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Response functions of gapped spin systems in high magnetic field

Alexei K. Kolezhuk, Hans-Jürgen Mikeska

Published Jan 1, 2008• 8 citations
OpenAlex

Abstract

We study the dynamical structure factor of gapped one-dimensional spin systems in the critical phase in high magnetic field. It is shown that the presence of a “condensate ” in the ground state in the high-field phase leads to interesting signatures in the response functions. Recently, there has been an increasing interest in the physics of gapped onedimensional spin systems subject to an external magnetic field which is strong enough to close the gap and bring the system into a new critical phase with finite magnetization and gapless excitations. 1)- 13) The low-energy physics of the critical phase is usually

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We study the dynamical structure factor of gapped one-dimensional spin systems in the critical phase in high magnetic field. It is shown that the presence of a “condensate ” in the ground state in the high-field phase leads to interesting signatures in the response functions. Recently, there has been an increasing interest in the physics of gapped onedimensional spin systems subject to an external magnetic field which is strong enough to close the gap and bring the system into a new critical phase with finite magnetization and gapless excitations. 1)- 13) The low-energy physics of the critical phase is usually
We use the dynamical density matrix renormalization group technique to show that the high-energy part of the spectrum of an S=1S=1S=1 Heisenberg chain, placed in a strong external magnetic field HHH exceeding the Haldane gap $\ensuremath{\Delta}$, contains edge singularities, similar to those known to exist in the low-energy spectral response. It is demonstrated that in the frequency range $\ensuremath{\omega}\ensuremath{\gtrsim}\ensuremath{\Delta}$ the longitudinal (with respect to the applied field) dynamical structure factor is dominated by the power-law singularity ${S}^{\ensuremath{\Vert}}(q=\ensuremath{\pi},\ensuremath{\omega})\ensuremath{\propto}{(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{0})}^{\ensuremath{-}{\ensuremath{\alpha}}^{\ensuremath{'}}}$. We study the behavior of the high-energy edge exponent ${\ensuremath{\alpha}}^{\ensuremath{'}}$ and the edge ${\ensuremath{\omega}}_{0}$ as functions of the magnetic field. The existence of edge singularities at high energies is directly related to the Tomonaga-Luttinger liquid character of the ground state at $H>\ensuremath{\Delta}$ and is expected to be a general feature of one-dimensional gapped spin systems in high magnetic fields.
An interesting feature of spin- chains with a gap is that they undergo a commensurate-incommensurate transition in the presence of an external magnetic field H. The system is in a gapless incommensurate regime for all values of the magnetic field between the lower critical field Hc1{\mathrm{H}}_{\mathrm{c}1}Hc1​ and an upper critical field Hc2{\mathrm{H}}_{\mathrm{c}2}Hc2​, where it is gapless and has power-law correlations. We calculate the critical exponents for such a generic gapped system in the incommensurate regime at the critical field Hc1{\mathrm{H}}_{\mathrm{c}1}Hc1​ and in its vicinity. Our analysis also applies to the spin- ladder. We compute the full dynamical susceptibilities at finite temperature. We use the same to discuss the thermal broadening of various modes and obtain the low-temperature behavior of the nuclear spin relaxation rate ${\mathrm{T}}_{1}^{\mathrm{\ensuremath{-}}1}$. We discuss the results obtained here for the special cases of the dimerized chain, frustrated chain, and the spin- ladder.
The subject of the talk is the dynamics of gapped spin systems in presence of an external magnetic field sufficiently high to close the energy gap. The focus will be on anisotropic S = 1 Haldane chain, although applications to weakly coupled dimer system of the TlCuCl3 type and spin ladders with four-spin (ring) interaction will be briefly considered. Existing effective field-theoretical descriptions [1, 2, 3] of anisotropic S = 1 Haldane chain in applied field are reviewed and their predictions are compared with the results of recent inelastic neu-tron scattering experiments on the Haldane chain material NDMAP [4]. It is shown that none of the known theories can provide a consistent description of the experimental data. A new effective theory, based on the dimer field description for alternated S = 1 2 chain, is proposed. In the isotropic case it coincides with the theory of Affleck [1] but differs from it considerably when a finite anisotropy is present. It is shown that the theory is capable of a quantitative description of the data on NDMAP with a satisfactory accuracy, in the whole range of the applied fields both below and above the critical field [4, 5, 6].
The excitation spectrum of the one-dimensional spin-orbital model in a magnetic field is studied, using a recently developed dynamical density-matrix renormalization-group technique. The method is applied to chains with up to 80 sites, and examined for test cases such as the one-dimensional spin-1/2 Heisenberg antiferromagnet whose excitation spectrum is known exactly from the Bethe ansatz. In the spin-orbital chain, the characteristic dynamical response depends strongly on the model parameters and the applied magnetic field. The coupling between the spin and orbital degrees of freedom is found to influence the incommensuration at finite magnetizations. In the regions of the phase diagram with only massive spin and orbital excitations, a finite field is required to overcome the spin gap. An orbital mode is found to become massless in this partially spin-polarized regime, indicating a strong coupling between the two degrees of freedom. In the critical region with three elementary gapless excitations, a prominent particle-hole excitation is observed at higher energies, promoted by the biquadratic term in the model Hamiltonian of the spin-orbital chain.
Quantum dynamics of strongly correlated systems is a challenging problem. Although the low energy fractional excitations of one-dimensional integrable models are often well understood, exploring quantum dynamics in these systems remains challenging in the gapless regime, especially at intermediate and high energies. Based on the algebraic Bethe ansatz formalism, we study spin dynamics in a representative one-dimensional strongly correlated model, i.e., the antiferromagnetic spin-12\frac{1}{2}21​ XXZ chain with the Ising anisotropy, via the form-factor formulas. Various excitations at different energy scales are identified crucial to the dynamic spin structure factors under the guidance of sum rules. At small magnetic polarizations, gapless excitations dominate the low energy spin dynamics arising from the magnetic-field-induced incommensurability. In contrast, spin dynamics at intermediate and high energies is characterized by the two- and three-string states, which are multiparticle excitations based on the commensurate N\'eel ordered background. Our work is helpful for experimental studies on spin dynamics in both condensed matter and cold atom systems beyond the low energy effective Luttinger liquid theory. Based on an intuitive physical picture, we speculate that the dynamic feature at high energies due to the multiparticle antibound state excitations can be generalized to nonintegrable spin systems.