Calogero-Sutherland model (CSM) is a quantum integrable many-body system having a long-range interaction in one spatial dimension. The replica trick that is commonly used in the disordered system [NK02] is borrowed to compute the thermodynamic particle density in the vicinity of a boundary or an impurity [GNK02]. Combining this with the result for the bulk correlators, complete expansion formulae of the density and anyon creation operators in terms of vertex operators of a free boson are found. This enabled the computation of any thermodynamic density correlators and Green's functions of the CSM at any (irrational) values of couplings, in any 1+1D geometry.
Strongly coupled gauge theories have been a subject of extensive study, as solvable prototypes of realistic gauge field theories. A lattice version of such systems, with the gauge group generalized to any of classical Lie groups is considered in the large-Nc limit [NN01]. Exploitation of a novel identity called color-flavor transformation enables to demonstrate the homogeneity of the ground state configuration of mesons, thereby proving the chiral symmetry breaking from the first principle, and to derive non-linear sigma models over three types of symmetric spaces describing the Goldstone mesons. An effective Lagrangian derived from the global symmetry of 3D two-color QCD is used to uncover its phase structure in the T-mu plane [DN03].
It has been numerically established that the spectral fluctuation of Anderson Hamiltonians (AHs) at the metal-insulator transition (MIT) is universalily governed by a statistics that is a hybrid of Wigner-Dyson (long-range correlation) and Poisson (no correlation). [N99] pursues a possibility of describing this 'critical' statistics by non-standard random matrix ensembles that share the same multifractality of the wave functions as the critical AHs. By tuning the inverse-conductance parameter that characterizes the weak multifractality, analytically obtained expressions for the energy level spacing distributions are observed to be in perfect agreement with the numerical data from the AHs at the MIT.
The resemblance between the Dirac operator of QCD in the strong coupling regime and a disordered Hamiltonian was elevated into a conjecture that the spectral fluctuation of the former is described by 'chiral' random matrix ensembles (chRMEs) sharing the global symmetries of QCD. Universality of the spectral correlation functions of the chRMEs that underlies the said conjecture is established in [ADMN97]. Various spectral correlators of three symmetry classes of the chRMEs at finite quark masses, which are more realistic than the original massless or quenched model, are computed in [DN98, NN00], providing basis for comparison with lattice simulations.
Nonperturbative methods of large-N field theories are applied to describe statistical systems of geometrical objects, including Wilsonian renormalization group for matrix and vector-valued scalar field theories [N96], Brezin-Zinn-Justin method as a renormalization group on random surfaces [HINS95], and the double scaling limit of linear sigma models describing randomly branching polymers [NY90].
It is my privilege to have had joint papers with and learned much from researchers whom I respect, including Professors T. Yoneya, N. Sakai, M. Moshe, P. Damgaard, T. Nagao, J. Verbaarschot, A. Kamenev, and G. Dunne.
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