Research Article

String Theory, Quantum Phase Transitions, and the Emergent Fermi Liquid

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Science  24 Jul 2009:
Vol. 325, Issue 5939, pp. 439-444
DOI: 10.1126/science.1174962

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  1. Fig. 1

    (A) The phase diagram near a quantum critical point. Gray lines depict lines of constant μ0/T; the spectral function of fermions is unchanged along each line if the momenta are appropriately rescaled. As we increase μ0/T we cross over from the quantum critical regime to the Fermi liquid. (B) The trajectories in parameter space (μ0/T, ΔΨ) studied here. We show the crossover from the quantum critical regime to the Fermi liquid by varying μ0/T while keeping ΔΨ fixed; we cross back to the critical regime varying ΔΨd/2 for μ0/T fixed. The boundary region is not an exact curve but only a qualitative indication.

  2. Fig. 2

    (A) The spectral function A(ω,k) for μ0/T = 0.01 and m = –1/4. The spectral function has the asymptotic branchcut behavior of a conformal field of dimension ΔΨ = d/2 + m = 5/4: It vanishes for ω < k, save for a finite T tail, and for large ω it scales as Embedded Image. (B) The emergence of the quasi-particle peak as we change the chemical potential to μ0/T = –30.9 for the same value ΔΨ = 5/4. The three displayed momenta k/T are rescaled by a factor Teff/T for the most meaningful comparison with those in (A) (25). The insets show the full scales of the peak heights and the dominance of the quasi-particle peak for k ~ kF. (C) Vanishing of the spectral function at EF for ΔΨ = 1.05 and μ0/T = –30.9. The deviation of the dip location from EF is a finite temperature effect; it decreases with increasing μ0/T.

  3. Fig. 3

    Maxima in the spectral function as a function of k0 for ΔΨ = 1.35 and μ0/T = –30.9. Asymptotically for large k the negative-k branchcut recovers the Lorentz-invariant linear dispersion with unit velocity, but with the zero shifted to –μ0. The peak location of the positive-k branchcut that changes into the quasi-particle peak changes noticeably. It gives the dispersion relation of the quasi-particle near (EF, kF). The change of the slope from unity shows renormalization of the Fermi velocity. This is highlighted in the inset. Note that the Fermi energy EF is not located at ωAdS = 0. The AdS calculation visualizes the renormalization of the bare chemical potential μ0 = μAdS to the effective chemical potential μF = μ0EF felt by the low-frequency fermions.

  4. Fig. 4

    (A) Temperature dependence of the quasi-particle peak for ΔΨ = 5/4 and k/kF ≈ 0.5; all curves have been shifted to a common peak center. (B) The quasi-particle peak width δ ~ Re Σ(ω, k = kF) for ΔΨ = 5/4 as a function of T2; it reflects the expected behavior δ ~ T2 up to a critical temperature Tc0, beyond which the notion of a quasi-particle becomes untenable. (C and D) The imaginary part of the self-energy Σ(ω, k) near EF, kF for ΔΨ = 1.4, μ0/T = –30.9. The defining Im Σ(ω, k) ~ (ω – EF)2 + … dependence for Fermi-liquid quasi-particles is faint in (C) but obvious in (D). It shows that the intercept of ∂ω Im Σ(ω, k) vanishes at EF, kF.

  5. Fig. 5

    The quasi-particle characteristics as a function of μ0/T for ΔΨ = 5/4. (A) The change of kF, vF, mF, EF, and the pole strength Z (the total weight between half-maxima) as we change μ0/T. Beyond a critical value (μ0/T)c we lose the characteristic T2 broadening of the peak and there is no longer a real quasi-particle, although the peak is still present. For the Fermi liquid, kF/T rather than μ0/T is the defining parameter. (B) We can invert this relation, and (B) shows the quasi-particle characteristics as a function of kF/T. Note the linear relationships of mF and EF to kF and that the renormalized Fermi energy E(ren)kF2/(2mF) matches the empirical value EF remarkably well.

  6. Fig. 6

    The quasi-particle characteristics as a function of the Dirac fermion mass –½ < m < 0 corresponding to 1 < ΔΨ < Embedded Image for μ0/T = –30.9. The upper panel shows the independence of kF of the mass. This indicates Luttinger’s theorem if the anomalous dimension ΔΨ is taken as an indicator of the interaction strength. Note that vF and EF both approach finite values as ΔΨEmbedded Image. The lower panel shows the exponential vanishing pole strength Z (the integral between the half-maxima) as m → 0.

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