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

TL;DR

It is found that the mathematics of string theory is capable of describing fermionic quantum critical states, and the spectral functions of fermions in the field theory are computed.

Abstract

A central problem in quantum condensed matter physics is the critical theory governing the zero temperature quantum phase transition between strongly renormalized Fermi-liquids as found in heavy fermion intermetallics and possibly high Tc superconductors. We present here results showing that the mathematics of string theory is capable of describing such fermionic quantum critical states. Using the Anti-de-Sitter/Conformal Field Theory (AdS/CFT) correspondence to relate fermionic quantum critical fields to a gravitational problem, we compute the spectral functions of fermions in the field theory. By increasing the fermion density away from the relativistic quantum critical point, a state emerges with all the features of the Fermi-liquid.

Figures (6)

• Figure 1: The phase diagram near a quantum-critical point. Gray lines depict lines of constant $\mu_0/T$: the spectral function of fermions is unchanged along each line if the momenta are appropriately rescaled. As we increase $\mu_0/T$ we crossover from the quantum-critical regime to the Fermi-liquid. (B) The trajectories in parameter space $(\mu_0/T, \Delta_{\Psi})$ studied here. We show the crossover from the quantum critical regime to the Fermi liquid by varying $\mu_0/T$ keeping $\Delta_{\Psi}$ fixed; we cross back to the critical regime varying $\Delta_{\Psi}\rightarrow d/2$ for $\mu_0/T$ fixed. The boundary region is not an exact curve, but only a qualitative indication.
• Figure 2: (A) The spectral function $A(\omega,k)$ for $\mu_0/T=0.01$ and $m=-1/4$. The spectral function has the asymptotic branch cut behavior of a conformal field of dimension $\Delta_{\Psi}=d/2+m=5/4$: it vanishes for $\omega<k$, save for a finite $T$ tail, and for large $\omega$ scales as $\omega^{2\Delta_{\Psi}-d}$. (B) The emergence of the quasiparticle peak as we change the chemical potential to $\mu_0/T=-30.9$ for the same value $\Delta_{\Psi}=5/4$. The three displayed momenta $k/T$ are rescaled by a factor $T_{eff}/T$ for the most meaningful comparison with those in (A); see supp. The insets show the full scales of the peak heights and the dominance of the quasiparticle peak for $k\sim k_F$. (C) Vanishing of the spectral function at $E_F$ for $\Delta_{\Psi}=1.05$ and $\mu_0/T =-30.9$. The deviation of the dip-location from $E_F$ is a finite temperature effect. It decreases with increasing $\mu_0/T$.
• Figure 4: (A) Temperature dependence of the quasiparticle peak for $\Delta_{\Psi}=5/4$ and $k/k_F \simeq 0.5$; all curves have been shifted to a common peak center. (B) The quasiparticle peak width $\delta \sim {\rm Re}\Sigma(\omega,k=k_F)$ for $\Delta_{\Psi}=5/4$ as a function of $T^2$: it reflects the expected behavior $\delta \sim T^2$ up to a critical temperature $T_c/\mu_0$, beyond which the notion of a quasiparticle becomes untenable. (C) The imaginary part of the self-energy $\Sigma(\omega,k)$ near $E_F,~k_F$ for $\Delta_{\Psi}=1.4,~\mu_0/T=-30.9$ . The defining ${\rm Im} \Sigma(\omega,k)\sim (\omega-E_F)^2+\ldots$-dependence for Fermi-liquid quasiparticles is faint in panel (C) but obvious in panel (D). It shows that the intercept of $\partial_{\omega}{\rm Im} \Sigma(\omega,k)$ vanishes at $E_F,k_F$.
• Figure 5: The quasiparticle characteristics as a function of $\mu_0/T$ for $\Delta_{\Psi}=5/4$. Panel (A) shows the change of $k_F, v_F, m_F, E_F$ and the pole strength $Z$ (the total weight between half-maxima) as we change $\mu_0/T$. Beyond a critical value $(\mu_0/T)_c$ we lose the characteristic $T^2$ broadening of the peak and there is no longer a real quasiparticle, though the peak is still present. For the Fermi-liquid $k_F/T$ rather than $\mu_0/T$ is the defining parameter. We can invert this relation and panel (B) shows the quasiparticle characteristics as a function of $k_F/T$. Note the linear relationships of $m_F,E_F$ to $k_F$ and that the renormalized Fermi energy $E^{(ren)}\equiv k_F^2/(2m_F)$ matches the empirical value $E_F$ remarkably well.
• Figure 6: The quasiparticle characteristics as a function of the Dirac fermion mass $-1/2<m<0$ corresponding to $1 <\Delta_{\Psi} <3/2$ for $\mu_0/T=-30.9$. The upper panel shows the independence of $k_F$ of the mass. This indicates Luttinger's theorem if the anomalous dimension $\Delta_{\Psi}$ is taken as an indicator of the interaction strength. Note that $v_F, E_F$ both asymptote to finite values as $\Delta_{\Psi} \rightarrow 3/2$. The lower panel shows the exponential vanishing pole strength $Z$ (the integral between the half-maxima) as $m\rightarrow 0$.