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Strange metals and the AdS/CFT correspondence

Subir Sachdev

arXiv:1010.0682v3cond-mat.str-elcond-mat.stat-mechhep-th

TL;DR

The paper examines how quantum impurity and Kondo lattice models illuminate strange metals and their holographic descriptions. It finds a close correspondence between fractionalized Fermi-liquid mean-field solutions and holographic theories with AdS2 geometries, while noting missing emergent gauge fields in the latter.

  • Problem

    Strange metals depart from conventional Fermi-liquid behavior, motivating theories connecting their critical physics to conformal field theories and holographic descriptions.

    Source: §1 · p. 2

  • Method

    The paper reviews quantum antiferromagnet CFTs, quantum impurities, and Kondo lattice mean-field mappings, then compares their properties with AdS/CFT constructions.

    Source: §1 · pp. 2, 4

  • Results

    Kondo lattice mean-field theory yields a fractionalized Fermi liquid with conduction-electron Fermi surfaces coupled to a critical spin liquid, closely corresponding to holographic AdS2 × R^(D−1) theories.

    Source: §1 · p. 4, §5 · p. 17

  • Takeaways & Limitations

    The FL* phase is an attractive mean-field candidate for strange metals and provides a physical interpretation of similarities between Kondo lattice and holographic theories.

    Source: §1 · p. 4, §5 · p. 17

    The holographic theory may lack emergent gauge fields needed for a complete description of spatial correlations.

    Source: §5 · p. 18

Abstract

from arXiv · show

I begin with a review of quantum impurity models in condensed matter physics, in which a localized spin degree of freedom is coupled to an interacting conformal field theory in d = 2 spatial dimensions. Their properties are similar to those of supersymmetric generalizations which can be solved by the AdS/CFT correspondence; the low energy limit of the latter models is described by a AdS2 geometry. Then I turn to Kondo lattice models, which can be described by a mean- field theory obtained by a mapping to a quantum impurity coupled to a self-consistent environment. Such a theory yields a 'fractionalized Fermi liquid' phase of conduction electrons coupled to a critical spin liquid state, and is an attractive mean-field theory of strange metals. The recent holographic description of strange metals with a AdS2 x R2 geometry is argued to be related to such mean-field solutions of Kondo lattice models.

1. Introduction

The paper situates strange metals and holographic metals within strongly interacting conformal field theories, then connects their AdS2-based description to Kondo-lattice mean-field theories of fractionalized Fermi liquids.

1. Introduction

  • Strange metals show resistance linear in T and other observables that deviate from conventional Fermi-liquid behavior across several correlated-electron compounds.Examples include cuprates, ruthenium oxides, iron pnictides, organic metals, and heavy-fermion compounds.Source: §1 · p. 2
  • AdS/CFT relates strongly coupled conformal field theories to gravitational theories in higher-dimensional anti-de Sitter spaces, with the emergent dimension interpretable as an RG energy scale.In the large-M limit, SYM4 maps to classical Einstein gravity on AdS5.Source: §1 · p. 2
  • At low temperature and nonzero density, phenomenological AdS theories produce metallic states with Fermi surfaces, non-Fermi-liquid damping, and, for a particular parameter, resistance linear in T.These states form one of two broad classes obtained in the nonzero-density construction; the other involves scalar condensation and is not studied here.Source: §1 · p. 3
  • The low-energy holographic theory factorizes into AdS2 × R^{D−1}, making frequency-dependent quantum criticality central while leaving spatial fluctuations, zero-temperature entropy, and field content unresolved.The paper identifies these issues as key condensed-matter mysteries of the holographic strange metal.Source: §1 · pp. 3–4
  • The paper adds matter degrees of freedom to a zero-density CFT sequentially, finding a close single-impurity correspondence and connecting Kondo-lattice FL* mean field to holographic AdS2 metals.The Kondo-lattice solution describes conduction-electron Fermi surfaces coupled to a critical spin liquid, while the holographic theory reaches a similar mean-field structure without infinite-range interactions.Source: §1 · p. 4

2. Quantum antiferromagnets and CFT3s

Two-dimensional quantum antiferromagnets provide CFT3s near quantum critical points: even-spin unit cells yield the Wilson-Fisher theory, while odd-spin unit cells motivate a fractionalized gauge-field description whose N = 2 criticality remains unsettled.

2.1. Even number of S = 1/2 spins per unit cell

  • The dimer antiferromagnet tunes between Néel order at g = 1 and a singlet valence-bond state at large g, requiring a quantum phase transition at g = gc.The tuning parameter changes nearest-neighbor exchanges between J and J/g, and the large-g state preserves the Hamiltonian’s symmetries.Source: §2.1 · pp. 5–6
  • Evidence indicates that the dimer-antiferromagnet transition is governed by the three-dimensional Wilson-Fisher CFT3.Neutron-scattering measurements in TlCuCl3 agree quantitatively with the predicted excitation-spectrum transformation across the transition [32, 33].Source: §2.1 · p. 6

2.2. Odd number of S = 1/2 spins per unit cell

  • In a square-lattice antiferromagnet, continuously tuning a coupling can destroy Néel order and produce a valence-bond solid with four equivalent columnar states.A VBS is a spin-singlet state with nonzero degeneracy from spontaneously broken lattice symmetry, and has been observed in EtMe3P[Pd(dmit)2]2.Source: §2.2 · p. 6, Figure 2 · p. 7
  • A proposed direct Néel–VBS transition uses fractionalized bosonic spinors z_α and an emergent U(1) gauge field A_μ rather than either order parameter directly.The Higgs phase corresponds to Néel order, while the gapped-spinor Coulomb phase is a spin liquid that is ultimately destabilized toward VBS order by monopole effects.Source: §2.2 · pp. 7–8
  • Whether the proposed Néel–VBS transition is a CFT3 remains unsettled for N = 2, despite fixed-point evidence order-by-order in the 1/N expansion.The theories discussed are nonsupersymmetric analogs of the CFT3 realized by SYM3, whose holographic insights are treated elsewhere [9] [10].Source: §2.2 · p. 8

3. Quantum impurity in a CFT

Quantum impurities coupled to interacting CFTs flow to nontrivial universal fixed points with power-law correlations, irrational susceptibility and entropy, and close AdS2 counterparts. A large-N treatment makes this connection explicit through conformal low-energy correlators.

3. Quantum impurity in a CFT

  • The impurity coupling flows to a stable intermediate fixed point J = J∗ when the bulk is at the Wilson-Fisher CFT3 fixed point.Source: §3 · p. 9
  • The fixed point has power-law impurity correlations with non-trivial exponents, a Curie susceptibility χimp = C/T with irrational C, and finite irrational zero-temperature entropy.Source: §3 · pp. 9–10
  • Numerical studies agree well with the theoretical expectations, while alternative strong-coupling formulations recover the same universal fixed point and nontrivial impurity properties.Source: §3 · p. 10
  • For a bulk CPN−1 theory, bulk spinon excitations screen the impurity more efficiently, producing a distinct impurity theory studied with large-N and numerical methods.Source: §3 · p. 11
  • The supersymmetric impurity model has a universal gauge-like coupling to SYM4 fields, is solvable at large M, and maps at low energy to an AdS2 geometry with nonzero impurity entropy.Source: §3 · p. 11

3.1. Large N solution

  • The large-N solution sums rainbow graphs through self-consistency equations for the fermion Green’s function and self-energy, using the constraint nχ = N/2.Source: §3.1 · pp. 12–13
  • At low energies, assuming critical D(τ) ∼τ^−γ and 1/|τ|, T ≪J, the solution has conformal power-law behavior with an exponent fixed by the self-consistency condition.Source: §3.1 · pp. 13–14
  • The conformal impurity correlators parallel those inherited from AdS2, strengthening the proposed connection between quantum impurity models and AdS2 gravity.Source: §3.1 · p. 14

4. From a quantum impurity to lattice models

Kondo lattice models map to self-consistent quantum impurity problems, producing FL* states with a critical spin liquid and conduction-electron Fermi surface. Their resulting marginal-Fermi-liquid self-energy and transport agree with holographic metals, supporting a correspondence with AdS2 × R^{D−1} factorization.

From lattice models to impurity models

  • Self-consistent impurity mappings provide mean-field theories for Kondo lattice models, with analogous equations arising in random infinite-range and large-dimension non-random systems.The environment is composed of the same degrees of freedom as the impurity, so self-consistency supplements the impurity solution.Source: §4 · pp. 15–16

Comparison with holography

  • The shared impurity–environment and AdS2 × R^{D−1} factorizations motivate a correspondence between Kondo-lattice mean fields and holographic strange metals.The paper emphasizes that similar results arise from different underlying approximations.Source: §4 · pp. 15, 17

Mean-field phases

  • The FL* phase has a small conduction-electron Fermi surface while localized spins form a spin liquid, unlike the large-Fermi-surface FL phase.In the FL* solution, the spin-liquid correlator must equal the impurity correlator through an additional self-consistency condition.Source: §4 · p. 16

Critical spin liquid

  • Self-consistency requires a gapless power-law spectrum and fixes an exponent whose value corresponds to marginal Fermi-liquid behavior.The same exponent is obtained in large-dimension solutions of non-random lattice models.Source: §4 · pp. 16–17
  • The mean-field state is a critical spin liquid with non-zero ground-state entropy density, proposed as the low-energy realization of AdS^{D+1} matter at non-zero chemical potential.This state applies to both random and non-random models and can be viewed as a gapless bosonic liquid without Bose condensation.Source: §4 · p. 17

Comparison with holography

  • For h < 2, the conduction-electron self-energy is non-Fermi-liquid, and its form and transport properties agree with holographic results [22].At the self-consistent value h = 1, the self-energy has marginal-Fermi-liquid form.Source: §4 · p. 17

5. Conclusion

The reviewed FL* theory is an attractive candidate for strange metals and is supported phenomenologically, while its quantum-impurity description remains incomplete because emergent gauge excitations are missing. The holographic metal may share this omission, although string theory has also shown a transition from FL* to FL [59].

5. Conclusion

  • The FL* theory obtained from quantum-impurity models is an attractive candidate for describing strange-metal phases and satisfactorily describes experimental observations at a semi-phenomenological level.The paper emphasizes the surprising similarity between this theory and one arising from AdS/CFT.Source: §5 · p. 17
  • The quantum-impurity description of FL* is incomplete because its spin-liquid component should contain emergent gauge excitations essential for spatial correlations.The paper identifies gauge fields such as the Aµ field in Eq. (7) as missing ingredients.Source: §5 · pp. 17–18
  • The holographic metal may likewise be missing emergent gauge fields, motivating efforts to identify the appropriate gauge fields in the holographic theory.Nickel and Son argue that the holographic metal may have this deficiency [82].Source: §5 · p. 18
  • String theory has demonstrated a transition from an FL* phase to an FL phase [59].Source: §5 · p. 18
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