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Fermi surfaces and gauge-gravity duality

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Liza Huijse, Subir Sachdev

High Energy Physics - Theory · Submitted 26 Apr 2011 · 81 references

Pith's verdict: A useful survey of compressible quantum matter with two new mean-field toy models; the bridge to full ABJM and N=4 SYM is explicitly a conjecture, not a derivation, and the paper mostly says so.

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arXiv ID1104.5022 (version 2)
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CitationLiza Huijse, Subir Sachdev. "Fermi surfaces and gauge-gravity duality." arXiv:1104.5022 (2011). https://pith.science/paper/QUPAD5UU
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The paper

The paper claims

The compressible phases of condensed matter — Fermi liquid, non-Fermi liquid, and fractionalized Fermi liquid — should also appear in the ABJM and $N=4$ super Yang-Mills theories once a chemical potential is turned on.

The reading

The paper asks a bridging question: when a chemical potential is applied to the two gauge theories that anchor gauge-gravity duality — the ABJM theory in two spatial dimensions and $N=4$ super Yang-Mills in three — what zero-temperature phases result? Its organizing principle is the Luttinger theorem, which says that in any compressible state where the global charge symmetry and translation invariance survive, the total charge density is fixed by the momentum-space volumes enclosed by the Fermi surfaces of all fermions, whether or not those fermions are the original microscopic particles. Using that constraint, the authors survey the known compressible phases of condensed matter models — the doublon metal, Bose-Fermi mixtures, and the fractionalized Fermi liquid — and then write down simplified toy Lagrangians modelled on the particle content and symmetries of the two supersymmetric theories at nonzero chemical potential. Minimizing the mean-field free energy of those toys produces phase diagrams containing the same cast: ordinary Fermi liquids, non-Fermi liquids with Fermi surfaces coupled to gapless gauge fields, and fractionalized phases where some Fermi surfaces are hidden from charge-carrying probes. The claim a sympathetic reader should take away is that these are not curiosities of condensed-matter Lagrangians but a plausible guide to what the holographic theories do at finite density.

What carries the argument

The load-bearing tool is the generalized Luttinger theorem of Eq. (1.4), $langle Qrangle = sum_{ell,in,mathrm{fermions}} q_ell V_ell$, which ties the conserved charge density to the momentum-space volumes enclosed by every fermionic Fermi surface, including composite or fractionalized fermions introduced through a Hubbard-Stratonovich decoupling. A Fermi surface is defined precisely as the locus where the inverse Green's function vanishes at zero frequency, $G_ell^{-1}(k=k_F,omega=0)=0$, together with the mild condition that the imaginary part of the inverse Green's function vanish away from that locus as $omega to 0$. This definition is what lets the theorem apply beyond Landau Fermi liquid theory, to surfaces with non-Fermi liquid singularities of the form $G^{-1}(k,omega)=q^{1-eta}Phi(omega/q^{z/2})$. The theorem does the bookkeeping work of the whole paper: it converts a single number, the charge density, into a constraint on how many Fermi surfaces can exist and how large they are, and it is what makes the FL* phase distinctive, because there the Fermi volume seen by gauge-invariant probes is smaller than the total density, with the deficit carried by hidden gauge-charged surfaces.

Extended reading notes

Core claim

The paper's central claim is that the compressible phases already catalogued in condensed matter — Fermi liquid (FL), non-Fermi liquid (NFL) with Fermi surfaces blurred by gapless gauge fluctuations, and fractionalized Fermi liquid (FL*) with coexisting gauge-charged and gauge-neutral Fermi surfaces — are also present in models built from the symmetries and particle content of the ABJM theory in $d=2$ and $N=4$ super Yang-Mills in $d=3$ after a chemical potential is applied. The argument runs through two toy Lagrangians, $L_1$ in Eq. (5.2) and $L_3$ in Eq. (6.2), in which the gauge group is reduced to $U(1)$ or $SU(2)$, flavor indices are dropped, and the particles are treated non-relativistically. Minimizing the mean-field free energy subject to gauge-charge neutrality yields the phase diagrams of Figs. 4 through 12, and those diagrams contain FL, NFL, and FL* phases in the ABJM-inspired case, and FL plus superfluid phases in the SYM-inspired case. A sharp structural result separates the two: in $L_1$ the fermion pairing term couples to a boson pair $b_+b_-$, so the energy correction behaves as $g_1^2|b_+|^2|b_-|^2$ times a logarithm and can have a stable minimum at $b_+=b_-=0$, which is what permits NFL and FL* phases to survive; in $L_3$ the pairing couples to a single boson $g_1^2|Phi^a|^2$, giving the standard BCS logarithm that makes any phase with $lambda^a$ Fermi surfaces unstable to superfluidity, so no NFL or FL* appears there.

Load-bearing premise

The bridge from condensed matter to the supersymmetric theories runs entirely through simplified toy Lagrangians that reduce the gauge group to $U(1)$ or $SU(2)$, drop flavor indices, make the particles non-relativistic, and are analyzed at mean-field level with gauge fluctuations omitted; the paper assumes, without deriving it, that this reduction preserves the qualitative phase structure of the full theories.

Editorial extensions

If this is right

  • If the toy models are faithful, the zero-temperature phase diagrams of the ABJM theory in $d=2$ and $N=4$ SYM in $d=3$ at nonzero chemical potential contain NFL and FL* phases, not merely the superconducting or superfluid phases that holographic studies had already found.
  • The FL* phase offers a concrete target for gravity duals: gauge-neutral fermions scatter strongly off a gauge-charged sector, so a small Fermi volume seen by charge-carrying probes becomes a prediction to look for in a classical gravity solution.
  • The Luttinger count gives an experimentally checkable bookkeeping rule for any compressible state whose global charge symmetry is unbroken, even when the quasiparticles near the Fermi surface are not Fermi liquid-like.
  • Which chemical potential is turned on matters qualitatively: with $mu_1neq 0$ the scalar is doubly charged and cannot mix with fermion pairs, so no deconfined NFL or FL* phases appear, whereas with $mu_1=mu_2neq 0$ the scalar carries unit charge, the pairing term $g_1$ is forbidden, and deconfined FL* and NFL phases reappear in Figs. 11 and 12.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The pairing instability the paper flags as an affliction of every NFL and FL* phase — gauge-boson exchange driving Cooper pairing that gaps the Fermi surfaces — is the natural first thing a dual gravity computation should be checked against, since a classical bulk solution with a charged scalar condensate would be the gravity-side image of that instability.
  • Because the Luttinger constraint is stated for any unbroken global $U(1)$, it supplies a consistency check on holographic models independent of their details: any proposed gravity dual of a compressible state that reports a charge density inconsistent with its fermionic Fermi volumes is either not in the phase it claims or has broken the symmetry.
  • The paper's reduction to non-relativistic toy Lagrangians is a controlled-looking but undemonstrated step; a natural follow-up is to ask whether the same phase diagrams survive keeping the full $U(N)times U(N)$ or $SU(N)$ gauge group and relativistic dispersion at leading order in large $N$.
  • The distinction between the two models turns on whether the pairing term couples to one boson or a boson pair, which suggests a general selection rule: whether a holographic theory at finite density admits deconfined NFL phases depends on the charge assignment of its scalar sector under the residual symmetry, a criterion that could be applied to other supergravity truncations.
Original 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.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,451 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.13 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

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.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 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. contribution 0.88 0.44 0.12 asserted 0.92
s1 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>. background 1.00 0.06 0.09 hedged possibility 0.94
s2 We survey models of interacting bosons and/or fermions and/or gauge fields which realize such phases. contribution 0.92 0.55 0.06 asserted 0.76
s3 Some phases have Fermi surfaces with the singularities of Landau's Fermi liquid theory, while other Fermi surfaces have non-Fermi liquid singularities. result 0.69 0.16 0.09 asserted 0.97
s4 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. result 0.43 0.46 0.61 hedged possibility 0.80

What kind of paper the classifier reports this is: review 1.00 · evidence from literature 0.52.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. FIG. 1: The non-Fermi liquid (NFL) doublon metal phase. The blue blurry shading of the Fermi
Figure 2. FIG. 2: Schematic phase diagram of the theory
Figure 3. FIG. 3: Schematic phase diagram of the theory
Figure 4. FIG. 4: Mean field phase diagram of the theory
Figure 5. FIG. 5: The phase diagram of
Figure 6. FIG. 6: As in Fig. 5, with
Figure 7. FIG. 7: As in Fig. 5, but with
Figure 8. FIG. 8: As in Fig. 7, with
Figure 9. FIG. 9: Mean field phase diagram of the SYM-inspired theory
Figure 10. FIG. 10: As in Fig. 9, but with a different value of
Figure 11. FIG. 11: Mean field phase diagram of the SYM-inspired theory
Figure 12. FIG. 12: As in Fig. 11, but with a different value of
Figure 13. FIG. 13: Mean-field phase diagram of
Figure 14. FIG. 14: We plot

Figures reproduced from arXiv:1104.5022 by the authors.

References
How later papers used it

65 later papers cite it

Evidence: substantial. 53 of the citing papers share no author with it, and 5 of those carry a confident label from Pith's citation classifier.

Pith read 25 of the 119 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Compares with it 1 of 4 papers with this label
Cites it as support 1 of 3 papers with this label
How the literature uses it

62 arXiv papers cite this in 116 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 75 sentences without a confident label

Builds on it 1 paper
Uses its method or tool 1 paper
Compares with it 4 papers
  • The three phases that appear have important similarities with the FL, FL$^*$ and NFL phases discussed in [ref] .

    Fractionalization of holographic Fermi surfaces · Context · 1.00

  • With the truncation $A_μ^(1)=-A_μ^(2)=A_μ$ and recalling that the covariant derivative $D_μ= (_̣μ-iA_μ^(i))$ acting on both $ϕ_i$ and $ψ_i$, we obtain the action S = N(N-1)∫d^3x[kħ/4πϵ^μνρA_μF_νρ+f_+^†((_̣τ-iA_τ)-(∇⃗-iA⃗)^2/ 2m-F_12/2m)f_+. .+f_-^†((_̣τ+iA_τ)-(∇⃗+iA⃗)^2/ 2m+F_12/2m)f_- +b^†_+((_̣τ-iA_τ)-(∇⃗-iA⃗)^2/2m)b_+. +b^†_-((_̣τ+iA_τ)-(∇⃗+iA⃗)^2/2m)b_- -2πħ^2/mkb^†_+b^†_-b_+b_- .-πħ^2/km[b^†_+b_+f^†_-f_-+b^†_-b_-f^†_+f_+]], with an extra CS term, and where to facilitate comparison to ([ref]) we have denoted $ψ_1=f_+, ψ_2=f_-,ϕ_1=b_+,ϕ_2=b_-$ and changed to the conventions of [ref] .

    Nonrelativistic limit of the abelianized ABJM model and the ADS/CMT correspondence · Comparison with nonrelativistic abelian toy models for ABJM · 1.00

  • This behavior is reminiscent of various theoretical models based on a large spatial dimension mean field approximation [ref] , such as the gapless quantum liquids of [ref] and the “local quantum critical point” of [ref] [1]Some suggestions for a connection have been made in [ref] ..

    Lectures on holographic non-Fermi liquids and quantum phase transitions · Semi-local quantum liquids · 0.86

Cites it as support 2 papers
Cites it as background 24 papers

What it cites · 116 sentences without a confident label

Uses the method or tool of 2 papers
  • We therefore make the following mode expansion [ref] in terms of canonical Fermi operators $a_i^a (k)$ and $b_i^a (k)$ λ_i^a (x) = ∫d^3k/(2 π)^3 ( [ - sin( θ_k/2) e^-i ϕ_k; cos( θ_k/2) ] ) [ a_i^a (k) e^- i k ·x + b_i^a † (k) e^ i k ·x ], where $θ_k$ and $ϕ_k$ are the polar and azimuthal angles of $k$, so under $k →- k$, $θ→π- θ$ and $ϕ→ϕ+ π$.

    Color-Flavor Locking and Chiral Symmetry Breaking in High Density QCD · Lagrangian for $𝒩=4$ Super Yang-Mills theory · 0.90

  • Now it is useful to introduce a fermionic `molecular' field $c_σ$ by a Hubbard-Stratonovich decoupling [ref] of the two-body interaction: ℒ_bf = f_σ^†[ ∂_τ- ∇^2/2 m_f - μ] f_σ+ b^†[ ∂_τ- ∇^2/2 m_b - μ_b ] b + u/2 (b^† b )^2 + 1/g c_σ^†c_σ- c_σ^†b f - f^†b^†c_σ.

    Sum Rules and Ward Identities in the Kondo Lattice · Boson-fermion mixture · 0.99

Compares with 4 papers
  • Instead, the boson and gauge sector of $ℒ_0$ is closely related to theories of doped antiferromagnets [ref] in which the global charge is carried by the bosons, and the spin is carried by the fermions.

    Competing orders II: the doped quantum dimer model · Theory similar to the ABJM model in $d=2$ · 0.83

  • Instead, the boson and gauge sector of $ℒ_0$ is closely related to theories of doped antiferromagnets [ref] in which the global charge is carried by the bosons, and the spin is carried by the fermions.

    Doping a Mott Insulator: Physics of High Temperature Superconductivity · Theory similar to the ABJM model in $d=2$ · 0.83

  • Instead, the boson and gauge sector of $ℒ_0$ is closely related to theories of doped antiferromagnets [ref] in which the global charge is carried by the bosons, and the spin is carried by the fermions.

    Dual vortex theory of doped Mott insulators · Theory similar to the ABJM model in $d=2$ · 0.83

Cites as background 7 papers

Discussion

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