Two interacting particles in a disordered chain II: Critical statistics and maximum mixing of the one body states

For two particles in a disordered chain of length L with on-site interaction U , a duality transformation maps the behavior at weak interaction onto the behavior at strong interaction. Around the fixed point of this transformation, the interaction yields a maximum mixing of the one body states. When L ≈ L1 (the one particle localization length), this mixing results in weak chaos accompanied by multifractal wave functions and critical spectral statistics, as in the one particle problem at the mobility edge or in certain pseudo-integrable billiards. In one dimension, a local interaction can only yield this weak chaos but can never drive the two particle system to full chaos with Wigner-Dyson statistics. PACS. 05.45.+b Theory and models of chaotic systems – 72.15.Rn Quantum localization – 71.30.+h Metal-insulator transitions and other electronic transitions The competition between two body (electron-electron) interaction and one body kinetic energy in disordered systems is a fundamental problem of permanent interest. We denote by U, t and W the parameters characterizing the interaction, the one body kinetic energy and the fluctuations of the random potential for a d-dimensional system of size L. When U is small, the N -body eigenstates are close to the symmetrized products of one body states (Slater determinants for spinless fermions) which contain the effects of t and W completely. The effect of U can be treated as a perturbation, yielding a mixing of those symmetrized products. When U increases, the consequence of this mixing is that an increasing number of one body states is needed to describe the exact N -body states. If the one body states are localized by the disorder, delocalization in real space results from this mixing. This is why the interaction can induce in certain cases metallic behavior in a system which would be an insulator otherwise. When U is large and dominates, one can get on the contrary a correlated insulator which might be metallic at weaker interaction. A Wigner crystal pinned by disorder is a good example of such an interaction-induced insulator. In the large-U limit, t becomes the small parameter, and one expands in powers of t/U . The issue is to know the range of validity of these perturbative approaches, and to describe how the system goes from the first limit to the second when U increases. For this purpose, we consider a one-dimensional disordered lattice with on-site interactions. a e-mail: [email protected] We discuss the simple case of two electrons with opposite spins (the orbital part of the wavefunction is symmetric as in the case of two bosons). For the main sub-band of states centered around E = 0, both in the limits where U = 0 (free bosons) and U = ∞ (hard-core bosons), the two body states can be described in terms of two one body states. We use a duality transformation U → at/U to map the small U -limit onto the large U -limit (a ≈ √ 24). We first prove that the lifetimes of the free boson states and of the hard-core boson states are equal at the fixed point Uc of the duality transformation. At Uc one has the maximummixing of the one body states by the interaction and the enhancement factor [1] is maximum for the two particle localization length L2. Far from Uc, L2 is smaller and satisfies the duality relation. L2 → L1 (the one particle localization length) both for U → 0 and U → ∞. The study of the signature of this duality transformation on the spectral fluctuations is very interesting. For E = 0, taking L = L1 and increasing U , one gets two thresholds defining a range of interaction UF ≤ U ≤ UH. Outside this range, the levels are almost uncorrelated. Inside this range, the level repulsion is maximum, but does not reach the universal Wigner-Dyson (W-D) repulsion. The two particle system is not fully chaotic, but exhibits a weak chaos which is not arbitrarily situated between Poisson (integrable) and Wigner (chaos). The spacing distribution p(s) between consecutive energy levels and the statistics Σ2(E) (variance of the number of energy levels inside an energy interval E) are characteristic of the third known universality class [2]. One finds p(s) ≈ 4s exp(−2s) and Σ2(E) ≈ 0.16 + 0.41E for periodic boundary con2 Xavier Waintal et al.: TIP II: Critical statistics and maximum mixing of the one body states ditions. This is very close, if not identical, to the distributions found in many “critical” one body systems, such as an electron in a 3d random potential at the mobility edge [2,3,4] or in certain pseudo-integrable quantum billiards (rational triangles [5], rough billiards [6] and Kepler problem [7]). Furthermore, p(s) saturates to 4s exp(−2s) for U ≈ Uc only when the ratio 1 ≤ L1/L ≤ 10. We conclude that a local interaction can never drive the two particle system to full quantum chaos with Wigner-Dyson statistics in one dimension, but can at most yield weak critical chaos in a certain domain of interaction and of the ratios L1/L. We show in addition that this weak chaos is accompanied by multifractal wavefunctions, in agreement with Ref. [8]. Each one particle Hamiltonian is given by