Singular forward scattering and the renormalized Bethe ansatz ground state of the two-dimensional Hubbard model

In 1990, P. W. Anderson proposed that Pauli exclusion of intermediate scattering states makes the on-shell forward scattering of opposite-spin quasiparticles in the two-dimensional (2D) Hubbard model singular at finite density: the two-particle eigenvalue equation modified by the Fermi sea yields a finite forward-scattering phase shift, which through an orthogonality-catastrophe mechanism drives the quasiparticle residue $Z$ to zero, replaces the Fermi surface by a broader, singularity-free distribution of Bethe-ansatz charge rapidities, and separates charge and spin into modes with distinct dynamics and elastic-scattering-immune charge transport. The paper left the derivation of the phase shift, the construction of the 2D renormalized Bethe-ansatz ground state, its observable consequences, the spin-scattering origin of residual resistance, and the connection to gauge-field/projective formulations explicitly open. Later work disputed the dilute limit (Engelbrecht-Randeria, Fukuyama et al., Yarlagadda-Kurihara; strong-coupling T-matrix analyses find the Fermi liquid not destroyed), rigorously established the 2D Fermi liquid only at weak short-range coupling, and continued Anderson's program phenomenologically (hidden Fermi liquid) and in gauge-theoretic form (phase-string theory). The microscopic status of the singular forward-scattering mechanism and its consequences at finite density in the no-double-occupancy-projected regime remains unresolved. This problem asks for a controlled microscopic derivation or refutation of that mechanism and of its observable content.

Background

The two-dimensional (2D) Hubbard model — lattice fermions with hopping $t$ and on-site repulsion $U$,

$$H=-t\sum_{\langle ij\rangle\sigma}\left(c^{\dagger}_{i\sigma}c_{j\sigma}+\mathrm{h.c.}\right)+U\sum_i n_{i\uparrow}n_{i\downarrow},$$

where the sums $\langle ij\rangle$ run over nearest-neighbor site pairs and $n_{i\sigma}=c^{\dagger}_{i\sigma}c_{i\sigma}$ counts electrons of spin $\sigma$ on site $i$, is the minimal model of the CuO$_2$ planes of the cuprate high-temperature superconductors, and whether its normal (non-superconducting) state is a Landau Fermi liquid has been disputed since 1987. In a Fermi liquid, low-energy excitations are long-lived quasiparticles whose residue $Z$ (the weight of the free-electron pole in the single-particle Green's function) remains finite in the thermodynamic limit. In one dimension (1D) interactions instead produce a Luttinger liquid: $Z=0$, charge and spin propagate with different velocities, and the exact many-body eigenstates are given by the Bethe ansatz — a constructive wavefunction $\Psi=\sum_P \exp(\sum_l k_{P_l}x_l)[P]$, the sum running over permutations $P$ of the particles' momenta, where the $k_l$ are quasi-momenta (rapidity-like charge quantum numbers), which is exact because 1D two-body scattering is nondiffractive: a collision can only exchange the two particles' momenta. In two companion 1990 papers, P. W. Anderson asserted that the normal metallic state of the 2D Hubbard model is likewise not a Fermi liquid ('Luttinger-liquid' behavior, PRL 64, 1839), and in 'Singular forward scattering in the 2D Hubbard model and a renormalized Bethe ansatz ground state' (PRL 65, 2306) supplied the mechanism, as follows.

In many-body theory the two-particle scattering matrix $T$ — the on-shell two-body amplitude — serves as the effective pseudopotential acting between quasiparticles. For two opposite-spin particles at total momentum $2k_F$ ($k_F$ the Fermi momentum), the Fermi sea modifies the two-particle eigenvalue equation by excluding intermediate states: $L^2/U=\sum'_Q 1/(E-E_Q)$, where $L$ is the linear system size and the prime omits intermediate momenta lying inside the occupied Fermi surface. This Pauli blocking removes all but a fraction $\sim Q^2/(\pi k_F^2)$ of the small-momentum phase space and, Anderson argued, leaves a finite on-shell isotropic forward-scattering phase shift $\delta$ at the Fermi momentum, scaling at small $k_F$ as $\delta\sim\pi/(2\ln(k_Fa))$ with $a$ the lattice (short-distance) cutoff. A finite forward phase shift then acts like the Anderson orthogonality catastrophe — the loss of overlap between two many-body ground states when a scattering potential is switched on — suppressing the quasiparticle residue as $Z\sim e^{-(\delta/\pi)^2\ln L^2}\to0$, with frequency dependence $Z(\omega)\sim\omega^{(\delta/\pi)^2}$ at scale $L_{\mathrm{eff}}\sim v_F/\omega$ (where $v_F$ is the Fermi velocity), so the self-energy is non-analytic at the Fermi surface and Landau Fermi-liquid theory fails. Such a forward phase shift requires a long-range effective interaction, which Anderson identified as a $\sim1/r$ pseudopotential, interpreting it as the effect of Gutzwiller projection — the elimination of doubly occupied sites that moves spectral weight into the upper Hubbard band (the high-energy split band characteristic of a Mott insulator). With real scattering thereby nondiffractive (only spins exchanged), the charge sector was proposed to admit a renormalized Bethe ansatz ground state $\Psi=\sum_{P,Q}\exp(\sum_l k_{Q_l}x_{P_l})[Q,P]$, the double sum running over permutations $P$ of particle positions and $Q$ of rapidities, whose charge rapidities occupy a momentum-space volume larger than the conventional Luttinger Fermi surface and lack a Fermi-surface singularity, so that charge moves by collective 'sliding' of the rapidity distribution rather than by quasiparticles. Rediagonalizing the density-wave channels separates charge and spin velocities (charge-spin separation), and the collective charge modes ('holons') were argued, by analogy with dirty superconductors, to be immune to elastic impurity scattering, so that residual electrical resistance arises from spin scattering alone — a '$T_c=0$ superconductor' absent spin scattering.

The paper left this program explicitly incomplete, and the immediate literature split: J. R. Engelbrecht and M. Randeria computed 2D phase shifts and scattering amplitudes in the dilute regime and found no breakdown of Fermi-liquid behavior, while H. Fukuyama and coworkers analyzed the singularities of perturbation theory for the low-density 2D Hubbard model; the disagreement over which on-shell phase shift is the physical object was, as later commentators recorded, never formally reconciled. The question inherited from the source is the microscopic status of the entire construction at finite density.

Problem Statement

For the two-dimensional Hubbard model at finite density away from half filling (i.e. not at one electron per site), in the strong-coupling regime in which an upper Hubbard band exists — the regime for which the mechanism was proposed — together with its large-$U$ descendant with doubly occupied sites projected out, the $t$-$J$ model of projected hopping $t$ plus antiferromagnetic superexchange $J=4t^2/U$, which the source invokes through the projective transformation eliminating the upper Hubbard band, establish microscopically, by a controlled derivation or a rigorous result, whether the on-shell forward scattering of opposite-spin quasiparticles is singular in the sense of the 1990 proposal, and derive the consequences. Because the source's two-particle kernel is written for the unprojected Hubbard model at finite $U$, the two formulations are not interchangeable targets, and a resolution should state in which formulation it is established. The scientific components, all flowing from this single mechanism, are: (i) a controlled derivation and quantitative evaluation of the on-shell forward-scattering phase shift $\delta$ as a function of density and coupling for the two-particle problem with Pauli-excluded intermediate states (the source gives only the heuristic small-$k_F$ scaling $\delta\sim\pi/(2\ln(k_Fa))$), or a controlled demonstration that no such singular finite phase shift arises at finite density; (ii) the decision this implies between a Landau Fermi liquid (finite quasiparticle residue $Z$, the pole weight defined above) and a $Z=0$, Luttinger-like state — specifically, whether the orthogonality-type suppression $Z\sim e^{-(\delta/\pi)^2\ln L^2}$ survives a controlled treatment of the thermodynamic limit; (iii) the construction (or exclusion) of a renormalized Bethe-ansatz ground state in two dimensions given the claimed nondiffractive kinematics, together with the demonstration that its charge-rapidity distribution replaces the Fermi-surface singularity by collective motion; (iv) the observable consequences of the resulting state — single-particle spectral functions, response functions, and transport coefficients as functions of frequency, temperature, and impurity content, including a microscopic derivation or refutation of the claims that the charge modes are unaffected by elastic impurity scattering and that residual resistance arises only from spin scattering; and (v) the validation or refutation of the source's asserted relation that the exclusion-induced long-range $\sim1/r$ pseudopotential represents the major effect of the Gutzwiller projection that eliminates the upper Hubbard band and is compatible with the gauge-field/projective formulations that enforce the same projection. The common objective is a controlled microscopic theory that decides Fermi liquid versus singular-forward-scattering $Z=0$ ground state for the finite-density 2D Hubbard model and derives its observable content. The source paper explicitly deferred detailed discussion of specific properties to later publication and additionally flagged the treatment of the $l=0$ scattering channel in three dimensions as a further direction ('which we have not yet done'); that item is the paper's own acknowledged extension beyond the 2D core and lies outside this question.

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