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arXiv:1802.02854

Numerical Algorithm for Exact Finite Temperature Spectra and Its Application to Frustrated Quantum Spin Systems

Youhei Yamaji, Takafumi Suzuki, Mitsuaki Kawamura

Published Feb 8, 2018• 10 citations
arXivHTMLPDF

Abstract

A numerical algorithm to calculate exact finite-temperature spectra of many-body lattice Hamiltonians is formulated by combining the typicality approach and the shifted Krylov subspace method. The combined algorithm, which we name finite-temperature shifted Krylov subspace method for simulating spectra (FTK$ω$), efficiently reproduces the canonical-ensemble probability distribution at finite temperatures with the computational cost proportional to the Fock space dimension. The present FTK$ω$ enables us to exactly calculate finite-temperature spectra of many-body systems whose system sizes are twice larger than those handled by the canonical ensemble average and allows us to access the frequency domain without sequential real-time evolution often used in previous studies. By employing the reweighting method with the present algorithm, we obtain significant reduction of the numerical costs for temperature sweeps. Application to the Kiteav-Heisenberg model (KHM) on a honeycomb lattice demonstrates the capability of the FTK$ω$. The KHM shows quantum phase transitions from the quantum spin liquid (QSL) phase to magnetically ordered phases when the finite Heisenberg exchange coupling is introduced. We examine temperature dependence of dynamical spin structure factors of the KHM in proximity to the QSL. It is clarified that the crossover from a spin-excitation continuum, which is a characteristic of the QSL, to a damped high-energy magnon mode occurs at temperatures higher than the energy scale of the Heisenberg couplings or the spin gap that is a signature of the QSL at zero temperature. The crossover and the closeness to the Kitaev's QSL are quantitatively measured by the width of the excitation continuum or the magnon spectrum. The present results shed new light on analysis of neutron scattering and other spectroscopy measurements on QSL candidates.

DOI
10.48550/arxiv.1802.02854

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A numerical algorithm to calculate exact finite-temperature spectra of many-body lattice Hamiltonians is formulated by combining the typicality approach and the shifted Krylov subspace method. The combined algorithm, which we name finite-temperature shifted Krylov subspace method for simulating spectra (FTKωωω), efficiently reproduces the canonical-ensemble probability distribution at finite temperatures with the computational cost proportional to the Fock space dimension. The present FTKωωω enables us to exactly calculate finite-temperature spectra of many-body systems whose system sizes are twice larger than those handled by the canonical ensemble average and allows us to access the frequency domain without sequential real-time evolution often used in previous studies. By employing the reweighting method with the present algorithm, we obtain significant reduction of the numerical costs for temperature sweeps. Application to the Kiteav-Heisenberg model (KHM) on a honeycomb lattice demonstrates the capability of the FTKωωω. The KHM shows quantum phase transitions from the quantum spin liquid (QSL) phase to magnetically ordered phases when the finite Heisenberg exchange coupling is introduced. We examine temperature dependence of dynamical spin structure factors of the KHM in proximity to the QSL. It is clarified that the crossover from a spin-excitation continuum, which is a characteristic of the QSL, to a damped high-energy magnon mode occurs at temperatures higher than the energy scale of the Heisenberg couplings or the spin gap that is a signature of the QSL at zero temperature. The crossover and the closeness to the Kitaev's QSL are quantitatively measured by the width of the excitation continuum or the magnon spectrum. The present results shed new light on analysis of neutron scattering and other spectroscopy measurements on QSL candidates.
The S=1/2S=1/2S=1/2 Kitaev model provides an exactly solvable setting in which spin fractionalization produces itinerant Majorana fermions coupled to an emergent Z2\mathbb{Z}_2Z2​ gauge field. In realistic Kitaev magnets, however, non-Kitaev interactions break integrability and make their finite-temperature dynamics difficult to calculate. We develop a finite-temperature parton mean-field theory based on the Abrikosov-fermion representation and incorporate the effects of exchange interactions through the random phase approximation to calculate the dynamical spin structure factor. For the pure Kitaev model, the method reproduces a strong, nearly dispersionless peak at a finite low energy together with a broad continuum at higher energies. It also reproduces the momentum dependence of the dynamical spin structure factor and its temperature evolution, including the downward shift of the low-energy spectral weight, in qualitative agreement with continuous-time quantum Monte Carlo calculations. Within the present theory, the contraction of the continuum is traced to the temperature-induced reduction of the bond amplitudes, which narrows the parton bands and compresses the two-parton phase space. For ferromagnetic Kitaev coupling, the energy of the low-energy peak is minimized at the ΓΓΓ point for a ferromagnetic Heisenberg interaction and maximized there for an antiferromagnetic interaction. In the ferromagnetic Kitaev-ΓΓΓ model, the static response develops a finite-momentum softening, indicating an instability toward the period-three magnetic order reported in numerical studies. For uniform-ferromagnetic states selected by a [111][111][111] magnetic field, we find that a broad parton continuum coexists with magnonlike dispersive structures, producing spectral features consistent with a spinon-to-magnon crossover.
Accurately evaluating finite-temperature properties of quantum many-body systems remains a central challenge. Many existing quantum approaches require thermal-state preparation at each target temperature, making low-temperature calculations especially demanding in terms of circuit depth and accuracy. In this work, we present the finite-temperature quantum Krylov (FTQK) method, which combines stochastic trace estimation with a Hermitian generalized eigenvalue problem based on cos⁡(H~)\cos(\tilde H)cos(H~). The only quantum-side input required is the temperature-independent real-time overlap sequence gn=⟨φ∣e−inτH∣φ⟩g_n=\langleφ|e^{-inτH}|φ\ranglegn​=⟨φ∣e−inτH∣φ⟩. All benchmark results presented in this work were obtained from classical numerical simulations of the quantum-circuit protocol. For periodic spin-12\frac{1}{2}21​ Heisenberg chains with N=14N=14N=14 and 181818, FTQK accurately reproduces the specific heat, magnetic susceptibility, and entropy in the noiseless case. It also accurately reproduces the specific heat of the N=14N=14N=14 honeycomb Kitaev model, demonstrating that FTQK can be applied beyond simple one-dimensional chains to a model on a two-dimensional lattice with bond-dependent interactions. Furthermore, in a proof-of-principle finite-shot study for the N=14N=14N=14 Heisenberg chain, the main thermodynamic features remain well preserved even at σ=10−3σ=10^{-3}σ=10−3. Here, σ=1/Nshotσ=1/\sqrt{N_{\mathrm{shot}}}σ=1/Nshot​​, where NshotN_{\mathrm{shot}}Nshot​ is the number of shots used to estimate an overlap. These results demonstrate the potential of FTQK for finite-temperature calculations using reusable, temperature-independent real-time overlap data.
The proliferation of quantum fluctuations and long-range entanglement presents an outstanding challenge for the numerical simulation of interacting spin systems with exotic ground states. Here, we present a toolset of Chebyshev polynomial-based iterative methods that provides a unified framework to study the thermodynamical properties, critical behavior and dynamics of frustrated quantum spin models with controlled accuracy. Similar to previous applications of the Chebyshev spectral methods to condensed matter systems, the algorithmic complexity scales linearly with the Hilbert space dimension and the Chebyshev truncation order. Using this approach, we study two paradigmatic quantum spin models on the honeycomb lattice: the Kitaev-Heisenberg (K-H) and the Kitaev-Ising (K-I) models. We start by applying the Chebyshev toolset to compute nearest-neighbor spin correlations, specific heat and entropy of the K-H model on a 24-spin cluster. Our results are benchmarked against exact diagonalization and a popular iterative method based on thermal pure quantum states. The transitions between a variety of magnetic phases, namely ferromagnetic, Néel, zigzag and stripy antiferromagnetic and quantum spin liquid phases are obtained accurately and efficiently. We also determine the temperature dependence of the spin correlations, over more than three decades in temperature, by means of a finite temperature Chebyshev polynomial method introduced here. Finally, we report novel dynamical signatures of the quantum phase transitions in the K-I model. Our findings suggest that the efficiency, versatility and low-temperature stability of the Chebyshev framework developed here could pave the way for previously unattainable studies of quantum spin models in two dimensions.
The proliferation of quantum fluctuations and long-range entanglement presents an outstanding challenge for the numerical simulation of interacting spin systems with exotic ground states. Here, we present a Chebyshev iterative method that gives access to the thermodynamic properties and critical behavior of frustrated quantum spin models with good accuracy. The computational complexity scales linearly with the Hilbert space dimension and the number of Chebyshev iterations used to approximate the eigenstates. Using this approach, we calculate the spin correlations of the Kitaev-Heisenberg model, a paradigmatic model of honeycomb iridates that exhibits a rich phase diagram including a quantum spin liquid phase. Our results are benchmarked against exact diagonalization and a popular iterative method based on thermal pure quantum (TPQ) states. All methods accurately predict a transition to a stripy (spin-liquid) phase for the critical value of the Kitaev coupling $ J_K \approx -1.3 J_H $ (JK≈−8.0JHJ_K \approx -8.0 J_HJK​≈−8.0JH​) for honeycomb layers with ferromagnetic Heisenberg interactions (JH>0J_{H}>0JH​>0). Our findings suggest that a hybrid Chebyshev-TPQ approach could open the door to previously unattainable studies of quantum spin models in two dimensions.
Quantum spin liquids are long-range entangled states of matter with emergent gauge fields and fractionalized excitations. While candidate materials, such as the Kitaev honeycomb ruthenate $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$, show magnetic order at low temperatures TTT, here we demonstrate numerically a dynamical crossover from magnonlike behavior at low TTT and frequencies $\ensuremath{\omega}$ to long-lived fractionalized fermionic quasiparticles at higher TTT and $\ensuremath{\omega}$. This crossover is akin to the presence of spinon continua in quasi-1D spin chains. It is further shown to go hand in hand with persistent typicality down to very low TTT. This aspect, which has also been observed in the spin-1/2 kagome Heisenberg antiferromagnet, is a signature of proximate spin liquidity and emergent gauge degrees of freedom more generally, and can be the basis for the numerical study of many finite-TTT properties of putative spin liquids.
Quantum spin simulations of frustrated lattices remain challenging for both classical and quantum algorithms, particularly in parameter regimes relevant to quantum spin liquid (QSL) phases. In this work, we apply Sample-based Krylov Quantum Diagonalization (SKQD) to estimate the ground state of the antiferromagnetic XXZ Heisenberg model on the J1J_1J1​--J2J_2J2​ square lattice, the Kagome lattice, and a 1D chain, studying system sizes from 12 to 72 spins. In our application of SKQD, we identify a ZZ deformation of Δ=2Δ=2Δ=2 as a sufficiently sparse Hamiltonian and introduce two modifications to the SKQD framework tailored to spin models: a canonical bitstring compression scheme that preserves the effectiveness of configuration recovery under spin-flip degeneracy, and the use of multiple Krylov subspaces to improve ground state coverage without any increase in quantum resources. For the 1D chain and Kagome lattice, SKQD achieves sub-percent ground-state energy errors at system sizes up to 24 spins, including a relative error of 0.002%0.002\%0.002% on the 12-site Kagome lattice, surpassing the best prior VQE result of 0.01%0.01\%0.01% on the same system while requiring no variational optimization. SKQD further extends to system sizes well beyond the reach of prior quantum algorithm studies, reaching 72 spins across all three geometries. Beyond 24 spins, accuracy degrades to relative errors of 19%19\%19%--36%36\%36% at 72 sites, but the gradual scaling of error with system size suggests these limits are set by available shot budgets and circuit depth rather than fundamental algorithmic constraints. Although classical tensor network methods remain state-of-the-art for these models, this work establishes a new benchmark for quantum simulation of the frustrated Heisenberg model and demonstrates SKQD as a scalable, hardware-compatible approach for studying strongly correlated spin systems.