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Physical review. B./Physical review. B

Gauge-gravity duality comes to the laboratory: Evidence of momentum-dependent scaling exponents in the nodal electron self-energy of cuprate strange metals

Enea Mauri, Steef Smit, M. S. Golden, Henk T. C. Stoof

Published Apr 15, 2024• 4 citations
Publisher

Abstract

We show that the momentum-dependent scaling exponents of the holographic fermion self-energy of the conformal-to-${\mathrm{AdS}}_{2}$ 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.

DOI
10.1103/physrevb.109.155140

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Recent photoemission experiments [1] reveal that the excitations along the nodal region in the strange metal phase of the cuprates, rather than corresponding to poles in the single-particle Green function, exhibit power-law scaling as a function of frequency and temperature. Because such power-law scaling is indicative of a scale-invariant sector, as a first step, we perturbatively evaluate the electron self-energy due to interactions with scale-invariant unparticles. We focus on a ${G}_{0}W$-type diagram with an interaction $W$ mediated by a bosonic scalar unparticle. We find that in the high-temperature limit, the imaginary part of the self-energy $\mathrm{Im}\mathrm{\ensuremath{\Sigma}}$ is linear in temperature. In the low-temperature limit, $\mathrm{Im}\mathrm{\ensuremath{\Sigma}}$ exhibits the same power law in both temperature and frequency, with an exponent that depends on the scaling dimension of an unparticle operator. Such behavior is qualitatively consistent with the experimental observations. We then expand the unparticle propagator into coherent and incoherent contributions and study how the incoherent part violates the density of states (DOS) and density-density correlation function sum rules (f-sum rule). Such violations, in principle, can be observed experimentally. Our work indicates that the physical mechanism for the origin of the power-law scaling is the incoherent background, which is generated from the Mott-scale physics.

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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)2Sr2\ensuremath−xLaxCuO6+\ensuremathδ{}_{2}{\mathrm{Sr}}_{2\ensuremath{-}x}{\mathrm{La}}_{x}{\mathrm{CuO}}_{6+\ensuremath{\delta}}2​Sr2\ensuremath−x​Lax​CuO6+\ensuremathδ​ 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 \ensuremathΣPLL\ensuremath′\ensuremath′(\ensuremathω)\ensuremath∝(\ensuremathω2)\ensuremathα{\mathrm{\ensuremath{\Sigma}}}_{\text{PLL}}^{\ensuremath{'}\ensuremath{'}}(\ensuremath{\omega})\ensuremath{\propto}{({\ensuremath{\omega}}^{2})}^{\ensuremath{\alpha}}\ensuremathΣPLL\ensuremath′\ensuremath′​(\ensuremathω)\ensuremath∝(\ensuremathω2)\ensuremathα. 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 \ensuremathΣ(\ensuremathω,k)\ensuremath∝\ensuremathω(\ensuremath−\ensuremathω2)\ensuremathα(1\ensuremath−(k\ensuremath−kF)/kF)\ensuremath−1/2\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}\ensuremathΣ(\ensuremathω,k)\ensuremath∝\ensuremathω(\ensuremath−\ensuremathω2)\ensuremathα(1\ensuremath−(k\ensuremath−kF​)/kF​)\ensuremath−1/2, with \ensuremathℏkF\ensuremath{\hbar}{k}_{F}\ensuremathℏkF​ the Fermi momentum.
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 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 develop a Neural-ODE framework to infer effective bulk metric functions and the charge-weighted gauge potential from boundary fermionic spectra within a static, planar black-hole ansatz. Controlled benchmarks recover the known bulk profiles of the Einstein--Maxwell and Gubser--Rocha models. We then use normalized interaction self-energy targets generated from an extended power-law liquid model with parameters calibrated by angle-resolved photoemission measurements to test the proposed geometric explanation of cuprate strange-metal spectra in a semi-holographic setting. For this application, we use a massless probe in backgrounds with conformal-to-AdS2×R2\mathrm{AdS}_2\times\mathbb{R}^2AdS2​×R2 asymptotics and derive a spectral extraction formula from the conserved radial Dirac flux. We identify coordinate/Weyl redundancy and a degeneracy in the scaled Hawking temperature of the normalized inverse problem. The conformal factor remains undetermined, so bulk entropy and the associated contribution to electronic specific heat require independent input. Low-temperature targets are fitted accurately across the studied doping range. In the high-temperature test, however, the present ansatz and optimization search have not yielded a refined geometry that accurately reproduces the thermal spectral shape while preserving frequency symmetry. These results may guide the further development of semi-holographic models for cuprates, and the framework may also find applications in other strongly correlated electron systems.
Recent photoemission spectroscopy measurements (T. J. Reber et al., arXiv:1509.01611) on cuprate superconductors have inferred that over a wide range of doping, the imaginary part of the electron self-energy scales as ${\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{''}}\ensuremath{\sim}{({\ensuremath{\omega}}^{2}+{\ensuremath{\pi}}^{2}{T}^{2})}^{a}$ with a=1a=1a=1 in the overdoped Fermi-liquid state and a<0.5a<0.5a<0.5 in the optimal to underdoped regime. We show that this non-Fermi-liquid scaling behavior can naturally be explained by the presence of a scale-invariant state of matter known as unparticles. We evaluate analytically the electron self-energy due to interactions with fermionic unparticles. We find that, in agreement with experiments, the imaginary part of the self-energy scales with respect to temperature and energy as ${\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{''}}\ensuremath{\sim}{T}^{2+2\ensuremath{\alpha}}$ and ${\ensuremath{\omega}}^{2+2\ensuremath{\alpha}}$, where $\ensuremath{\alpha}$ is the anomalous dimension of the unparticle propagator. In addition, the calculated occupancy and susceptibility of fermionic unparticles, unlike those of normal fermions, have significant spectral weights even at high energies. This unconventional behavior is attributed to the branch cut in the unparticle propagator which broadens the unparticle spectral function over a wide energy range and nontrivially alters the scattering phase space by enhancing (suppressing) the intrinsic susceptibility at low energies for negative (positive) $\ensuremath{\alpha}$. Our work presents evidence suggesting that unparticles might be important low-energy degrees of freedom in strongly coupled systems such as the cuprate superconductors.
Recent photoemission experiments [1] reveal that the excitations along the nodal region in the strange metal phase of the cuprates, rather than corresponding to poles in the single-particle Green function, exhibit power-law scaling as a function of frequency and temperature. Because such power-law scaling is indicative of a scale-invariant sector, as a first step, we perturbatively evaluate the electron self-energy due to interactions with scale-invariant unparticles. We focus on a G0W{G}_{0}WG0​W-type diagram with an interaction WWW mediated by a bosonic scalar unparticle. We find that in the high-temperature limit, the imaginary part of the self-energy $\mathrm{Im}\mathrm{\ensuremath{\Sigma}}$ is linear in temperature. In the low-temperature limit, $\mathrm{Im}\mathrm{\ensuremath{\Sigma}}$ exhibits the same power law in both temperature and frequency, with an exponent that depends on the scaling dimension of an unparticle operator. Such behavior is qualitatively consistent with the experimental observations. We then expand the unparticle propagator into coherent and incoherent contributions and study how the incoherent part violates the density of states (DOS) and density-density correlation function sum rules (f-sum rule). Such violations, in principle, can be observed experimentally. Our work indicates that the physical mechanism for the origin of the power-law scaling is the incoherent background, which is generated from the Mott-scale physics.