We study the short-time behavior of the probability distribution P(H, t) of the surface height h(x = 0, t) = H in the Kardar-Parisi-Zhang (KPZ) equation in 1 + 1 dimension. The process starts from a stationary interface: h(x, t = 0) is given by a realization of two-sided Brownian motion constrained by h(0, 0) = 0. We find a singularity of the large deviation function of H at a critical value H = Hc. The singularity has the character of a second-order phase transition. It reflects spontaneous breaking of the reflection symmetry x ↔ −x of optimal paths h(x, t) predicted by the weak-noise theory of the KPZ equation. At |H| ≫ |Hc| the corresponding tail of P(H) scales as − ln P ∼ |H| 3/2 /t 1/2 and agrees, at any t > 0, with the proper tail of the Baik-Rains distribution, previously observed only at long times. The other tail of P scales as − ln P ∼ |H| 5/2 /t 1/2 and coincides with the corresponding tail for the sharp-wedge initial condition.

Statistical mechanics of 1D multivalent Coulomb gas may be mapped onto non-Hermitian quantum mechanics. We use this example to develop instanton calculus on Riemann surfaces. Borrowing from the formalism developed in the context of Seiberg-Witten duality, we treat momentum and coordinate as complex variables. Constant energy manifolds are given by Riemann surfaces of genus g ≥ 1. The actions along principal cycles on these surfaces obey ODE in the moduli space of the Riemann surface known as Picard-Fuchs equation. We derive and solve Picard-Fuchs equations for Coulomb gases of various charge content. Analysis of monodromies of these solutions around their singular points yields semiclassical spectra as well as instanton effects such as Bloch bandwidth. Both are shown to be in perfect agreement with numerical simulations.

A number of physical problems, including statistical mechanics of 1D multivalent Coulomb gases, may be formulated in terms of non-Hermitian quantum mechanics. We use this example to develop a non-perturbative method of instanton calculus for non-Hermitian Hamiltonians. This can be seen as an extension of semiclassical methods in conventional quantum mechanics. Treating momentum and coordinate as complex variables yields a Riemann surface of constant complex energy. The classical and instanton actions are given by periods of this surface; we show how to obtain these via methods from algebraic topology. We demonstrate the accuracy of this analytic procedure in comparison with numerical simulations for a class of periodic non-Hermitian Hamiltonians, as well as the validity of the Bohr–Sommerfeld quantization and Gamow's formula in these cases.

The critical point of a topological phase transition is described by a conformal field theory, where finite-size corrections to energy are uniquely related to its central charge. We investigate the finitesize scaling away from criticality and find a scaling function, which discriminates between phases with different topological indexes. This function appears to be universal for all five Altland-Zirnbauer symmetry classes with non-trivial topology in one spatial dimension. We obtain an analytic form of the scaling function and compare it with numerical results.Since the introduction of topological order in condensed matter physics, the field of topological insulators received constantly growing attention 1-4 . Although noninteracting topological phases were fully classified 5-7 and a plethora of topological edge states characterized 2,4,8-11 , little attention was given so far to finite-size effects around the topological transition. An important question is whether finite-size scaling is capable to distinguish between topological indexes and may be used as an indicator of the topological nature of the transition. One may also ask whether such scaling is universal or specific to a particular symmetry class, e.g. sensitive to Z vs. Z 2 topological index.In this paper we discuss the finite-size scaling of the ground state energy across topological phase transitions in 1 + 1 dimensional models. The critical point in such models is described by a conformal field theory 12 (CFT). The finite-size, N , scaling of the ground state energy E(N, 0) for an open system at criticality is known 13,14 to bewhere¯ (0) is the average bulk energy per particle, b(0) the size-independent boundary term and argument (0) specifies the exact critical point. Here length is measured in units of lattice spacing and energy in units of the Fermi velocity over the lattice spacing. The 1/N term appears to be universal and depends only on c -the central charge of the Virasoro algebra 12 .A relevant perturbation drives the system away from criticality, creating a spectral gap 2m and a corresponding correlation length ξ = 1/m. Our main observation is that the CFT expansion (1) may be generalized as arXiv:1508.03646v2 [cond-mat.stat-mech]

In addition, Cuffaro dedicates his efforts to the memory of Giuseppe, his father and Janssen dedicates his efforts to the memory of Heinrich, his father (1929Heinrich, his father ( -2015.