Breaking of the overall permutation symmetry in nonlinear optical susceptibilities of one-dimensional periodic dimerized Hückel model

Based on one-dimensional single-electron infinite periodic models of transpolyacetylene, we show analytically that the overall permutation symmetry of nonlinear optical susceptibilities is, although preserved in bound-state molecular systems, no longer generally held in periodic systems. The overall permutation symmetry breakdown provides a natural explanation of the widely observed large deviations of Kleinman symmetry in off-resonant regions of periodic systems. Finally, physical conditions to experimentally test the overall permutation symmetry breakdown are discussed. The nth-order optical susceptibility is generally defined as a rank-n tensor χ(n) μα1α2···αn (−ωσ ; ω1, ω2, . . ., ωn), where ωσ ≡ ∑i=1 ωi is the sum of incoming frequencies and μα1 · · ·αn are the indices of spatial directions. The intrinsic permutation symmetry, as described in Butcher and Cotter’s book [1], implies that the nth-order susceptibility is invariant under all n! permutations of pairs (α1, ω1), (α2, ω2), . . ., (αn, ωn). Intrinsic symmetry is a fundamental property of the nonlinear susceptibilities which arises from the principles of time invariance and causality, and applies universally to all physical systems. For the medium that is transparent and lossless for all relevant frequencies, i.e., far away from all transition frequencies, it is generally believed that the optical susceptibilities have a much more interesting property, namely, the overall permutation symmetry (or the full permutation symmetry in Boyd’s book [2]), in which the susceptibilities are invariant when the permutation includes the additional pair (μ,−ωσ ). Therefore, the nth-order susceptibility is invariant under all (n + 1)! permutations of the pairs (μ,−ωσ ), (α1, ω1), . . ., (αn, ωn) [3]. Furthermore, when the optical frequencies are much smaller than any of the transition frequencies, the dispersion of the medium at the relevant frequencies is negligible. It follows that the susceptibility is asymptotically invariant under all 3 Author to whom any correspondence should be addressed. 0953-8984/06/398987+07$30.00 © 2006 IOP Publishing Ltd Printed in the UK 8987 8988 M Xu and S Jiang permutations of the subscripts μ, α1, . . ., αn in the low-frequency limit. This property is known as Kleinman symmetry [1, 2, 4]. However, as observed by Simpson and his coauthors [5], the overwhelming majority of recent optical experiments on organic materials and crystals showed large deviations from Kleinman symmetry, even in the low-frequency off-resonant regions. However, the deviations from Kleinman symmetry in molecular systems are fairly small. In this paper, based on the theoretical framework developed in our previous work [6–10], we prove that the overall permutation symmetry of nonlinear optical susceptibilities is broken in one-dimensional (1D) periodic systems. On the other hand, the overall permutation symmetry remains valid in boundstate molecular systems. Since the overall permutation symmetry is the basis of Kleinman symmetry, this provides a natural explanation of why the deviations from Kleinman symmetry are much larger in periodic systems than in bound-state molecular systems. Indeed, despite the wide acceptance of the overall permutation symmetry in the nonlinear optics [1–3], no direct measurement has tested the validity of the assertion. We will suggest physical conditions to experimentally test the overall permutation symmetry breakdown. The analytical derivations of the overall permutation symmetry of nonlinear susceptibilities are rigorous and correct in molecular systems [1–3] where the position operator r is welldefined in real space. However, for periodic systems, the usual definition of r is no longer valid over all space. Instead a ‘saw-like’ position operator must be introduced to maintain the periodic property of the system [11, 12]. If periodic boundary conditions are applied to a physical system, the average electronic position could be anywhere for delocalized states [11]. This is clearly not the case for most molecular systems with only bound states. For periodic systems, the position operator r is conveniently defined in momentum space [13]: rnk,n′k′ = i∇kζn,n′(k,k′)+ n,n′(k)δ(k − k′), (1)