 12, and l = 7, 9, 11, respectively. The renormalization of the c band energy dispersion ε˜c (q) in equation (24) results from the finite-range attractive

14 ˜ c,c (±2kF , q). It tends to render slightly more negative the interaction associated with the renormalized phase shifts Φ energy of the c band hole at momentum q relative to the zero-energy level corresponding to the c pseudofermions at the Fermi points ±2kF . Since the c band hole emerges at momentum values that refer to the range q ∈] − 2kF , 2kF [ associated with the 1DHM particle bandwidth Wcp = εc (2kF ) − εc (0), that energy effect leads in average to a slight increase of that energy bandwidth. The overall c band energy bandwidth Wcp + Wch = 4t remains though invariant under the UT. Hence the hole bandwidth Wch = εc (π) − εc (2kF ) is slightly decreased by such effects. The renormalization of the 1DHM energy dispersion εc (q) follows for |q| < 2kF and |q| > 2kF from the renormalization of Wcp and Wch , respectively, which is addressed in the Supplementary Section IV. This gives, ε˜c (q) = (1 + γc βc ) εc (q) for q ∈ [−2kF , 2kF ]   p  Wc εc (q) for |q| ∈ [2kF , π] ε˜c (q) = 1 − γc βc Wch ε˜c (±2kF ) = 0 ,

(26)

ε˜c (q) = (1 + βc ) εc (q) for q ∈ [−2kF , 2kF ]   p  Wc ε˜c (q) = 1 − βc εc (q) for |q| ∈ [2kF , π] Wch ε˜c (±2kF ) = 0 ,

(27)

for ξ˜c ∈]1, ξc ] and,

for ξ˜c ∈]1/2, 1[. Here Wcp = |εc (0)| and Wch = 4t − |εc (0)|. As justified in the Supplementary Section IV, the parameters βc and γc appearing in equations (26) and (27) read,   1 ξc ξc − ξ˜c βc = . (28) 1− √ and γc = ξc ξc − 1 2 The renormalized ε˜c (q) expressions given in these equations and corresponding Wcp and Wch renormalization are not valid too near half filing. Their validity is restricted by the inequality, Wch > βc . 2Wcp

(29)

It ensures that the decrease of the hole c band energy bandwidth must be smaller than half that bandwidth. This condition is not obeyed very near the electronic density ne = 1. c

eff ˜R ˜ The universal ER expression. The ER appears in the phase-shift term Φ c,c (kr ) of Φc,c (±2kF , q). Such a 1 c ˜ c,c (±2kF , q)) expression given for small relative momentum term is behind the contribution 2 kr Reff in the cot(−2π Φ c eff ˜ ˜R in equation (2). From the general properties of the 1DHM phase shifts from which Φ c,c (kr ) emerges for ξc < ξ, one c c Reff Reff 1 c ˜ ˜ finds that 2π Φc,c (kr ) ∈ [−π, π] with Φc,c (kr ) = π arctan(kr Reff C) for the branch arctan(x) ∈ [−π/2, π/2] where kr = (q − 2kF ) ∈] − 4kF , 0[ and kr = (q + 2kF ) ∈]0, 4kF [. The constant C is uniquely determined by imposing that ˜ c,c (±2kF , q)) behaves as given in equation (2) in what the term in the limit of small relative momentum cot(−2π Φ  

1 2 kr

c Reff is concerned. This uniquely gives C =

1 4

sin2

c 1 eff ˜R arctan Φ c,c (kr ) = π

(ξ˜c −1)2 π ξ˜c

and thus,

c kr Reff sin2 4

(ξ˜c − 1)2 π ξ˜c

!! ,

(30)

for both kr = (q − 2kF ) ∈] − 4kF , 0[ and kr = (q + 2kF ) ∈]0, 4kF [. c expression, equation (6), that refers to the important range ξ˜c ∈]1/2, 1[ Our goal is the derivation of the universal Reff for which α > 1/8. That expression only involves the parameters ξc , ξ˜c , and l = 6, 7, .... Here l is the integer number in the large-distance expression of the potential Vc (x), equation (4). Such a distance is between the c pseudofermion ˜ c,c (±2kF , q). For the corresponding electronic potentials under and the c band hole involved in the phase shift Φ consideration for which Ve (r) = U F (r)/r for r > 0 where limr→∞ F (r) = 0, that ER must be finite. A universality occurs for some scattering problems of neutral particles for which the ER is of the general form R∞ re = 2 0 dr((ψ 0 (r))2 − (ψc (r))2 ). Here r is the distance of the two particles. It can be a 1D variable x > 0 or a radial coordinate r > 0 for 3D neutral particles. The wave function ψ 0 (r) is given in terms of the SL a by ψ 0 (r) = 1 − r/a

15 for r ∈ [0, ∞]. The wave function ψ(r) obeys in turn a Schrödinger equation at zero energy that involves the potential 2 1 d ψ(r) U (r) associated with the two neutral particles interaction. It is of general form − 2µ dx2 + U (r) ψ(r) = 0 with boundary condition ψ(0) = 0 and normalization at r → ∞ as ψ(r) = ψ 0 (r) = 1 − r/a. Here µ is a non-universal reduced mass. The potential U (r) has for large r the universal form U (r) = −C/(r/2rl )l where l > 5 is an integer number, rl is a well-defined length scale, and C = (2rl )l−2 /µ. An additional property that is behind the universality of the class of scattering problems considered here is that the contribution of the wave function ψ(r) at small distances to re is negligible. Their ψ(r) large-r expression has a √ 0 0 1 (r ) + B2 J 1 (r )). Here B1 and B2 are constants universal form in terms of Bessel functions, ψ(r) = r (B1 J l−2 − l−2 √ l−2 and the variable r0 in the Bessel functions argument reads r0 = (2 2/l R− 2)/(r/2rl )− 2 . ∞ For such problems, the large-r expression of ψ(r) can be used in 2 0 dr((ψ 0 (r))2 − (ψc (r))2 ) for all r ∈ [0, ∞] provided that U (r) is replaced at small r by an energy-independent boundary condition of universal form given below. It uniquely relates the ratio B2 /B1 to a ratio a ¯/a. Here a ¯ is a length scale whose value is non-universal yet is well-defined for each specific scattering problem. It can be viewed as a second SL. This applies to scattering problems of neutral particles whose SL absolute value can be either finite (yet parametrically large) or infinite. In the latter UL case, it is required that the ratio a ¯/a is finite, so that both 1/a = 0 and 1/¯ a=0 in the TL. Examples of scattering problems belonging to this universal class are the present UL c pseudofermion - c band hole scattering problem and the 3D atomic problem of ref. 35 for which a and a ¯ are finite yet parametrically large. For the present case of the neutral c pseudofermion and c band hole, the ER is for ξ˜c ∈]1/2, 1[ and thus α ∈ ]1/8, 49/32[ indeed given by46 , Z ∞
c Reff =2 dx (ψc0 (x))2 − (ψc (x))2 . (31) 0

where the functions ψc0 (x) and ψc (x) are such that this integral converges provided that l > 5 in equation (4). On the one hand, for the URT initial 1DHM one has that Vc (x) = 0. This initial URT boundary condition corresponds to the wave function ψc0 (x) in equation (31). Since Vc (x) = 0, it is the zero-energy solution of the Schrödinger equation for the free motion, −

1 d2 ψc0 (x) = 0. 2µl dx2

(32)

Here µl is a non-universal reduced mass. The function ψc0 (x) is of the following form for all x ∈ [0, ∞], ψc0 (x) = 1 −

x . a

(33)

On the other hand, the wave function ψc (x) in equation (31) is associated with the potential Vc (x) 6= 0 felt by the c pseudofermion and c band hole under the UT. It is induced by the electronic potential Ve (r). That wave function is the solution of a corresponding Schrödinger equation at zero energy, −

1 d2 ψc (x) + Vc (x) ψc (x) = 0 , 2µl dx2

(34)

with the boundary condition ψc (0) = 0. It is normalized at x → ∞ as ψc (x) = ψc0 (x) = 1 − x/a. The non-universal reduced mass µl also appears in the large-x Vc (x) expression, equation (4). However, it does not appear in the final ER expression. The present neutral fractionalized particles scattering problem corresponds to the extreme UL in which the SL absolute value is infinite. And this applies both to the SLs a and a ˜ such that 1/a = 0 and 1/˜ a = 0. However, their πnF a ˜

˜ tan(2π Φ

(2k ,2k −2π/L))

F F ratio aa˜ = πncF a = tan(2π Φc,c is finite. In the scattering UL under consideration, the contribution to the c,c (2kF ,2kF −2π/L)) c integral, equation (31), comes in the TL only from ψc (x) at large distances. This though implies that Vc (x) is replaced at small x by the universal energy-independent boundary condition mentioned above, which involves the ratio a ˜/a. The integral on the right-hand side of equation (31) converges provided that at large distance x the potential Vc (x) decreases faster than x−5 . At such a large distance the wave function ψc (x) is the solution of equation (34) with the attractive potential given by its large-distance asymptotic form Vc (x) = −Cc /xl , equation (4). This gives the differential equation,

d2 ψc (x) 2(2rl )l−2 + ψc (x) = 0 . dx2 xl

(35)

16 In the following this equation can be used for all x ∈ [0, ∞] with the potential Vc (x) replaced at small distances x by the universal energy-independent boundary √ √condition. Introducing a new function Jc = ψc (x)/ x and expressing it in terms of the above suitable variable, y = (2 2/l − l−2 2)/(x/2rl )− 2 , equation (35) is transformed into the Bessel equation, d2 Jc (y) 1 dJc (y) + dy 2 y dy   1 Jc (y) = 0 . + 1− (l − 2)2 y 2 The general solution of this equation leads to a function ψc (x) of the form,  √  1 (x) + B2 φ 1 (x) ψc (x) = x B1 φ l−2 , − l−2

(36)

(37)

1 (x) in terms of the above variable where B1 and B2 are x independent constants. Upon expressing the functions φ± l−2 1 y, they equal a Bessel function J± l−2 (y),

√  − l−2 ! 2 2 2 x . l − 2 2rl

1 (x) ≡ J 1 φ± l−2 ± l−2

(38)

The use in the expression, equation (37), of the following asymptotic behaviour of the Bessel functions for x  1 and thus y  1, 1

φ

1 l−2

(x) = J

1 l−2

y l−2   (y) ≈ 1 l−1 2 l−2 Γ l−2 1

1 (x) = J 1 (y) ≈ φ− l−2 − l−2

Γ

2 l−2   l−3 l−2

1

,

1

(39)

y l−2

leads to, √

ψc (x) ≈ 

1  √  l−2

2 l−2

2rl

Γ 

× 1 +

x 2rl





l−1 l−2

 B1

   2 l−1  l−2 Γ l−2 B2 l−2 ,   √ B1 2 Γ l−3 l−2

(40)

for large x. The normalization at x → ∞ as ψc (x) = ψc0 (x) = 1 − x/a then requires that, 1 B1 = √ 2rl



l−2 √ 2

1  l−2

 Γ

l−1 l−2

 ,

(41)

and B2 =

B20

√

2rl =− a

1 √ ! l−2   2 l−3 Γ . l−2 l−2

(42)

√ − l−2 2 , of the The use in ψc2 (x) × dx dy when expressed in terms of the above variable, y = (2 2/l − 2)/(x/2rl ) 0 2 2 mathematical relations provided in the Supplementary Section V.A allows the development of (ψc (x)) − ψc (x) in the following form, (ψc0 (x))2 − ψc2 (x) = gcvirtual (x) + gc (x) .

(43)

gcvirtual (x) = (ψc0 (x))2 − fc (x) ,

(44)

Here,

17 where,  2 x d { [B12 φ2 1 (x) fc (x) = (2rl ) l−2 dx 2rl 1 (x) φ 1 (x) + B1 B2 φ l−2 − l−2 2

+

B22 2 φ 1 (x)]} 3 − l−2

(45)

and  gc (x) =

x 2rl

− (l−2) +1 2

√ 1 (x) φ 1 4 2 rl {B12 φ l−2 (x) l−2 +1

B1 B2 1 1 (x) φ 1 1 (x) φ (x)] [φ l−2 − l−2 +1 (x) + φ− l−2 l−2 +1 2 B2 1 1 (x) φ + 2 φ− l−2 − l−2 +1 (x)} . 3

+

(46)

The divergences all appear in the functions (ψc0 (x))2 and fc (x). In the case of fc (x) and irrespectively on whether |a| is finite or infinite, such divergences are singled out within the three surface terms. Those are the first terms on the right-hand side of the Supplementary Section V.A three equations. As for scattering problems for which the SL is finite, the constants B1 and B2 , equations (41) and (42), respectively, in the expression of fc (x) are uniquely determined by the requirement of theRdivergences from fc (x) and (ψc0 (r))2 exactly canceling each other under the ∞ integration in equation (31), so that 2 0 dx gcvirtual (x) = 0. Here gcvirtual (x) = (ψc0 (x))2 − fc (x), equation (44). The first term in the integral of the function gcvirtual (x), Z ∞ dx gcvirtual (x) = 2 0 Z ∞
2 dx (ψc0 (x))2 − fc (x) , (47) 0

can be written as, Z 2 0

∞

dx (ψc0 (x))2

1 x3 x2 + = lim 2 x − x→∞ a 3 a2 

 .

(48)

The expressions of B1 and B2 given in equations (41) and (42) have been inherently constructed to ensure that the normalization at x → ∞ of ψc (x), equation (40), is ψc (x) = ψc0 (x) = 1 − x/a, so that,   Z ∞ 1 x3 x2 , (49) 2 + dx fc (x) = lim 2 x − x→∞ a 3 a2 0 R∞ which exactly equals 2 0 dx(ψc0 (x))2 , equation (48). For completeness and for the sake of consistency, in the Supplementary Section V.B such a behaviour is confirmed in terms of the function fc (x), equation (45). However, in the present UL for which a → −∞ the ratios x2 /a and x3 /a2 are nor well defined. The correct limits to be taken are, Z ∞ Z ∞ 0 2 2 dx (ψc (x)) = 2 dx fc (x) 0 0   1 x3 x2 = lim lim 2 x − + = lim 2x , (50) x→∞ x→∞ a→−∞ a 3 a2 which leads again to the exact vanishing of the integral, equation (47). The ER in equation (31) can then be calculated for ξ˜c ∈]1/2, 1[ and thus α > 1/8 as, Z ∞ c Reff =2 dx gc (x) .

(51)

0

Both for 1/a finite and in the UL, 1/a = 0, the function gcvirtual (x) = (ψc0 (x))2 − fc (x), equation (44), has a virtual character as it does not contribute to the ER, equation (51). The only role of the function fc (x) in ψc (x) = p fc (x) − gc (x) is to cancel (ψc0 (x))2 in the large x limit. There is though a qualitative difference between the scattering

18 problems for which 1/a is finite and vanishes, respectively. For the former problems, the constants B1 and B2 to be used in the expression of gc (x) must be the same as those used in the expression of fc (x). On the one hand, this also applies to the present UL in what concerns the finite constant B1 . On the other hand, in the UL the constant B2 vanishes for fc (x). The use of the mathematical solution B2 = 0 in the expression of gc (x) has though no physical meaning in the UL if the ratio a ˜/a is finite. That would provide a unphysical constant ER, independent of that ratio and thus independent ˜ of ξc , ξc , ueff , and ne . In the UL there are though other mathematical solutions in the case that a ˜/a is finite. This follows from in equation (50), limx→∞ lima→−∞ 2(x−x2 /a+ 31 x3 /a2 ) = limx→∞ 2x. Hence only 2x stemming from the term 1 in ψc0 (x) = 1 − x/a, equation (33), must be cancelled by fc (x). The existence of several mathematical solutions stems for any SL asl such that asl → −∞ doing it in limx→∞ limasl →−∞ 2(x − x2 /asl + 13 x3 /a2sl ) = limx→∞ 2x. The main issue is thus to choose the mathematical solution that is physical. A first requirement is mathematical. It is that the constant B2 has for fc (x) and gc (x) the same general expression in terms of a SL asl whose unique inverse value is 1/asl = 0 for both fc (x) and gc (x). However, the common required 1/asl → 0 limit can be reached under the use of different physically acceptable mathematical boundary conditions suitable to each of these functions. That suitable to the function gc (x) must account for the energy-independent boundary condition that replaces Vc (x) at small distances x. The physically acceptable mathematical solution for the constant B2 suitable to the present UL for which a ˜/a is finite has the general form,   a ˜ B  1  where B2 = − asl cos π l−2

1 1 1  = 0 for fc (x) and =  = ˜ asl afc a a˜aeff 1 1 (52) = = 0 for gc (x) . asl a On the one hand, for fc (x) the equality of this B2 expression to that given in equation (42) is reached provided that afc = a (˜ a/˜ aeff ) where,   B0 π a ˜eff = −a 2 cos B1 l−2   2 √ ! l−2 l−3   Γ l−2 π 2  . = 2rl cos (53) l−2 l−2 Γ l−1 l−2 The constant B20 is here that given in equation 42. On the other hand, the boundary condition that replaces Vc (x) at small distances x is universal for the class of scattering problems considered above. It is such that for the function gc (x) one must use the actual SL asl = a = −∞ in equation (52), which gives,   a ˜ B  1  6= B20 . B2 = − (54) a cos π l−2  
a ˜ π 2 It is confirmed below that such a universal boundary condition is indeed B cos B1 l−2 = − a . This is why it requires in the UL that a ˜/a is finite and thus 1/˜ a = 0 in the TL. The use in the gc (x) expression, equation (46), of the constant B1 , equation (41), and of the constant B2 , equation (54), leads then to,  − (l−2)    2 +1 √ 2 x l − 2 l−2 l−1 √ gc (x) = 2 2 Γ2 2rl l−2 2 1 1 × {φ l−2 (x) φ l−2 +1 (x)   φ 1 (x) φ 1 1 (x) φ 1 (x) − l−2 +1 (x) + φ− l−2 a ˜ l−2 l−2 +1   − π a 2 cos l−2  2 φ 1 (x) φ 1 − l−2 − +1 (x) a ˜  l−2  + }. (55) π a 3 cos2 l−2

19 In what the function gc (x), equation (55), is concerned there is an additional requirement for it to be physically acceptable. It must be the part of a corresponding general wave function ψc (x) that obeys the Schrödinger equation at zero energy, equation (35). Moreover, the function gc (x) must be the only part of that wave function that contributes to the ER, equation (51). Its remaining part is therefore virtual in what that ER is concerned. This requirement reveals that concerning the function gc (x) the choice of the boundary conditions behind equation (54) maps the UL scattering problem pinto an effective 1/aeff finite scattering problem. Specifically, its solution is a wave function of the form ψc (x) = feff (x) − gc (x) that obeys the Schrödinger equation at zero energy, equation (35), and whose function gc (x) remains being that given in equation (55). The goal of feff (x) is merely cancelling in 0 the large x limit an effective wave function ψeff (x) = 1 − x/aeff that plays the role of that in equation (33). Its SL aeff is finite and reads aeff = a (˜ aeff /˜ a) with a ˜eff provided in equation (53). Hence the use in the UL of the universal energy-independent boundary condition that replaces Vc (x) at small x 0 is equivalent to a transformation whose mappings ψ 0 (x) = 1 − x/a → ψeff (x) = 1 − x/aeff and a ˜ → a ˜eff involve eff the 1DHM Vc (x) = 0 function ψ 0 (x) in equation (33) and the SL a ˜. The important point is that the ratio aa˜eff = R ˜ ∞ tan(2π Φ (2k ,2k −2π/L)) c,c F F a ˜ 0 2 a = tan(2π Φc,c (2kF ,2kF −2π/L)) remains invariant under it. The corresponding obtained integral 2 0 dx((ψeff (x)) − 0 feff (x)) = p 0 again vanishes. The only role of the effective functions feff (x) and ψeff (x) is indeed to ensure that ψc (x) = feff (x) − gc (x) obeys the Schrödinger equation at zero energy, equation (35). Before calculating the ER,  we confirm that the universal energy-independent boundary condition under considera
B2 π tion indeed reads B1 cos l−2 = − aa˜ , as given above. This universal boundary condition also applies to scattering problems of neutral particles belonging to the above universal class whose SL is finite. It is easier to directly extract it from them. One starts by accounting for the parameters that describe the non-universal short-distance part of the corresponding potential curve. However, in the end of the suitable procedures to reach the universal energy-independent boundary condition such parameters disappear and that boundary condition naturally emerges. We consider the atomic scattering problem studied in ref. 35 mentioned above that refers to neutral atoms. It has all reported general properties of the universal class os scattering problems under consideration. While 1/a = 0 and 1/˜ a = 0 in the case of the present scattering problem, the corresponding SLs a and a ¯, respectively, of the neutral atoms considered in that reference are finite yet parametrically large. The short-distance part of that scattering problem atomic potential curve determines the actual magnitude of a non-universal zero-energy phase denoted by Φ. From the use of equations (4) and (16) of that reference, one finds that such a zero-energy phase Φ obeys the equations,

 tan Φ −

 π and l−2   π   B + B cos 1 2 l−2 π A   tan Φ − = = π 2(l − 2) B B2 sin l−2 π 2(l − 2)



a = 1− cot a ¯ 



(56)

    π π respectively. The use of the relations A = B1 + B cos l−2 and B = B2 sin l−2 of the constants A and B of ref. 35 to the constants B1 and B2 used here, reveals that the equality of the two expressions on side of  the  right-hand
π a ¯ 2 equation (56) leads indeed to the universal boundary condition under consideration, B cos = − . B1 l−2 a Consistently with its universality, it is independent of the non-universal phase Φ associated with the specific shortdistance part of the atomic potential curve under consideration. Moreover, it is independent of the non-universal individual values of a and a ¯, respectively. Such a universal boundary condition indeed applies to all scattering problems of neutral particles that obey the above general properties. This applies independently of the specific form of the corresponding potential short-distance part and of the a and a ¯ values provided that a ¯/a is finite. c In order to derive the universal expression of the ER Reff , equation (6), which is valid for ξ˜c ∈]1/2, 1[ and thus

20 α ∈]1/8, 49/32[, the gc (x) expression, equation (55), is used in equation (51). This gives,    2 √ 2 l−1 l − 2 l−2 c √ Reff = 2 2 Γ l−2 2  − l−2 Z ∞ +1 2 x 1 (x) φ 1 φ l−2 ×{ dx − l−2 +1 (x) 2rl 0  − l−2  Z ∞ 2 +1 x a ˜ dx − a 2rl 0 1 (x) φ 1 1 (x) φ 1 φ l−2 (x) − l−2 +1 (x) + φ− l−2 l−2 +1   × π 2 cos l−2  2 Z ∞  − l−2 2 +1 a ˜ x + dx a 2rl 0 1 (x) φ φ− l−2 − 1 +1 (x)  l−2  × }. π 3 cos2 l−2 c After performing the integrations, one finally reaches the Reff expression, equation (6), where ˜ c,c (2kF ,2kF −2π/L)) tan(2π Φ tan(2π Φc,c (2kF ,2kF −2π/L))

and the constants c1 and c2 are found to read,       2 l−1 l−4 Γ Γ Γ l−2 l−2 l−2 2(l − 2)       c1 = l−3 π 1 2 cos l−2 Γ l−2 Γ l−2     2 Γ l−2 Γ l−4 l−2 2       and = π 1 cos l−2 Γ l−2 Γ l−3 l−2       3 l−1 l−5 2 Γ l−2 Γ l−2 Γ l−2 (l − 1)       c2 = π 1 Γ l−3 cos2 l−2 Γ3 l−2 l−2     3 l+1 Γ Γ − l−2 l−2 3 (l + 1) 1      , = (l − 1) cos2 π Γ − 1 Γ − l−1 l−2

l−2

(57)

a ˜ a

=

πnF ˜ c a πnF c a

=

(58)

l−2

respectively. They are functions of l that decrease from c1 = c2 = 2 at l = 6 to c1 = 1 and c2 = 1/3 for l → ∞. Moreover, the overall coefficient a0 on the right-hand side of equation (6) is the lattice spacing whose relation to the length scale rl is found to be given by,   1   l−2     2 1 4 Γ Γ 2 (l−2) l−2 l−2 2        . a0 = 2rl  (59)  2 3 3π sin π 2 Γ Γ l−2 l−2 l−2 This expression contains important physical information. Its inversion gives the expression, equation (4), of rl in units of lattice spacing. For l = 6, rl is the well known van der Waals length. Its l dependence provided in equation (4) is another universal result of the present analysis. Finally, the ER expression used for the regime for which α ∈ [α0 , 1/8[ and thus ξ˜c ∈ [ξc , 1[ reads,   a ˜ c Reff = a0 1 − . (60) a c This linear behaviour in the ratio a ˜/a respects the physically required boundary conditions limξ˜c →ξc Reff = 0 and c limξ˜c →1 Reff = a0 . This is though a problem of little physical interest for the present study. Indeed, for α < 1/8 the ER is smaller than the lattice spacing. Low-energy TLL α values such that α < 1/8 can actually be reached by the 1DHM itself.

21  
B2 π cos l−2 = − aa¯ , justifies the direct relation to the atomic The universality behind the boundary condition B 1 scattering ER expression obtained in ref. 35 to that given in equation (6). The former is expressed in terms of quantities defined in that reference as re = Fn − Gn /(Fn a) + Hn /(Fn a2 ). If one maps Fn onto the lattice spacing of the present problem and the quantities Gn /(Fn a) and Hn /(Fn a2 ) are expressed in terms of the ratio a ¯/a of ref. 35, one confirms the direct relations Gn /(Fn a) = c1 (¯ a/a) and Hn /(Fn a2 ) = c2 (¯ a/a)2 . Here c1 and c2 have exactly the universal form, equation (58), under the replacement of n by l. This is in spite of the very different physics within which the SLs a ¯ and a are for the atomic scattering problem finite (yet parametrically large) and Fn has no relation whatsoever to a lattice spacing. Limits of the URT validity. As for the 1DHM momentum dependent exponents expressions18 , those of the s, c, and c0 branch-line exponents given in equation (25) are not valid at the low-energy limiting values at which the excitation wave vector in the spectral-function expression, equation (23), reads k ≈ ±kF . While in the TL this refers only to k = ±kF , for finite-size systems it corresponds to two small low-energy regions in the vicinity of k = ±kF . Both this property and the positivity of the s branch exponent for α ∈ [0.60, 0.70] in these momentum regions, specifically k ∈ [−kF , −kF + δks ] and k ∈ [kF − δks , kF ] where δks /kF ≈ 0.15, are consistent with the lack of low-energy spectral weight in the ARPES images shown in 1e,f. Concerning the validity of both the spectral functions expressions in equation (23) for the URT and that for the 1DHM PDT (see Supplementary Section III.A), when for an electron removal spectral function τ branch line there is for (˜ ω (k) − ω) < 0 (ii) no spectral weight and (ii) a very small amount of weight, they are (i) exact and (ii) a very good approximation for an energy window corresponding to a small energy deviation (˜ ω (k) − ω) > 0 from the high-energy branch-line spectrum ω ˜ (k). In the present case, the electron removal spectral function expressions are exact for the s branch line and a very good approximation for the k ranges of the c and c0 branch lines for which the corresponding power-law exponents in equation (25) are negative, respectively. This though does not apply concerning the vicinity of the c branch line in the two above small low-energy regions near k = ±kF , again consistently with the lack of low-energy spectral weight in such low-energy regions in the ARPES images of 1e,f. Some 1D quantum liquids with finite-range interactions have broken symmetry ground states associated with charge density waves. However, their metallic states emerge at low temperatures. In such cases, the analysis of this paper refers to the corresponding metallic states. Those are the states suitable for the description of the 1D metallic states that refer to the ARPES images at finite temperature. References

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Acknowledgments This work is dedicated to the memory of the late Adilet Imambekov whom J. M. P. C. acknowledges for discussions that were very helpful in producing it. We thank Jan Zaanen for useful discussions. Y. O. and S.-i. K. thank the kind support and illuminating discussions to obtain and analyze the ARPES dataset by Patrick Le Fèvre, François Bertran, and Amina Taleb-Ibrahimi. J. M. P. C. and T. Č. acknowledge the support from NSAF U1530401 and computational resources from CSRC (Beijing), the Portuguese FCT through the Grant UID/FIS/04650/2013, and the NSFC Grant 11650110443. Author contributions The theoretical description has been conceived by J. M. P. C. with important support from T. Č., who performed the needed numerical calculations, and the corresponding theoretical analysis was carried out by J. M. P. C. and T. Č.. Y. O. and S.-i. K. have provided experimental data beyond those reported in Refs. 33 and 34 and contributed to the clarification of key scientific issues concerning the interplay of the theory and experiments. All authors contributed to the scientific discussion, contributed to and agreed on the manuscript. Additional information Supplementary Information accompanies this paper. Competing financial interests The authors declare no competing financial interests.

24

Supplementary Information The universal renormalized theory (URT) results from a universal transformation (UT) that has as starting point the 1D Hubbard model (1DHM). In this Supplementary Information results on the 1DHM useful for our studies are provided. Moreover, the slight renormalization of the particle and hole c band energy bandwidths by the finite-range interactions is a related issue also addressed in the following. Indeed, the corresponding small deviations of such energy bandwidths are limited or even prevented by several properties of the 1DHM c band energy dispersion itself that remain invariant under the UT. Finally, some useful mathematical side results are also provided.

I.

THE PSEUDOFERMIONS AS NEUTRAL PARTICLES

We consider first the 1DHM in its complete Hilbert space spanned by 4L energy eigenstates. Here we have used units of lattice spacing one for which L  1 denotes both the number of lattice sites and the lattice length. The corresponding general analysis goal is to identify the non-neutral fractionalized particles that couple to charge and spin probes and those that are neutral. This is needed to confirm that the scattering processes under consideration in our study refer to neutral fractionalized particles. The relation of the 1DHM pseudofermion representation to the electrons involves as an intermediate step the rotated electrons. The electron - rotated-electron unitary operator is uniquely defined by its matrix elements between all 4L 1DHM energy eigenstates1 . Rotated electrons are inherently constructed to their numbers of singly occupied sites and doubly occupied sites being good quantum numbers for u = U/4t > 0. (Here t is the transfer integral and U the onsite repulsion). For electrons this applies only in the u−1 → 0 limit. Rotated electrons have for u > 0 the same charge and spin 1/2 as the electrons. Since they become electrons in the u−1 → 0 limit and thus u → ∞ limit, they are quasiparticles of a type of turned upside-down Fermi liquid. An energy eigenstate η-spin projection and spin projection are given by Sηz = − 21 (L − Ne ) and Ssz = − 21 (Ne↑ − Ne↓ ), respectively. Here Ne = Ne↑ + Ne↓ and Neσ is the number of spin-projection σ =↑, ↓ electrons. It equals that of σ rotated electrons. An energy eigenstate η-spin and spin are denoted in the following by Sη and Ss , respectively. The rotated-electron occupancy configurations that generate the exact u > 0 1DHM energy eigenstates naturally decouple into fractionalized particles occupancies1 : A doubly occupied site gives rise to one η-spin 1/2 of projection −1/2 plus one neutral c band hole. A unoccupied site gives rise to one η-spin 1/2 of projection +1/2 plus one neutral c band hole. A site occupied by one rotated electron of spin projection ±1/2 gives rise to one spin 1/2 of projection ±1/2 plus one neutral c pseudofermion. Hence the spins 1/2 are those of the rotated electrons that singly occupy sites. The η-spins 1/2 with projections +1/2 and −1/2 refer to the η-spin degrees of freedom of the rotated-electron unoccupied and double occupied sites, respectively. The numbers (i) Nc of c pseudofermions and (ii) Nch = L − Nc of c band holes equal those of (i) rotated-electron singly occupied sites and (ii) rotated-electron unoccupied plus doubly occupied sites, respectively. General u > 0 energy eigenstates are populated by Nc c pseudofermions whose momentum band has Nch holes, Lη = Nch η-spins 1/2, and Ls = Nc spins 1/2. There are Mη = 2Sη unpaired η-spins 1/2 and Ms = 2Ss unpaired spins 1/2. The remaining even number Lη − Mη of η-spins 1/2 and Ls − Ms of spins 1/2 are paired within Lη /2 − Sη η-spin1 singlet pairs and P Ls /2 − Ss spin-singlet pairs, respectively . Within the u > 0 1DHM Bethe-ansatz solution, one has P∞ ∞ that Lη − Mη = n=1 2n Nηn and Ls − Ms = n=1 2n Nsn . The latter expressions are valid in the thermodynamic limit (TL), within which Nηn and Nsn are the numbers of charge and spin strings of length n = 1, 2, ..., ∞, respectively. For general u > 0 energy eigenstates, the number Ms,±1/2 of unpaired spins 1/2 of projection ±1/2 and Mη,±1/2 of unpaired η-spins 1/2 of projection ±1/2 are good quantum numbers. They read, Mα,±1/2 = (Sα ∓ Sαz ) and Mα = Mα,−1/2 + Mα,+1/2 = 2Sα where α = η, s .

(S1)

The set of an energy eigenstate Lη /2 − Sη η-spin-singlet pairs and Ls /2 − Ss spin-singlet pairs contains an equal number of η-spins 1/2 and spins 1/2, respectively, of opposite projection. Hence the total number Lα,±1/2 of η-spins 1/2 of projection ±1/2 (α = η) and spins 1/2 of projection ±1/2 (α = s) is given by, Lα,±1/2 =

1 (Lα ∓ 2Sαz ) where α = η, s . 2

(S2)

The unpaired η-spins 1/2 are the fractionalized particles that couple to charge probes. In addition, the unpaired spins 1/2 are the fractionalized particles that couple to spin probes. This is confirmed in the following. A side important result is the neutral character of the 1DHM c pseudofermions and c band holes, which remains invariant under the UT.

25 The 1DHM in a uniform vector potential Φ/L remains exactly solvable by the Bethe ansatz. The flux Φ reads Φ = Φ↑ = Φ↓ and Φ = Φ↑ = −Φ↓ for coupling to the charge/η-spin and spin degrees of freedom, respectively. One first considers lowest weight states (LWSs) of both the η-spin and spin SU (2) algebras for which Sη = −Sηz and Ss = −Ssz . The LWSs momentum eigenvalues, P (Φ↑ , Φ↓ ), are found in the TL to have the general exact form, P (Φ/L) = P (Φ↑ /L, Φ↓ /L) = P (0) + (Nc −

X n

2n Nsn )

X Φ↑ + Φ ↓ Φ ↑ − Φ↓ − (Nch − 2n Nηn ) 2L 2L n

Φ ↑ − Φ↓ Φ↑ + Φ ↓ = P (0) + Ms − Mη . 2L 2L

(S3)

Here P (0) is the LWSs Φ = 0 momentum eigenvalue whose form1 is unimportant for the present analysis. The non-LWSs can be generated from the LWSs by application onto the latter of off-diagonal generators of the η-spin and spin SU (2) algebras1 . One then finds that for general u > 0 energy eigenstates, which includes both LWSs and non-LWSs, equation (S3) becomes, P (Φ/L) = P (0) + (Ms,+1/2 − Ms,−1/2 )

Φ↑ − Φ ↓ Φ↑ + Φ ↓ − (Mη,+1/2 − Mη,−1/2 ) . 2L 2L

(S4)

The charge and spin current operators expectation values of the Φ → 0 and u > 0 energy eigensates can then be derived from the Φ/L dependence of the energy eigenvalues E(Φ/L). Specifically, hJˆηz i = dE(Φ/L)/d(Φ/L)|Φ=Φ↑ =Φ↓ =0 and hJˆsz i = dE(Φ/L)/d(Φ/L)|Φ=Φ↑ =−Φ↓ =0 , respectively. Moreover, dP (Φ/L)/d(Φ/L)|Φ=Φ↑ =Φ↓ =0 gives the number of charge/η-spin fractionalized particles and dP (Φ/L)/d(Φ/L)|Φ=Φ↑ −Φ↓ =0 that of spin fractionalized particles that couple to the vector potential Φ/L. The use of the exact momentum eigenvalues, Eqs. (S3) and (S4), then confirms that such fractionalized particles are the Mη = 2Sη unpaired η-spins 1/2 and Ms = 2Ss unpaired spins 1/2, respectively. The c momentum band neutral pseudofermions and neutral holes occupancy configurations merely describe the translational degrees of freedom of the unpaired η-spins 1/21 .

II.

THE SUBSPACE OF INTEREST FOR THE PRESENT SCATTERING PROBLEM

The one-electron spectral function near and at the singular features refers to a quantum problem that corresponds to the 1DHM in a well defined subspace. It is spanned by LWSs of the η-spin and spin SU (2) algebras. Furthermore, the u > 0 energy eigenstates that span such a subspace are no populated by paired η-spins 1/2. The number of unpaired η-spins 1/2 of projection +1/2 then reads Mη = L − Ne . The states under consideration are in addition populated by Ls = Ne↑ spins 1/2, of which Ne↓ are paired within Ns = Ne↓ /2 spin-neutral composite s pseudofermions. (A s pseudofermion is the simplified designation used here and in the article for a s1 pseudofermion of ref. 1). The internal structure of a neutral s pseudofermion thus corresponds to a single spin-singlet pair. The remaining Ms = Ne↑ − Ne↓ spins 1/2, if any, are unpaired spins 1/2 of projection +1/2. Moreover, the u > 0 energy eigenstates that span the above subspace are also populated by Nsh = 2Ss s band holes, Nc = Ne c pseudofermions, and Nch = L − Ne c band holes. For the 1DHM in its complete Hilbert space, the numbers of unpaired η-spins 1/2 and unpaired spins 1/2 are in general different from those of c and s band holes, respectively. P∞ In the TL the former numbers P∞read Mη = 2Sη and Ms = 2Ss whereas the latter are given by Nch = 2Sη + n=1 2n Nηn and Nsh = 2Ss + n=2 2(n − 1) Nsn , respectively. However, in the present 1DHM subspace such numbers are equal. This could suggest that the c and s band holes contain the η-spin and spin internal degrees of freedom of the unpaired η-spins 1/2 and unpaired spins 1/2, respectively. If so, they would not be neutral particles. In contrast, the occupancy configurations of the c and s pseudofermions in their momentum bands only describe the translational degrees of freedom of the Mη = 2Sη unpaired η-spins and Ms = 2Ss unpaired spins, respectively. The zero spin-density ground states of interest for the scattering problem studied in this paper have an even number Ne0 of electrons and rotated electrons. For them Sη = −Sηz = 21 (L − Ne0 ) and Ss = −Ssz = 0. Hence such ground sates are occupied by Mη,+1/2 = L − Ne0 η-spins 1/2 of projection +1/2, Nc = Ne0 c pseudofermions, Nch = L − Ne0 c band holes, and Ns = Ne0 /2 s pseudofermions. They are not populated by unpaired spins 1/2. The one-electron removal spectral weight distribution near the (k, ω)-plane branch lines is controlled by transitions to excited u > 0 energy eigenstates with Mη = L−Ne0 +1 η-spins 1/2 of projection +1/2, Nc = Ne0 −1 c pseudofermions, Nch = L − Ne0 + 1 c band holes, Ns = (Ne0 − 1)/2 s pseudofermions, a single Nsh = 2Ss = 1 s band hole, and a single Ms = 2Ss = 1 unpaired spin 1/2. The emerging c band hole and the emerging s band hole describe the translational degrees of freedom of the emerging unpaired η-spin 1/2 and of the emerging unpaired spin 1/2, respectively.

26 III.

THE 1D HUBBARD MODEL PDT SPECTRAL-FUNCTION EXPRESSIONS NEAR THE BRANCH LINES AND RELATION TO THE MQIM A.

Spectral-function expressions near the branch lines

For integrable models such as the 1DHM, the PDT1–3 is fully equivalent to the mobility quantum impurity model (MQIM)4,5 . (See ref. 6 for the integrable model studied by means of the MQIM in ref. 7). Within the 1DHM PDT, the one-electron removal spectral function has three main types of singularities, called here s, c, and c0 branch lines, respectively. In their (k, ω)-plane vicinity, such a spectral function has the universal form, B(k, ω) ∝ (ωτ (k) − ω)ζτ (q) for small (ωτ (k) − ω) > 0 where τ = s, c, c0 , ωτ (k) < 0, and ω < 0. The (k, ω)-plane τ = s, c, c0 spectra ωτ (k) and corresponding relations to the momentum q of the pseudofermion holes associated with the three τ = s, c, c0 branch lines under consideration have the same form as in equation (24) with the renormalized spectra ω ˜ τ (k) and β = c, s band energy dispersions ε˜β (q) replaced by the 1DHM spectra ωτ (k) and dispersions εβ (q), respectively. The latter are defined in the following.

The c and s band energy dispersions for zero spin density

B.

For zero spin density the εc (q) and εs (q) pseudofermion energy dispersions are defined by the equations, εc (q) = ε¯c (kc (q)) for q ∈ [−π, π] and εs (q) = ε¯s (Λ(q)) for q ∈ [−kF , kF ] where Z k Z Λ ε¯c (k) = dk 0 2t ηc (k 0 ) and ε¯s (Λ) = dΛ0 2t ηs (Λ0 ) .

(S5)

∞

Q

Moreover, the real charge momentum rapidity function kc (q) for q ∈ [−π, π] and real spin rapidity function Λs (q) for q ∈ [−kF , kF ] are defined in terms of the following integral equations obeyed by their inverse functions qc (k) and qs (Λ), respectively,   Z sin k − Λ 1 ∞ qc (k) = k + for k ∈ [−π, π] and dΛ 2πσ(Λ) arctan π −∞ u   Z 1 Q Λ − sin k qs (Λ) = dk 2πρ(k) arctan π −Q u   Z ∞ 1 Λ − Λ0 0 0 − dΛ 2πσ(Λ ) arctan for Λ ∈ [−∞, ∞] such that π −∞ 2u qc (±Q) = ±2kF , qc (±π) = ±π and qs (±∞) = ±kF where ± Q = kc (±2kF ) . (S6) Finally, the distributions 2t ηc (Λ) and 2t ηs (Λ) in equation (S5) are the solutions of the coupled integral equations, Z cos k ∞ 2t ηs (Λ) (S7) 2t ηc (k) = 2t sin k + dΛ
2 , π u −∞ 1 + sin k−Λ u

and 1 2t ηs (Λ) = πu

Z

C.

Q

1 dk
2 − Λ−sin k 2π u −Q 1+ u 2t ηc (k)

Z

∞

−∞

dΛ0

2t ηs (Λ0 ) . 0
2 1 + Λ−Λ 2u

(S8)

The phase shifts in the momentum dependent exponents

P F The diluted quantum liquid considered in the article involves a number NβF = ι Nβ,ι of β = c, s pseudofermions F with momentum q near the ±qF β pseudofermions. The corresponding numbers Nβ,ι  1 of right (ι = +1) and left F F (ι = −1) β = c, s pseudofermions refer to small densities nF β,ι = Nβ,ι /L such that nβ,ι → 0 for L → ∞. The numbers F F F F F F deviations δNβ = δNβ+1 + δNβ,−1 and current numbers deviations δJβ = (δNβ+1 − δNβ,−1 )/2 that result from transitions from the ground state to the one-electron excited u > 0 energy eigenstates play an important role in the PDT expression of the spectral-function exponent ζτ (q)1–3 . To start with we consider that the initial ground state has electronic density ne and spin density m in the intervals ne ∈ [0, 1] and m ∈ [0, ne ], respectively.

27 Within the TL, one distinguishes the low-energy deviations δNβF and δJβF of such numbers at and very near the β = c, s Fermi points ±qF c = ±2kF and ±qF s = ±kF ↓ from the high-energy c or s band hole that emerges at a momentum q ∈] − qF β , qF β [ away from such points. The τ = s and τ = c, c0 branch lines display singularities for the ranges of the momentum k corresponding to those of such a s and c band hole momentum q, respectively, for which the spectral-function exponent ζτ (q) is negative. Only for such k ranges do such branch lines contribute to the corresponding (k, ω)-plane singular peaks distribution. That exponent is within the PDT given by, X X ζτ (q) = −1 + 2∆ιβ (q) . (S9) β=c,s ι=±1

In the present case of the one-electron removal spectral function of interest for angle-resolved-photoemission spectroscopy (ARPES), the functionals 2∆ιβ (q) in that expression have the general form, 2  ! F X δN 0 β − Φc βτ (ι2kF , q) 2∆ιc =  ξc1 β 0 δJβF0 + ι ξc0 β 0 2 0 β =c,s

 2∆ιs = 

X

ξs1 β 0 δJβF0 + ι ξs0 β 0

β 0 =c,s

δNβF0 2

2

!

− Φs βτ (ιkF ↓ , q) .

(S10)

Here q is the momentum of the s band hole for τ = s in βτ or c band hole for τ = c and τ = c0 in βτ created under the transition from the ground state to the one-electron excited u > 0 energy eigenstate. (The τ = c and τ = c0 branch lines refer to different q subintervals). The contributions involving the β = c, s and j = 0, 1 parameters ξβj β 0 stem from low-energy β 0 = c, s band particlehole processes near the β 0 = c, s Fermi points that dress the single βτ band hole creation processes. These parameters are the entries of the conformal-field theory 2×2 dressed-charge matrix and of the transposition of its inverse matrix3,8 , # # " " ξc0 c ξc0 s ξc1 c ξc1 s 0 1 −1 T 1 , (S11) and Z = ((Z ) ) = Z = ξs0 c ξs0 s ξs1 c ξs1 s respectively. For the PDT, they are the following combinations of phase shifts, X ξβj β 0 = δβ,β 0 + (ι)j Φβ,β 0 (qF β , ιqF β 0 ) where β = c, s ,

β 0 = c, s ,

j = 0, 1 .

(S12)

ι=±1

Within the MQIM, the branch-line functionals, equation (S10), specific to the one-electron removal spectral function of the 1DHM are written as,  2  2 cβτ sβτ X X δ δ 2∆ιc =  ξc1 β 0 δJβF0 + ι  and 2∆ιs =  ξs1 β 0 δJβF0 + ι  . (S13) 2π 2π 0 0 β =c,s

β =c,s

Here διsβτ /2π and διcβτ /2π are the ι = ±1 MQIM shift functions associated with the τ = s, c, c0 branch lines. Their expression in terms of PDT pseudofermion phase shifts is, X δNβF0 διcβτ = ι ξc0 β 0 − Φc βτ (ι2kF , q) 2π 2 0 β =c,s

διsβτ 2π

=

X

ι ξs0 β 0

δNβF0 − Φs βτ (ιkF ↓ , q) . 2

D.

The pseudofermion phase shifts at zero spin density

β 0 =c,s

In the present zero spin density m = 0 case, the matrices entries in equation (S11) simplify to, " # " # ξc ξc /2 1/ξc 0 1 0 √ √ √ lim Z = and lim Z = . m→0 m→0 0 1/ 2 −1/ 2 2

(S14)

(S15)

28 √ 9,10 2K The parameter ξc appearing here is directly related to the Tomonaga-Luttinger liquid (TLL) parameter  c  √ as ξc = 2Kc . Its dependence on the electronic density ne ∈ [0, 1] and u = U/4t is given by ξc = ξc sinuQ . The

corresponding function ξc (r) obeys the integral equation, Z rQ sin Q dr0 D(r − r0 ) ξc (r0 ) where rQ = ξc (r) = 1 + and u −rQ     1 r r Z ∞ Γ + i Γ 1 − i 2 4 4 cos(ω r) i d 1   . dω = ln  D(r) = π 0 1 + e2ω 2π dr Γ 1 − i r Γ 1 + i r 2

4

(S16)

4

√ The parameter ξc has limiting values ξc = 2 for u → 0 and ξc = 1 for u → ∞. The criterion behind using in our analysis the parameter ξc rather than Kc is that it is directly related to charge c pseudofermion phase shifts in units of 2π. The corresponding relations have the same form as those provided in ˜ c,β where β = c, s replaced by Φc,β . Inversion equation (11) with the parameter ξ˜c replaced by ξc and the phase shifts Φ of such relations leads under the same replacements to Φc,β expressions of the same form as those provided in equation (12). The numbers deviations in the functionals, equation (S10), to be used for the one-electron removal s, c and c0 branch lines are those provided in equation (20). The resulting functionals and corresponding MQIM shift functions have the same general form as those given in equations (21) and (22) under the replacement of the renormalized quantities by those of the present 1DHM. The same applies to the general form of the s, c, and c00 branch line exponents, which under such replacements is the same as that given in equation (25). Such 1DHM exponents are plotted as a function of the wave vector k for electronic density ne = 0.176 and u = ueff = 0.3 in Figs. 3-4 where they correspond to the black solid lines. The 1DHM β = c, s pseudofermion phase shifts in units of 2π can be expressed as,   0 ¯ β,β 0 Λβ (q) , Λβ 0 (q ) Φβ,β 0 (q, q 0 ) = Φ for β = c, s and β 0 = c, s , (S17) u u where Λc (q) = sin k(q)/u. On the one hand, for zero spin density the s pseudofermion phase shifts under consideration are independent of u = u/4t. They are given in units of 2π in equation (9). On the other hand, the c pseudofermion phase shifts in units ¯ c,c (r, r0 ) and Φ ¯ c,s (r, r0 ) the solutions of of 2π, Φc,c (q, q 0 ) and Φc,s (q, q 0 ), are for m → 0 given by equation (S17) with Φ the integral equations, Z rQ ¯ c,c (r, r0 ) = −Dc,c (r − r0 ) + ¯ c,c (r00 , r0 ) , Φ dr00 D(r − r00 ) Φ (S18) −rQ

and ¯ c,s (r, r0 ) = −Dc,s (r − r0 ) + Φ

Z

rQ

−rQ

¯ c,s (r00 , r0 ) , dr00 D(r − r00 ) Φ

(S19)

respectively. The kernel in these equations is given in equation (S16) and the free terms are defined by the expressions,     1 r r Z ∞ Γ + i Γ 1 − i 2 4 4 sin(ω r) i 1    and dω = ln  Dc,c (r) = π 0 ω(1 + e2ω ) 2π Γ 1 − i r Γ 1 + i r 2

4

4

  π  1 Dc,s (r) = arctan sinh r , 2π 2

(S20)

respectively, where Γ(x) is the Gamma function.

E.

The c band particle and hole energy bandwidths

The momentum rapidity c energy dispersion ε¯c (k), equation (S5), can be expressed in terms of the momentum ¯ c,c in equation (S17). After some integral equations manipulations we find the following exact rapidity phase shift Φ

29 expression, ε¯c (k) = ε¯0c (k) − ε¯0c (Q) where   Z Q sin k 0 sin k U 0 ¯ 0 dk Φc,c ε¯c (k) = − − 2t cos k + 2t , sin k 0 2 u u −Q      U sin k sin k ¯ ¯ = − − 2t cos k − 2t cos Q Φc,c rQ , − Φc,c −rQ , 2 u u   Z Q 0 d ¯ sin k sin k dk 0 cos k 0 0 Φ + 2t , . c,c dk u u −Q

(S21)

Hence the c band energy dispersion εc (q) = ε¯c (kc (q)) can be written as, εc (q) = ε0c (q) − ε0c (2kF ) where U ε0c (q) = − − 2t cos kc (q) + 2t ∆Φc (q) cos kc (2kF ) 2   Z kc (2kF ) sin k 0 sin kc (q) d ¯ , . + 2t dk 0 cos k 0 0 Φ c,c dk u u −kc (2kF )

(S22)

where ∆Φc (q) =

X

(−ι) Φc,c (ι2kF , q) .

(S23)

ι=±1

¯ c,c (r, r0 ) = −Φ ¯ c,c (−r, −r0 ), we have rewritten in equation (S21) the integral in the first Accounting for that Φ expression by parts and solved the surface term. P This has singled out the part of that dispersion associated with the corresponding 1DHM phase shifts, ∆Φc (q) = ι=±1 (−ι) Φc,c (ι2kF , q). (See equations (S22) and (S23)). Specifically, ∆Φc (q) = Φc,c (−2kF , q) − Φc,c (+2kF , q) is the difference of the phase shifts in units of 2π acquired by the c pseudofermions at the opposite c band Fermi points −2kF and +2kF , respectively, due to the emergence under a one-electron excitation of one c band hole at momentum q ∈] − 2kF , 2kF [. Here we consider again the m → 0 limit of interest for the article studies. The particle energy bandwidth is the same εc (0). It refers to their ranges q ∈ [−2kF , 2kF ] and k ∈ [−Q, Q], respectively. for εc (q) and ε¯c (k), Wcp = −εc (0) = −¯ Similarly, the hole energy bandwidth is given by Wch = εc (π) = ε¯c (π) for |q| ∈ [2kF , π] and |k| ∈ [Q, π], respectively. From the use of equation (S21), they can be written as, ε¯0c (k)

Wcp = −¯ εc (0) = 2t(1 − cos Q) + 2t (∆ΦF c − ∆Φc (0)) cos Q      Z Q sin k 0 sin k 0 0 0 d ¯ ¯ + 2t dk cos k Φc,c , rQ − Φc,c ,0 dk 0 u u −Q Wch = 4t − Wcp where ∆ΦF c ≡ ±∆Φc (±2kF ) = ∓Φc,c (2kF , ±2kF ) ± Φc,c (−2kF , ±2kF ) =

(ξc − 1) . ξc

(S24)

¯ c,c (±rQ , rQ ) and Φc,c (±2kF , 0) = To express Wcp in terms of ∆ΦF c − ∆Φc (0), we have used that Φc,c (±2kF , 2kF ) = Φ ¯ c,c (±rQ , 0). Φ Since Wch = 4t − Wcp , one can consider only Wcp . In units of 2t(1 − cos Q), it can be written as, Wcp = 1 + Gc − ∆ΦF c , 2t(1 − cos Q)

(S25)

where 1 Gc = (1 − cos Q)

( ∆ΦF c +

"Z

Q

d dk cos k 0 dk −Q 0

0

#)      0 0 sin k sin k ¯ c,c ¯ c,c Φ , rQ − Φ ,0 − ∆Φc (0) . u u

(S26)

The dimensionless ratio in equation (S25) has limiting behaviors, lim

ξc →ξcmin

Wcp = 1 and 2t(1 − cos Q)

limmax

ξc →ξc

Wcp = 2, 2t(1 − cos Q)

(S27)

30 √ √ to u → 0 and u → ∞, respectively. It where ξcmin = 1 and ξcmax = 2 and the limits ξc → 2 and ξc → 1 correspond √ is a decreasing function of u ∈]0, ∞[ and an increasing function of ξc ∈]1, 2[ such that, Wcp /2t(1 − cos Q) ∈ [1, 2], or equivalently, Wcp ∈ [1, 1 + Gmax − ∆Φmax c Fc ] , 2t(1 − cos Q)

(S28)

where Gmax − ∆Φmax ξc →√1 related c F c = 1. The minimum value 1 corresponds √ limits and the √to the u → ∞ and 2 limits. Here ∆Φmax maximum value 1 + Gmax − ∆Φmax c F c = ( 2 − 1)/ 2 is the largest F c = 2 to the u → 0 and ξc → ∆ΦF c value reached in the latter related limits. IV.

SLIGHT RENORMALIZATION OF THE PARTICLE AND HOLE c BAND ENERGY BANDWIDTHS BY THE FINITE-RANGE INTERACTIONS

The small positive deviation δWcp associated with the slight increase of Wcp induced by the finite-range interaction whose positivity is justified in the article tends to suppress the effect of the 1DHM negative phase-shift contribution max max −∆Φmax ]. However, F c in the overall interval, equation (S28). This would map that interval onto [1 + ∆ΦF c , 1 + Gc p h the energy bandwidth deviations δWc and δWc are limited or even prevented by several properties of the 1DHM c band energy dispersion that remain invariant under the UT. One then finds below that only in the u → ∞ and ξc → 1 related limits is the maximum deviation δWcp /2t(1 − cos Q) = ∆Φmax F c allowed. The properties under consideration are the following: I - The overall c band dispersion energy bandwidth reads 4t. As for the 1DHM, due to symmetry that bandwidth is for the URT independent of the electronic interactions and also given by ε˜c (π) − ε˜c (0) = 4t. That it remains invariant under the UT then implies that δWcp = −δWch ≥ 0. II - The dimensionless ratio Wcp /2t(1 − cos Q) ≥ 0, equation (S25),√obeys the inequality Wcp /2t(1 − cos Q) ≤ 2. As given in equation (S27), Wcp /2t(1 − cos Q) = 2 in the ξc → ξcmax = 2 limit. As justified below, such an inequality ˜ cp = Wcp + δWcp implies that ˜ cp /2t(1 − cos Q) = 2 where W remains invariant under the UT. Hence that limξc →ξcmax W √ max h p δWc = −δWc = 0 in the ξc → ξc = 2 limit. That this inequality is also obeyed by the URT stems from the form Ve (r) = U F (r)/r of the electronic potential for r > 0. Although Ueff 6= U for ξ˜c < ξc and finite Ueff values, for very small ueff = Ueff /4t  1 one has that Ueff ≈ U and Ueff = U as ueff → 0 for ξ˜c < ξc . This implies that Ve (r) → 0 as ueff → 0. That such a limit is smooth and all quantitive effects of Ve (r) continuously decrease upon decreasing ueff , justifies why as reported in point II the ˜ cp /2t(1 − cos Q) ≤ 2 also holds within the URT. inequality W As mentioned above, only in the ξc → ξcmin = 1 limit and thus ueff → ∞ limit is the above absolute maximum value of max{δWcp } permitted, so that, √ p p max p lim max{δWcp } = ∆Φmax (S29) F c Wc and thus max{δWc } ∈ [0, ∆ΦF c Wc ] for ξc ∈ [1, 2] . ξc →ξcmin =1

√ The max{δWcp } minimum value, max{δWcp } = 0, is reached in the ξc → ξcmax = 2 limit. Indeed, limξc →ξcmax δWcp = 0, √ as justified above. The simplest form of max{δWcp } for the whole interval ξc ∈ [1, 2] consistent with the limiting values of its range in equation (S29) reads, √ p p max{δWcp } = (∆Φmax (S30) F c − ∆ΦF c ) Wc = βc Wc for ξc ∈ [1, 2] , where, βc =

∆Φmax Fc

− ∆ΦF c

√   ( 2 − 1) (ξc − 1) 1 ξc √ = − = 1− √ . ξc ξc 2 2

(S31)

˜ cp /2t(1−cos Q)} = Wcp (1+βc )/2t(1−cos Q) smoothly and continuously increases upon increasing It is such that max{W √ min the initial UT ξc value from 1 + ∆Φmax = 1 to 2 for ξc → ξcmax = 2. Hence both the range F c < 2 for ξc → ξc p ˜p max{δWcp } ∈ [0, ∆Φmax F c Wc ] in equation (S29) and the inequality Wc /2t(1 − cos Q) ≤ 2 reported in the point II are obeyed. √ On the one hand, the phase-shift parameter βc = (1 − ξc / 2)/ξc only depends on the initial UT parameter ξc . It defines the maximum allowed deviations, max{δWcp }, that can be induced by the finite-range interactions. On the other hand, such deviations are of the general form δWcp = γc βc where the parameter γc ≤ 1 depends on the UT ξc → ξ˜c renormalized parameter ξ˜c . As given in equation (28), it is such that γc = 0 for ξ˜c → ξc and γc = 1 for

31 ξ˜c → 1. This ensures that in the ξ˜c → ξc limit that recovers the 1DHM the boundary condition δWcp = 0 is satisfied. Moreover, the maximum allowed value for the deviation, max{δWcp } = βc Wcp , is reached in the ξ˜c → 1 limit. Since the deviation δWcp cannot increase beyond βc Wcp , one has that its value saturates for the ξ˜c range ξ˜c ∈]1/2, 1[, for which it is thus given by δWcp = βc Wcp . The renormalization of the energy dispersion ε˜c (q) reported in equations (26) and (27) follows for (i) |q| < 2kF ˜ cp and (ii) W ˜ ch , respectively. Specifically, W ˜ cp = (1 + βc )Wcp for ξ˜c ∈]1/2, 1[, and (ii) |q| > 2kF from that of (i) W ˜ cp = (1 + γc βc )Wcp for ξ˜c ∈]1, ξc [, and the corresponding renormalization of the hole energy bandwidth W ˜ ch uniquely W h p ˜ ˜ determined by the relation Wc = 4t − Wc , which remains unchanged under the UT. V.

USEFUL MATHEMATICAL RELATIONS AND CONFIRMATION OF THE LARGE-x BEHAVIOUR OF THE FUNCTION fc (x) A.



2 1 (y) J l−2 4

y l−2 +1



1 (y) J− l−2

Useful mathematical relations



1 (y) (l − 2) d  J l−2 =−  4 2 dy y l−2

2

4

y l−2 +1

4

1 J 1 J l−2 (y) l−2 +1

2  1 (y) J (l − 2) d  − l−2  (l − 2) =−  − 4 6 dy 3 y l−2

(l − 2) d =− 4 dy (l − 2) − 4

B.

  − (l − 2)



1 (y) J 1 (y) J l−2 − l−2

y l−2 +1

2 

1 (y) J 1 (y) J l−2 − l−2

!

4

,

(S32)

y l−2

1 J 1 J− l−2 +1 (y)) − l−2

!

4

,

(S33)

y l−2

!

4

y l−2

1 (y) J 1 1 (y) J 1 J− l−2 (y) + J l−2 − l−2 +1 (y) l−2 +1 4

! .

(S34)

y l−2

Confirmation of the large-x behaviour of the function fc (x)

The use in equation (45) of the constants B1 and B2 = B20 , equations (41) and (42), respectively, leads to,  2    2  x l − 2 l−2 d l−1 √ fc (x) = 2rl { φ2 1 (x) [Γ2 l−2 dx 2rl l−2 2 2   √ ! l−2     2rl 2 l−1 l−3 1 (x) φ 1 (x) − Γ Γ φ l−2 − l−2 a l−2 l−2 l−2 4 √ ! l−2  2   2rl 1 2 l−3 Γ2 φ2− 1 (x)]} . + (S35) l−2 a 3 l−2 l−2 One then straightforwardly finds that,   2  2   Z ∞ l − 2 l−2 x l−1 2 √ 2 dx fc (x) = lim 4rl { [Γ φ2 1 (x) l−2 x→∞ 2rl l−2 2 0 2 !   √     l−2 2rl 2 l−1 l−3 1 (x) φ 1 (x) − Γ Γ φ l−2 − l−2 a l−2 l−2 l−2 4 √ ! l−2  2   2rl 1 2 l−3 2 + Γ φ2− 1 (x)]} . l−2 a 3 l−2 l−2

(S36)

32 The use in this expression of the asymptotic behaviour of the Bessel functions for x  1 and thus y  1, equation (39), leads to the desired result in equation (49). References

1 2 3 4 5 6 7 8 9 10

J. M. P. Carmelo, T. Čadež, Nucl. Phys. B 914, 461-552 (2017). J. M. P. Carmelo, K. Penc, D. Bozi, Nucl. Phys. B 725, 421-466 (2005); Nucl. Phys. B 737, 351 (2006), Erratum. J. M. P. Carmelo, L. M. Martelo, K. Penc, Nucl. Phys. B 737, 237 (2006). A. Imambekov, L. I. Glazman, Science 323, 228-231 (2009). A. Imambekov, T. L. Schmidt, L. I. Glazman, Rev. Mod. Phys. 84, 1253-1306 (2012). J. M. P. Carmelo, P. D. Sacramento, Ann. Phys. 369, 102-127 (2016). A. Imambekov, L. I. Glazman, Phys. Rev. Lett. 100, 206805 (2008). J. M. P. Carmelo, P. Horsch, A. A. Ovchinnikov, Phys. Rev. B 45, 7899 (1992). J. Voit, Rep. Prog. Phys. 58, 977-1116 (1995). H. J. Schulz, Phys. Rev. Lett. 64, 2831-2834 (1990).