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arXiv:1507.08624 · Physical Review B

Quasi-soliton scattering in quantum spin chains

Rogier Vlijm, Martin Ganahl, Davide Fioretto, Michael Brockmann, Masudul Haque, Hans Gerd Evertz, Jean-Sébastien Caux

Published Dec 18, 2015• 23 citations
arXivHTMLPDF

Abstract

The quantum scattering of magnon bound states in the anisotropic Heisenberg spin chain is shown to display features similar to the scattering of solitons in classical exactly solvable models. Localized colliding Gaussian wave packets of bound magnons are constructed from string solutions of the Bethe equations and subsequently evolved in time, relying on an algebraic Bethe ansatz based framework for the computation of local expectation values in real space-time. The local magnetization profile shows the trajectories of colliding wave packets of bound magnons, which obtain a spatial displacement upon scattering. Analytic predictions on the displacements for various values of anisotropy and string lengths are derived from scattering theory and Bethe ansatz phase shifts, matching time-evolution fits on the displacements. The time-evolved block decimation algorithm allows for the study of scattering displacements from spin-block states, showing similar scattering displacement features.

DOI
10.1103/physrevb.92.214427

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The quantum scattering of magnon bound states in the anisotropic Heisenberg spin chain is shown to display features similar to the scattering of solitons in classical exactly solvable models. Localized colliding Gaussian wave packets of bound magnons are constructed from string solutions of the Bethe equations and subsequently evolved in time, relying on an algebraic Bethe ansatz based framework for the computation of local expectation values in real space-time. The local magnetization profile shows the trajectories of colliding wave packets of bound magnons, which obtain a spatial displacement upon scattering. Analytic predictions on the displacements for various values of anisotropy and string lengths are derived from scattering theory and Bethe ansatz phase shifts, matching time-evolution fits on the displacements. The time-evolved block decimation algorithm allows for the study of scattering displacements from spin-block states, showing similar scattering displacement features.
The recent advance of techniques in controlling ultra-cold gases in optical lattice provides a ideal experimental platform to study quantum many-body physics. The almost isolated and highly tunable systems created in these experimental setups are perfect for studying coherent time evolution of closed quantum many-body systems. In this thesis I present an analytical approach on the time evolution of the anisotropic Heisenberg spin chain (XXZ model), which is a fundamental model but has rich dynamics. This approach is developed through generalizing the Yudson contour representation. With this approach I obtain the exact time-dependent wave functions after a quantum quench in an integral form for generic initial states and for any anisotropy ∆. I reformulate the integrals and show their physical interpretation as propagating magnons and their bound states, which were predicted by the “string” solutions of Bethe Ansatz equations. To apply the new approach to specific problems, I calculate the quench dynamics of various observables from some particular initial states: starting from a local N ́eel state I calculate the time evolution of the antiferromagnetic order parameter–staggered magnetization; starting from a state with consecutive flipped spins I calculate the evolution of the local magnetization, and predict the evolution of the induced spin currents. These results and predictions can be confronted with recently experiments in ultra-cold gases in optical lattices, and are expected to contribute to the understanding of non-equilibrium physics of isolated many-body systems.
Abstract This chapter verifies the conjecture for the wave function, the Bethe ansatz wave function, of the anisotropic Heisenberg quantum spin chain by examining first the cases for one, two, and three spin deviations. The equations determining the quasi- momenta are the Bethe ansatz equations, now obtained from the coordinate Bethe ansatz. The Bethe ansatz equations derive from the eigenvalue equation in combination with boundary conditions, here periodic boundary conditions. These quasi-momenta also determine the energy eigenvalue. However, solving the Bethe ansatz equations to obtain a particular state requires more considerations. New variables, called rapidities, are useful. The consideration of the thermodynamic limit then allows to extract information about the ground state and low-lying excitations of the anisotropic quantum spin chain from the Bethe ansatz equations. Furthermore, complex solutions of the Bethe ansatz equations, called strings, are considered.
By circumventing the difficulty of obtaining exact string state solutions to Bethe ansatz equations, we devise a truncated string state space approach for investigating spin dynamics in a nonintegrable spin-12\frac{1}{2}21​ Heisenberg chain subjected to a staggered field at various magnetizations. The obtained dynamical spectra reveal a series of elastic peaks at integer multiples of the ordering wave vector QQQ, indicating the presence of multi-QQQ Bethe string states within the ground state. The spectrum exhibits a separation between different string continua as the strength of the staggered field increases at low magnetization, reflecting the confinement of the Bethe strings. This approach provides a unified string-state-based framework for understanding spin dynamics in low-dimensional nonintegrable Heisenberg models, which has a successful application to observations across various phases of the quasi-one-dimensional antiferromagnet YbAlO3{\mathrm{YbAlO}}_{3}YbAlO3​.
We consider the nonequilibrium evolution in the spin-1/2 XXZ Heisenberg chain for fixed magnetization after a local quantum quench. This model is equivalent to interacting spinless fermions. Initially an infinite magnetic field is applied to n consecutive sites and the ground state is calculated. At time t=0 the field is switched off and the time evolution of observables such as the z component of spin is computed using the time evolving block decimation algorithm. We find that the observables exhibit strong signatures of linearly propagating spinon and bound state excitations. These persist even when integrability-breaking perturbations are included. Since bound states ("strings") are notoriously difficult to observe using conventional probes such as inelastic neutron scattering, we conclude that local quantum quenches are an ideal setting for studying their properties. We comment on implications of our results for cold atom experiments.
In this article we review recent developments in the one-dimensional Bethe ansatz solvable multi-chain quantum models. The algebraic version of the Bethe ansatz (the quantum inverse scattering method) permits us to construct new families of integrable Hamiltonians using simple generalizations of the well known constructions of the single-chain model. First we consider the easiest example (`basic' model) of this class of models: the antiferromagnetic two-chain spin-1/2 model with the nearest-neighbour and next-nearest-neighbour spin-frustrating interactions (zigzag chain). We show how the algebra of the quantum inverse scattering method works for this model, and what are the important features of the Hamiltonian (which reveal the topological properties of two dimensions together with the one-dimensional properties). We consider the solution of the Bethe ansatz for the ground state (in particular, commensurate-incommensurate quantum phase transitions present due to competing spin-frustrating interactions are discussed) and construct the thermal Bethe ansatz (in the form of the `quantum transfer matrix') for this model. Then possible generalizations of the basic model are considered: an inclusion of a magnetic anisotropy, higher-spin representations (including the important case of a quantum ferrimagnet), the multi-chain case, internal degrees of freedom of particles at each site, etc. We observe the similarities and differences between this class of models and related exactly solvable models: other groups of multi-chain lattice models, quantum field theory models and magnetic impurity (Kondo-like) models. Finally, the behaviour of non-integrable (less constrained) multi-chain quantum models is discussed.