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arXiv:cond-mat/0212485 · Physical Review Letters

Algorithm for Linear Response Functions at Finite Temperatures: Application to ESR Spectrum of s=12 Antiferromagnet Cu Benzoate

Toshiaki Iitaka, Toshikazu Ebisuzaki

Published Jan 31, 2003• 54 citations
arXivHTMLPDF

Abstract

We introduce an efficient and numerically stable method for calculating linear response functions chi(q,omega) of quantum systems at finite temperatures. The method is a combination of numerical solution of the time-dependent Schrödinger equation, random vector representation of trace, and Chebyshev polynomial expansion of Boltzmann operator. This method should be very useful for a wide range of strongly correlated quantum systems at finite temperatures. We present an application to the ESR spectrum of s=1 / 2 antiferromagnet Cu benzoate.

DOI
10.1103/physrevlett.90.047203

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We introduce an efficient and numerically stable method for calculating linear response functions chi(q,omega) of quantum systems at finite temperatures. The method is a combination of numerical solution of the time-dependent Schrödinger equation, random vector representation of trace, and Chebyshev polynomial expansion of Boltzmann operator. This method should be very useful for a wide range of strongly correlated quantum systems at finite temperatures. We present an application to the ESR spectrum of s=1 / 2 antiferromagnet Cu benzoate.
An O(N)O(N)O(N) algorithm is proposed for calculating linear response functions of noninteracting electrons. This algorithm is simple and suitable to parallel and vector computation. Since it avoids O(N3){O(N}^{3})O(N3) computational effort of matrix diagonalization, it requires only O(N)O(N)O(N) computational efforts, where NNN is the dimension of the state vector. The use of this O(N)O(N)O(N) algorithm is very effective since, otherwise, we have to calculate a large number of eigenstates, i.e., the occupied one-electron states up to the Fermi energy and the unoccupied states with higher energy. The advantage of this method compared to the Chebyshev polynomial method recently developed by Wang and Zunger [L. W. Wang, Phys. Rev. B 49, 10 154 (1994); L. W. Wang and A. Zunger, Phys. Rev. Lett. 73, 1039 (1994)] is that our method can calculate linear response functions without any storage of huge state vectors on external storage.
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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.
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