Optical Integral and Sum Rule Violation Saurabh Maiti, Andrey V. Chubukov Department of Physics, University of Wisconsin, Madison, Wisconsin 53706, USA (Dated: November 9, 2018) The purpose of this work is to investigate the role of the lattice in the optical Kubo sum rule in the cuprates. We compute conductivities, optical integrals W , and Δ W between superconducting and normal states for 2-D systems with lattice dispersion typical of the cuprates for four different models – a dirty BCS model, a single Einstein boson model, a marginal Fermi liquid model, and a collective boson model with a feedback from superconductivity on a collective boson. The goal of the paper is two-fold. First, we analyze the dependence of W on the upper cut-off ( ω c ) placed on the optical integral because in experiments W is measured up to frequencies of order bandwidth. For a BCS model, the Kubo sum rule is almost fully reproduced at ω c equal to the bandwidth. But for other models only 70%-80% of Kubo sum rule is obtained up to this scale and even less so for Δ W , implying that the Kubo sum rule has to be applied with caution. Second, we analyze the sign of Δ W . In all models we studied Δ W is positive at small ω c , then crosses zero and approaches a negative value at large ω c , i.e. the optical integral in a superconductor is smaller than in a normal state. The point of zero crossing, however, increases with the interaction strength and in a collective boson model becomes comparable to the bandwidth at strong coupling. We argue that this model exhibits the behavior consistent with that in the cuprates. The analysis of sum rules for optical conductivity has a long history. Kubo, in an extensive paper 1 in 1957, used a general formalism of a statistical theory of irreversible processes to investigate the behavior of the conductivity in electronic systems. For a system of interacting elec- trons, he derived the expression for the integral of the real part of a (complex) electric conductivity σ (Ω) and found that it is independent on the nature of the interactions and reduces to Here n is the density of the electrons in the system and m is the bare mass of the electron. This expression is exact provided that the integration extends truly up to infinity, and its derivation uses the obvious fact that at energies higher than the total bandwidth of a solid, elec- trons behave as free particles. The independence of the r.h.s. of Eq. (1) on temper- ature and the state of a solid (e.g., a normal or a super- conducting state – henceforth referred to as NS and SCS respectively) implies that, while the functional form of σ (Ω) changes with, e.g., temperature, the total spectral weight is conserved and only gets redistributed between different frequencies as temperature changes. This con- servation of the total weight of σ (Ω) is generally called a sum rule. One particular case, studied in detail for conventional superconductors, is the redistribution of the spectral weight between normal and superconducting states. This is known as Ferrel-Glover-Tinkham (FGT) sum rule: 2,3 where n$_{s}$ is the superfluid density, and πn$_{s}$e $^{2}$/ (2 m ) is In reality, as already pointed out by Hirsch, there is no true violation as the change of the total spectral weight where n$_{⃗}$ k is the electronic distribution function and ε$_{⃗}$ k is the band dispersion. Prime in the upper limit of the inte- gration has the practical implication that the upper limit is much larger than the bandwidth of a given band which crosses the Fermi level, but smaller than the frequencies of interband transitions. Interactions with external ob- jects, e.g., phonons or impurities, and interactions be- tween fermions are indirectly present in the distribution function which is expressed via the full fermionic Green’s function as n$_{⃗}$ k = T ∑ $_{m}$G ( ⃗ k, ω$_{m}$ ). For ϵ$_{k}$ = k $^{2}$/ 2 m , ∇ 2 ⃗ $_{k$_{x}$}$ε$_{⃗}$ k = 1 /m , W$_{K}$ = πne $^{2}$/ (2 m ), and Kubo sum rule reduces to Eq. (1). In general, however, ε$_{⃗}$ k is a lattice dispersion, and Eqs. (1) and (3) are different. Most im- portant, W$_{K}$ in Eq. (3) generally depends on T and on the state of the system because of n$_{⃗}$ $_{k}$. In this situation, the temperature evolution of the optical integral does not reduce to a simple redistribution of the spectral weight – the whole spectral weight inside the conduction band changes with T . This issue was first studied in detail by Hirsch 4 who introduced the now-frequently-used nota- tion “violation of the conductivity sum rule”. In practice, the integration up to an infinite frequency is hardly possible, and more relevant issue for practical applications is whether a sum rule is satisfied, at least ap- proximately, for a situation when there is a single electron band which crosses the Fermi level and is well separated from other bands. Kubo considered this case in the same paper of 1957 and derived the expression for the “band”, or Kubo sum rule the spectral weight under the δ -functional piece of the conductivity in the superconducting state. arXiv:1001.0764v2 [cond-mat.str-el] 13 Jan 2010 I. INTRODUCTION ∫ ∞ 0 Re σ (Ω) d Ω = π 2 ne 2 m (1) ∫ ‘ ∞ ′ 0 Re σ (Ω) d Ω = W$_{K}$ = πe 2 2 N ∑ ⃗ k ∇ 2 ⃗ $_{k$_{x}$}$ε$_{⃗}$ $_{k}$n$_{⃗}$ k (3) ∫ ∞ 0+ Re σ$_{NS}$ (Ω) = ∫ ∞ 0+ Re σ$_{sc}$ (Ω) + πn$_{s}$e 2 2 m (2)