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arXiv:1812.07316

Analytical solutions of inhomogeneous transverse field Ising models

Abhijit P. Chaudhari, Rajeev Singh, S. K. Mishra

Published Dec 18, 2018• 0 citations
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Abstract

The inhomogeneous transverse field Ising models mainly impurity based and the joint chain are analysed analytically using Jordan-Wigner transformations. The effects of inhomogeneities on the phase transition have been discussed in detail. We constructed an ansatz to diagonalize the two models which are taken into consideration. The inhomogeneity is quantified by a coupling parameter, which can be tuned to control the occurrence of quantum phase transition in these models. We have shown a systematic setup using which we can generalise the solution to a system with an arbitrary number of impurity sites and junctions, which are separated by at least two lattice sites. We have analysed the quantum critical point by calculating the correlation functions, transverse magnetization and the gap between the ground state and the first excited state.

DOI
10.48550/arxiv.1812.07316

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The inhomogeneous transverse field Ising models mainly impurity based and the joint chain are analysed analytically using Jordan-Wigner transformations. The effects of inhomogeneities on the phase transition have been discussed in detail. We constructed an ansatz to diagonalize the two models which are taken into consideration. The inhomogeneity is quantified by a coupling parameter, which can be tuned to control the occurrence of quantum phase transition in these models. We have shown a systematic setup using which we can generalise the solution to a system with an arbitrary number of impurity sites and junctions, which are separated by at least two lattice sites. We have analysed the quantum critical point by calculating the correlation functions, transverse magnetization and the gap between the ground state and the first excited state.
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Phase transitions in Ising models are a broadly studied subject within physics.In this thesis we will study the quantum mechanical transverse-field Ising model in order to determine analytical critical points for quantum phase transitions.This will be done by introducing the concept of fermionic operators and the Jordan-Wigner transformation.Hereafter it is possible to Fourier transform the transverse Ising Hamiltonian and derive an exact eigenenergy spectrum.With these eigenenergies it is possible to derive the condition(s) for which a phase transition can occur.This is done by looking when the eigenenergy solutions are equal to zero.This causes the system to become unstable and thus indicates a point of phase transition.Further in this thesis we will cover two different variations of the transverse-field Ising model which are the di-atomic transverse Ising chain and the frustrated Ising chain.The frustrated Ising chain will be a modification of the mono-atomic Ising chain where there is an extra interaction term introduced which will require an assumption of lowest order approximation of perturbation theory to solve analytically.I would like to thank Lars Fritz for the great learning experience as a physicist within my bachelor thesis.
Abstract Advance in quantum simulations using trapped ions or superconducting elements allows detailed analysis of the transverse field Ising model (TFIM), which can exhibit a quantum phase transition and has been a paradigm in exactly solvable quantum systems. The Jordan–Wigner transformation maps the one-dimensional TFIM to a fermion model, but additional complications arise in finite systems and introduce a fermion-number parity constraint when periodic boundary condition is imposed. By constructing the free energy and spin correlations with the fermion-number parity constraint and comparing the results to the TFIM with open boundary condition, we show that the boundary effects can become significant for the anti-ferromagnetic TFIM with odd number of sites at low temperature.
We investigate the non-equilibrium dynamics of the transverse field quantum Ising chain evolving from an inhomogeneous initial state given by joining two macroscopically different semi-infinite chains. We obtain integral expressions for all two-point correlation functions of the Jordan--Wigner Majorana fermions at any time and for any value of the transverse field. Using this result, we analytically compute the profiles of various physical observables in the space-time scaling limit and show that they can be obtained from a hydrodynamic picture based on ballistically propagating quasiparticles. Going beyond the hydrodynamic limit, we analyze the approach to the non-equilibrium steady state and find that the leading late time corrections display a lattice effect. We also study the fine structure of the propagating fronts which are found to be described by the Airy kernel and its derivatives. Near the front we observe the phenomenon of energy back-flow where the energy locally flows from the colder to the hotter region.
Abstract We study the quantum Ising model in the transverse inhomogeneous magnetic field. Such a system can be approached numerically through exact diagonalization and analytically through the renormalization group techniques. Basic insight into its physics, however, can be obtained by adopting the Kibble–Zurek theory of non-equilibrium phase transitions to description of spatially inhomogeneous systems at equilibrium. We employ all these approaches and focus on derivatives of longitudinal and transverse magnetizations, which have extrema near the critical point. We discuss how these extrema can be used for locating the critical point and for verification of the Kibble–Zurek scaling predictions in the spatial quench.
The isotropic XYXYXY model (s=1/2)(s=1/2)(s=1/2) in a transverse field, with uniform long-range interactions among the transverse components of the spins, on the inhomogeneous periodic chain, is studied. The model, composed of NNN segments with nnn different exchange interactions and magnetic moments, is exactly solved by introducing the integral Gaussian transformation and the generalized Jordan--Wigner transformation, which reduce the problem to the diagonalization of a finite matrix of nthn\text{th}nth order. The quantum transitions induced by the transverse field are determined by analyzing the induced magnetization of the cell and the equation of state. The phase diagrams for the quantum transitions, in the space generated by the transverse field and the interaction parameters, are presented. As expected, the model presents multiple, first- and second-order quantum transitions induced by the transverse field, and it corresponds to an extension of the models recently considered by the authors. Detailed results are also presented, at T=0T=0T=0, for the induced magnetization and isothermal susceptibility ${\ensuremath{\chi}}_{T}^{zz}$ as a function of the transverse field.