Complete the finite-temperature phase diagram of the commensurate spin-density-wave metal–insulator transition in layered cuprates: disorder effects, incommensurate competitors, and tricritical robustness

In lightly doped layered cuprates the metal–insulator transition coincides with the onset of antiferromagnetism. Abrikosov (Physica C 391, 147 (2003)) modeled this with a planar tight-binding band hosting commensurate spin-density-wave (SDW) order at the copper-sublattice wavevectors, finding a Néel temperature with a non-BCS square-root-in-coupling exponent inherited from saddle-point logarithms, a phase diagram in the temperature–doping plane containing second-order lines, a tricritical point, and a first-order line, a $T=0$ first-order criterion $\mu_1 = V_{N0}/\sqrt{2}$, and an Abrikosov–Gor'kov-type pair-breaking suppression of the transition temperature by impurity scattering. However, the disordered case was completed only at $T=0$ and in limiting cases: the full finite-temperature free energy with disorder, the possible stabilization of incommensurate (FFLO-like) antiferromagnetic states under weak disorder, and the quantitative robustness of the tricritical coordinates to subleading logarithmic terms were left open. This problem asks for the completion of that analysis: the full finite-$T$ disordered phase diagram (including the shifted tricritical point and first-order line), a controlled determination of whether incommensurate/inhomogeneous SDW phases intervene at low disorder, and the quantitative correction of the tricritical coordinates beyond leading logarithmic accuracy.

Background

The parent compounds of the high-temperature superconducting cuprates (for example La$_{2-x}$Sr$_x$CuO$_4$ near half filling) are antiferromagnetic insulators, and upon light doping they cross over to metals; understanding this metal–insulator transition (MIT) is a prerequisite for any complete theory of the cuprate phase diagram. Two limiting theoretical pictures existed: an itinerant-band picture, in which antiferromagnetism is a spin-density wave (SDW) — a periodic modulation of the electron spin density that opens an energy gap on parts of the Fermi surface, analogous to an antiferromagnetic version of a charge-density wave — and a localized-spin (Heisenberg) picture valid deep in the insulating state. Angle-resolved photoemission spectroscopy (ARPES; a technique measuring the occupied electronic states as a function of momentum) shows that a large Fermi surface persists in the metallic phase up to the transition, which favors a band description near the transition.

Abrikosov's paper 'Metal–insulator transition in layered cuprates (SDW model)' (Physica C 391, 147 (2003)) built such a band description. The single-particle spectrum comes from a planar LCAO (linear combination of atomic orbitals) tight-binding band — the Emery-type band model of the CuO$_2$ plane — with O $2p$ and Cu $3d_{x^2-y^2}$ orbitals, nearest-neighbor $p$–$d$ hopping $t$, in-plane lattice period $a$, and a large Cu–O level offset $\varepsilon_0 \gg t$, following the tight-binding parametrization of Mishonov and Penev. The key feature is extended saddle points (van Hove points, where the electron group velocity vanishes and the two-dimensional density of states becomes logarithmically singular) on the Fermi contour near the Brillouin-zone diagonals, which produce double-logarithmic enhancements of the SDW instability. The SDW is taken commensurate with the copper sublattice, i.e. with ordering wavevectors $\mathbf{q} = \pi(1,\pm 1)$, so the transition is a band insulator–metal transition driven by an exchange interaction of strength $g$, not by Mott-Hubbard physics; the staggered mean field $V$ gaps the saddle regions and reconstructs the large Fermi contour. Doping is represented by the chemical-potential shift $\mu_1 = \mu - (\varepsilon_0 + 4t^2/\varepsilon_0)$ from the half-filled (saddle-point) value. Within static mean-field theory the paper derived: (1) the Néel temperature (the antiferromagnetic ordering temperature) at half filling, $T_{N0} = (2\pi\gamma t^2/\varepsilon_0)\exp[-\sqrt{8}\,\pi a t/(g\varepsilon_0)^{1/2}]$ with $\gamma = e^C \approx 1.78$ ($C$ the Euler–Mascheroni constant), whose exponent contains the interaction only under a square root — a non-BCS feature inherited from the saddle-point logarithms, and the same parametric structure found earlier for van-Hove-driven instabilities by Dzyaloshinskii, Hirsch–Scalapino, and Labbé–Bok; (2) an implicit self-consistency equation for $V(T, \mu_1)$ as a sum over discrete imaginary fermionic (Matsubara) frequencies — the standard finite-temperature many-body representation — with dimensionless parameters $c = (V/2\pi T)^2$ and $b = (\mu_1/2\pi T)^2$; (3) a mean-field phase diagram in the $(T, \mu_1)$ plane with a continuous (second-order) line at small $\mu_1$, a tricritical point (where the line changes character) at reduced coordinates $m \approx 1.073$ and $t \approx 0.561$ (with $m \equiv \mu_1/T_{N0}$ and $t \equiv T_N/T_{N0}$, where $T_N$ is the transition temperature at the given $\mu_1$; this reduced-temperature symbol is distinct from the hopping integral $t$), and a first-order line beyond it, with the analytic zero-temperature criterion $\mu_1 = V_{N0}/\sqrt{2}$, where $V_{N0} = (\pi/\gamma)T_{N0}$ is the zero-temperature staggered field; (4) explicit logarithmic-accuracy maps from $\mu_1$ to the excess carrier density $n_e$ and hence to the observed Fermi-surface volume; and (5) disorder effects via impurity scattering in the Born approximation (the leading weak-scattering self-energy treatment), which acts like Abrikosov–Gor'kov magnetic-impurity pair breaking in superconductors and gives $\ln(T_{N0}/T_N) = \Psi(1/2 + 1/(2\pi T_N\tau)) - \Psi(1/2)$ ($\Psi$ the digamma function, $1/\tau$ the quasiparticle scattering rate), plus a $T=0$ free-energy estimate showing that disorder shrinks the antiferromagnetic domain but preserves both the first- and second-order regimes. Two earlier band-picture lineages feed into this problem. Dzyaloshinskii's van Hove analysis treats the doubly logarithmic Cooper and density-wave polarization loops by parquet summation (their systematic mutual insertion), finding superconducting — and, at zone-boundary singularities, mixed superconducting–antiferromagnetic–charge-ordered — instabilities with the same non-BCS $\exp(-\mathrm{const}/\sqrt{g})$ scale that the source paper, citing it together with Hirsch–Scalapino and Labbé–Bok, recovers in $T_{N0}$. The nesting-based analyses in the Friedel–Kohmoto line, by contrast, predict incommensurate antiferromagnetism — an ordering wavevector drifting roughly linearly with doping — and continuous transitions away from half filling, at odds with the commensurate order observed in layered cuprates.

The paper is explicit about what the analysis does not do. Spin fluctuations in the metallic phase (and their interplay with superconductivity) are omitted; the model is strictly two-dimensional, single-band, nearest-neighbor; incommensurate (FFLO-like, i.e. spatially modulated like the Fulde–Ferrell–Larkin–Ovchinnikov superconducting states) antiferromagnetism is argued to be destroyed by disorder but is not treated; the disordered free energy is evaluated only at $T=0$ and in limiting cases, so the full finite-$T$ disordered phase diagram — the fate of the tricritical point and first-order line under disorder — was not computed; and the numerical tricritical coordinates carry unquantified sensitivity to the discarded subleading logarithmic terms.

Problem Statement

Within Abrikosov's commensurate SDW model of the metal–insulator transition in layered cuprates (planar LCAO band with extended saddle points, exchange interaction $g$, staggered mean field $V$ at the ordering wavevectors $\mathbf{q} = \pi(1,\pm 1)$, chemical-potential shift $\mu_1$ as the doping variable, and Born-approximation impurity scattering with rate $1/\tau$), complete the phase-diagram analysis that the source paper left unfinished, specifically:

1. Compute the full finite-temperature free-energy difference between the antiferromagnetic (insulating) and paramagnetic (metallic) phases in the presence of impurity scattering, and determine the complete disordered phase diagram in the $(T, \mu_1, 1/\tau)$ parameter space: the second-order line, the shifted tricritical point, the first-order line, and the $T \to 0$ criterion generalizing $\mu_1 = V_{N0}/\sqrt{2}$, establishing whether disorder changes the transition order (e.g. removes the first-order regime) or only shifts boundaries.

2. Decide, by controlled calculation rather than assertion, whether incommensurate (FFLO-like) or other spatially modulated SDW phases occur in this band structure at finite $\mu_1$, and under what disorder/cleanliness conditions they are stabilized or destroyed — i.e. test the paper's claim that disorder destroys incommensurate antiferromagnetism.

3. Quantify the sensitivity of the tricritical coordinates ($m \approx 1.073$, $t \approx 0.561$ in the clean case) and of the first-order thresholds to the leading-logarithmic approximations (retaining only $\ln^2$ and leading $\ln$ terms) by evaluating the subleading corrections.

The three sub-questions above jointly characterize the finite-temperature, finite-disorder phase topology of the SDW-driven metal–insulator transition in this model. (The separate open-ended directions the paper lists — dynamical spin fluctuations, their interplay with superconductivity, and extensions to three-dimensional or multiband models — lie outside the scope of this problem.)

References

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