Finite-temperature Lanczos Method Research Articles

Magnetism of the S=12 J1−J2 square-kagome lattice antiferromagnet

The spin-$1/2$ Heisenberg antiferromagnet on the square-kagome (SK) lattice has attracted growing attention as a model system of highly frustrated quantum magnetism. A further motivation for theoretical studies comes from the recent discovery of SK spin-liquid compounds. The SK antiferromagnet exhibits two non-equivalent nearest-neighbor bonds $J_1$ and $J_2$. One may expect that in SK compounds $J_1$ and $J_2$ are of different strength. We present a numerical study of finite systems by means of the finite-temperature Lanczos method. We discuss the temperature dependence of the specific heat $C(T)$, the entropy $S(T)$, and of the susceptibility $X(T)$ of the $J_1$-$J_2$ SK Heisenberg antiferromagnet varying $J_2/J_1$ in the range $0 \le J_2/J_1 \le 4$. We also discuss the zero-field ground state of the model. We find indications for a magnetically disordered singlet ground state for $0 \le J_2/J_1 \lesssim 1.65$. Beyond $J_2/J_1 \sim 1.65$ the singlet ground state gives way for a ferrimagnetic ground state. In the region $0.77 \lesssim J_2/J_1 \lesssim 1.65$ the low-temperature thermodynamics is dominated by a finite singlet-triplet gap filled with low-lying singlet excitations leading to an exponentially activated low-temperature behavior of $X(T)$. On the other hand, the low-lying singlets yield an extra maximum or a shoulder-like profile below the main maximum in the $C(T)$ curve. For $J_2/J_1 \lesssim 0.7$ the low-temperature thermodynamics is characterized by a large fraction of $N/3$ weakly coupled spins leading to a sizable amount of entropy at very low temperatures. In an applied magnetic field the magnetization process features plateaus and jumps in a wide range of $J_2/J_1$.

Fe-Gd Ferromagnetic Cyclic Coordination Cluster [FeIII4GdIII4(teaH)8(N3)8(H2O)] with Magnetic Anisotropy─Theory and Experiment.

The synthesis, structural, and magnetic characterization of [FeIII4LnIII4(teaH)8(N3)8(H2O)] (Ln = Gd and Y) and the previously reported isostructural Dy analogue are discussed. The commonly held belief that both FeIII and GdIII can be regarded as isotropic ions is shown to be an oversimplification. This conclusion is derived from the magnetic data for the YIII analogue in terms of the zero-field splitting seen for FeIII and from the fact that the magnetic data for the new GdIII analogue can only be fit employing an additional anisotropy term for the GdIII ions. Furthermore, the Fe4Gd4 ring shows slow relaxation of magnetization. Our analysis of the experimental magnetic data employs both density functional theory as well as the finite-temperature Lanczos method which finally enables us to provide an almost perfect fit of magnetocaloric properties.

Thermal conductivity and heat diffusion in the two-dimensional Hubbard model

We study the electronic thermal conductivity $\kappa_\textrm{el}$ and the thermal diffusion constant $D_\textrm{Q,el}$ in the square lattice Hubbard model using the finite-temperature Lanczos method. We exploit the Nernst-Einstein relation for thermal transport and interpret the strong non-monotonous temperature dependence of $\kappa_\textrm{el}$ in terms of that of $D_\textrm{Q,el}$ and the electronic specific heat $c_\textrm{el}$. We present also the results for the Heisenberg model on a square lattice and ladder geometries. We study the effects of doping and consider the doped case also with the dynamical mean-field theory. We show that $\kappa_\textrm{el}$ is below the corresponding Mott-Ioffe-Regel value in almost all calculated regimes, while the mean free path is typically above or close to lattice spacing. We discuss the opposite effect of quasi-particle renormalization on charge and heat diffusion constants. We calculate the Lorenz ratio and show that it differs from the Sommerfeld value. We discuss our results in relation to experiments on cuprates. Additionally, we calculate the thermal conductivity of overdoped cuprates within the anisotropic marginal Fermi liquid phenomenological approach.

Topological Quantum Phase Transitions of Anisotropic Antiferromagnetic Kitaev Model Driven by Magnetic Field

AbstractThe evolution of quantum spin liquid states (QSL) of the anisotropic antiferromagnetic (AFM) Kitaev model with the [001] magnetic field by utilizing the finite‐temperature Lanczos method (FTLM) is investigated. In this anisotropic Kitaev model with and (K is the energy unit), due to the competition between anisotropy and magnetic field, the system emerges four exotic quantum phase transitions (QPTs) when and , while only two QPTs when . At these magnetic‐field tuning quantum phase transition points, the low‐energy excitation spectrums appear level crossover, and the specific heat, magnetic susceptibility and Wilson ratio display anomalies; accordingly, the topological Chern number may also change. These results demonstrate that the anisotropic interacting Kitaev model with modulating magnetic field displays more rich phase diagrams, in comparison with the isotropic Kitaev model.

Thermodynamics of the spin-half square kagome lattice antiferromagnet

Over the last decade, the interest in the spin-$1/2$ Heisenberg antiferromagnet (HAF) on the square kagome (also called shuriken) lattice has been growing as a model system of quantum magnetism with a quantum paramagnetic ground state, flat-band physics near the saturation field, and quantum scars. A further motivation to study this model comes from the recent discovery of a gapless spin liquid in the square kagome magnet $\mathrm{K}{\mathrm{Cu}}_{6}\mathrm{Al}\mathrm{Bi}{\mathrm{O}}_{4}{(\mathrm{S}{\mathrm{O}}_{4})}_{5}\mathrm{Cl}$ [M. Fujihala et al., Nat. Commun. 11, 3429 (2020)]. Here, we present large-scale numerical investigations of the specific heat $C(T)$, the entropy $S(T),$ as well as the susceptibility $\ensuremath{\chi}(T)$ by means of the finite-temperature Lanczos method for system sizes of $N=18,24,30,36,42,48$, and $N=54$. We find that the specific heat exhibits a low-temperature shoulder below the major maximum which can be attributed to low-lying singlet excitations filling the singlet-triplet gap, which is significantly larger than the singlet-singlet gap. This observation is further supported by the behavior of the entropy $S(T)$, where a change in curvature is present just at about $T/J=0.2$, the same temperature where the shoulder in $C$ sets in. For the susceptibility the low-lying singlet excitations are irrelevant, and the singlet-triplet gap leads to an exponentially activated low-temperature behavior. The maximum in $\ensuremath{\chi}(T)$ is found at a pretty low temperature ${T}_{\mathrm{max}}/J=0.146$ (for $N=42$) compared to ${T}_{\mathrm{max}}/J=0.935$ for the unfrustrated square-lattice HAF signaling the crucial role of frustration also for the susceptibility. We find a striking similarity of our square kagome data with the corresponding ones for the kagome HAF down to very low $T$. The magnetization process featuring plateaus and jumps and the field dependence of the specific heat that exhibits characteristic peculiarities attributed to the existence of a flat one-magnon band are discussed as well.

Gauss quadrature based finite temperature Lanczos method

The finite temperature Lanczos method (FTLM), which is an exact diagonalization method intensively used in quantum many-body calculations, is formulated in the framework of orthogonal polynomials and Gauss quadrature. The main idea is to reduce finite temperature static and dynamic quantities into weighted summations related to one- and two-dimensional Gauss quadratures. Then lower order Gauss quadrature, which is generated from Lanczos iteration, can be applied to approximate the initial weighted summation. This framework fills the conceptual gap between FTLM and kernel polynomial method, and makes it easy to apply orthogonal polynomial techniques in the FTLM calculation.

Isothermal and adiabatic magnetization processes of the spin- 12 Heisenberg model on an anisotropic triangular lattice

In this study, we investigate the magnetic susceptibility, entropy, and isothermal magnetization curve of the spin-1/2 Heisenberg model on an anisotropic triangular lattice using the orthogonalized finite-temperature Lanczos method. In addition, we investigate the adiabatic magnetization curve and magnetocaloric effect. We estimate these physical quantities with sufficient accuracy in the thermodynamic limit, except at low temperatures. We observe a 1/3 magnetization plateau in the isothermal magnetization process, whereas the plateau is observed to have a slope in the adiabatic process. We show that the magnetocaloric effect can be used to detect the signature of phase transitions. We believe that these results will be useful for understanding the magnetism of anisotropic triangular lattice compounds through a comparison with experimental results in the future.

Frustrated magnetism of spin- 12 Heisenberg diamond and octahedral chains as a statistical mechanical monomer-dimer problem

It is evidenced that effective lattice-gas models of hard-core monomers and dimers afford a proper description of low-temperature features of spin-$\frac{1}{2}$ Heisenberg diamond and octahedral chains. In addition to monomeric particles assigned within the localized-magnon theory to bound one- and two-magnon eigenstates, the effective monomer-dimer lattice-gas model includes dimeric particles assigned to a singlet-tetramer (singlet-hexamer) state as a cornerstone of dimer-tetramer (tetramer-hexamer) ground state of a spin-$\frac{1}{2}$ Heisenberg diamond (octahedral) chain. The feasibility of the effective description is confirmed through the exact diagonalization and finite-temperature Lanczos methods. Both quantum spin chains display rich ground-state phase diagrams including discontinuous as well as continuous field-driven phase transitions, whereby the specific heat shows in vicinity of the former phase transitions an extraordinary low-temperature peak coming from a highly degenerate manifold of low-lying excitations.

Accuracy of the typicality approach using Chebyshev polynomials

Abstract Trace estimators allow us to approximate thermodynamic equilibrium observables with astonishing accuracy. A prominent representative is the finite-temperature Lanczos method (FTLM) which relies on a Krylov space expansion of the exponential describing the Boltzmann weights. Here we report investigations of an alternative approach which employs Chebyshev polynomials. This method turns out to be also very accurate in general, but shows systematic inaccuracies at low temperatures that can be traced back to an improper behavior of the approximated density of states with and without smoothing kernel. Applications to archetypical quantum spin systems are discussed as examples.

Spin diffusion and spin conductivity in the two-dimensional Hubbard model

We study the spin diffusion and spin conductivity in the square lattice Hubbard model by using the finite-temperature Lanczos method. We show that the spin diffusion behaves differently from the charge diffusion and has a nonmonotonic $T$ dependence. This is due to a progressive liberation of charges that contribute to spin transport and enhance it beyond that active at low temperature due to the Heisenberg exchange. We further show that going away from half-filling and zero magnetization increases the spin diffusion, but that the increase is insufficient to reconcile the difference between the model calculations and the recent measurements on cold-atoms.

Dynamical spin correlations of the kagome antiferromagnet

Temperature-dependent dynamical spin correlations, which can be readily accessed via a variety of experimental techniques, hold the potential of offering a unique fingerprint of quantum spin liquids and other intriguing dynamical states. In this work we present an in-depth study of the temperature-dependent dynamical spin structure factor $S({\bf q}, \omega)$ of the antiferromagnetic (AFM) Heisenberg spin-1/2 model on the kagome lattice with additional Dzyaloshinskii--Moriya (DM) interactions. Using the finite-temperature Lanczos method on lattices with up to $N = 30$ sites we find that even without DM interactions, chiral low-energy spin fluctuations of the $120^\circ$ AFM order parameter dominate the dynamical response. This leads to a nontrivial frequency dependence of $S({\bf q}, \omega)$ and the appearance of a pronounced low-frequency mode at the M point of the extended Brillouin zone. Adding an out-of-plane DM interactions $D^z$ gives rise to an anisotropic dynamical response, a softening of in-plane spin fluctuations, and, ultimately, the onset of a coplanar AFM ground-state order at $D^z > 0.1 J$. Our results are in very good agreement with existing inelastic neutron scattering and temperature-dependent NMR spin-lattice relaxation rate ($1/T_1$) data on the paradigmatic kagome AFM herbertsmithite, where the effect of its small $D^z$ on the dynamical spin correlations is shown to be rather small, as well as with $1/T_1$ data on the novel kagome AFM YCu$_3$(OH)$_6$Cl$_3$, where its substantial $D^z \approx 0.25 J$ interaction is found to strongly affect the spin dynamics.

Anomalous thermodynamics of a quantum spin system with large residual entropy

In contrast to strongly frustrated classical systems, their quantum counterparts typically have a non-degenerate ground state. A counterexample is the celebrated Heisenberg sawtooth spin chain with ferromagnetic zigzag bonds $J_1$ and competing antiferromagnetic basal bonds $J_2$. At a quantum phase transition point $|J_2/J_1|=1/2$, this model exhibits a flat one-magnon excitation band leading to a massively degenerate ground-state manifold which results in a large residual entropy. Thus, for the spin-half model, the residual entropy amounts to exactly one half of its maximum value $\lim_{T\to\infty} S(T)/N = \ln2$. In the present paper we study in detail the role of the spin quantum number $s$ and the magnetic field $H$ in the parameter region around the transition (flat-band) point. For that we use full exact diagonalization up to $N=20$ lattice sites and the finite-temperature Lanczos method up to $N=36$ sites to calculate the density of states as well as the temperature dependence of the specific heat, the entropy and the susceptibility. The study of chain lengths up to $N=36$ allows a careful finite-size analysis. At the flat-band point we find extremely small finite-size effects for spin $s=1/2$, i.e., the numerical data virtually correspond to the thermodynamic limit. In all other cases the finite-size effects are still small and become visible at very low temperatures. In a sizeable parameter region around the flat-band point the former massively degenerate ground-state manifold acts as a large manifold of low-lying excitations leading to extraordinary thermodynamic properties at the transition point as well as in its vicinity such as an additional low-temperature maximum in the specific heat. Moreover, there is a very strong influence of the magnetic field on the low-temperature thermodynamics including an enhanced magnetocaloric effect.

Loop-gas description of the localized-magnon states on the kagome lattice with open boundary conditions

The high-field regime of the spin-s XXZ antiferromagnet on the kagome lattice gives rise to macroscopically degenerate ground states thanks to a completely flat lowest single-magnon band. The corresponding excitations can be localized on loops in real space and have been coined "localized magnons". Thus, the description of the many-body ground states amounts to characterizing the allowed classical loop configurations and eliminating the quantum mechanical linear relations between them. Here, we investigate this loop-gas description on finite kagome lattices with open boundary conditions and compare the results with exact diagonalization for the spin-1/2 XY model on the same lattice. We find that the loop gas provides an exact account of the degenerate ground-state manifold while a hard-hexagon description misses contributions from nested loop configurations. The densest packing of the loops corresponds to a magnon crystal that according to the zero-temperature magnetization curve is a stable ground state of the spin-1/2 XY model in a window of magnetic fields of about 4% of the saturation field just below this saturation field. We also present numerical results for the specific heat obtained by the related methods of thermal pure quantum (TPQ) states and the finite-temperature Lanczos method (FTLM). For a field in the stability range of the magnon crystal, one finds a low-temperature maximum of the specific heat that corresponds to a finite-temperature phase transition into the magnon crystal at low temperatures.

Finite-temperature density-matrix renormalization group method for electron-phonon systems: Thermodynamics and Holstein-polaron spectral functions

We investigate the thermodynamics and finite-temperature spectral functions of the Holstein polaron using a density-matrix renormalization group method. Our method combines purification and local basis optimization (LBO) as an efficient treatment of phonon modes. LBO is a scheme which relies on finding the optimal local basis by diagonalizing the local reduced density matrix. By transforming the state into this basis, one can truncate the local Hilbert space with a negligible loss of accuracy for a wide range of parameters. In this work, we focus on the crossover regime between large and small polarons of the Holstein model. Here, no analytical solution exists and we show that the thermal expectation values at low temperatures are independent of the phonon Hilbert space truncation provided the basis is chosen large enough. We then demonstrate that we can extract the electron spectral function and establish consistency with results from a finite-temperature Lanczos method. We additionally calculate the electron emission spectrum and the phonon spectral function and show that all the computations are significantly simplified by the local basis optimization. We observe that the electron emission spectrum shifts spectral weight to both lower frequencies and larger momenta as the temperature is increased. The phonon spectral function experiences a large broadening and the polaron peak at large momenta gets significantly flattened and merges almost completely into the free-phonon peak.

Charge transport in the Hubbard model at high temperatures: Triangular versus square lattice

High-temperature bad-metal transport has been recently studied both theoretically and in experiments as one of the key signatures of strong electronic correlations. Here we use the dynamical mean field theory (DMFT) and its cluster extensions, as well as the finite-temperature Lanczos method (FTLM) to explore the influence of lattice frustration on the thermodynamic and transport properties of the Hubbard model at high temperatures. We consider the triangular and the square lattice at half-filling and at 15\% hole-doping. We find that for $T \gtrsim 1.5t$ the self-energy becomes practically local, while the finite-size effects become small at lattice-size $4 \times 4$ for both lattice types and doping levels. The vertex corrections to optical conductivity, which are significant on the square lattice even at high temperatures, contribute less on the triangular lattice. We find approximately linear temperature dependence of dc resistivity in doped Mott insulator for both types of lattices.

Flat-band physics in the spin-1/2 sawtooth chain

We consider the strongly anisotropic spin-1/2 XXZ model on the sawtooth-chain lattice with ferromagnetic longitudinal interaction Jzz = ΔJ and aniferromagnetic transversal interaction Jxx = Jyy = J > 0. At Δ = −1∕2 the lowest one-magnon excitation band is dispersionless (flat) leading to a massively degenerate set of ground states. Interestingly, this model admits a three-coloring representation of the ground-state manifold [H.J. Changlani et al., Phys. Rev. Lett. 120, 117202 (2018)]. We characterize this ground-state manifold and elaborate the low-temperature thermodynamics of the system. We illustrate the manifestation of the flat-band physics of the anisotropic model by comparison with two isotropic flat-band Heisenberg sawtooth chains. Our analytical consideration is complemented by exact diagonalization and finite-temperature Lanczos method calculations.Graphical abstract

Thermodynamic properties of an S=12 ring-exchange model on the triangular lattice

By using a numerically exact diagonalization technique and a block-extended version of the finite-temperature Lanczos method, we study thermodynamic properties of an $S=1/2$ Heisenberg model on the triangular lattice with an antiferromagnetic nearest-neighbor interaction $J$ and a four-spin ring-exchange interaction $J_{\rm c}$. Calculations are performed on small clusters under the periodic-boundary conditions. In contrast to the purely triangular case with $J_{\rm c}=0$, the specific heat exhibits a characteristic double-peak structure for $J_{\rm c}/J \gtrsim 0.04$. From the calculation of the entropy and the uniform magnetic susceptibility, it is shown that non-magnetic excitations exist below the magnetic excitation for $J_{\rm c}/J \gtrsim 0.04$.

Adapting Planck's route to investigate the thermodynamics of the spin-half pyrochlore Heisenberg antiferromagnet

The spin-half pyrochlore Heisenberg antiferromagnet (PHAF) is one of the most challenging problems in the field of highly frustrated quantum magnetism. Stimulated by the seminal paper of M. Planck [M. Planck, Verhandl. Dtsch. phys. Ges. 2, 202 (1900)] we calculate thermodynamic properties of this model by interpolating between the low- and high-temperature behavior. For that we follow ideas developed in detail by B. Bernu and G. Misguich and use for the interpolation the entropy exploiting sum rules [the ``entropy method'' (EM)]. We complement the EM results for the specific heat, the entropy, and the susceptibility by corresponding results obtained by the finite-temperature Lanczos method (FTLM) for a finite lattice of $N=32$ sites as well as by the high-temperature expansion (HTE) data. We find that due to pronounced finite-size effects the FTLM data for $N=32$ are not representative for the infinite system below $T\ensuremath{\approx}0.7$. A similar restriction to $T\ensuremath{\gtrsim}0.7$ holds for the HTE designed for the infinite PHAF. By contrast, the EM provides reliable data for the whole temperature region for the infinite PHAF. We find evidence for a gapless spectrum leading to a power-law behavior of the specific heat at low $T$ and for a single maximum in $c(T)$ at $T\ensuremath{\approx}0.25$. For the susceptibility $\ensuremath{\chi}(T)$ we find indications of a monotonous increase of $\ensuremath{\chi}$ upon decreasing of $T$ reaching ${\ensuremath{\chi}}_{0}\ensuremath{\approx}0.1$ at $T=0$. Moreover, the EM allows us to estimate the ground-state energy to ${e}_{0}\ensuremath{\approx}\ensuremath{-}0.52$.

Finite-temperature properties of the Kitaev-Heisenberg models on kagome and triangular lattices studied by improved finite-temperature Lanczos methods

Frustrated quantum spin systems such as the Heisenberg and Kitaev models on various lattices, have been known to exhibit various exotic properties not only at zero temperature but also for finite temperatures. Inspired by the remarkable development of the quantum frustrated spin systems in recent years, we investigate the finite-temperature properties of the $S=1/2$ Kitaev-Heisenberg models on kagome and triangular lattices by means of finite-temperature Lanczos methods with improved accuracy. In both lattices, multiple peaks are confirmed in the specific heat. To find the origin of the multiple peaks, we calculate the static spin structure factor. The origin of the high-temperature peak of the specific heat is attributed to a crossover from the paramagnetic state to a short-range ordered state whose static spin structure factor has zigzag or linear intensity distributions in momentum space. In the triangular Kitaev model, the "order by disorder" due to quantum fluctuation occurs. On the other hand, in the kagome Kitaev model it does not occur even with both quantum and thermal fluctuations.

Accuracy of the finite-temperature Lanczos method compared to simple typicality-based estimates

We study trace estimators for equilibrium thermodynamic observables that rely on the idea of typicality and derivatives thereof such as the finite-temperature Lanczos method (FTLM). As numerical examples quantum spin systems are studied. Our initial aim was to identify pathological examples or circumstances, such as strong frustration or unusual densities of states, where these methods could fail. Instead we failed with the attempt. All investigated systems allow such approximations, only at temperatures of the order of the lowest energy gap the convergence is somewhat slower in the number of random vectors over which observables are averaged.