Spin-charge decoupling and the photoemission line-shape in one dimensional insulators
aa r X i v : . [ c ond - m a t . s t r- e l ] S e p Spin-charge decoupling and the photoemission line-shape in one dimensional insulators
Valeria Lante ∗ and Alberto Parola CNISM and Dipartimento di Fisica e Matematica,Universit`a dell’Insubria, Via Valleggio 11, I-22100 Como, Italy (Dated: December 5, 2018)The recent advances in angle resolved photoemission techniques allowed the unambiguous experi-mental confirmation of spin charge decoupling in quasi one dimensional (1D) Mott insulators. Thisopportunity stimulates a quantitative analysis of the spectral function A ( k, ω ) of prototypical onedimensional correlated models. Here we combine Bethe Ansatz results, Lanczos diagonalizationsand field theoretical approaches to obtain A ( k, ω ) for the 1D Hubbard model as a function of theinteraction strength. By introducing a single spinon approximation , an analytic expression is ob-tained, which shows the location of the singularities and allows, when supplemented by numericalcalculations, to obtain an accurate estimate of the spectral weight distribution in the ( k, ω ) plane.Several experimental puzzles on the observed intensities and line-shapes in quasi 1D compounds,like SrCuO , find a natural explanation in this theoretical framework. PACS numbers: 71.10.Fd, 79.60.-i
I. INTRODUCTION
Since the theoretical prediction of the decoupling ofspin and charge excitations in one dimensional (1D)models , many experiments have long sought to verifythis effect . According to the spin-charge separation sce-nario, the vacancy ( e + ) created by removing an electronin a photoemission experiment decays into two collectiveexcitations (or quasi-particles ), known as spinon ( s ) and holon ( h ), carrying spin and charge degrees of freedomrespectively. The recent observation of a well definedtwo-peak structure in the angle-resolved photoemissionspectra (ARPES) of the quasi-1D materials SrCuO andSr CuO is deemed a significant clue of spin-chargedecoupling, confirming previous expectations.However, other quasi one dimensional materials failto show distinct holon and spinon peaks, casting somedoubt on the interpretation of ARPES experiments basedon spin charge decoupling. A number of puzzling featuresalso suggest that more physics, beyond the simple decay e + → s + h , is involved in the photoemission process:the spectral functions of SrCuO and Sr CuO reportedby Kim et al. and by Kidd et al. systematically displaybroad line-shapes in contrast to the sharp edges expectedon the basis of the available calculations on model sys-tems. The spectral intensity also appears considerablyweaker in a half of the Brillouin zone, a feature oftenascribed to cross section effects .A quantitative theoretical understanding of ARPESin low dimensional systems is important and deservesa careful investigation because ARPES provides a directexperimental probe to the single particle excitation spec-trum, allowing for reliable estimates of the key param-eters governing the physics of strongly correlated elec-trons: the electron bandwidth and the Coulomb repul-sion. Here we will focus on the 1D Hubbard model, asimple lattice model defined by just two coupling con-stants: the nearest neighbor hopping integral t and the on-site Coulomb repulsion U : H = − t X i,σ h c † i +1 ,σ c i,σ + h.c. i + U X i n i ↑ n i ↓ (1)Although several other terms, such as next-nearest hop-ping, further orbital degrees of freedom, temperature,disorder or lattice instabilities, would be necessary in arealistic model of these materials, we believe that an ac-curate investigation of the simplest hamiltonians shouldbe performed before facing more challenging problems.The theoretical studies aimed at the investigation ofthe spectral properties of one dimensional models areeither fully numerical, like Lanczos diagonalizations and density matrix renormalization group (DMRG)techniques , or are carried out in the limiting casesof infinite or vanishing interaction U/t . In the for-mer case, they suffer from severe finite size effects, inthe latter the interplay between charge fluctuations andstrong correlations is not satisfactorily taken into ac-count. Monte Carlo studies of dynamical properties ofquantum systems are instead hampered by the necessityto perform an analytic continuation to real times.In this paper, we provide the first quantitative evalua-tion of the full spectral function A ( k, ω ) of the 1D Hub-bard model at half filling for intermediate and strongcoupling U/t . A formalism based on the Bethe Ansatzsolution , and supplemented by Lanczos diagonaliza-tions, is developed and is shown to provide a transparentdescription of the dynamical properties of mobile chargesin Mott insulators. From this analysis we find that the1D Hubbard model does indeed contain the physics re-quired for a quantitative interpretation of photoemissionexperiments. In particular: i ) the underlying free elec-tron Fermi surface plays a key role in defining the shapeand the intensity of the ARPES signal, up to fairly largeeffective couplings U/t ; ii ) the power-law singularitieswhich characterize the spectral function in one dimen-sion give rise to intrinsically broad peaks, whose width isproportional to the intensity of the line; iii ) ARPES dataare extremely sensitive to the Hubbard parameters andallow for a direct determination of the effective couplingconstants in quasi 1D materials. As a working example,we apply our method to SrCuO , where accurate ARPESdata are available , and we derive reliable estimates for t and U .The plan of the paper is as follows. In Section II weintroduce and motivate the single spinon approximationwhich lies at the basis of our method, deriving the pre-dicted formal structure of the spectral function in onedimensional models. Section III shows how Lanczos di-agonalizations provide a precise quantitative estimate ofthe quasi-particle weight required for the evaluation ofthe spectral function. Then, in Section IV we discussthe weak coupling limit, where a thorough field theoret-ical analysis is available. The application to the case ofSrCuO is performed in Section V, while in the Conclu-sions we briefly discuss the generalization of our methodto more complex one dimensional hamiltonians. II. THE ANALYTICAL STRUCTURE OF THESPECTRAL FUNCTION
The dynamical properties of one (spin down) hole inthe half filled Hubbard model are embodied in the spec-tral function A ( k, ω ) which, at zero temperature, can bewritten as: A ( k, ω ) = X {| Ψ i} |h Ψ | c k, ↓ | Ψ i| δ ( ω − E + E ) (2)where | Ψ i is the ground state of the model at zero dop-ing, i.e. when the number of electrons N equals thenumber of sites of the lattice L , E is the correspond-ing energy and {| Ψ i} represents a complete set of one-hole intermediate states, of energies E . The whole en-ergy spectrum of the Hubbard hamiltonian (1) can beobtained from the Lieb and Wu equations . In the ther-modynamic limit, its structure has been thoroughly in-vestigated in a series of papers by Woynarovich (seealso the comprehensive book by Essler et al. Ref. ).In summary, the exact excitation spectrum at half fill-ing and at arbitrary coupling U/t depends on two setsof “rapidities” describing the charge and spin degreesof freedom respectively. The excitation energy is al-ways written as the sum of contributions involving justtwo elementary excitations, representing collective quasi-particles: “holons” (of momentum k h and energy ǫ h ( k h ))and “spinons”(of momentum Q ∈ ( π , π ) and energy ǫ s ( Q )). The simplest physical excitation created by theremoval of an electron of momentum k gives rise to oneholon and one spinon satisfying the momentum conserva-tion equation k = k h + Q . The total energy of this state is E = E + ǫ h ( k h )+ ǫ s ( Q ). Besides this suggestive “decay”mechanism of the electron, other excited states also ap-pear in the exact spectrum: they are either multi-spinonand multi-holon states, or states involving the creationof double occupancies . However, it is remarkable that the full excitation spectrum can be always expressed interms of ǫ h ( k h ) and ǫ s ( Q ), showing that spin charge de-coupling holds, in the Hubbard model, at all values of U/t and at all energy scales . The two quasi-particles,holon and spinon, are both collective excitations involv-ing an extensive number of degrees of freedom and canbe approximately related to simple real space pictures ofa “hole” and an unpaired spin only in the strong couplinglimit, where spin-charge decoupling acquires a more in-tuitive meaning. As U → , closely related to thefree particle band structure: ǫ h ( k h ) = 4 t cos( k h /
2) and ǫ s ( Q ) = 2 t | cos Q | .While the whole energy spectrum of the Hubbardhamiltonian is known in detail, the matrix elements ap-pearing in Eq. (2) are of difficult evaluation. Moreoverthe summation over the intermediate states formally in-volves a number of terms exponentially large in N , mak-ing the exact implementation of the definition (2) im-practical. Our approach, which allows for the evaluationof the full spectral function in the thermodynamic limit,is based on the single spinon approximation : i.e. we ne-glect the contribution to the spectral function comingfrom all multi-spinon excited states and all excitationswith complex rapidities, but we evaluate exactly the ma-trix elements involving one holon and one spinon. Theaccuracy of this method is tested a posteriori by use of acompleteness sum rule and can be estimated of the orderof few percents. Such a remarkable performance of thesingle spinon approximations is not unusual in one di-mensional physics: a known example is provided by theHaldane-Shastry spin model (HSM) , where each inter-mediate state contributing to the dynamical spin corre-lation is completely expressible in terms of eigenstatesof the HSM with only two spinons. In this case, only asmall O ( L ) number of eigenstates contribute to the exact dynamical spin correlation function as proved in Ref. .Similarly, in our approach, the most relevant intermedi-ate states are expressible is terms of eigenstates of theHubbard model with only one spinon and one holon ex-citations.A first clue on the structure of the spectral functionin one dimensional models can be obtained by analyzingthe U → ∞ limit, where double occupancies are inhib-ited and several exact results are available . At half fill-ing ( N = L ) the Hubbard hamiltonian is mapped ontoa Heisenberg hamiltonian: each site is singly occupiedand the ground state is a non-degenerate singlet of zeromomentum . When a hole of momentum k h is created,all the eigenfunctions of the Hubbard hamiltonian (withperiodic boundary conditions) can be written as : | Ψ > = 1 √ L X x, { y i } e ik h x φ H ( y , . . . , y M ) | x, { y i }i (3)where | x, { y i }i represents the configuration of L − M = L/ { y i } ) and of the hole ( x ). The amplitude φ H is ageneric eigenfunction of the Heisenberg hamiltonian onthe “squeezed chain”, i.e. on the L − | Ψ > entering the spectral function (2) havemomentum − k relative to the ground state at half fill-ing. Due to the factorized form of the eigenfunctions (3)the total momentum of the state satisfies k = k h + Q where Q is the momentum of the Heisenberg eigenfunc-tion φ H , expressed in integer multiples of 2 π/ ( L − E = E + ǫ h ( k h ) + ǫ s ( Q ),where, to lowest order in J = 4 t /U , the first (holon)contribution is just the kinetic energy of a free particle( ǫ h ( k h ) = 2 t cos k h ) and the second one (spinon) is theenergy of the eigenstate φ H referred to the ground stateenergy of the Heisenberg ring of L sites . This analysisshows, in an intuitive way, the origin of momentum andenergy conservation in the decay process of the vacancyand suggests that, in the U → ∞ limit, the most rele-vant contributions to the sum of intermediate states inEq. (2) come from the lowest energy eigenstates φ H ofthe Heisenberg hamiltonian for the L − Q = πL − n ( n = 0 . . . L − {| Ψ > } inEq. (2) can be (approximately) replaced by a sum over L − single spinon states . This special set of intermediatestates | Ψ i , which we argue provides the dominant contri-bution to the spectral weight for each spinon momentum Q , will be referred to as | − k, Q i in order to emphasizethe two quantum numbers which uniquely identify them.The single spinon approximation can be easily tested inthe U → ∞ limit where it proves extremely accurate. Inthe next Section we will show that it remains fully satis-factory also at finite coupling. In fact, it is known thatthe eigenstate structure of the Hubbard model displaysa remarkable continuity in U/t , the only singular pointbeing the (trivial) free particle limit U = 0. However,when charge fluctuations are allowed for, by lowering thestrength of the on-site repulsion U , the identification ofthe single spinon states | − k, Q i is not easy, because thespinon momentum Q is not a good quantum number anymore, although it can be still formally defined on the ba-sis of the Bethe Ansatz solution of the Hubbard model .The key observation, which will be exploited in the nextSection, is that the correct single spinon states can beidentified at finite U via Lanczos or DMRG calculationsby following adiabatically the evolution of the Heisenbergstates as U is gradually decreased.Keeping only the single spinon states in the summationof Eq. (2), the spectral function in the thermodynamiclimit becomes A ( k, ω ) = Z dQ π Z k ( Q ) δ ( ω − ǫ h ( k − Q ) − ǫ s ( Q ))= 12 π X Q ∗ Z k ( Q ∗ ) | v h ( Q ∗ − k ) + v s ( Q ∗ ) | (4)Where we have defined the quasi-particle weight as the matrix element Z k ( Q ) ≡ lim L →∞ ( L − |h− k, Q | c k, ↓ | Ψ i| (5)The sum in Eq. (4) runs over all the solutions Q ∗ ( k, ω )of the algebraic equation ω = ǫ h ( k − Q ) + ǫ s ( Q ) (6)where ǫ h ( k h ) [ ǫ s ( Q )] is the known holon [spinon] exci-tation energy and v h ( k h ) = dǫ h dk h [ v s ( Q ) = dǫ s dQ ] theassociated velocity. Equation (4) is the main result ofthis work: an explicit and computable expression for thespectral function of one dimensional models. In the spe-cial case of the Hubbard model, the Bethe Ansatz so-lution directly provides spinon and holon dispersions inthe thermodynamic limit further simplifying the evalu-ation of the spectral function. Due to the presence ofa spinon Fermi surface, the dispersion relation ǫ s ( Q ) isdefined only in the interval π < Q < π , it van-ishes at the boundaries and has a single maximum at Q = π , while ǫ h ( k h ) is an even and periodic functionin the whole range − π < k h < π with maximum at k = 0 . The only missing ingredient in Eq. (4) isthe quasi-particle weight Z k ( Q ) which defines the line-shape and intensity of the spectral function. Previousstudies have shown that in spin isotropic models, likethe Hubbard model, the quasi-particle weight is a regu-lar function with square root singularities at the spinonFermi surface Q = π ± π/
2. This implies that A ( k, ω ) haspower law singularities too, whenever either Q ∗ definedby Eq. (6) lies at the spinon Fermi surface, or when thetotal excitation velocity v h ( Q ∗ − k ) + v s ( Q ∗ ) vanishes. Inboth instances, square root divergences are expected :in the former case the location of the singularity iden-tifies the holon dispersion via (6) ω = ǫ h ( k + π ± π );in the latter case the singularity is trivially due to bandstructure effects and does not necessarily corresponds toa pure spinon contribution as often assumed. However,at small to moderate interactions U/t , the holon veloc-ity | v h ( k h ) | displays an abrupt drop around k h ∼ π placing the band lower edge close to Q ∼ π + k , i.e. at ω ∼ ǫ h ( π ) + ǫ s ( π + k ), thereby following the spinon bandfor 0 < k < π . This particular feature of the Hubbardmodel dispersion is apparent in the shape of the holonspectrum which sharply bends at k h ∼ ± π so to dis-play a vanishing charge velocity at band edges. This alsoagrees with the “relativistic” form of the holon spectrumpredicted by bosonization at weak coupling , as reportedin Eq. (8). The expected location of the square root sin-gularities of the spectral function in the ( k, ω ) plane isshown for few values of the coupling in Fig. 1. The holonbranch (shown as full circles in the figure) marks preciselythe holon excitation spectrum ǫ h ( k h ) while the locationof the singularities due to the band structure (shown ascrosses in the figure) differs from the spinon ǫ s ( Q ) dis-persion by less than 0 . t . Note also that the curvatureof the “spinon branch” displays a significant dependence FIG. 1: Location of the singularities of the spectral functionin the thermodynamic limit for three values of U in the plane( k , E = − ω ). The (green) circles correspond to the singulari-ties of the Z k ( Q ) ( holon branch ), while the black stars to theextrema of the excitation spectrum.FIG. 2: Panel (a): holon bandwidth W h = ǫ h ( π/ − ǫ h ( π )as a function of U/t . Panel (b): ratio between the spinonbandwidth W s = ǫ s ( π ) and W h as a function of U/t . on U/t , allowing for a rather precise experimental deter-mination of the effective coupling ratio. Therefore weconclude that precise photoemission data, able to iden-tify the singularities of the spectral function, do providedirect information on both holon and, within a good ap-proximation, also spinon excitations.The full holon bandwidth is always 4 t at all couplings,due to the particle-hole symmetry of the Hubbard modelbut the upper and lower branches of the holon band arenot symmetrical at finite U . This observation is relevantfor the correct interpretation of photoemission experi-ments, because an estimate of the effective hopping inte-gral t is usually performed by measuring the half band-width of the upper holon branch leading to a sizableoverestimate of t . In Fig. 2 we show the bandwidth W h of the upper holon branch (i.e. ǫ h ( π/ − ǫ h ( π )) andthe ratio between the spinon and the holon bandwidths W s /W h as a function of the coupling U/t . Both quanti-ties, which allow for a direct estimate of t and U/t fromARPES, show a remarkable (even non monotonic) depen-dence on the coupling constants. By comparing theseresults with the dispersion curves for SrCuO reportedin Ref. we can estimate the Hubbard effective couplingconstants appropriate of this material: t ∼ .
53 eV and
U/t ∼ III. THE QUASI-PARTICLE WEIGHT FROMLANCZOS DIAGONALIZATION
Unfortunately, the formal Bethe Ansatz solution doesnot lead to a practical way for the evaluation of the quasi-particle weight (5) at arbitrary couplings and thereforewe resort to Lanczos diagonalizations in lattices up to L = 14 sites. As previously noticed, we first have to de-vise a method to select the correct single spinon states atfinite coupling U/t , for these states are not identified by agood quantum number at finite U . Our method is basedon an adiabatic procedure starting from the strong cou-pling limit. We first perform a Lanczos diagonalizationon the ( L − Q , leading to the numericaldetermination of φ H and of the exact eigenstates of theone-hole Hubbard model for U → ∞ via Eq. (3). In thislimit, the single spinon states are indeed the lowest en-ergy eigenstates at fixed spinon momentum Q and can beeasily obtained by Lanczos (or DMRG) technique, whilstat finite U the relevant intermediate states are not nec-essarily in the low excitation energy portion of the Hub-bard spectrum. Then we take advantage of the continu-ity of the one spinon states between the weak and strongcoupling limit by adiabatically lowering the interactionstrength U and performing successive Lanczos diagonal-izations for smaller and smaller couplings U n . At the n th step we keep the exact eigenstate having the largest over-lap with the eigenstate at the ( n − th level. In this waywe are able to identify the single spinon states down tosmall values of U ∼ t , each state being uniquely identifiedby Q , i.e. by the momentum of the “parent” Heisenbergeigenstate.A check on the validity of the single-spinon approx-imation comes from the completeness condition on theintermediate states: n ↓ ( k ) = h Ψ | c † k, ↓ c k, ↓ | Ψ i = X {| Ψ i} |h Ψ | c k, ↓ | Ψ i| ≥ X Q |h− k, Q | c k, ↓ | Ψ i| = 1 L − X Q Z k ( Q ) (7)where n ↓ ( k ) is the momentum distribution of the downspins at half filling and the equality holds if and only ifthe single spinon states included in the sum via the defi-nition of the quasi-particle weight Z k ( Q ) (5) exhaust thespectral weight at each k . The amount of violation ofthis sum rule quantifies the weight of all the neglectedstates in the Hilbert space due to the single-spinon ap-proximation. In Fig.3 we plot n ↓ ( k ) and L − P Q Z k ( Q )restricted to the one spinon states: the violation of thecompleteness condition is smaller than 0 .
01 at all k ’s .Note how, even at fairly large values of U/t , the momen-tum distribution is considerably depressed for k largerthan the free electron Fermi momentum k F = π , strongly FIG. 3: Momentum distribution n ↓ ( k ) (black open circles)and L − P Q Z k ( Q ) (red triangles) versus the hole momentum k from Lanczos diagonalization, for U = 7 t in a L = 14 ring.FIG. 4: Panel (a): Z k ( Q ) versus spinon momentum Q for U = 100 t and different lattice sizes ( × : L=6, N : L=8, (cid:4) :L=10, ∗ : L=12, • : L=14). Open (green) symbols for totalmomentum k = π/ k = 0. Linesare polynomial fit to Lanczos data. Skewed boundary con-ditions are used in order to fix the same total momentum ofthe state − k (relative to half filling) for all L ’s. Panel (b):same as (a) for U = 7 t . Panel (c): binding energy at fixed k referred to half filling versus spinon momentum Q in thethermodynamic limit E = E − E for U = 100 t . Panel (d):same as (c) for U = 7 t . reducing the spectral weight in the second half of the Bril-louin zone. This feature is consistent with the photoemis-sion experiments performed with high energy photons .Conversely, in the strong coupling limit U → ∞ , the mo-mentum distribution becomes flat, n ↓ ( k ) = 1 /
2, washingout this effect.The dependence of the quasi-particle weight on thestrength of the Coulomb repulsion has been investigatedand is summarized in Fig.4 for strong and intermedi-ate
U/t and for different lattice sizes. The quasi-particleweight has been evaluated by Lanczos diagonalization onlattices ranging from L = 6 to L = 14 sites. By usingstandard periodic boundary conditions, the total momen-tum of the state would be quantized in units of 2 π/L ,making size scaling impractical. In order to avoid thisproblem we have adopted skewed boundary conditions: given an arbitrary hole momentum k we choose the fluxat the boundary in such a way to match k with the quan-tization rule. Figure 4 reveals an astoundingly negligiblesize dependence and the expected vanishing of the quasi-particle spectral weight outside the spinon Fermi surface,with singularities at the Fermi momenta. While Z k ( Q )is almost independent on k at large U , as expected , itshows more structure for realistic values of U/t . The fur-ther peak (or shoulder) present for k ≤ π is indeed remi-niscent of the free Fermi nature of the electrons at U = 0.In the free particle limit , only one state provides a finitecontribution to the spectral function: the holon sits atthe bottom of the band ( k h = π ) and the quasi-particleweight Z k ( Q ) reduces to a delta function at Q = π + k .When such a form of Z k ( Q ) is substituted in Eq. (4),the known free particle result is recovered. Remarkably,a remnant of the free particle peak in Z k ( Q ) is still visibleat U = 7 t , as shown in Fig. 4b. IV. WEAK COUPLING LIMIT
The Green’s function of one dimensional models hasbeen thoroughly investigated by bosonization methods:while in the Luttinger liquid regime its asymptotic formis characterized by power law tails , precisely at half fill-ing the Green’s function is known to display a more com-plex behavior due to the presence of a gap in the holonspectrum. At weak coupling, the holon dispersion nearthe bottom of the band shows a “relativistic ” structure: ǫ h ( k h ) = q v h δk h + m (8)where m is the charge gap and δk h = k h ± π is the holonmomentum measured from the bottom of the band. Notethat the holon spectrum (8) is shifted by µ = U/ U = 3 t and theform (8) predicted by bosonization with suitably chosenparameters m , v h .In order to compare the results of our single spinon ap-proximation with the bosonization form, it is convenientto introduce the single hole Green’s function in imaginarytime: G ↓ ( k, τ ) = < Ψ | c † k, ↓ e − ( H − µ ) τ c k, ↓ | Ψ > (9)According to bosonization , the Green’s function G ↓ ( k, τ ) of a hole of momentum close to k F = π/ G R ↓ ( x, τ ) ≡ Z dk π G R ↓ ( k, τ ) e ikx = e i π x G h ( x, τ ) G s ( x, τ ) (10)where the superscript R identifies the contribution to theGreen’s function due to right moving holes. Here, G h and G s just depend on holon and spinon degrees of freedomrespectively. The spinon term is simply given by G s ( x, τ ) = 1 √ v s τ + ix (11)while the holon contribution is predicted, by the formfactor approach, to behave as G h ( x, τ ) = Γ r mv h Z ∞−∞ dθ e h θ/ − mτ cosh θ − im xvh sinh θ i (12)with Γ ∼ . ... . The question now arises whetherour single spinon approximation is consistent with sucha factorized form. By inserting a complete set of inter-mediate states into the definition (9) and adopting thesingle spinon approximation, the full Green’s function inimaginary time can be written as G ↓ ( k, τ ) = 1 L X Q Z k ( Q ) e − [ ǫ h ( k h )+ ǫ s ( Q )] τ (13)where the momentum conservation relation k = k h + Q is understood and the holon spectrum ǫ h ( k h ) is nowreferred to the chemical potential µ . Notice that, due tomomentum conservation, the combined requirements ofhaving k ∼ k F = π and k h ∼ − π (i.e. the hole sits nearthe bottom of the band) force Q ∼ π . By substitutingthe asymptotic forms (8) and ǫ s (cid:0) π − q (cid:1) = v s q for q & G R ↓ ( x, τ ) = e i π x mv h Z π dq π e − [ iqx + v s qτ ] (14) × Z ∞−∞ dθ π Z k ( Q ) cosh θ e h − im xvh sinh θ − mτ cosh θ i where we set δk h ≡ − mv h sinh θ . This form does indeedfactorize in a holon and spinon part, as predicted bybosonization, provided the quasi-particle weight does: Z k ( Q ) ∼ Z h ( k h ) Z s ( Q ) (15)Notice that our approach, being based on a numericalevaluation of the quasi-particle weight, does not allow foran independent demonstration of such a factorized form.We just observe that the bosonization approach and thesingle-spinon approximation lead to the same result if weassume that Eq. (15) holds. Following Ref. we arguethat the factorization of the quasi-particle weight at lowenergies (15) reflects the trivial structure of the holon-spinon scattering matrix in this limit.As previously noticed, the spinon contribution to thequasi-particle weight gives rise to the square root diver-gence at the spinon Fermi surface, with leading behavior Z s (cid:0) π − q (cid:1) ∼ q − / for q & Z h ( k h ) = √ v h p ǫ h ( k h ) − v h δk h ǫ h ( k h ) (16) FIG. 5: Panel (a): Holon spectrum ǫ h ( k h ) of the Hubbardmodel at U = 3 t from Bethe Ansatz (shifted by µ = U/ m = 0 . t and v h = 2 . t . Panel (b): Dimensionless holonquasi-particle weight ¯ Z h = q mv h Z h in the single spinon ap-proximation for U = 3 t and lattice sizes L = 10 (full squares), L = 12 (empty circles), L = 14 (full circles), compared to thebosonization result (16) (line). with ǫ h ( k h ) given by Eq. (8). The scale factor in Eq.(16) has been fixed by evaluating the Green’s function inthe τ → Z h ( k h ) obtained by Lanczos diag-onalization at U = 3 t . No fitting parameters have beenused: In order to obtain Z h ( k h ) we first evaluated Z k ( Q ),as discussed in Section III, then we divided the result by Z s ( Q ) ∼ q − / evaluated at the spinon momentum Q closest to the spinon Fermi point Q F = π . The twoparameters v h and m are independently obtained fromthe holon spectrum (also shown in Fig. 5a). As usualthe Lanczos data display a very small size dependenceand allow for a precise identification of the holon quasi-particle weight Z h ( k h ). The agreement between the twoexpressions is remarkable for δk h > δk h . Note however that theasymptotic form of the holon quasi-particle weight (16)holds only at low energies and weak coupling, while thecomparison shown in Fig. 5 is performed for U = 3 t .The results at lower values of U/t are plagued by severefinite size effects: in the U → m vanishes exponentially and the dimensionless momen-tum scale m/v h vanishes as well. Therefore, at very weakcoupling, the relevant holon momenta are constrained inan extremely small interval around k h = ± π , a range noteasily accessible due to the momentum quantization rulein finite Hubbard rings. V. RESULTS FOR SrCuO We are now ready to compare our results for the spec-tral function of the 1D Hubbard model with precise pho-toemission data recently obtained for SrCuO . A pre-liminary study, based on the strong coupling limit of FIG. 6: Locus of singularities of the spectral function forthe Hubbard model at U = 3 . t = 0 .
53 eV (opencircles) compared to the experimental results by Kim et al. (full circles) for k ⊥ = 0 . the Hubbard model, pointed out some discrepancies, re-lated to the peak heights and widths . Fig. 6 showsthe singularity loci of the 1D Hubbard model with theparameters t = 0 .
53 eV and U = 3 . et al. . The niceagreement suggests that this material indeed representsa good experimental realization of the simple one dimen-sional Hubbard model. The effects due to inter-chaincoupling, phonons, finite temperature and other pertur-bations appears rather small and mostly limited to thespinon branch. We remark that the same material hasbeen already theoretically investigated on the basis of theHubbard and t-J model by several groups leadingto different sets of parameters both for the hopping in-tegral 0 . . t . . . U . . U/t .In Fig.7 the spectral function has been plotted versusthe binding energy E = − ω for three representative val-ues of the total hole momentum k . The experimental linebroadening reported in Ref. has been also included inthe Hubbard model results, leading to a merging of closepeaks. The density plot clearly reproduces the overallshape defined by the singularities of the spectral functionshown in Fig.6. As expected, most of the spectral weightis indeed concentrated in the first half of the Brillouinzone between the holon and the spinon band. Althoughthe relative intensity of the ARPES signal at the two sin-gularities depends on the details of the band structure,the power-law nature of the divergences implies that theintrinsic width of each peak is always comparable withthe separation between the holon and the spinon branch∆ ω ∼ ǫ h ( k + π ) − [ ǫ h ( π ) + ǫ s ( π + k )]. The average in-tensity can be estimated on the basis of the sum rule(7) and scales as (∆ ω ) − / , getting smaller when the twobranches separate, as shown both in experiments and innumerical calculations . k/ π E /t FIG. 7: Upper panels: A ( k, ω ) calculated via Eq. (4) for threerepresentative values of the momentum k and U = 7 t . Thebinding energy is E = − ω . The lower curves are the convo-lution of the spectral function with a Gaussian with FWHMequal to 0 . t corresponding to an experimental resolution of60 meV . Lower panel: density plot of the spectral functionon the ( k, E ) plane with the experimental broadening. VI. CONCLUSIONS
The single spinon approximation, combined to BetheAnsatz results and Lanczos diagonalizations allows toobtain very accurate results for the dynamic propertiesof a single hole in the one dimensional Hubbard model.The Lehmann representation of the spectral function (2)shows that two separate ingredients combine to define theoverall shape of A ( k, ω ): the excitation spectrum andthe quasi-particle weight. The idea at the basis of ourmethod is to limit the size effects that plague numericalresults by dealing with these two quantities separately: inthe 1D Hubbard model the excitation spectrum is givenexactly by the Bethe Ansatz equations in the thermody-namic limit , while the quasi-particle weight is obtained,in the single spinon approximation, by Lanczos diagonal-ization. Size effects are shown to be negligible and theaccuracy of the approximation can be checked a posteri-ori by a frequency sum rule (7). Our expression for thespectral function of the 1D Hubbard model (4) is con-sistent with the structure predicted by bosonization atweak coupling, provided the quasi-particle weight Z k ( Q )factorizes as shown in Eq. (15). A numerical test carriedout at U = 3 t does not show a convincing quantitativeagreement with the result obtained by the form factorapproach , possibly due to the difficulty to achieve the U → shows that ourmethod allows for a direct comparison between theory and ARPES experiments and for an accurate determi-nation of the Hubbard parameters which best describethe hole dynamics in the material. The spectral func-tion derived here provides a natural explanation of theobserved reduction of the spectral weight in a half of theBrillouin zone and of the broad line-shape detected in ex-periments. Future applications of this method to the caseof cold atoms in optical traps may help in pointing outthe peculiar features of one dimensional physics in otherexperimental realizations of correlated one dimensionalFermi gases.We thank C. Kim and F.H.L Essler for stimulatingcorrespondence. ∗ Electronic address: [email protected] See for instance J. Solyom, Adv. Phys. , 201 (1979). C.Kim, A.Y.Matsuura, Z.-X.Shen, N.Motoyama, H.Eisaki,S.Uchida, T.Tohyama and S. Maekawa, Phys. Rev. Lett. , 4054 (1996). B.J.Kim, H.Koh, E.Rotenberg, S.-J.Oh, H.Eisaki,N.Motoyama, S.Uchida, T. Tohyama, S. Maekawa, Z.-X.Shen and C.Kim, Nature Physics, , 397 (2006). T.E.Kidd, T.Valla, P.D.Johnson, K.W.Kim, G.D.Gu andC.C.Homes, Phys. Rev. B , 054503 (2008). M.Hoinkis, M.Sing, S.Glawion, L.Pisani, R.Valenti, S. vanSmaalen, M.Klemm, S.Horn and R.Claessen, Phys. Rev. B , 245124 (2007). S.Suga, A.Shigemoto, A.Sekiyama, S.Imada, A.Yamasaki,A.Irizawa and S.Kasai, Phys. Rev. B , 155106 (2004). H.Matsueda, N.Bulut, T.Tohyama and S. Maekawa, Phys.Rev. B , 075136 (2005). S.Sorella and A.Parola, J. Phys.:Cond. Matt. ,3589(1992). F.H.L.Essler and A.M.Tsvelik, Phys. Rev. B , 115117(2002). The limitation to intermediate and strong coupling is atechnical one: at weak coupling size effects become morerelevant and the adiabatic procedure employed in this workmight fail. E.H.Lieb and F.Y.Wu, Phys. Rev. Lett. , 1445 (1968). F.Woynarovich, J. Phys. C: Solid State Phys, , 85,(1982). F.Woynarovich, J. Phys. C: Solid State Phys, , 5293(1983). F.H.L.Essler, H.Frahm, F.G¨ohmann, A.Kl¨umper andV.E.Korepin,
The One-Dimensional Hubbard Model , Cam-bridge University Press, (2005). F.D.M. Haldane Phys. Rev. Lett. , 635 (1988). F.D.M.Haldane and M.R.Zirnbauer, Phys. Rev. Lett. ,4055 (1993). In one dimension and exactly at U = ∞ all spin configura-tions are degenerate: however in the U → ∞ limit the spindegeneracy is lifted and the ground state of the Heisenbergmodel is singled out. M.Ogata and H.Shiba, Phys. Rev. B , 2326 (1990). With the notation adopted in this work, at U → ∞ , ǫ s ( Q ) = J π | cos( Q ) | for π < Q < π . Our definition of the spinon momentum Q , which for U → ∞ must reduce to the total momentum of the Heisen-berg wave function in Eq. (3), differs from the one usuallyadopted in Bethe Ansatz studies ( p s ) : Q = π − p s . S.Sorella and A.Parola, Phys. Rev. Lett. , 4604 (1996);Phys. Rev. B , 6444 (1998). At the special values of k defined by the coincidence of thetwo singularities the critical exponent changes from 1 / / In experiments the lower branch is not detected becauseit usually overlaps with other valence bands of the com-pound. The accuracy of the single spinon approximation is simi-lar also at weaker coupling: for U = 4 t the completenesscondition (7) is violated at most by 0 .
015 at all k ’s. Z.V.Popovi´c, V.A.Ivanov, M.J.Kostantinovi´c,A.Cantarero, J.Mart´ınez-Pastor, D.Olgu´ın, M.I.Alonso,M.Garriga, O.P.Khuong, A.Vietkin and V.V.Moshchalkov,Phys. Rev. B , 165105 (2001). A.Koitzsch, S.V. Borisenko, J.Geck, V.B.Zabolotnyy,M.Knupfer, J.Fink, P.Ribeiro, B.B¨uchner and R.Follath,Phys. Rev. B73