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Applications of the gauge/gravity duality to the cuprate strange metal

Enea Mauri

Published Nov 30, 2022• 0 citations
Publisher

Abstract

We studied the mysterious behavior of a class of materials known as cuprates, of particular practical interests due to their high transition temperature to a superconducting phase, where electricity is carried without resistance. Due to the strong interactions among the electrons, these systems cannot be described in terms of any conventional theory of metals and new approaches are needed. One such approach is the holographic duality, that allows us to model strongly interacting quantum system through an higher-dimensional, but easier to treat, theory of gravity. In this thesis we used the duality to study certain features of the non-superconducting phase of cuprates, known as the strange metal. We presented how to introduce the long-range Coulomb interaction, that allowed us to model plasmon excitations experimentally observed in multilayer cuprates, and to study a phenomenon of a two-layer system known as Coulomb drag. Furthermore, we showed that a prediction from the duality of a peculiar momentum-dependet scaling of the Fermionic self-energy is able to very accurately model a recently-observed feature of the spectral function measured in high-quality photoemission-spectroscopy measurements on a cuprate.

DOI
10.33540/1585

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The anomalous strange metal phase found in high-T c cuprates does not follow the conventional condensed-matter principles enshrined in the Fermi liquid and presents a great challenge for theory.Highly precise experimental determination of the electronic self-energy can provide a test bed for theoretical models of strange metals, and angle-resolved photoemission can provide this as a function of frequency, momentum, temperature and doping.Here we show that constant energy cuts through the nodal spectral function in (Pb,Bi) 2 Sr 2-x La x CuO 6+δ have a non-Lorentzian lineshape, consistent with a selfenergy that is k dependent.This provides a new test for aspiring theories.Here we show that the experimental data are captured remarkably well by a power law with a k-dependent scaling exponent smoothly evolving with doping, a description that emerges naturally from anti-de Sitter/conformal-field-theory based semi-holography.This puts a spotlight on holographic methods for the quantitative modelling of strongly interacting quantum materials like the cuprate strange metals.The rich temperature versus doping diagram of the cuprate high-T c superconductors presents a cornucopia of non-conformity 1 .The superconductivity does not follow BCS theory anywhere in the phase diagram 2,3 , and in their normal state these unusual metals display qualitatively different experimental behaviour compared to the Fermi liquid, with the linear-in-T resistivity and quadratic-in-T Hall angle being among the most strange.One intriguing possible explanation of strange metal phenomenology is that it is controlled by a strongly interacting quantum critical phase.If so, the question follows whether their properties can be

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Strange connections to strange metals

Hong Liu

Liu replies: While many ingenious ideas have been advanced to explain the strange-metal phase, the community has not reached a consensus, and many of the phase’s properties remain mysterious. Philip W. Anderson’s hidden Fermi liquid theory, arguably a most ambitious attempt, has provided a useful lens to view various aspects of the phase, but further developments are worthwhile.As described in my Quick Study, a key obstacle in the pursuit to explain the strange metal phase is the lack of a suitable mathematical framework to characterize “quantum soups”—strongly interacting, dense quantum matter whose constituents have lost their individuality. Holographic duality has now provided such a framework, although only for the class of systems that have gravity duals. It does for those quantum soups what Boltzmann equations do for a dilute gas of quasiparticles: It enables one to calculate reliably all equilibrium and nonequilibrium properties.At short distances, holographic systems are conformal, Lorentz invariant, and often supersymmetric, all of which make them seem poor models for real-life condensed-matter systems. Those symmetries, however, are all destroyed at long distances if a temperature or a chemical potential is turned on. The surprise of the past few years is that despite significant differences in short-distance details, at long distances most states of matter that have been understood—superfluids, Fermi liquids, magnets, stripes, and so on—can be described in terms of dual gravity systems. Such insensitivity to microscopic detail gives hope that the striking parallels in macroscopic behavior found between black holes and strange metals are not accidents, and gravity may help us decipher the mysteries of strange metals.I thank Anderson for emphasizing other anomalies exhibited by the strange-metal phase of cuprates that I did not have space to mention. He is correct that the holographic approach has not yet produced a model that could account for all anomalous properties. However, the power-law dependence of photoemission spectra on energy, his last bullet point, is a hallmark of holographic strange metals. I discussed that point in the last part of the Quick Study, in terms of the power-law temperature dependence of scattering rates. Such power laws follow from the semilocal property emphasized in the second-to-last paragraph.Whether or not one finds a “conventional” explanation for strange metals, connections between the physics of strange metals and black holes are worth exploring. They hint at a new paradigm for thinking about strongly correlated quantum soups. As an added bonus, we may also obtain new insights into quantum gravity from advances in condensed-matter physics.© 2013 American Institute of Physics.

Physics Today•Mar 29, 2013•1 citation
We studied the mysterious behavior of a class of materials known as cuprates, of particular practical interests due to their high transition temperature to a superconducting phase, where electricity is carried without resistance. Due to the strong interactions among the electrons, these systems cannot be described in terms of any conventional theory of metals and new approaches are needed. One such approach is the holographic duality, that allows us to model strongly interacting quantum system through an higher-dimensional, but easier to treat, theory of gravity. In this thesis we used the duality to study certain features of the non-superconducting phase of cuprates, known as the strange metal. We presented how to introduce the long-range Coulomb interaction, that allowed us to model plasmon excitations experimentally observed in multilayer cuprates, and to study a phenomenon of a two-layer system known as Coulomb drag. Furthermore, we showed that a prediction from the duality of a peculiar momentum-dependet scaling of the Fermionic self-energy is able to very accurately model a recently-observed feature of the spectral function measured in high-quality photoemission-spectroscopy measurements on a cuprate.
We show that the momentum-dependent scaling exponents of the holographic fermion self-energy of the conformal-to-AdS2_22​ Gubser-Rocha model can describe new findings from angle-resolved photoemission spectroscopy experiments on a single layer (Pb,Bi)2_{2}2​Sr2−x_{2-x}2−x​Lax_xx​CuO6+δ_{6+δ}6+δ​ copper-oxide. In particular, it was recently observed, in high-precision measurements on constant energy cuts along the nodal direction, that the spectral function departs from the Lorentzian line shape that is expected from the power-law-liquid model of a nodal self-energy, with an imaginary part featureless in momentum as ΣPLL′′(ω)∝(ω2)αΣ''_{\text{PLL}}(ω) \propto (ω^2)^αΣPLL′′​(ω)∝(ω2)α. By direct comparison with experimental results, we provide evidence that this departure from either a Fermi liquid or the power-law liquid, resulting in an asymmetry of the spectral function as a function of momentum around the central peak, is captured at low temperature and all dopings by a semi-holographic model that predicts a momentum-dependent scaling exponent in the electron self-energy as Σ(ω,k)∝ω(−ω2)α(1−(k−kF)/kF)−1/2Σ(ω,k) \propto ω(-ω^2)^{α(1 - (k - k_F)/k_F) - 1/2}Σ(ω,k)∝ω(−ω2)α(1−(k−kF​)/kF​)−1/2, with ℏkF\hbar k_FℏkF​ the Fermi momentum.
The anomalous strange metal phase found in high-T c cuprates does not follow the conventional condensed-matter principles enshrined in the Fermi liquid and presents a great challenge for theory.Highly precise experimental determination of the electronic self-energy can provide a test bed for theoretical models of strange metals, and angle-resolved photoemission can provide this as a function of frequency, momentum, temperature and doping.Here we show that constant energy cuts through the nodal spectral function in (Pb,Bi) 2 Sr 2-x La x CuO 6+δ have a non-Lorentzian lineshape, consistent with a selfenergy that is k dependent.This provides a new test for aspiring theories.Here we show that the experimental data are captured remarkably well by a power law with a k-dependent scaling exponent smoothly evolving with doping, a description that emerges naturally from anti-de Sitter/conformal-field-theory based semi-holography.This puts a spotlight on holographic methods for the quantitative modelling of strongly interacting quantum materials like the cuprate strange metals.The rich temperature versus doping diagram of the cuprate high-T c superconductors presents a cornucopia of non-conformity 1 .The superconductivity does not follow BCS theory anywhere in the phase diagram 2,3 , and in their normal state these unusual metals display qualitatively different experimental behaviour compared to the Fermi liquid, with the linear-in-T resistivity and quadratic-in-T Hall angle being among the most strange.One intriguing possible explanation of strange metal phenomenology is that it is controlled by a strongly interacting quantum critical phase.If so, the question follows whether their properties can be
We show that the momentum-dependent scaling exponents of the holographic fermion self-energy of the conformal-to-AdS2{\mathrm{AdS}}_{2}AdS2​ Gubser-Rocha model can describe new findings from angle-resolved photoemission spectroscopy experiments on a single-layer (Pb,Bi)${}_{2}{\mathrm{Sr}}_{2\ensuremath{-}x}{\mathrm{La}}_{x}{\mathrm{CuO}}_{6+\ensuremath{\delta}}$ copper oxide. In particular, it was recently observed in high-precision measurements on constant energy cuts along the nodal direction that the spectral function departs from the Lorentzian line shape that is expected from the power-law-liquid model of a nodal self-energy, with an imaginary part featureless in momentum as ${\mathrm{\ensuremath{\Sigma}}}_{\text{PLL}}^{\ensuremath{'}\ensuremath{'}}(\ensuremath{\omega})\ensuremath{\propto}{({\ensuremath{\omega}}^{2})}^{\ensuremath{\alpha}}$. By direct comparison with experimental results, we provide evidence that this departure from either a Fermi liquid or the power-law liquid, resulting in an asymmetry of the spectral function as a function of momentum around the central peak, is captured at low temperature and all dopings by a semiholographic model that predicts a momentum-dependent scaling exponent in the electron self-energy as $\mathrm{\ensuremath{\Sigma}}(\ensuremath{\omega},k)\ensuremath{\propto}\ensuremath{\omega}{(\ensuremath{-}{\ensuremath{\omega}}^{2})}^{\ensuremath{\alpha}(1\ensuremath{-}(k\ensuremath{-}{k}_{F})/{k}_{F})\ensuremath{-}1/2}$, with $\ensuremath{\hbar}{k}_{F}$ the Fermi momentum.
Since the mid-eighties there has been an accumulation of metallic materials whose thermodynamic and transport properties differ significantly from those predicted by Fermi liquid theory. Examples of these so-called non-Fermi liquids include the strange metal phase of high transition temperature cuprates, and heavy fermion systems near a quantum phase transition. We report on a class of non-Fermi liquids discovered using gauge/gravity duality. The low energy behavior of these non-Fermi liquids is shown to be governed by a nontrivial infrared (IR) fixed point which exhibits nonanalytic scaling behavior only in the temporal direction. Within this class we find examples whose single-particle spectral function and transport behavior resemble those of strange metals. In particular, the contribution from the Fermi surface to the conductivity is inversely proportional to the temperature. In our treatment these properties can be understood as being controlled by the scaling dimension of the fermion operator in the emergent IR fixed point.
Black Holes as Tools When confronted with a difficult problem, physicists often resort to mapping it to a more tractable one. A good example of this strategy is provided by new developments linking string theory and condensed-matter physics to make theoretical connections between gravity and complex systems of interacting electrons. This theoretical convergence provides a description of Fermi liquids, which can be thought of as interacting systems of electrons whose excitations can be expressed in terms of non-interacting quasiparticles. Several interesting systems elude quasiparticle description, but Faulkner et al. (p. 1043 , published online 5 August) have now developed a mathematical framework that describes the non-Fermi liquid represented by the strange metal phase of cuprate high-temperature superconductors. They calculate the electronic response and, for a particular value of a tunable parameter, recover the linear resistivity. Further development of this framework may allow elucidation of other exotic properties of the cuprates and similar complex systems.
Liu replies: While many ingenious ideas have been advanced to explain the strange-metal phase, the community has not reached a consensus, and many of the phase’s properties remain mysterious. Philip W. Anderson’s hidden Fermi liquid theory, arguably a most ambitious attempt, has provided a useful lens to view various aspects of the phase, but further developments are worthwhile.As described in my Quick Study, a key obstacle in the pursuit to explain the strange metal phase is the lack of a suitable mathematical framework to characterize “quantum soups”—strongly interacting, dense quantum matter whose constituents have lost their individuality. Holographic duality has now provided such a framework, although only for the class of systems that have gravity duals. It does for those quantum soups what Boltzmann equations do for a dilute gas of quasiparticles: It enables one to calculate reliably all equilibrium and nonequilibrium properties.At short distances, holographic systems are conformal, Lorentz invariant, and often supersymmetric, all of which make them seem poor models for real-life condensed-matter systems. Those symmetries, however, are all destroyed at long distances if a temperature or a chemical potential is turned on. The surprise of the past few years is that despite significant differences in short-distance details, at long distances most states of matter that have been understood—superfluids, Fermi liquids, magnets, stripes, and so on—can be described in terms of dual gravity systems. Such insensitivity to microscopic detail gives hope that the striking parallels in macroscopic behavior found between black holes and strange metals are not accidents, and gravity may help us decipher the mysteries of strange metals.I thank Anderson for emphasizing other anomalies exhibited by the strange-metal phase of cuprates that I did not have space to mention. He is correct that the holographic approach has not yet produced a model that could account for all anomalous properties. However, the power-law dependence of photoemission spectra on energy, his last bullet point, is a hallmark of holographic strange metals. I discussed that point in the last part of the Quick Study, in terms of the power-law temperature dependence of scattering rates. Such power laws follow from the semilocal property emphasized in the second-to-last paragraph.Whether or not one finds a “conventional” explanation for strange metals, connections between the physics of strange metals and black holes are worth exploring. They hint at a new paradigm for thinking about strongly correlated quantum soups. As an added bonus, we may also obtain new insights into quantum gravity from advances in condensed-matter physics.© 2013 American Institute of Physics.