arXiv subjects

Alexios P. Polychronakos

Publications and source records attributed to Alexios P. Polychronakos.

At least 37 records · Page 2Linked to original sources

arXiv2022

Length and area generating functions for height-restricted Motzkin meanders

We derive the length and area generating function of planar height-restricted forward-moving discrete paths of increments +1, 0, or -1 with arbitrary starting and ending points, the so-called Motzkin meanders, and the more general length-area generating functions for Motzkin paths with markers monitoring the number of passages from the two height boundaries ('floor' and 'ceiling') and the time spent there. The results are obtained by embedding Motzkin paths in a two-step anisotropic Dyck path process and using propagator, exclusion statistics and bosonization techniques. We also present a cluster expansion of the logarithm of the generating functions that makes their polynomial structure explicit. These results are relevant to the derivation of statistical mechanical properties of physical systems such as polymers, vesicles, and solid-on-solid interfaces.

math-ph↗

arXiv2021

Algebraic area enumeration of random walks on the honeycomb lattice

We study the enumeration of closed walks of given length and algebraic area on the honeycomb lattice. Using an irreducible operator realization of honeycomb lattice moves, we map the problem to a Hofstadter-like Hamiltonian and show that the generating function of closed walks maps to the grand partition function of a system of particles with exclusion statistics of order $g=2$ and an appropriate spectrum, along the lines of a connection previously established by two of the authors. Reinterpreting the results in terms of the standard Hofstadter spectrum calls for a mixture of $g=1$ (fermion) and $g=2$ exclusion whose physical meaning and properties require further elucidation. In this context we also obtain some unexpected Fibonacci sequences within the weights of the combinatorial factors appearing in the counting of walks.

math-ph↗

arXiv2021

Hamiltonian and exclusion statistics approach to discrete forward-moving paths

We use a Hamiltonian (transition matrix) description of height-restricted Dyck paths on the plane in which generating functions for the paths arise as matrix elements of the propagator to evaluate the length and area generating function for paths with arbitrary starting and ending points, expressing it as a rational combination of determinants. Exploiting a connection between random walks and quantum exclusion statistics that we previously established, we express this generating function in terms of grand partition functions for exclusion particles in a finite harmonic spectrum and present an alternative, simpler form for its logarithm that makes its polynomial structure explicit.

math-ph↗

arXiv2020

A shifted binomial theorem and trigonometric series

We introduce a shifted version of the binomial theorem, and use it to study some remarkable trigonometric integrals and their explicit rewriting in terms of binomial multiple sums. Motivated by the expressions of area generating functions arising in the counting of closed walks on various lattices, we propose similar sums involving fractional values of the area and show that they are closely related to their integer counterparts and lead to rational sequences converging to powers of $π$. Our results, other than their mathematical interest, could be relevant to generalizations of statistical mechanical models of the Heisenberg chain type involving higher spins or $SU(N)$ degrees of freedom.

math-ph↗

arXiv2020

Lyapunov exponents and entanglement entropy transition on the noncommutative hyperbolic plane

We study quantum dynamics on noncommutative spaces of negative curvature, focusing on the hyperbolic plane with spatial noncommutativity in the presence of a constant magnetic field. We show that the synergy of noncommutativity and the magnetic field tames the exponential divergence of operator growth caused by the negative curvature of the hyperbolic space. Their combined effect results in a first-order transition at a critical value of the magnetic field in which strong quantum effects subdue the exponential divergence for {\it all} energies, in stark contrast to the commutative case, where for high enough energies operator growth always diverge exponentially. This transition manifests in the entanglement entropy between the `left' and `right' Hilbert spaces of spatial degrees of freedom. In particular, the entanglement entropy in the lowest Landau level vanishes beyond the critical point. We further present a non-linear solvable bosonic model that realizes the underlying algebraic structure of the noncommutative hyperbolic plane with a magnetic field.

hep-th↗

arXiv2020

Exchange interactions, Yang-Baxter relations and transparent particles

We introduce a class of particle models in one dimension involving exchange interactions that have scattering properties satisfying the Yang-Baxter consistency condition. A subclass of these models exhibits reflectionless scattering, in which particles are "transparent" to each other, generalizing a property hitherto only known for the exchange Calogero model. The thermodynamics of these systems can be derived using the asymptotic Bethe-ansatz method.

hep-th↗

arXiv2020

Solitons in fluctuating hydrodynamics of diffusive processes

We demonstrate that fluid mechanical systems arising from large fluctuations of one-dimensional statistical processes generically exhibit solitons and nonlinear waves. We derive the explicit form of these solutions and examine their properties for the specific cases of the Kipnis-Marchioro-Presutti model (KMP) and the Symmetric Exclusion Process (SEP). We show that the two fluid systems are related by a nonlinear transformation, which reveals an additional symmetry in the SEP system, but still have markedly different properties. In particular, the KMP fluid has a nontrivial sound wave spectrum exhibiting birefringence, while sound waves for the SEP fluid are essentially trivial. The appearance of sound waves and soliton configurations in the KMP model is related to the onset of instabilities.

cond-mat.stat-mech↗

arXiv2019

Two anyons on the sphere: nonlinear states and spectrum

We study the energy spectrum of two anyons on the sphere in a constant magnetic field. Making use of rotational invariance we reduce the energy eigenvalue equation to a system of linear differential equations for functions of a single variable, a reduction analogous to separating center of mass and relative coordinates on the plane. We solve these equations by a generalization of the Frobenius method and derive numerical results for the energies of non-analytically derivable states.

hep-th↗

arXiv2019

Exclusion statistics and lattice random walks

We establish a connection between exclusion statistics with arbitrary integer exclusion parameter $g$ and a class of random walks on planar lattices. This connection maps the generating function for the number of closed walks of given length enclosing a given algebraic area on the lattice to the grand partition function of particles obeying exclusion statistics $g$ in a particular single-particle spectrum, determined by the properties of the random walk. Square lattice random walks, described in terms of the Hofstadter Hamiltonian, correspond to $g=2$. In the $g=3$ case we explicitly construct a corresponding chiral random walk model on a triangular lattice, and we point to potential random walk models for higher $g$. In this context, we also derive the form of the microscopic cluster coefficients for arbitrary exclusion statistics.

cond-mat.stat-mech↗

arXiv2019

Anyons on the sphere: analytic states and spectrum

We analyze the quantum mechanics of anyons on the sphere in the presence of a constant magnetic field. We introduce an operator method for diagonalizing the Hamiltonian and derive a set of exact anyon energy eigenstates, in partial correspondence with the known exact eigenstates on the plane. We also comment on possible connections of this system with integrable systems of the Calogero type.

hep-th↗

arXiv2018

Feynman's proof of the commutativity of the Calogero integrals of motion

During his last year of life Feynman became interested in integrable models. In the course of his study of one dimensional particles with inverse square-type potentials (the Calogero class of models) he came up with a field theory-inspired proof of the commutativity of the integrals of motion of the system, which remained unpublished. I present an exact transcript of the proof together with a scan of the first page of Feynman's original manuscript and some historical background.

physics.hist-ph↗

arXiv2017

Emergence of Calogero family of models in external potentials: Duality, Solitons and Hydrodynamics

We present a first-order formulation of the Calogero model in external potentials in terms of a generating function, which simplifies the derivation of its dual form. Solitons naturally appear in this formulation as particles of negative mass. Using this method, we obtain the dual form of Calogero particles in external quartic, trigonometric and hyperbolic potentials, which were known to be integrable but had no known dual formulation. We derive the corresponding soliton solutions, generalizing earlier results for the harmonic Calogero system, and present numerical results that demonstrate the integrable nature of the soliton motion. We also give the collective fluid mechanical formulation of these models and derive the corresponding fluid soliton solutions in terms of meromorphic fields, commenting on issues of stability and integrability.

math-ph↗

arXiv2016

Composition of many spins, random walks and statistics

The multiplicities of the decomposition of the product of an arbitrary number $n$ of spin $s$ states into irreducible $SU(2)$ representations are computed. Two complementary methods are presented, one based on random walks in representation space and another based on the partition function of the system in the presence of a magnetic field. The large-$n$ scaling limit of these multiplicities is derived, including nonperturbative corrections, and related to semiclassical features of the system. A physical application of these results to ferromagnetism is explicitly worked out. Generalizations involving several types of spins, as well as spin distributions, are also presented. The corresponding problem for (anti-)symmetric composition of spins is also considered and shown to obey remarkable duality and bosonization relations and exhibit qualitatively different large-$n$ scaling properties.

hep-th↗

arXiv2015

Cosmic Four-Fermion Neutrino Secret Interactions, Enhancement and Total Cross Section

The scattering of neutrinos assuming a "secret" interaction at low energy is considered. To leading order in energy, the two-body potential is a $δ$-potential, and it is used as a motivation to study generic short-range elastic interactions between neutrinos. The scattering cross section and Sommerfeld enhancement depend on two phenomenological parameters deriving from the exact form of the potential, akin to "renormalized" coupling constants. Repulsive potentials lead to a decrease in the total cross section, resulting in an enhancement of the neutrino density. For attractive potentials of the right form, substantial Sommerfeld enhancement can appear.

hep-ph↗

arXiv2014

Exact operator Hamiltonians and interactions in the droplet bosonization method

We derive the exact form of the bosonized Hamiltonian for a many-body fermion system in one spatial dimension with arbitrary dispersion relations, using the droplet bosonization method. For a single-particle Hamiltonian polynomial in the momentum, the bosonized Hamiltonian is a polynomial of one degree higher in the bosonic "boundary" field and includes subleading lower-order and derivative terms. This generalizes the known results for massless relativistic and nonrelativistic fermions (quadratic and cubic bosonic Hamiltonians, respectively). We also consider two-body interactions and demonstrate that they lead to interesting collective behavior and phase transitions in the Fermi sea.

hep-th↗

arXiv2013

Effective action and phase transitions of scalar field on the fuzzy sphere

Scalar field theory on the fuzzy two-sphere, represented as a hermitian matrix model that includes kinetic, mass and quartic interaction terms, is studied. The effective action in the symmetric large-N regime is analyzed using a self-consistent bootstrap method which fixes its form up to sixth order in the eigenvalues and gives a closed expression to all orders in the quadratic invariant of the matrix, valid close to semicircular distributions. Using this action the eigenvalue distribution is calculated for the interacting theory in the appropriate scaling limit and the phase transition from the disordered to the symmetric ordered phase is identified, including nonperturbative effects.

hep-th↗

arXiv2013

Brane compactifications and 4-dimensional geometry in the IKKT model

We study in detail certain brane solutions with compact extra dimensions M^4 x K in the IKKT matrix model, with K being a two-dimensional rotating torus embedded in R^6. We focus on the compactification moduli and the fluctuations of K \subset R^6 and their physical significance. Mediated by the Poisson tensor, they contribute to the effective 4-dimensional metric on the brane, and thereby become gravitational degrees of freedom. We show that the zero modes corresponding to the global symmetries of the model lead to Ricci-flat 4-dimensional metric perturbations, wherever the energy-momentum tensor vanishes. Their coupling to the energy momentum tensor depends on the extrinsic curvature of the brane.

hep-th↗

arXiv2013

The quantum angular Calogero-Moser model

The rational Calogero-Moser model of n one-dimensional quantum particles with inverse-square pairwise interactions (in a confining harmonic potential) is reduced along the radial coordinate of R^n to the `angular Calogero-Moser model' on the sphere S^{n-1}. We discuss the energy spectrum of this quantum system, its degeneracies and the eigenstates. The spectral flow with the coupling parameter yields isospectrality for integer increments. Decoupling the center of mass before effecting the spherical reduction produces a `relative angular Calogero-Moser model', which is analyzed in parallel. We generalize our considerations to the Calogero-Moser models associated with Coxeter groups. Finally, we attach spin degrees of freedom to our particles and extend the results to the spin-Calogero system.

math-ph↗