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arXiv:1506.00649 · Physical Review B

Power-law optical conductivity from unparticles: Application to the cuprates

Kridsanaphong Limtragool, Philip Phillips

Published Oct 16, 2015• 16 citations
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Abstract

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.

DOI
10.1103/physrevb.92.155128

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I am concerned in these lectures with the breakdown of the particle concept in strongly correlated electron matter. I first show that the standard procedure for counting particles, namely Luttinger's theorem, breaks down anytime pole-like excitations are replaced by ones that have a divergent self-energy. Such a breakdown obtains in electronic systems whose pole-like excitations do not extend to the edge of the Brillouin zone, as in Fermi arcs in the cuprates. Since any non-trivial infrared dynamics in strongly correlated electron matter must be controlled by a critical fixed point, unparticles are the natural candidate to explain the presence of charged degrees of freedom that have no particle content. The continuous mass formulation of unparticles is recast as an action in anti de Sitter space. Such an action serves as the generating functional for the propagator. This mapping fixes the scaling dimension of the unparticle to be $d_U=d/2+\sqrt{d^2+4}/2$ and ensures that the corresponding propagator has zeros with $d$ the spacetime dimension of the unparticle field. The general {\it dynamical} mechanism by which bulk operators, such as the Pauli term, couple to the scaling dimension of the boundary operator and thereby lead to a vanishing of the spectral weight at zero energy is reviewed in the context of unparticles and zeros. The analogue of the BCS gap equations with unparticles indicates that the transition temperature increases as the attractive interaction strength decreases, indicating that unparticles are highly susceptible to a superconducting instability.

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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 \ensuremathω\ensuremath−2/3{\ensuremath{\omega}}^{\ensuremath{-}2/3}\ensuremathω\ensuremath−2/3 with the associated phase angle between the imaginary and real parts of arctan\ensuremathσ2\ensuremathσ1=60\ensuremath∘arctan\frac{{\ensuremath{\sigma}}_{2}}{{\ensuremath{\sigma}}_{1}}={60}^{\ensuremath{\circ}}arctan\ensuremathσ1​\ensuremathσ2​​=60\ensuremath∘, 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 \ensuremathω\ensuremath−2/3{\ensuremath{\omega}}^{\ensuremath{-}2/3}\ensuremathω\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.
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.
Understanding the origin of electron incoherence is believed to be the first step toward the resolution of the mysteries of the high-Tc_{c}c​ cuprate superconductors. Such electron incoherence manifests itself most evidently in the non-Drude form of the optical absorption spectrum of the system. The spectral weight transfer related to such dissipative response, which is absent in conventional Fermi liquid metal, has direct consequence on the dc transport property of the system. However, a theoretical study of the optical conductivity of a strongly correlated model is a formidable task. Here we present an exact formula for the optical conductivity of the 2D Hubbard model from the low energy effective theory perspective. We show that the optical conductivity in Matsubara frequency of the 2D Hubbard model can be represented as the ensemble average of the optical conductivity of non-interacting systems in the background of fluctuating local moment. We find that such an ensemble average can be done exactly with a sign-problem-free Monte Carlo simulation if we assume the widely adopted Millis-Monien-Pines spin susceptibility for the fluctuating local moment. For thermal fluctuation of the local moment, our formula can be used to calculate directly the optical conductivity in real frequency which can be compared with the result of optical measurements in the cuprate superconductors.
Many properties of strongly correlated materials behave differently from those obeying the standard theory of metals. Explaining such anomalous properties may require one to consider a non-traditional model. In this dissertation, we construct several phenomenological models based on the notions of unparticles and nonlocalities to explain some of the exotic properties of the high temperature cuprate superconductors. The properties of the cuprates that we investigate include the fractional power-law in optical conductivity, the power-law liquid form of the scattering life-time, the violation of the conductivity sum rule, and the appearance of the anomalous dimension for the current. Our results suggest that the correct description of the low-energy theory of the normal state of the cuprates could involve unparticles and nonlocal actions.
The optical conductivity of the 2D t−Jt-Jt−J model and the origin of electron incoherence in the high-Tc_{c}c​ cuprate superconductors: a variational study
Understanding the origin of electron incoherence is the first step toward a theoretical description of the non-Fermi liquid behavior of the high-Tc_{c}c​ cuprate superconductors. Such electron incoherence manifests itself most evidently in the non-Drude behavior of the optical response of the system and the anomalous density fluctuation behavior in the long wave length limit. The spectral weight transfer related to such dissipative response, which is absent in conventional Fermi liquid metal, has direct consequence on the dc transport property of the system in the normal state and the superfluid stiffness in the superconducting state. It is found that such electron incoherence remains significant even in the clean limit and at low temperature and thus must be attributed to the strong electron correlation effect in the cuprate superconductors. Here we study such an intrinsic effect in the 2D t−Jt-Jt−J model through the variational calculation of its optical conductivity σ(ω)σ(ω)σ(ω). We assume a resonating valence bond ground state as our starting point and find that a significant portion of the total optical spectral weight remains incoherent throughout the phase diagram. The optical absorption is found to extend all the way to an energy of the order of the bare band width. We find that both the total optical weight Kˉ\bar{K}Kˉ and the integrated incoherent optical weight III increase monotonically with doping, with their ratio Rincoh=I/KˉR_{incoh}=I/\bar{K}Rincoh​=I/Kˉ decreasing monotonically with doping. Our results indicate that the majority part of electron incoherence in the 2D t−Jt-Jt−J model can be attributed to the electron fractionalization mechanism assumed in such a treatment. We also find that the Drude weight deduced from D=Kˉ−ID=\bar{K}-ID=Kˉ−I scales linearly with hole doping, without any sign of a non-monotonic behavior in the overdoped regime.
I am concerned in these lectures with the breakdown of the particle concept in strongly correlated electron matter. I first show that the standard procedure for counting particles, namely Luttinger's theorem, breaks down anytime pole-like excitations are replaced by ones that have a divergent self-energy. Such a breakdown obtains in electronic systems whose pole-like excitations do not extend to the edge of the Brillouin zone, as in Fermi arcs in the cuprates. Since any non-trivial infrared dynamics in strongly correlated electron matter must be controlled by a critical fixed point, unparticles are the natural candidate to explain the presence of charged degrees of freedom that have no particle content. The continuous mass formulation of unparticles is recast as an action in anti de Sitter space. Such an action serves as the generating functional for the propagator. This mapping fixes the scaling dimension of the unparticle to be dU=d/2+d2+4/2d_U=d/2+\sqrt{d^2+4}/2dU​=d/2+d2+4​/2 and ensures that the corresponding propagator has zeros with ddd the spacetime dimension of the unparticle field. The general {\it dynamical} mechanism by which bulk operators, such as the Pauli term, couple to the scaling dimension of the boundary operator and thereby lead to a vanishing of the spectral weight at zero energy is reviewed in the context of unparticles and zeros. The analogue of the BCS gap equations with unparticles indicates that the transition temperature increases as the attractive interaction strength decreases, indicating that unparticles are highly susceptible to a superconducting instability.