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Quantum Integrability in Systems with Finite Number of Levels

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Abstract

We consider the problem of defining quantum integrability in systems with finite number of energy levels starting from commuting matrices and construct new general classes of such matrix models with a given number of commuting partners. We argue that if the matrices depend on a (real) parameter, one can define quantum integrability from this feature alone, leading to specific results such as exact solvability, Poissonian energy level statistics and to level crossings.

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Notes

  1. Imagine that we are given a specific Hermitean 6×6 matrix, can we determine if it is the descendant of a quantum integrable model in a particular number-symmetry subsector? An explicit example might be from the numerical study in Ref. [17], where a number of matrices are found by restricting the Hubbard model to a particular number sector with all space time symmetries factored out. We provide a systematic method of looking at such questions in this work.

  2. For example, the Anderson localization of electrons in a random potential is often viewed as a transition from Poisson (localized) to GOE (extended) statistics. In this context it is not obvious if or how the concept of quantum integrability plays a role.

  3. For example, in the 1d Hubbard model H i(u) represent blocks of the Hamiltonian and linear in u conserved currents corresponding to a certain complete set of u-independent symmetry quantum numbers, see [5] for more details.

  4. More precisely it reads \(T_{km}^{i}(d_{k}^{j}-d_{m}^{j})=T_{km}^{j}(d_{k}^{i}-d_{m}^{i})\). If \(d_{k}^{i}=d_{m}^{i}\) for some, but not all i, S km is defined through the non-degenerate eigenvalues. If \(d_{k}^{i}=d_{m}^{i}\) for all i, matrices V i share a common 2×2 identity block. Then with a u-independent unitary transform one can go to a basis where \(T_{km}^{i}=0\) for all i. In either case Eq. (4) still holds.

  5. Degenerate d k are regarded as level crossings at u=∞. That such definition is necessary is seen e.g. by redefining the parameter u→1/x and multiplying the matrices by x. We have the same commuting family now linear in x, but with the crossing moved to x=0. Note also that we cannot fix the order of d k without loss of generality, because we have already chose our indexing so that ε k are ordered.

  6. If we set γ i =1, this construction is a simple extension of the permutation operator for SU(N) in Yang’s well known work [34] into matrix space, but seems to be new for the general case γ i ≠1.

  7. To obtain f(ε) in this limit, we scale ε k and d k with N so that they fill finite intervals when N→∞.

  8. Equivalently, one can redefine x→x/N as in the BCS model for which x=g=λδ, where λ is the dimensionless coupling and δ∝1/N is the mean level spacing in ε k .

  9. Such examples are available analytically (type N−2 matrices) or can be generated numerically by solving Eq. (5), see [5] for details.

  10. The “rule of three”, a term coined by McGuire in [40] originates from the fact that for three particles in one dimension, we have only two constants of motion generically. These are the total energy and the total momentum, and thus exact solvability is not to be expected. If the three particle case is solvable in some standard form, such as Bethe’s Ansatz, the rule empirically suggests the existence of sufficient further conservation laws and hence of exact solvability with arbitrary number of particles.

  11. Individual sectors do not have additional sector-specific u-dependent integrals. This is the case for the fermion (42) and Gaudin models, but not for the Hubbard model where there are numerous such sector-specific integrals [5, 38].

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Acknowledgements

E.A.Y. is grateful to UCSC Physics Department, where part of this research was conducted, and especially to Sriram Shastry for hospitality. The work at UCSC was supported in part by DOE under Grant No. FG02-06ER46319. E.A.Y. also acknowledges financial support by the David and Lucille Packard Foundation and the National Science Foundation under Award No. NSF-DMR-0547769. We thank our collaborators Haile Owusu and Daniel Hansen for helpful discussions.

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Authors and Affiliations

  1. Center for Materials Theory, Department of Physics and Astronomy, Rutgers University, Piscataway, NJ, 08854, USA

    Emil A. Yuzbashyan

  2. Physics Department, University of California, Santa Cruz, CA, 95064, USA

    B. Sriram Shastry

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  1. Emil A. Yuzbashyan
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Correspondence to B. Sriram Shastry.

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Yuzbashyan, E.A., Shastry, B.S. Quantum Integrability in Systems with Finite Number of Levels. J Stat Phys 150, 704–721 (2013). https://doi.org/10.1007/s10955-013-0689-9

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