arXiv subjects

T. R. Vishnu

Publications and source records attributed to T. R. Vishnu.

8 recordsLinked to original sources

arXiv2026

Nonlinearity-induced transition in heat conduction through a topological metamaterial of rotors

We investigate heat conduction in a one-dimensional chain of rigid rotors. The rotors are constrained to rotate in a plane about fixed pivot points and coupled by springs, such that in equilibrium, the neighboring rotors lie on opposite sides of the chain axis. The linearized limit of this model valid for small angular displacements, was first introduced by Kane and Lubensky (KL) as a topological mechanical insulator hosting zero-energy vibrational edge modes. We show that the linearized KL chain behaves as a thermal insulator at low temperature in both the topological phases with a finite band gap, and the heat current falls exponentially with the chain length. When the gap vanishes at the topological phase transition, the KL chain becomes a good thermal conductor and conducts heat ballistically. The chain of rotors for arbitrary angular displacements hosts nonlinear solitary waves and distinct topological mechanical phases. Our numerical analysis shows normal (diffusive) heat conduction in all topological phases of the nonlinear chain. Nevertheless, a finite thermal conductivity is achieved for different system sizes in different topological phases of this nonlinear chain.

cond-mat.stat-mech↗

arXiv2025

From Chiral Topological Dynamics to Chiral Topological Amplification: Real vs Imaginary Parameters in a Hermitian Bosonic Chain

We propose a Hermitian quadratic bosonic model (QBH) whose dynamical matrix exhibits distinct topological and dynamical phenomena depending on whether the hopping and pairing amplitudes are real or purely imaginary. In the real-parameter regime, the dynamical matrix is unitarily equivalent to four decoupled copies of the sublattice-symmetric non-Hermitian Su-Schrieffer-Heeger (nSSH2) model, thereby inheriting its topological phases and energy spectrum-including the Möbius phase, a gapless topological phase with fractional winding number, having no Hermitian counterpart. We show that the dynamics generated by the QBH Hamiltonian naturally reproduce non-Hermitian time evolution, without invoking nonlinear Schrödinger dynamics or ad hoc normalization. It is demonstrated by analytically calculating the Loschmidt amplitude and computing the dynamical topological order parameter under periodic boundary conditions, which displays a distinct chiral response in the Möbius phase. In contrast, when the hopping and pairing terms are taken to be purely imaginary, the dynamical matrix becomes unitarily equivalent to a different version of the sublattice-symmetric non-Hermitian Su-Schrieffer-Heeger (nSSH1) model that supports only two topological phases: trivial and non-trivial, and the Möbius phase disappears. The latter system exhibits sublattice-dependent chiral amplification under open boundary conditions. We show that this amplification arises from the non-trivial topology of the dynamical matrix, establishing a clear link between topological phase and amplification behavior in the imaginary-parameter regime.

quant-ph↗

arXiv2024

Screwon spectral statistics and dispersion relation in the quantum Rajeev-Ranken model

The Rajeev-Ranken (RR) model is a Hamiltonian system describing screw-type nonlinear waves (screwons) of wavenumber $k$ in a scalar field theory pseudodual to the 1+1D SU(2) principal chiral model. Classically, the RR model based on a quadratic Hamiltonian on a nilpotent/Euclidean Poisson algebra is Liouville integrable. Upon adopting canonical variables in a slightly extended phase space, the model was interpreted as a novel 3D cylindrically symmetric quartic oscillator with a rotational energy. Here, we examine the spectral statistics and dispersion relation of quantized screwons via numerical diagonalization validated by variational and perturbative approximations. We also derive a semiclassical estimate for the cumulative level distribution which compares favorably with the one from numerical diagonalization. The spectrum shows level crossings typical of an integrable system. The $i^{\rm th}$ unfolded nearest neighbor spacings are found to follow Poisson statistics for small $i$. Nonoverlapping spacing ratios also indicate that successive spectral gaps are independently distributed. After displaying universal linear behavior over energy windows of short lengths, the spectral rigidity saturates at a length and value that scales with the square-root of energy. For strong coupling $λ$ and intermediate $k$, we argue that reduced screwon energies can depend only on the product $λk$. Numerically, we find power law dependences on $λ$ and $k$ with an approximately common exponent $2/3$ provided the angular momentum quantum number $l$ is small compared to the number of nodes $n$ in the radial wavefunction. On the other hand, for the ground state $n = l = 0$, the common exponent becomes 1.

nlin.SI↗

arXiv2024

Heat transport through an open coupled scalar field theory hosting stability-to-instability transition

We investigate heat transport through a one-dimensional open coupled scalar field theory, depicted as a network of harmonic oscillators connected to thermal baths at the boundaries. The non-Hermitian dynamical matrix of the network undergoes a stability-to-instability transition at the exceptional points as the coupling strength between the scalar fields increases. The open network in the unstable regime, marked by the emergence of inverted oscillator modes, does not acquire a steady state, and the heat conduction is then unbounded for general bath couplings. In this work, we engineer a unique bath coupling where a single bath is connected to two fields at each edge with the same strength. This configuration leads to a finite steady-state heat conduction in the network, even in the unstable regime. We also study general bath couplings, e.g., connecting two fields to two separate baths at each boundary, which shows an exciting signature of approaching the unstable regime for massive fields. We derive analytical expressions for high-temperature classical heat current through the network for different bath couplings at the edges and compare them. Furthermore, we determine the temperature dependence of low-temperature quantum heat current in different cases. Our study will help to probe topological phases and phase transitions in various quadratic Hermitian bosonic models whose dynamical matrices resemble non-Hermitian Hamiltonians, hosting exciting topological phases.

cond-mat.stat-mech↗

arXiv2023

Spectral solutions for the Schrödinger equation with a regular singularity

We propose a modification in the Bethe-like ansatz to reproduce the hydrogen atom spectrum and the wave functions. Such a proposal provided a clue to attempt the exact quantization conditions (EQC) for the quantum periods associated with potentials V (x) which are singular at the origin. In a suitable limit of the parameters, the potential can be mapped to |x| potential. We validate our EQC proposition by numerically computing the Voros spectrum and matching it with the true spectrum for |x| potential. Thus we have given a route to obtain the spectral solution for the one dimensional Schrödinger equation involving potentials with regular singularity at the origin.

quant-ph↗

arXiv2020

An introduction to Lax pairs and the zero curvature representation

Lax pairs are a useful tool in finding conserved quantities of some dynamical systems. In this expository article, we give a motivated introduction to the idea of a Lax pair of matrices $(L,A)$, first for mechanical systems such as the linear harmonic oscillator, Toda chain, Eulerian rigid body and the Rajeev-Ranken model. This is then extended to Lax operators for one-dimensional field theories such as the linear wave and KdV equations and reformulated as a zero curvature representation via a $(U,V)$ pair which is illustrated using the nonlinear Schrödinger equation. The key idea is that of realizing a (possibly) nonlinear evolution equation as a compatibility condition between a pair of linear equations. The latter could be an eigenvalue problem for the Lax operator $L$ and a linear evolution equation generated by $A$, for the corresponding eigenfunction. Alternatively, they could be the first order linear system stating the covariant constancy of an arbitrary vector with respect to the 1+1 dimensional gauge potential $(V,U)$. The compatibility conditions are then either the Lax equation $\dot L = [L, A]$ or the flatness condition $U_t - V_x + [U, V] = 0$ for the corresponding gauge potential. The conserved quantities then follow from the isospectrality of the Lax and monodromy matrices.

nlin.SI↗

arXiv2019

Invariant tori, action-angle variables and phase space structure of the Rajeev-Ranken model

We study the classical Rajeev-Ranken model, a Hamiltonian system with three degrees of freedom describing nonlinear continuous waves in a 1+1-dimensional nilpotent scalar field theory pseudodual to the SU(2) principal chiral model. While it loosely resembles the Neumann and Kirchhoff models, its equations may be viewed as the Euler equations for a centrally extended Euclidean algebra. The model has a Lax pair and r-matrix leading to four generically independent conserved quantities in involution, two of which are Casimirs. Their level sets define four-dimensional symplectic leaves on which the system is Liouville integrable. On each of these leaves, the common level sets of the remaining conserved quantities are shown, in general, to be 2-tori. The non-generic level sets can only be horn tori, circles and points. They correspond to measure zero subsets where the conserved quantities develop relations and solutions degenerate from elliptic to hyperbolic, circular or constant functions. A geometric construction allows us to realize each common level set as a bundle with base determined by the roots of a cubic polynomial. A dynamics is defined on the union of each type of level set, with the corresponding phase manifolds expressed as bundles over spaces of conserved quantities. Interestingly, topological transitions in energy hypersurfaces are found to occur at energies corresponding to horn tori, which support purely homoclinic orbits. The dynamics on each horn torus is non-Hamiltonian, but expressed as a gradient flow. Finally, we discover a family of action-angle variables for the system that apply away from horn tori.

nlin.SI↗

arXiv2019

On the Hamiltonian formulation, integrability and algebraic structures of the Rajeev-Ranken model

The integrable 1+1-dimensional SU(2) principal chiral model (PCM) serves as a toy-model for 3+1-dimensional Yang-Mills theory as it is asymptotically free and displays a mass gap. Interestingly, the PCM is 'pseudodual' to a scalar field theory introduced by Zakharov and Mikhailov and Nappi that is strongly coupled in the ultraviolet and could serve as a toy-model for non-perturbative properties of theories with a Landau pole. Unlike the 'Euclidean' current algebra of the PCM, its pseudodual is based on a nilpotent current algebra. Recently, Rajeev and Ranken obtained a mechanical reduction by restricting the nilpotent scalar field theory to a class of constant energy-density classical waves expressible in terms of elliptic functions, whose quantization survives the passage to the strong-coupling limit. We study the Hamiltonian and Lagrangian formulations of this model and its classical integrability from an algebraic perspective, identifying Darboux coordinates, Lax pairs, classical r-matrices and a degenerate Poisson pencil. We identify Casimirs as well as a complete set of conserved quantities in involution and the canonical transformations they generate. They are related to Noether charges of the field theory and are shown to be generically independent, implying Liouville integrability. The singular submanifolds where this independence fails are identified and shown to be related to the static and circular submanifolds of the phase space. We also find an interesting relation between this model and the Neumann model allowing us to discover a new Hamiltonian formulation of the latter.

hep-th↗