Fermi surfaces and gauge-gravity duality

TL;DR

The paper investigates zero-temperature compressible quantum matter, where a conserved U(1) charge $\mathcal{Q}$ yields Fermi surfaces constrained by the Luttinger theorem $\langle \mathcal{Q} \rangle = \sum_{ll\text{ fermions}} q_\u001ell V_\u001ell$. It surveys condensed-matter models (doublon metal, Bose-Fermi mixtures, and fractionalized phases) and then extends the discussion to holographic contexts by constructing ABJM-like models in $d=2$ and ${\cal N}=4$ SYM in $d=3$ under nonzero chemical potential, revealing FL, NFL, and FL* phases with varying gauge dynamics. The analysis highlights how gauge fluctuations and deconfinement yield non-Fermi-liquid behavior and how some NFL/FL* phases can persist as intermediate-energy descriptions before superconducting instabilities set in. The work thereby strengthens the bridge between compressible quantum matter and gauge-gravity duality, suggesting holographic realizations for FL* and related phases and outlining paths for future embedding in gravity duals.

Abstract

We give a unified overview of the zero temperature phases of compressible quantum matter: i.e. phases in which the expectation value of a globally conserved U(1) density, Q, varies smoothly as a function of parameters. Provided the global U(1) and translational symmetries are unbroken, such phases are expected to have Fermi surfaces, and the Luttinger theorem relates the volumes enclosed by these Fermi surfaces to <Q>. We survey models of interacting bosons and/or fermions and/or gauge fields which realize such phases. Some phases have Fermi surfaces with the singularities of Landau's Fermi liquid theory, while other Fermi surfaces have non-Fermi liquid singularities. Compressible phases found in models applicable to condensed matter systems are argued to also be present in models obtained by applying chemical potentials (and other deformations allowed by the residual symmetry at non-zero chemical potential) to the paradigmic supersymmetric gauge theories underlying gauge-gravity duality: the ABJM model in spatial dimension d=2, and the N=4 SYM theory in d=3.

Figures (14)

• Figure 1: The non-Fermi liquid (NFL) doublon metal phase. The blue blurry shading of the Fermi surface indicates the coupling of the $f_\pm$ fermions to a fluctuating gapless gauge field, so that the fermion Green's function has the singular behavior of Eq. (\ref{['nfl']}) near the Fermi surface (in $d=2$). Despite the non-Fermi liquid character of the fermion excitations, the value of $k_F$, and so the location of the Fermi surface, is sharply defined. The global SU(2) spin is carried by the bosons $b_{\pm \sigma}$ which are gapped in this phase.
• Figure 2: Schematic phase diagram of the theory $\mathcal{L}_{bf}$ in Eq. (\ref{['Lbf']}) for a strong interaction $g$; with weak interactions, the two intermediate phases are not present---see Ref. powell for more details. The Fermi liquid (FL) phases have no Bose condensate, and the two global U(1) symmetries constrain the two Fermi surfaces of the $f_\sigma$ and $c_\sigma$ fermions via the Luttinger relation in Eq. (\ref{['Vcf']}). Unlike the model of Section \ref{['sec:doublon']}, the Fermi surface excitations are not coupled to a fluctuating gauge field, and Fermi liquid-like quasiparticles survive near the Fermi surface; this is indicated by the uniform shading within the Fermi surface. The case with only a $c_\sigma$ Fermi surface is allowed only for $\left\langle Q \right\rangle = \left\langle Q_b \right\rangle$. The SF+FL phases have both a Bose condensate and Fermi surfaces; the non-zero $\langle b \rangle$ hybridizes the $f_\sigma$ and $c_\sigma$ fermions, the Fermi surface quasiparticles are therefore linear combinations of $f_\sigma$ and $c_\sigma$. There can be one or two such Fermi surfaces as shown above, depending upon parameters. There is only one Luttinger constraint on the volumes of the Fermi surfaces in the SF+FL phases. Here and in the following figures, we follow the convention of shading $f_\sigma$ Fermi surfaces blue, $c_\sigma$ Fermi surfaces red, and Fermi surfaces of hybridized fermions purple.
• Figure 3: Schematic phase diagram of the theory $\mathcal{L}_{*}$ in Eq. (\ref{['Ls']}). This is similar to the phase diagram in Fig. \ref{['pd_bf']}, but some of the Fermi surface excitations are now coupled to a fluctuating gapless gauge field: such Fermi surfaces are indicated by the blurry shading, as in Fig. \ref{['pd_nfl']}. The colors of the Fermi surfaces are chosen as in Fig. \ref{['pd_bf']}. The fractionalized Fermi liquid (FL*) phase has Fermi surfaces of both gauge-neutral and gauge-charged fermions. The spin liquid (SL) phase has only a gauge-charged Fermi surface and is incompressible, and is the only incompressible phase in our phase diagrams. Unlike Fig. \ref{['pd_bf']}, the phases with a $b$ condensates are not superfluids because there is no gauge-invariant condensate which violates a global U(1) conservation.
• Figure 4: Mean field phase diagram of the theory $\mathcal{L}_1$ in Eq. (\ref{['l1']}) in the limit of very large $\epsilon_1$, when we have $\langle b_\pm \rangle =0$, and there is no possibility of SF order. All phases are compressible, the global U(1) symmetry is preserved, and the phases are distinguished by the configurations of the Fermi surfaces. The phase boundaries in this limit are at $\epsilon_2=2\mu$ and $\mu=0$. The Fermi surfaces are colored as in Fig. \ref{['pd_ffl']}. Fermi surfaces whose volumes are degenerate by symmetry are shown by a single circle, while inequivalent Fermi surfaces are shown separately.
• Figure 5: The phase diagram of $\mathcal{L}_1$ for $g_1=0$ and $g_2 = 1$. The other parameters are shown, or described in the text. All the phases labeled SF have $\langle b_\pm \rangle \neq 0$, while the remainder have $\langle b_\pm \rangle =0$. The Fermi surfaces are colored as in Fig. \ref{['pd_ffl']} and \ref{['pd_abjm0']}. For $g_1 =0$, all but the $v$ term in the energy depend only upon $|b_+|^2 + |b_-|^2$; we have assumed a small $v < 0$, so that degeneracy of the condensate is lifted, and we have $\langle b_+ \rangle = \langle b_- \rangle \neq 0$ in all the phases with a SF label.