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arXiv:1503.06111 · Physical Review B

Computation of ESR spectra from the time evolution of the magnetization: Comparison of autocorrelation and Wiener-Khinchin-relation-based methods

Hiroki Ikeuchi, Hans A. De Raedt, Sylvain Bertaina, Seiji Miyashita

Published Dec 21, 2015• 7 citations
arXivHTMLPDF

Abstract

The calculation of finite temperature electron spin resonance (ESR) spectra for concrete specified crystal configurations is a very important issue in the study of quantum spin systems. Although direct evaluation of the Kubo formula by means of numerical diagonalization yields exact results, memory and CPU time restrictions limit the applicability of this approach to small system sizes. Methods based on the time evolution of a single pure quantum state can be used to study larger systems. One such method exploits the property that the expectation value of the autocorrelation function obtained for a few samples of so-called thermal typical states yields a good estimate of the thermal equilibrium value. In this paper, we propose a new method based on a Wiener-Khinchin-like theorem for quantum system. By comparison with exact diagonalization results, it is shown that both methods yield correct results. As the Wiener-Khinchin-based method involves sampling over thermal typical states, we study the statistical properties of the sampling distribution. Effects due to finite observation time are investigated and found to differ for the two methods but it is also found that, for both methods, the effects vanish as the system size increases. We present ESR spectra of the one-dimensional XXZ Heisenberg chain of up to 28 spins and discuss the dependence of separation of double peaks on the chain length.

DOI
10.1103/physrevb.92.214431

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The calculation of finite temperature electron spin resonance (ESR) spectra for concrete specified crystal configurations is a very important issue in the study of quantum spin systems. Although direct evaluation of the Kubo formula by means of numerical diagonalization yields exact results, memory and CPU time restrictions limit the applicability of this approach to small system sizes. Methods based on the time evolution of a single pure quantum state can be used to study larger systems. One such method exploits the property that the expectation value of the autocorrelation function obtained for a few samples of so-called thermal typical states yields a good estimate of the thermal equilibrium value. In this paper, we propose a new method based on a Wiener-Khinchin-like theorem for quantum system. By comparison with exact diagonalization results, it is shown that both methods yield correct results. As the Wiener-Khinchin-based method involves sampling over thermal typical states, we study the statistical properties of the sampling distribution. Effects due to finite observation time are investigated and found to differ for the two methods but it is also found that, for both methods, the effects vanish as the system size increases. We present ESR spectra of the one-dimensional XXZ Heisenberg chain of up to 28 spins and discuss the dependence of separation of double peaks on the chain length.
The electron spin resonance spectrum of the XXZ\mathit{XXZ}XXZ spin chain with finite length shows a double-peak structure at high temperatures around the electron paramagnetic resonance (EPR) frequency. This fact has been pointed out by direct numerical methods [S. El Shawish, O. C\'epas, and S. Miyashita, Phys. Rev. B 81, 224421 (2010); H. Ikeuchi, H. De Raedt, S. Bertaina, and S. Miyashita, Phys. Rev. B 92, 214431 (2015)]. The question of whether the double-peak structure survives in the thermodynamics is of particular interest. We study the size dependence of the line shape, including the even-odd effect. It is found that the peaks forming the double-peak structure are assigned to individual resonances, each of which is specified by the magnetizations of the resonating states $(M,M\ensuremath{-}1)$. To understand dependences, we decompose the spectrum into contributions from transitions specified by the magnetization, and we characterize the structure of the spectrum by individual contributions. We analyze the size dependence of each contribution individually by extending the moment method introduced by M. Brockman et al. to each component, and we find that the mean of each peak approaches the paramagnetic resonance point with 1/N1/N1/N (where NNN is the length of the chain), which indicates that the separation of the peaks of the double-peak structure also vanishes inversely with the system size. We also study the temperature dependence of the structure. At low temperatures, the spectrum has a single peak with a finite width at a position with a finite shift from the frequency of EPR, as pointed out by the analysis of field-theoretical works [M. Oshikawa and I. Affleck, Phys. Rev. Lett. 82, 5136 (1999)]. The study of the temperature dependence of the spectrum shows how the high-temperature spectrum changes to the low-temperature one with a drastic broadening of the spectrum.
We use the concept of typicality to study the real-time dynamics of spin and energy currents in spin-1/21/21/2 models in one dimension and at nonzero temperatures. These chains are the integrable XXZ chain and a nonintegrable modification due to the presence of a staggered magnetic field oriented in the zzz direction. In the framework of linear response theory, we numerically calculate autocorrelation functions by propagating a single pure state, drawn at random as a typical representative of the full statistical ensemble. By comparing to small-system data from exact diagonalization (ED) and existing short-time data from time-dependent density matrix renormalization group, we show that typicality is satisfied in finite systems over a wide range of temperature and is fulfilled in both integrable and nonintegrable systems. For the integrable case, we calculate the long-time dynamics of the spin current and extract the spin Drude weight for large systems outside the range of ED. We particularly provide strong evidence that the high-temperature Drude weight vanishes at the isotropic point. For the nonintegrable case, we obtain the full relaxation curve of the energy current and determine the heat conductivity as a function of magnetic field, exchange anisotropy, and temperature.
The essence of quantum magnetism lies in correlated many-body states. They determine the physical properties beyond paramagnetism. Even constructions as simple as singlets or triplets cannot be understood without the mathematical concept of superpositions. This is even more evident if time evolution of quantum states and phenomena such as quantum coherenceQuantum coherence are considered. The major approach to assess correlated many-body states in quantum magnetism of molecules is exact diagonalizationDiagonalization of the many-body Hamiltonian describing the physical system. Symmetries are of great help when tackling the numerical task of matrix diagonalization since they allow to decompose a problem of large Hilbert spaceHilbert space dimension into a number of smaller problems. This approach will be presented first. It rests on the identification of the available symmetries—e.g., U(1), SU(2), as well as point groupsPoint group—and their application. However, the applicability of exact diagonalizationDiagonalization is limited by the exponential growth of the Hilbert spaceHilbert space of the spin system with the number of spins: $$\dim \left( {\mathcal{H}} \right) = \mathop \prod \nolimits_{i} \left( {2 s_{i} + 1} \right)$$ . Nowadays, complex Hermitian matricesHermitian matrices of a linear dimension of about 100,000 can be completely diagonalized on supercomputers which limits the number of spins depending on their size sis_{i}si​ to just a few. I will therefore present a very accurate approximation to exact diagonalization that rests on Lanczos procedures and the concept of typicality. The most popular version, the finite-temperature Lanczos methodFinite-Temperature Lanczos Method (FTLM), allows to evaluate low-energy spectroscopic data, thermal averages as well as time evolutions for systems with Hilbert space dimensions of up to about 1011. Hands-on experience and examples will be provided.
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.
This paper presents numerical results of the ESR absorption spectrum for S=12S=\frac{1}{2}S=21​ XY\mathrm{XY}XY model in one dimension. We obtain detailed numerical results for a wide range of temperature and magnetic field for a large system of 400 sites, using the method developed by Derzhko et al. In the high-temperature regime, a similar behavior is observed for both the ferromagnet and the antiferromagnet; these line shapes have only one peak. On the other hand, in the low-temperature regime, line shapes show very different behaviors depending on the sign of the exchange coupling. While the line shape for the ferromagnet has only one peak as in the high-temperature regime, the line shape for antiferromagnet has two peaks and shows very strong magnetic-field dependence. We also discuss the physical meaning by comparing our numerical results with several theoretical results: the exact result in the high-temperature limit, Kubo-Tomita theory, field theory, mean-field theory, and elementary excitations.
We demonstrate that the concept of quantum typicality allows for significant progress in the study of real-time spin dynamics and transport in quantum magnets. To this end, we present a numerical analysis of the spin-current autocorrelation function of the antiferromagnetic and anisotropic spin-1/2 Heisenberg chain as inferred from propagating only a single pure state, randomly chosen as a "typical" representative of the statistical ensemble. Comparing with existing time-dependent density-matrix renormalization group data, we show that typicality is fulfilled extremely well, consistent with an error of our approach, which is perfectly under control and vanishes in the thermodynamic limit. In the long-time limit, our results provide for a new benchmark for the enigmatic spin Drude weight, which we obtain from chains as long as L=33 sites, i.e., from Hilbert spaces of dimensions almost O(104) larger than in existing exact-diagonalization studies.