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arXiv:1608.06637 · Physical review. B./Physical review. B

Realizing infrared power-law liquids in the cuprates from unparticle interactions

Kridsanaphong Limtragool, Chandan Setty, Zhidong Leong, Philip Phillips

Published Dec 8, 2016• 14 citations
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Abstract

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.

DOI
10.1103/physrevb.94.235121

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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 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 Im\ensuremathΣ\mathrm{Im}\mathrm{\ensuremath{\Sigma}}Im\ensuremathΣ is linear in temperature. In the low-temperature limit, Im\ensuremathΣ\mathrm{Im}\mathrm{\ensuremath{\Sigma}}Im\ensuremathΣ 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.
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.
We calculate the optical conductivity by using several models for unparticle or scale-invariant matter. Within a Gaussian action for unparticles that is gauged with Wilson lines, we find that the conductivity computed from the Kubo formalism with vertex corrections yields no nontrivial deviation from the free-theory result. This result obtains because, at the Gaussian level, unparticles are just a superposition of particle fields and hence any transport property must be consistent with free theory. Beyond the Gaussian approach, we adopt the continuous-mass formulation of unparticles and calculate the Drude conductivity directly. We show that unparticles in this context can be tailored to yield an algebraic conductivity that scales as ${\ensuremath{\omega}}^{\ensuremath{-}2/3}$ with the associated phase angle between the imaginary and real parts of $arctan\frac{{\ensuremath{\sigma}}_{2}}{{\ensuremath{\sigma}}_{1}}={60}^{\ensuremath{\circ}}$, as is seen in the cuprates. Given the recent results [J. High Energy Phys. 4, 40 (2014); J. High Energy Phys. 7, 24 (2015); arXiv:1506.06769] that gravitational crystals lack a power-law optical conductivity, this constitutes the first consistent account of the ${\ensuremath{\omega}}^{\ensuremath{-}2/3}$ conductivity and the phase angle seen in optimally doped cuprates. Our results indicate that, at each frequency in the scaling regime, excitations on all energy scales contribute. Hence, incoherence is at the heart of the power law in the optical conductivity in strongly correlated systems such as the cuprates.
Since any nontrivial infrared dynamics in strongly correlated electron matter must be controlled by a critical fixed point, we argue that the form of the single-particle propagator can be deduced simply by imposing scale invariance. As a consequence, the unparticle picture proposed by Georgi14 is the natural candidate to describe such dynamics. Unparticle stuff is scale-invariant matter with no particular mass. Scale invariance dictates that the propagator has an algebraic form which can admit zeros and hence is a candidate to explain the ubiquitous pseudogap state of the cuprates. We refer to the nonperturbative electronic state formed out of unparticles as an un-Fermi liquid. We show that the underlying action of the continuous mass formulation of unparticles can be recast as an action in anti--de Sitter space which serves as the generating functional for the propagator. We find that this mapping fixes the scaling dimension of the unparticle to be dU=d/2+d2+4/2{d}_{U}=d/2+\sqrt{{d}^{2}+4}/2dU​=d/2+d2+4​/2 and ensures that the corresponding propagator has zeros with ddd the space-time dimension of the unparticle field. Should d=2+1d=2+1d=2+1, unparticles acquire the nontrivial phase $2\ensuremath{\pi}{d}_{U}$ upon interchange. Because dU{d}_{U}dU​ is noninteger and in general not half integer, clockwise and counterclockwise interchange of unparticles do not lead to the same phase and time-reversal symmetry is broken spontaneously as reported in numerous experiments in the pseudogap phase of the cuprates. The possible relevance of this mechanism to such experiments is discussed. We then formulate the analogous BCS gap using unparticles and find that in contrast to the Fermi-liquid case, the transition temperature increases as the attractive interaction strength decreases, indicating that unparticles are highly susceptible to a superconducting instability.
Recent photoemission spectroscopy measurements (T. J. Reber et al., arXiv:1509.01611) of cuprate superconductors have inferred that the self-energy exhibits critical scaling over an extended doping regime, thereby calling into question the conventional wisdom that critical scaling exists only at isolated points. In particular, this new state of matter, dubbed a power-law liquid, has a self-energy whose imaginary part scales as ${\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{'}\ensuremath{'}}\ensuremath{\sim}{({\ensuremath{\omega}}^{2}+{\ensuremath{\pi}}^{2}{T}^{2})}^{\ensuremath{\alpha}}$, with $\ensuremath{\alpha}=1$ in the overdoped Fermi-liquid state and $\ensuremath{\alpha}\ensuremath{\le}0.5$ in the optimal to underdoped regime. Previously, we showed that this self-energy can arise from interactions between electrons and unparticles, a scale-invariant sector that naturally emerges from strong correlations. Here, taking the self-energy as a given, we first reconstruct the real part of the self-energy. We find that the resultant quasiparticle weight vanishes for any doping level less than optimal, implying an absence of particlelike excitations in the underdoped regime. Consequently, the Fermi velocity vanishes and the effective mass diverges for $\ensuremath{\alpha}\ensuremath{\le}\frac{1}{2}$, in agreement with earlier experimental observations. We then use the self-energy to reconstruct the spectral function and compute the superconducting Tc{T}_{c}Tc​ within the BCS formalism. We find that the Tc{T}_{c}Tc​ has a domelike structure, implying that broad scale invariance manifested in the form of a power-law liquid is the likely cause of the superconducting dome in the cuprates.
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_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.