We examine a parametric cycle in the N -body Lieb-Liniger model that starts from the free system and goes through Tonks-Girardeau and super-Tonks-Girardeau regimes and comes back to the free system. We show the existence of exotic quantum holonomy, whose detailed workings are analyzed with the specific sample of two-and three-body systems. The classification of eigenstates based on clustering structure naturally emerges from the analysis.

Entanglement of multipartite systems is studied based on exchange symmetry under the permutation group SN . With the observation that symmetric property under the exchange of two constituent states and their separability are intimately linked, we show that anti-symmetric (fermionic) states are necessarily globally entangled, while symmetric (bosonic) states are either globally entangled or fully separable and possess essentially identical states in all the constituent systems. It is also shown that there cannot exist a fully separable state which is orthogonal to all symmetric states, and that full separability of states does not survive under total symmetrization unless the states are originally symmetric. Besides, anyonic states permitted under the braid group BN should also be globally entangled. Our results reveal that exchange symmetry is actually sufficient for pure states to become globally entangled or fully separable.

We consider a series of massive scaling limits m 1 → ∞, q → 0, lim m 1 q = Λ 3 followed by2 of the β-deformed matrix model of Selberg type (N c = 2, N f = 4) which reduce the number of flavours to N f = 3 and subsequently to N f = 2. This keeps the other parameters of the model finite, which include n = N L and N = n+N R , namely, the size of the matrix and the "filling fraction". Exploiting the method developed before, we generate instanton expansion with finite g s , ǫ 1,2 to check the Nekrasov coefficients (N f = 3, 2 cases) to the lowest order. The limiting expressions provide integral representation of irregular conformal blocks which contains a 2d operator limn : e (1/2)α 2 φ(q) : and is subsequently analytically continued. *

We consider the half-genus expansion of the resolvent function in the β-deformed matrix model with three-Penner potential under the AGT conjecture and the 0d-4d dictionary. The partition function of the model, after the specification of the paths, becomes the DF conformal block for fixed c and provides the Nekrasov partition function expanded both in gs = √ − 1 2 and in = 1 + 2 . Exploiting the explicit expressions for the lower terms of the free energy extracted from the above expansion, we derive the first few corrections to the Seiberg-Witten prepotential in terms of the parameters of SU (2), N f = 4, N = 2 supersymmetric gauge theory.

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We study the inequivalent quantizations of the N = 3 Calogero model by separation of variables, in which the model decomposes into the angular and the radial parts. Our inequivalent quantizations respect the 'mirror-S 3 ' invariance (which realizes the symmetry under the cyclic permutations of the particles) and the scale invariance in the limit of vanishing harmonic potential. We find a two-parameter family of novel quantizations in the angular part and classify the eigenstates in terms of the irreducible representations of the S 3 group. The scale invariance restricts the quantization in the radial part uniquely, except for the eigenstates coupled to the lowest two angular levels for which two types of boundary conditions are allowed independently from all upper levels. It is also found that the eigenvalues corresponding to the singlet representations of the S 3 are universal (parameter-independent) in the family, whereas those corresponding to the doublets of the S 3 are dependent on one of the parameters. These properties are shown to be a consequence of the spectral preserving SU (2) (or its subrgoup U (1)) transformations allowed in the family of inequivalent quantizations.

An interplay of an exotic quantum holonomy and exceptional points is examined in onedimensional Bose systems. The eigenenergy anholonomy, in which Hermitian adiabatic cycle induces nontrivial change in eigenenergies, can be interpreted as a manifestation of eigenenergy's Riemann surface structure, where the branch points are identified as the exceptional points which are degeneracy points in the complexified parameter space. It is also shown that the exceptional points are the divergent points of the non-Abelian gauge connection for the gauge theoretical formulation of the eigenspace anholonomy. This helps us to evaluate anti-path-ordered exponentials of the gauge connection to obtain gauge covariant quantities.

Approximated formulas for real quasimomentum and the associated energy spectrum are presented for onedimensional Bose gas with weak attractive contact interactions. On the basis of the energy spectrum, we obtain the equation of state in the high-temperature region, which is found to be the van der Waals equation without volume correction.