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Physical Review Letters

Microscopic calculation of the dynamical correlations in two-dimensional quantum antiferromagnets at low temperatures:T=0results

Thomas Becher, George Reiter

Published Aug 28, 1989• 28 citations
Publisher

Abstract

We calculate the spin-spin relaxation function for a two-dimensional quantum antiferromagnet of arbitrary spin at low temperature from first principles. The calculation justifies the picture of the excitations of the system as a weakly interacting Bose gas of spin waves.
DOI
10.1103/physrevlett.63.1004

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Microscopic calculation of dynamical correlations in two-dimensional quantum antiferromagnets at low temperature (abstract)

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We present low-temperature dynamic properties of the quantum two-dimensional antiferromagnetic Heisenberg model with spin S=1∕2S=1∕2S=1∕2. The calculation of the dynamic correlation function is performed by combining a projection operator formalism and the modified spin-wave theory, which gives a gap in the dispersion relation for finite temperatures. The so calculated dynamic correlation function shows a double peak structure. We also obtain the spin-wave damping and compare our results to some experimental data and theoretical results obtained by other authors using different approaches.
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The dilute, two-dimensional Bose gas exhibits a novel regime of relaxational dynamics in the regime ${k}_{B}T\ensuremath{\gtrsim}\ensuremath{\mid}\ensuremath{\mu}\ensuremath{\mid}$ where TTT is the absolute temperature and $\ensuremath{\mu}$ is the chemical potential. This may also be interpreted as the quantum criticality of the zero density quantum critical point at $\ensuremath{\mu}=0$. We present a theory for this dynamic, to leading order in $1∕\mathrm{ln}[\ensuremath{\Lambda}∕({k}_{B}T)]$, where $\ensuremath{\Lambda}$ is a high energy cutoff. Although pairwise interactions between the bosons are weak at low energy scales, the collective dynamics are strongly coupled even when $\mathrm{ln}(\ensuremath{\Lambda}∕T)$ is large. We argue that the strong-coupling effects can be isolated in an effective classical model, which is then solved numerically. Applications to experiments on the gap-closing transition of spin-gap antiferromagnets in an applied field are presented.
We discuss the theory of the nonzero temperature (T)(T)(T) spin dynamics and transport in one-dimensional Heisenberg antiferromagnets with a gap $\ensuremath{\Delta}$. For $T\ensuremath{\ll}\ensuremath{\Delta}$, we develop a semiclassical picture of thermally excited particles. Multiple inelastic collisions between the particles are crucial, and are described by a two-particle S\mathcal{S}S matrix which is shown to have a superuniversal form at low momenta. This is established by computations on the O(3)O(3)O(3) $\ensuremath{\sigma}$ model, and strong- and weak-coupling expansions (the latter using a Majorana fermion representation) for the two-leg S=1/2S=1/2S=1/2 Heisenberg antiferromagnetic ladder. As an aside, we note that the strong-coupling calculation reveals an S=1S=1S=1, two-particle bound state which leads to the presence of a second peak in the T=0T=0T=0 inelastic neutron-scattering (INS) cross section for a range of values of momentum transfer. We obtain exact, or numerically exact, universal expressions for the thermal broadening of the quasiparticle peak in the INS cross section, the spin diffusivity, and for the field dependence of the NMR relaxation rate 1/T1{1/T}_{1}1/T1​ of the effective semiclassical model; these are expected to be asymptotically exact for the quantum antiferromagnets in the limit $T\ensuremath{\ll}\ensuremath{\Delta}$. The results for 1/T1{1/T}_{1}1/T1​ are compared with the experimental findings of Takigawa et al. [Phys. Rev. Lett. 76, 2173 (1996)] and the agreement is quite good. In the regime $\ensuremath{\Delta}<T<$(a typical microscopic exchange) and we argue that a complementary description in terms of semiclassical waves applies, and give some exact results for the thermodynamics and dynamics.
This thesis is concerned with the numerical study of one-dimensional (1D) spin-1/2 quantum magnets and related method development. Its focus is on the calculation of dynamical spin correlation functions both at zero and finite temperature. This is motivated by the accessibility of dynamical quantities in experiments such as inelastic neutron scattering (INS) and electron spin resonance (ESR). The numerical methods used in this thesis are based on extensions of the density-matrix renormalization group (DMRG) and are formulated in the framework of matrix product states (MPS). While zero-temperature dynamical correlation functions are computed with existing MPS frequency-domain methods, an MPS frequency-domain approach for their calculation at finite temperature is developed in this thesis. The new method combines the Liouville-space formulation of the dynamics with a moment expansion of the dynamical correlation function. The majority of the results are obtained via MPS-based Chebyshev expansions. These numerical techniques are applied to two different model systems describing real materials. The first one is the material copper pyrimidine dinitrate (Cu-PM) which is modeled by a 1D spin-1/2 Heisenberg antiferromagnet with Dzyaloshinskii-Moriya interactions. The spin dynamics of this model is studied in an applied magnetic field and compared to ESR experiments. Zero-temperature calculations for momentum- and frequency-resolved dynamical quantities give direct access to the intensity of the elementary excitations and go beyond the low-energy description by the quantum sine-Gordon model. Thus, a deviation from the Lorentz invariant dispersion for the single-soliton resonance is found. The presence of the strongest boundary bound state previously predicted from a boundary sine-Gordon field theory is confirmed, while composite boundary-bulk excitations have too low intensities to be found in the numerical results. At finite temperature, there is a temperature-induced crossover of the soliton. Moreover, additional temperature effects such as interbreather transitions emerge, which is confirmed by accompanying ESR experiments on Cu-PM over a wide range of the applied field strength. The second system studied in this thesis is the compound BaCu2_22​V2_22​O8_88​. It is shown that the magnetic properties of this quasi-1D material can be described by a strongly alternating antiferromagnetic-ferromagnetic spin-1/2 Heisenberg chain. As found for other dimerized systems, the strong correlations in BaCu2_22​V2_22​O8_88​ persist even at elevated temperatures. Moreover, these correlations lead to an asymmetric lineshape broadening of the magnetic excitations at finite temperature. Upon raising the temperature, an increasingly asymmetric lineshape is observed in high-resolution INS experiments for BaCu2_22​V2_22​O8_88​. In this thesis, the lineshape is calculated by the MPS finite-temperature method developed as a part of this work. Comparing these results to the INS data and the lineshape obtained by a diagrammatic approach, excellent agreement is found over a broad temperature range. This demonstrates that coherent quantum behavior persists at elevated temperatures in BaCu2_22​V2_22​O8_88​ and that it can be predicted quantitatively.