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arXiv:1302.0230 · Physical Review B

Multiparticle spectral properties in the transverse field Ising model by continuous unitary transformations

Benedikt Fauseweh, Götz Silvester Uhrig

Published May 9, 2013• 15 citations
arXivHTMLPDF

Abstract

The one-dimensional transverse field Ising model is solved by continuous unitary transformations in the high-field limit. A high accuracy is reached due to the closure of the relevant algebra of operators, which we call string operators. The closure is related to the possibility to map the model by Jordan-Wigner transformation to noninteracting fermions, but it is proven without referring to this mapping. The effective model derived by the continuous unitary transformations is used to compute the contributions of one, two, and three elementary excitations to the diagonal dynamic structure factors. The three-particle contributions have, so far, not been addressed analytically, except close to the quantum critical point.

DOI
10.1103/physrevb.87.184406

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The one-dimensional transverse field Ising model is solved by continuous unitary transformations in the high-field limit. A high accuracy is reached due to the closure of the relevant algebra of operators, which we call string operators. The closure is related to the possibility to map the model by Jordan-Wigner transformation to noninteracting fermions, but it is proven without referring to this mapping. The effective model derived by the continuous unitary transformations is used to compute the contributions of one, two, and three elementary excitations to the diagonal dynamic structure factors. The three-particle contributions have, so far, not been addressed analytically, except close to the quantum critical point.
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.
In this thesis we describe a novel perturbative approach to low-dimensional quantum many-particle systems, which is based on continuous unitary transformations. We consider systems, which are defined on a lattice and allow a perturbative decomposition. The unperturbed part must have an equidistant spectrum -- the difference between two successive levels is called a quasi-particle. In this case the perturbative transformation leads to an effective Hamiltonian, which conserves the number of particles. The same transformation is used to also derive the effective counterparts of other, experimentally relevant observables. The effective operators are obtained as series expansions in the (small) perturbation parameter. In each order we find a set of products of ladder-operators, which act on states uniquely defined by the number of quasi-particles and their position in the lattice. Thus all calculations can be done in real space. The mathematical structure of the effective operators is extensively analysed. We additionally give all details necessary to implement the method on a computer, which allows the calculation of the effective quantities up to high orders. The method facilitates quantitative calculations of multi-particle excitations and spectral densities of experimentally relevant observables. We include a comprehensive application of the method to the two-dimensional Shastry-Sutherland model, a strongly frustrated quantum spin system. The model has experience an experimental realization by SrCu2(BO3)2. The limit of strong dimerization serves as starting point. Here the ground state is given by singlets on all dimers -- a single triplet constitutes an elementary excitation, i.e. quasi-particle. We quantitatively calculate the one- and two-triplet energies as well as the spectral densities of the Raman and neutron scattering operators. The findings for the spectral densities in particular represent new results. Comparing them to experimental data leads to interesting insights into the spectral properties of low-dimensional quantum spin systems.
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.
One dimensional quantum Ising model with nearest neighbor interaction in transverse magnetic field is one of the simplest spin models which undergo quantum phase transition. This model has been precisely solved using different methods. In this paper, we solve this model in uniform magnetic field -Jgσxn precisely using a new method called Continuous Unitary Transformations (CUT) or flow equations, and derive its expectation values <σxn>, <σzn> and <σxnσxn+j>. Then, we apply this method on one dimensional quantum Ising model in staggered magnetic field (-1)nJgσxn. Results show that both models have the same critical properties as expected, and it was also found that spontaneous symmetry breaking cannot be derived from CUT.
We study the structure of the continuous matrix product operator (cMPO)[1] for the transverse field Ising model (TFIM). We prove TFIM’s cMPO is solvable and has the form <?CDATA T=e−12H^FT={{\rm{e}}}^{-\displaystyle \frac{1}{2}{\hat{H}}_{{\rm{F}}}}T=e−21​H^F​?> T = e − 1 2 H ^ F . <?CDATA H^F{\hat{H}}_{{\rm{F}}}H^F​?> H ^ F is a non-local free fermionic Hamiltonian on a ring with circumference β, whose ground state is gapped and non-degenerate even at the critical point. The full spectrum of <?CDATA H^F{\hat{H}}_{{\rm{F}}}H^F​?> H ^ F is determined analytically. At the critical point, our results verify the state–operator-correspondence[2] in the conformal field theory (CFT). We also design a numerical algorithm based on Bloch state ansatz to calculate the low-lying excited states of general (Hermitian) cMPO. Our numerical calculations coincide with the analytic results of TFIM. In the end, we give a short discussion about the entanglement entropy of cMPO’s ground state.
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.