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Stellar spectroscopy: Fermions and holographic Lifshitz criticality

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Sean A. Hartnoll, Diego M. Hofman, David Vegh

High Energy Physics - Theory · Submitted 16 May 2011 · 40 references

Pith's verdict: A careful, honest WKB computation of the electron-star fermion Green's function; the Luttinger framing is a bit stronger than the algebra supports, but the core results hold.

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arXiv ID1105.3197 (version 1)
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CitationDavid Vegh, Diego M. Hofman, Sean A. Hartnoll. "Stellar spectroscopy: Fermions and holographic Lifshitz criticality." arXiv:1105.3197 (2011). https://pith.science/paper/ETYWSOCG
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The paper

The paper claims

A holographic electron star turns out to hold a dense ladder of Fermi surfaces, not one.

The reading

The paper asks what a strongly interacting metal at finite charge density looks like when its gravity dual is an 'electron star': a fluid of charged fermions filling a region outside a horizon in anti-de Sitter space, with an emergent Lifshitz-scaling interior. The authors compute the retarded Green's function $G^R(ω,k)$ of the gauge-invariant fermion operator that makes up the star, by solving the bulk Dirac equation in a WKB approximation and matching across the turning points of an effective Schrödinger potential. What comes out is a dense ladder of Fermi surfaces rather than a single Fermi-liquid surface, with poles spaced closely in frequency and momentum. If the result holds, it gives a concrete non-Fermi liquid at finite density in which a sharp notion of Fermi volume survives, and it locates the energy at which quasiparticles stop being well defined: the dispersion $ω ∼ k^{z}$ set by the emergent Lifshitz exponent.

What carries the argument

The engine is a WKB treatment of the bulk Dirac equation, recast as a zero-energy Schrödinger problem $-Φ'' + γ^{2}VΦ = 0$ with $V = g(r^{2}k^{2} + m^{2} - (ω+h)^{2}/f)$, where $γ = eL/κ ≫ 1$ is the large parameter forced by the electron-star limit. The sign structure of $V$ is controlled by the local Fermi momentum $k_{F}(r)$, defined by $r^{2}k_{F}^{2} = h^{2}/f - m^{2}$, which vanishes at the star edge and peaks at an extremal radius $r_{⋆}$ giving $k^{⋆}_{F}$. Three regimes follow from the turning points: no oscillatory region (case I, $k > k^{⋆}_{F}$), an intermediate oscillatory well (case II, $k^{⋆}_{F} > k$ and $ω^{1/z}$), and direct tunneling into the Lifshitz throat (case III, small $k$). The Fermi momenta are read off from the Bohr-Sommerfeld condition $∫_{r_{1}}^{r_{2}} dr r√{g}√{k_{F}^{2}(r) - k_{F}^{(n)2}} = π(n+½)$, and the Luttinger count follows by differentiating this condition with respect to $n$ and exchanging the order of integration, which converts the sum over boundary Fermi volumes into the bulk integral $Q = (1/3π^{2})∫ dr r√{g} k_{F}^{3}(r)$ that defines the star's charge density. Near the minimum and maximum of the potential the matching is redone with parabolic cylinder functions, giving the Green's function at $k ∼ k^{⋆}_{F}$ and at $k ∼ ω^{1/z}$ respectively.

Extended reading notes

Core claim

The central claim is that the boundary fermion Green's function of an electron star contains a large number of closely spaced Fermi surface poles, one for each bound state of the effective Schrödinger problem, and that the $2+1$ dimensional volumes of these Fermi surfaces add up to the total charge density $Q$, so the Luttinger count is satisfied. The poles are not all equal: those at small Fermi momentum carry exponentially larger residue than the ones near the maximal Fermi momentum $k^{⋆}_{F}$, while the states near $k^{⋆}_{F}$ are exponentially more stable because they sit further from the horizon. The Green's function changes character at the critical dispersion $ω ∼ k^{z}$: for $ω ≲ k^{z}$ the excitations are long lived and the spectral density is a sum of near-delta peaks, while for $ω ≳ k^{z}$ the fermions dissipate efficiently into the critical Lifshitz interior and the spectral density becomes featureless. The authors also estimate that fermions near this critical dispersion contribute $σ(Ω) ∼ Ω^{2/z}$ to the optical conductivity, which beats the $Ω^{2}$ of the stable star fermions at small frequency.

Load-bearing premise

Everything rests on the double limit in which the bulk fermion fluid is a WKB problem, $e^{2} ∼ κ/L ≪ 1$ with $κ^{2}m^{2}/e^{2}$ of order one, so that $γ = eL/κ ≫ 1$; outside this regime the discrete pole ladder, the matching formulae, and the identification of boundary Fermi momenta with a WKB quantization condition all change qualitatively.

Editorial extensions

If this is right

  • Electron stars are not fractionalized Fermi liquids in the technical sense: all of the charge is visible in gauge-invariant boundary Fermi surfaces, so the sum of their $2+1$ dimensional volumes equals $Q$.
  • Quantum oscillations and spectral weight decouple: the extremal momentum $k^{⋆}_{F}$ fixes the de Haas–van Alphen period but has the smallest residue of all the Fermi surfaces, so an oscillation experiment and a photoemission-style measurement would rank the same surfaces in opposite order.
  • The critical fermions near $ω ∼ k^{z}$ dominate the low-frequency optical conductivity with $σ(Ω) ∼ Ω^{2/z}$, overtaking the $Ω^{2}$ contribution of the stable star fermions.
  • In the small-star limit with $z=2$ the Green's function is obtained in closed form, exhibiting an explicit ladder of stable poles, exponentially small widths, and a semicircle of quasinormal poles in the lower half frequency plane.

Reading between the lines

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

  • The whole ladder is a WKB artifact in the sense that it is discrete only because $γ$ is large; a genuinely quantum electron star at moderate $γ$ would be the natural place to check whether the ladder merges into a true continuum or survives as a resolvable fine structure.
  • Because the $ω ∼ k^{z}$ crossover follows only from the near-horizon Lifshitz geometry and the existence of a potential maximum, the same structure should appear in dilaton-gravity and superconducting $z=1$ backgrounds with the exponent and the lightcone's frequency support adjusted accordingly.
  • If Luttinger's count is restored as soon as backreacting bulk fermions carry the charge, then the 'fractionalized' reading of probe-fermion results in extremal Reissner–Nordström–AdS is an artifact of the probe limit rather than a property of the phase, which is a claim the authors themselves raise as plausible but do not settle.
Original abstract

Electron stars are fluids of charged fermions in Anti-de Sitter spacetime. They are candidate holographic duals for gauge theories at finite charge density and exhibit emergent Lifshitz scaling at low energies. This paper computes in detail the field theory Green's function G^R(w,k) of the gauge-invariant fermionic operators making up the star. The Green's function contains a large number of closely spaced Fermi surfaces, the volumes of which add up to the total charge density in accordance with the Luttinger count. Excitations of the Fermi surfaces are long lived for w <~ k^z. Beyond w ~ k^z the fermionic quasiparticles dissipate strongly into the critical Lifshitz sector. Fermions near this critical dispersion relation give interesting contributions to the optical conductivity.

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, 5,470 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.30 · compares with prior results 0.06 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Electron stars are fluids of charged fermions in Anti-de Sitter spacetime. They are candidate holographic duals for gauge theories at finite charge density and exhibit emergent Lifshitz scaling at low energies. This paper computes in detail the field theory Green's function G^R(w,k) of the gauge-invariant fermionic operators making up the star. The Green's function contains a large number of closely spaced Fermi surfaces, the volumes of which add up to the total charge density in accordance with the Luttinger count. Excitations of the Fermi surfaces are long lived for w <~ k^z. Beyond w ~ k^z the fermionic quasiparticles dissipate strongly into the critical Lifshitz sector. Fermions near this critical dispersion relation give interesting contributions to the optical conductivity.

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 Electron stars are fluids of charged fermions in Anti-de Sitter spacetime. background 1.00 0.05 0.03 asserted 1.00
s1 They are candidate holographic duals for gauge theories at finite charge density and exhibit emergent Lifshitz scaling at low energies. background 1.00 0.07 0.06 hedged possibility 0.60
s2 This paper computes in detail the field theory Green's function G^R(w,k) of the gauge-invariant fermionic operators making up the star. contribution 0.98 0.96 0.09 asserted 1.00
s3 The Green's function contains a large number of closely spaced Fermi surfaces, the volumes of which add up to the total charge density in accordance with the Luttinger count. result 0.99 0.83 0.25 asserted 1.00
s4 Excitations of the Fermi surfaces are long lived for w <~ k^z. result 0.99 0.82 0.13 asserted 1.00
s5 Beyond w ~ k^z the fermionic quasiparticles dissipate strongly into the critical Lifshitz sector. result 0.97 0.82 0.16 asserted 1.00
s6 Fermions near this critical dispersion relation give interesting contributions to the optical conductivity. interpretation 0.50 0.62 0.11 asserted 0.95

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.67.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
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. A nonzero charge density is imposed as a UV boundary condition. The IR physics
Figure 2. Illustrative radial dependence of the Schr¨odinger potential when
Figure 3. Illustrative radial dependence of the Schr¨odinger potential (with
Figure 4. Schematic dependence of the dual field theory fermion spectral density as a
Figure 5. Exponent of the residue of the Fermi surface poles as a function of Fermi mo
Figure 6. Typical distribution of the poles in the fermion Green’s function close to
Figure 7. Poles in the retarded Green’s function: Imaginary part of the retarded Green’s
Figure 8. Spectral density of many Fermi surfaces interacting with a dissipative critical
Figure 9. Dissolution of the Fermi surface pole in the critical region: Spectral density as a

Figures reproduced from arXiv:1105.3197 by the authors.

References
How later papers used it

64 later papers cite it

Evidence: substantial. 58 of the citing papers share no author with it, and 14 of those carry a confident label from Pith's citation classifier.

Pith read 16 of the 150 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

Uses its method or tool 3 of 6 papers with this label
Compares with it 1 of 8 papers with this label
How the literature uses it

61 arXiv papers cite this in 145 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 · 94 sentences without a confident label

Contests it 1 paper
  • This conclusion is not intrinsic to holographic states supported by fermions, but rather is an artifact of the Thomas-Fermi approximation used in the pioneering work on electron stars [ref] .

    A quantum electron star · Introduction · contest 0.77

Uses its method or tool 5 papers
Compares with it 8 papers
  • Let us compare the situation to the electron star phase, studied in detail in [ref] .

    Polarized solutions and Fermi surfaces in holographic Bose-Fermi systems · The Schrödinger equation in a standard form · 0.98

  • In our conventions $Q=∫dz√(-g)g^zzg^ttn=∫dzn(z)e^h/2$ thus giving the same result as in [ref] .]: [equation] with $q_WKB$ determined by [equation] Notice that this formula corresponds with common knowledge on the density of electron star [ref] .

    Quantum corrected phase diagram of holographic fermions · WKB density · 0.96

  • The method differs conceptually from earlier probe fermion computations of magnetic oscillations in an electron star background [ref] and it predicts a dependence on the magnetic field and temperature of the oscillation amplitude that departs from Fermi liquid theory.

    Magnetic oscillations in a holographic liquid · Introduction · 0.93

Cites it as support 3 papers
  • This relation was made precise in [ref] , where it was shown that in field theory phases dual to an electron star, where all the charge is sourced by bulk fermions, the Luttinger equality ([ref]) holds when the sum is restricted to gauge-invariant Fermi surfaces.

    Horizons, holography and condensed matter · Horizons, fractionalization and black hole information · 0.97

  • Indications from recent work [ref] are that these low-energy excitations arise from the presence of an infinite number of Fermi surfaces, with a Fermi wavevector which varies continuously with $u$.

    What can gauge-gravity duality teach us about condensed matter physics? · Beyond AdS$_2 ×R^2$ · 0.99

  • It can also develop certain “fermionic stabilities” to transition to a Fermi-liquid phase [ref] .

    Quantum many-body physics from a gravitational lens · The zoo of holographic phases · 0.96

Cites it as background 21 papers
  • Recently, there have been exciting progresses on the holographic realizations of Fermi surfaces using the AdS/CFT such as the ones using the Reissner-Nordström black holes [ref] and the star-like solutions, called electron stars [ref] .

    Holographic Fermi Surfaces and Entanglement Entropy · Introduction and Summary · 0.89

  • Recent years, AdS/CFT correspondence has been widely used to study strongly coupled fermionic systems [ref] .

    Holographic Entanglement Entropy and Hidden Fermi surfaces · Introduction · 1.00

  • Some progress has been made recently in using holography to learn about metallic states of matter which are not described by Landau's Fermi liquid theory [ref] .

    A quantum electron star · Introduction · 0.99

What it cites · 62 sentences without a confident label

Uses the method or tool of 3 papers
  • The Dirac equation can then be written [ref] √(g^rr) σ_3 _r Φ- m Φ- i √(g^xx) k σ_2 Φ+ √(|g^tt|) (+ A_t) σ_1 Φ= 0 .

    Emergent quantum criticality, Fermi surfaces, and AdS2 · The WKB Dirac equation · 0.81

  • This can be obtained in the small star limit as follows.[This approach originated in collaboration with Pavel Petrov and was used to prove some assertions made in [ref] .

    Electron star birth: A continuous phase transition at nonzero density · The `small star' limit · 0.89

  • Similar methods were also used in [ref] .] Firstly note that the definitions of $μ̂$ and $c$ appearing in ([ref]) can be extended away from the exterior Reissner-Nordström-AdS region by defining $c(r) = r^2 √(f g)$ and $c μ̂(r) = h - r h'$.

    Holographic metals at finite temperature · The `small star' limit · 0.96

Uses data or values from 1 paper
Compares with 7 papers
  • The answer agrees precisely with the `semi-holographic' result [ref] with a Lifshitz IR geometry, as it must.

    Semi-Holographic Fermi Liquids · The boundary fermion Green's function: generic $, k$ · 0.99

  • However we still capture the crossover from cases II to III, our primary interest.] Perhaps curiously, the Green's function ([ref]) bears some resemblance to the IR Green's functions of extremal Reissner-Nordström-AdS [ref] .

    Emergent quantum criticality, Fermi surfaces, and AdS2 · Solvable case: z=2 · 1.00

  • This plot is very similar to plots describing the fermion spectral density in a superconducting background [ref] .

    Normalizable fermion modes in a holographic superconductor · The WKB Dirac equation · 0.99

Cites as background 7 papers

Discussion

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