Open questions of the two-dimensional Luttinger liquid from forward-scattering phase shifts (fractional statistics without anyons)

P. W. Anderson's 1992 lecture 'Fractional Statistics without Anyons' proposed that the normal state of the two-dimensional Hubbard model is a two-dimensional Luttinger liquid generated by a finite on-shell forward-scattering phase shift — a fractional exclusion statistics acting without anyonic exchange phases — with spin-charge separation, power-law single-particle Green's functions, and superconductivity claimed as a consequence. The unresolved issues left open by the program the 1992 lecture initiated are: the effect of the neglected backward-scattering (exchange) terms on the fixed point and the exponents; the exact angular dependence of the Green's-function exponents beyond the angular average over the one-dimensional result; the extension to three dimensions; the interplay with longer-range Coulomb interactions; the quantitative comparison with cuprate experiments (resistivity, gap structure); and the derivation of the claimed superconducting consequence (its operative mechanism, a gap equation, and the transition temperature). The compound question remains open.

Background

In Landau's Fermi-liquid theory, the low-energy states of an interacting metal are weakly dressed single-particle excitations ('quasiparticles') with the electron's quantum numbers, and the single-particle Green's function has a pole of finite weight. In one spatial dimension this fails: the low-energy modes separate into independent spin and charge collective excitations ('spin-charge separation'), correlation functions decay as power laws with interaction-dependent exponents, and there is no quasiparticle pole; Haldane named this state the 'Luttinger liquid'. Haldane also formulated fractional exclusion statistics, a generalized Pauli principle in which inserting one particle excludes a non-integer number of single-particle states.

The cuprate high-temperature superconductors raised the question of whether such physics can survive in two dimensions. Their normal ('strange') metal possesses a Fermi surface whose volume obeys Luttinger's theorem (the statement that the Fermi volume counts the particle density), yet its resistivity, optics, and spectroscopy deviate strongly from Fermi-liquid expectations. Anderson proposed that the two-dimensional Hubbard model — the minimal repulsive model of the CuO2 planes, with on-site repulsion $U$ — is not a Fermi liquid but a two-dimensional Luttinger liquid, generated by a non-perturbative mechanism: the on-shell forward-scattering S-matrix of two opposite-spin particles embedded in the Fermi sea retains a finite phase shift $\eta$ (the K-matrix calculation, i.e. the two-particle scattering kernel with the exclusion principle imposed on intermediate states, giving $\tan\eta \approx 1/(1/(N(0)U) + \ln k_F a)$, with $N(0)$ the Fermi-level density of states, $k_F$ the Fermi wavevector, and $a$ a short-distance cutoff). Because the Pauli principle forbids the soft recoil processes that otherwise screen forward scattering, inserting one particle shifts the allowed momenta of all others by $\delta Q = \eta/(\pi L)$ in a system of linear size $L$, mimicking fractional statistics — 'fractional statistics without anyons', since no topological exchange phase is involved. The implied Landau interaction function diverges as $f_{kk'} \propto (\epsilon_k - \epsilon_{k'})/(k-k')^2$ when the two momenta approach the same point of the Fermi surface ($\epsilon_k$ is the single-particle energy), which the author argued destroys Fermi-liquid theory.

The low-energy theory uses tomographic (multidimensional) bosonization — an extension of one-dimensional bosonization (the rewriting of low-energy fermions in terms of collective density variables) that assigns density operators $\rho^\Omega_{Q\sigma}$ to each direction $\Omega$ on the Fermi surface. The phase shift enters as a singular forward interaction $\propto (\eta/\pi) v_F Q\, \rho^\Omega_{Q\uparrow} \rho^\Omega_{-Q\downarrow}$ of exactly the one-dimensional Luttinger form ($v_F$ is the Fermi velocity); a Bogoliubov transformation, a linear rotation that diagonalizes two coupled harmonic modes, then produces decoupled charge and spin modes with direction-dependent velocities $v_c(\Omega)$ and $v_s(\Omega)$ — spin-charge separation in two dimensions. The spin-carrying excitations (spinons) carry statistics parameter $\eta/\pi$ (semionic, i.e. intermediate between bosonic and fermionic), and the charge-carrying excitations (holons) are collective objects of characteristic momentum $\sim 2k_F$. The single-particle Green's function was taken as an angular average of the one-dimensional form $G_{1D} \sim e^{ik_F x}/[(x-v_s t)^{1/2}(x-v_c t)^{1/2}(x^2 - v_c^2 t^2)^{\alpha}]$, with $\alpha$ an anomalous exponent; no quasiparticle pole survives. Response functions to zero-frequency perturbations were argued to vanish ('confinement'), implying superconductivity at zero temperature without a pairing mechanism, and the state was argued to be stable against weak interlayer hopping $t_\perp$ below a critical value.

This construction is a proposed fixed-point theory — a renormalization-group fixed point, meaning a scale-invariant low-energy limit of the interacting theory — whose load-bearing steps — the phase shift, the decoupling of Fermi-surface patches, the neglect of exchange (backward-scattering) terms — are exactly the points that the program's own continuation left unresolved, as stated in the problem below. The problem is famous in condensed-matter theory as the question of whether a Luttinger liquid can exist in more than one dimension and whether it describes the cuprate strange metal; it remains a reference point for modern non-Fermi-liquid theory.

Problem Statement

Does the two-dimensional Hubbard model (equivalently, a generic two-dimensional Fermi system with repulsive interactions) realize the two-dimensional Luttinger liquid proposed in P. W. Anderson's 1992 lecture 'Fractional Statistics without Anyons' — a non-Fermi-liquid state with spin-charge separation and power-law single-particle correlations generated by the finite on-shell forward-scattering phase shift, i.e. fractional statistics without anyons — and what are its exact properties? The following issues remain unresolved in the program the lecture initiated, all bearing on this single objective: (i) determine the precise effect of the neglected backward-scattering (exchange) terms, which the lecture expected to induce only a small logarithmic renormalization, on the fixed-point properties and on the exponents; (ii) derive the exact angular dependence, on Fermi-surface position, of the single-particle Green's-function exponents, going beyond the angular average over the one-dimensional result and beyond the angular dependence already carried by the velocities $v_c(\Omega)$ and $v_s(\Omega)$; (iii) settle whether the Luttinger-liquid construction extends to three dimensions, where the lecture suggested only an unstable fixed point and provided neither a concrete model nor a proof; (iv) determine how longer-range Coulomb interactions modify the forward-scattering phase shift and the critical behavior of the putative fixed point; (v) make the quantitative comparison with the experimental properties of the cuprates, including the exact form of the normal-state resistivity and the gap structure; (vi) derive the superconducting transition that the lecture claimed as a consequence of this normal state: determine the operative mechanism (whether via the lecture's 'confinement' argument or a pairing mechanism), a gap equation if applicable, and the transition temperature $T_c$.

References

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