Lanczos Exact Diagonalization Research Articles

Disorder effects in the Kitaev-Heisenberg model

We study the interplay of disorder and Heisenberg interactions in the Kitaev model on a honeycomb lattice. The effect of disorder on the transition between Kitaev spin liquid and magnetic ordered states as well as the stability of magnetic ordering is investigated. Using Lanczos exact diagonalization we discuss the consequences of two types of disorder: (i) random-coupling disorder and (ii) singular-coupling disorder. They exhibit qualitatively similar effects in the pure Kitaev-Heisenberg model without long-range interactions. The range of spin-liquid phases is reduced and the transition to magnetic ordered phases becomes more crossoverlike. Furthermore, the long-range zigzag and stripy orderings in the clean system are replaced by their three domains with different ordering direction. Especially in the crossover range the coexistence of magnetically ordered and Kitaev spin-liquid domains is possible. With increasing the disorder strength the area of domains becomes smaller and the system goes into a spin-glass state. However, the disorder effect is different in magnetically ordered phases caused by long-range interactions. The stability of such magnetic ordering is diminished by singular-coupling disorder and, accordingly, the range of the spin-liquid regime is extended. This mechanism may be relevant to materials like $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ and ${\mathrm{H}}_{3}{\mathrm{LiIr}}_{2}{\mathrm{O}}_{6}$ where the zigzag ground state is stabilized by weak long-range interactions. We also find that the flux gap closes at a critical disorder strength and vortices appears in the flux arrangement. Interestingly, the vortices tend to form kinds of commensurate ordering.

Entanglement and quantum correlations in the XX spin-1/2 honeycomb lattice

The ground state phase diagram of the dimerized spin-1/2 XX honeycomb model in presence of a transverse magnetic field (TF) is known. With the absence of the magnetic field, two quantum phases, namely, the Néel and the dimerized phases have been identified. Moreover, canted Néel and the paramagnetic (PM) phases also emerge by applying the magnetic field. In this paper, using two powerful numerical exact techniques, Lanczos exact diagonalization, and Density matrix renormalization group (DMRG) methods, we study this model by focusing on the quantum correlations, the concurrence, and the quantum discord (QD) among nearest-neighbor spins. We show that the quantum correlations can capture the position of the quantum critical points in the whole range of the ground state phase diagram consistent with previous results. Although the concurrence and the QD are short-range, informative about long-ranged critical correlations. In addition, we address a ”magnetic-entanglement” behavior that starts from an entangled field around the saturation field.

Quantum many-body scars in spin-1 Kitaev chains

To provide a physical example of quantum scars, we study the many-body scars in the spin-1 Kitaev chain where the so-called PXP Hamiltonian is exactly embedded in the spectra. Regarding the conserved quantities, the Hilbert space is fragmented into disconnected subspaces and we explore the associated constrained dynamics. The continuous revivals of the fidelity and the entanglement entropy when the initial state is prepared in $\vert\mathbb{Z}_k\rangle$ ($k=2,3$) state illustrate the essential physics of the PXP model. We study the quantum phase transitions in the one-dimensional spin-1 Kitaev-Heisenberg model using the density-matrix renormalization group and Lanczos exact diagonalization methods, and determine the phase diagram. We parametrize the two terms in the Hamiltonian by the angle $\phi$, where the Kitaev term is $K\equiv\sin(\phi)$ and competes with the Heisenberg $J\equiv\cos(\phi)$ term. One finds a rich ground state phase diagram as a function of the angle $\phi$. Depending on the ratio $K/J\equiv\tan(\phi)$, the system either breaks the symmetry to one of distinct symmetry broken phases, or preserves the symmetry in a quantum spin liquid phase with frustrated interactions. We find that the scarred state is stable for perturbations which obey $\mathbb{Z}_2$-symmetry, while it becomes unstable against Heisenberg-type perturbations.\\ \textit{Accepted for publication in Physical Review Research}

Dynamical properties of the Haldane chain with bond disorder

By using Lanczos exact diagonalization and quantum Monte Carlo combined with stochastic analytic continuation, we study the dynamical properties of the S = 1 antiferromagnetic Heisenberg chain with different strengths of bond disorder. In the weak disorder region, we find weakly coupled bonds which can induce additional low-energy excitation below the one-magnon mode. As the disorder increases, the average Haldane gap closes at δΔ ∼ 0.5 with more and more low-energy excitations coming out. After the critical disorder strength δc ∼ 1, the system reaches a random-singlet phase with prominent sharp peak at ω = 0 and broad continuum at ω > 0 of the dynamic spin structure factor. In addition, we analyze the distribution of random spin domains and numerically find three kinds of domains hosting effective spin-1/2 quanta or spin-1 sites in between. These “spins” can form the weakly coupled long-range singlets due to quantum fluctuation which contribute to the sharp peak at ω = 0.

Ground state and low-energy excitations of the Kitaev-Heisenberg two-leg ladder

We study the ground state and low-lying excited states of the Kitaev-Heisenberg model on a ladder geometry using the density matrix renormalization group and Lanczos exact diagonalization methods. The Kitaev and Heisenberg interactions are parametrized as $K=\sin\phi$ and $J=\cos\phi$ with an angle parameter $\phi$. Based on the results for several types of order parameters, excitation gaps, and entanglement spectra, the $\phi$-dependent ground-state phase diagram is determined. Remarkably, the phase diagram is quite similar to that of the Kitaev-Heisenberg model on a honeycomb lattice, exhibiting the same long-range ordered states, namely rung-singlet (analog to N\'eel in 3D), zigzag, ferromagnetic, and stripy; and the presence of Kitaev spin liquids around the exactly solvable Kitaev points $\phi=\pm\pi/2$. We also calculate the expectation value of a plaquette operator corresponding to a $\pi$-flux state in order to establish how the Kitaev spin liquid extends away from the $\phi=\pm\pi/2$. Furthermore, we determine the dynamical spin structure factor and discuss the effect of the Kitaev interaction on the spin-triplet dispersion.

Metal-insulator transition in the two-dimensional Hubbard model: Dual fermion approach with Lanczos exact diagonalization

In this study the metal-insulator transition in the square-lattice Hubbard model at half-filling is revisited in relation to the DOS and spectral functions by means of the ladder dual fermion approximation (LDFA). For this purpose, a new expression of the two-body Green's function in the form of resolvents is proposed, which provides tractable and efficient means to calculate the local vertex function with the Lanczos exact diagonalization (ED) method. This makes it possible to use the Lanczos ED method as a solver of the effective impurity Anderson model for LDFA, opening up the way to access low temperatures for these perturbative extensions of the dynamical mean field theory and to obtain accurate DOS and spectral functions on the real frequency axis by a new variant of the maximum entropy method. It is found that for $U\le 3.5t$, as temperature decreases, the pseudogap formation due to antiferromagnetic correlations in the quasiparticle peak of the spectral function occurs at the X point $[k=(\pi,0)]$, spreads through the Fermi surface and ends at the M$_2$ point $[k=(\pi/2,\pi/2)]$. The almost simultaneous creation of the pseudogap and the loss of the Fermi liquid feature is consistent with that expected in the Slater regime. Although the pseudogap still appears in the quasi-particle-like single peak for $U\ge 4.0t$, the Fermi-liquid feature is partially lost on the Fermi surface already at higher temperatures as expected in the Mott-Heisenberg regime, in which local spins are preformed at high temperatures. A sharp crossover from a pseudogap phase to a Mott insulator at finite $U^{*} \approx 4.7t$ is found to occur below the temperature of the pseudogap formation similar to a previous study with the non-linear $\sigma$ model approach.

Spin-wave thermodynamics of square-lattice antiferromagnets revisited

Modifying the conventional spin-wave theory in a novel manner based on the Wick decomposition, we present an elaborate thermodynamics of square-lattice quantum antiferromagnets. Our scheme is no longer accompanied by the notorious problem of an artificial transition to the paramagnetic state inherent in modified spin waves in the Hartree-Fock approximation. In the cases of spin $\frac{1}{2}$ and spin $1$, various modified-spin-wave findings for the internal energy, specific heat, static uniform susceptibility, and dynamic structure factor are not only numerically compared with quantum Monte Carlo calculations and Lanczos exact diagonalizations but also analytically expanded into low-temperature series. Modified spin waves interacting via the Wick decomposition provide reliable thermodynamics over the whole temperature range of absolute zero to infinity. Adding higher-order spin couplings such as ring exchange interaction to the naivest Heisenberg Hamiltonian, we precisely reproduce inelastic-neutron-scattering measurements of the high-temperature-superconductor-parent antiferromagnet $\mathrm{La}_2\mathrm{CuO}_4$. Modifying Dyson-Maleev bosons combined with auxiliary pseudofermions also yields thermodynamics of square-lattice antiferromagnets free from thermal breakdown, but it is less precise unless temperature is sufficiently low. Applying all the schemes to layered antiferromagnets as well, we discuss the advantages and disadvantages of modified spin-wave and combined boson-pseudofermion representations.

Ordered states in the Kitaev-Heisenberg model: From 1D chains to 2D honeycomb

We study the ground state of the 1D Kitaev-Heisenberg (KH) model using the density-matrix renormalization group and Lanczos exact diagonalization methods. We obtain a rich ground-state phase diagram as a function of the ratio between Heisenberg (J = cosϕ) and Kitaev (K = sinϕ) interactions. Depending on the ratio, the system exhibits four long-range ordered states: ferromagnetic-z, ferromagnetic-xy, staggered-xy, Néel-z, and two liquid states: Tomonaga-Luttinger liquid and spiral-xy. The two Kitaev points {boldsymbol{phi }}{boldsymbol{=}}frac{{boldsymbol{pi }}}{{bf{2}}} and {boldsymbol{varphi }}{boldsymbol{=}}frac{{bf{3}}{boldsymbol{pi }}}{{bf{2}}} are singular. The ϕ-dependent phase diagram is similar to that for the 2D honeycomb-lattice KH model. Remarkably, all the ordered states of the honeycomb-lattice KH model can be interpreted in terms of the coupled KH chains. We also discuss the magnetic structure of the K-intercalated RuCl3, a potential Kitaev material, in the framework of the 1D KH model. Furthermore, we demonstrate that the low-lying excitations of the 1D KH Hamiltonian can be explained within the combination of the known six-vertex model and spin-wave theory.

Numerical study of the Kitaev-Heisenberg chain

We study the one-dimensional Kitaev-Heisenberg model as a possible realization of magnetic degrees of freedom of the K-intercalated honeycomb-lattice ruthenium trichloride α-RuCl3, denoted as K0.5RuClm. First, we discuss the possible charge ordering pattern in K0.5RuClm, where half of the j=12 spins are replaced by nonmagnetic ions in the honeycomb layer. Next, we investigate the low-energy excitations of the 1D Kitaev-Heisenberg model by calculating the dynamical spin structure factor using the Lanczos exact-diagonalization method. In the vicinity of Kitaev limit, there exist two well-separated dispersions. The bandwidth of each dispersion depends on the Heisenberg and Kitaev terms. This result may be relevant to the low-lying magnetic excitations of K0.5RuClm.

Noise-Induced Subdiffusion in Strongly Localized Quantum Systems.

We consider the dynamics of strongly localized systems subject to dephasing noise with arbitrary correlation time. Although noise inevitably induces delocalization, transport in the noise-induced delocalized phase is subdiffusive in a parametrically large intermediate-time window. We argue for this intermediate-time subdiffusive regime both analytically and using numerical simulations on single-particle localized systems. Furthermore, we show that normal diffusion is restored in the long-time limit, through processes analogous to variable-range hopping. With numerical simulations based on Lanczos exact diagonalization, we demonstrate that our qualitative conclusions are also valid for interacting systems in the many-body localized phase.

The spin-1 J1-J3 Heisenberg model on a triangular lattice

Motivated by the experimental data for NiGa2S4, the spin-1 Heisenberg model on a triangular lattice with the ferromagnetic nearest- and antiferromagnetic third-nearest-neighbor exchange interactions, J1 = −(1 − p)J and J3 = pJ, J > 0, is studied in the range 0 ≤ p ≤ 1. Mori’s projection operator technique and the Lanczos exact diagonalization are used. Mori’s method retains the rotation symmetry of spin components and does not anticipate any magnetic ordering. For zero temperature several phase transitions are observed. At pcr ≈ 0.2 the ground state is transformed from the ferromagnetic spin structure into a disordered state, which in its turn is changed to an antiferromagnetic long-range ordered state with the incommensurate ordering vector Q′ ≈ (1.16, 0) at p ≈ 0.31. With growing p the ordering vector moves along the X axis to the commensurate point Qc = (2π/3, 0) which is reached at p = 1. The final state with an antiferromagnetic long-range order can be conceived as four interpenetrating sublattices with the 120° spin structure on each of them. The model is able to describe the state with the incommensurate short-range order observed in NiGa2S4. To verify the used approach the ground state energy and corresponding spin-spin correlations are compared with exact-diagonalization results obtained with the SPINPACK code (the Lanczos exact diagonalization). Results of the two methods are in qualitative agreement.

Pulse and quench induced dynamical phase transition in a chiral multiferroic spin chain

Quantum dynamics of magnetic order in a chiral multiferroic chain is studied. We consider two different scenarios: Ultrashort terahertz (THz) excitations or a sudden electric field quench. Performing analytical and numerical exact diagonalization calculations we trace the pulse induced spin dynamics and extract quantities that are relevant to quantum information processing. In particular, we analyze the dynamics of the system chirality, the von Neumann entropy, the pairwise and the many body entanglement. If the characteristic frequencies of the generated states are non-commensurate then a partial loss of pair concurrence occurs. Increasing the system size this effect becomes even more pronounced. Many particle entanglement and chirality are robust and persist in the incommensurate phase. To analyze the dynamical quantum transitions for the quenched and pulsed dynamics we combined the Weierstrass factorization technique for entire functions and Lanczos exact diagonalization method. For a small system we obtained analytical results including the rate function of Loschmidt echo. Exact numerical calculations for a system up to 40 spins confirm phase transition. Quench- induced dynamical transitions have been extensively studied recently. Here we show that related dynamical transitions can be achieved and controlled by appropriate electric field pulses.

Ground state properties of an anisotropic bilinear-biquadratic model

In this paper, we consider anisotropic properties of a one-dimensional bilinear-biquadratic spin model with S = 1. The Hamiltonian with anisotropic parameter is studied numerically by using Lanczos exact diagonalization method at a critical point. It is found that the momentum corresponding to the ground state is affected by anisotropic parameter regularly, especially when total spin z-component is not zero.

Long-Period Charge Correlations in Charge-Frustrated Molecular θ-(BEDT-TTF)2X

Charge ordering (CO) with long-periodicities in a geometrically frustrated system, theta-(BEDT-TTF)2X, is numerically studied. We consider interacting fermion models on the anisotropic triangular lattice: the extended Hubbard model and the spinless fermion model, taking account of the inter-site Coulomb interactions on not only the nearest neighbor (NN) but also the next nearest neighbor (NNN) bonds. By applying Lanczos exact diagonalization method, we find that long-period charge correlations become dominant when NNN terms are included, in the charge-frustrated region. The calculated optical excitation spectra show that the charge transfer excitations in the long-period CO extend to a lower energy range compared to those in the short-period CO states including the 3-fold CO stabilized when NNN terms are absent. Our results suggest that the charge correlations experimentally observed in the title compounds near the CO insulating phase can be viewed as a tendency toward `tilted stripe'-type CO with large charge fluctuations.

A Numerical Study of the Effect of Disorder on Optical Conductivity in Inhomogeneous Superconductors

We present the effect of disorder on the optical conductivity of two-dimensional inhomogeneous superconductors by applying the kernel polynomial method to solve the Bogoliubov-de Gennes equations. By means of the lattice size scaling of the generalized inverse participation ratio, we find that the localization length of the quasiparticle decreases significantly with the increase of the disorder strength. Meanwhile, the weak disorder can readily restrain the Drude weight, while the superconducting gap has the tendency to suppress the low-energy optical conductivity. We also employ the Lanczos exact diagonalization method to study the competition between the on-site repulsive interactions and disorder. It is shown that the screening effect of repulsive interactions significantly enhances the Drude weight in the normal phase.

Ground-state phase diagram of the spin-12square-latticeJ1-J2model with plaquette structure

Using the coupled cluster method for high orders of approximation and Lanczos exact diagonalization we study the ground-state phase diagram of a quantum spin-1/2 J1-J2 model on the square lattice with plaquette structure. We consider antiferromagnetic (J1>0) as well as ferromagnetic (J1<0) nearest-neighbor interactions together with frustrating antiferromagnetic next-nearest-neighbor interaction J2>0. The strength of inter-plaquette interaction lambda varies between lambda=1 (that corresponds to the uniform J1-J2 model) and lambda=0 (that corresponds to isolated frustrated 4-spin plaquettes). While on the classical level (s \to \infty) both versions of models (i.e., with ferro- and antiferromagnetic J1) exhibit the same ground-state behavior, the ground-state phase diagram differs basically for the quantum case s=1/2. For the antiferromagnetic case (J1 > 0) Neel antiferromagnetic long-range order at small J2/J1 and lambda \gtrsim 0.47 as well as collinear striped antiferromagnetic long-range order at large J2/J1 and lambda \gtrsim 0.30 appear which correspond to their classical counterparts. Both semi-classical magnetic phases are separated by a nonmagnetic quantum paramagnetic phase. The parameter region, where this nonmagnetic phase exists, increases with decreasing of lambda. For the ferromagnetic case (J1 < 0) we have the trivial ferromagnetic ground state at small J2/|J1|. By increasing of J2 this classical phase gives way for a semi-classical plaquette phase, where the plaquette block spins of length s=2 are antiferromagnetically long-range ordered. Further increasing of J2 then yields collinear striped antiferromagnetic long-range order for lambda \gtrsim 0.38, but a nonmagnetic quantum paramagnetic phase lambda \lesssim 0.38.

Quantum entanglement and criticality of the antiferromagnetic Heisenberg model in an external field

By Lanczos exact diagonalization and the infinite time-evolving block decimation (iTEBD) technique, the two-site entanglement as well as the bipartite entanglement, the ground state energy, the nearest-neighbor correlations, and the magnetization in the antiferromagnetic Heisenberg (AFH) model under an external field are investigated. With increasing external field, the small size system shows some distinct upward magnetization stairsteps, accompanied synchronously with some downward two-site entanglement stairsteps. In the thermodynamic limit, the two-site entanglement, as well as the bipartite entanglement, the ground state energy, the nearest-neighbor correlations, and the magnetization are calculated, and the critical magnetic field hc = 2.0 is determined exactly. Our numerical results show that the quantum entanglement is sensitive to the subtle changing of the ground state, and can be used to describe the magnetization and quantum phase transition. Based on the discontinuous behavior of the first-order derivative of the entanglement entropy and fidelity per site, we think that the quantum phase transition in this model should belong to the second-order category. Furthermore, in the magnon existence region (h < 2.0), a logarithmically divergent behavior of block entanglement which can be described by a free bosonic field theory is observed, and the central charge c is determined to be 1.

Spin-Orbital Liquid on a Triangular Lattice

Using Lanczos exact diagonalization of finite clusters we demonstrate that the spin-orbital $d^1$ model for triply degenerate $t_{2g}$ orbitals on a triangular lattice provides an example of a spin-orbital liquid ground state. We also show that the spin-orbital liquid involves entangled valence bond states which violate the Goodenough-Kanamori rules, and modify effective spin exchange constants.

Spin frustration, charge ordering, and enhanced antiferromagnetism in TMTTF2SbF6

We theoretically investigate the effects of charge order and spin frustration on the spin ordering in TMTTF salts. Using first-principles band calculations, we find that a diagonal inter-chain transfer integral tq1, which causes spin frustration between the inter-chain dimers in the dimer-Mott insulating state, strongly depends on the choice of anion. Within the numerical Lanczos exact diagonalization method, we show that the ferroelectric charge order changes the role of tq1 from the spin frustration to the enhancement of the two-dimensionality in spin sector. The results indicate that tq1 assists the cooperative behavior between charge order and antiferromagnetic state observed in TMTTF2SbF6.

Entangled spin-orbital phases in the bilayer Kugel-Khomskii model

We derive the Kugel-Khomskii spin-orbital (SO) model for a bilayer and investigate its phase diagram depending on Hund's exchange $J_H$ and the $e_g$ orbital splitting $E_z$. In the (classical) mean-field approach with on-site spin $<S_i^z>$ and orbital $<\tau_i^z>$ order parameters and factorized spin-and-orbital degrees of freedom, we demonstrate a competition between the phases with either $G$-type or $A$-type antiferromagnetic (AF) or ferromagnetic long-range order. Next we develop a Bethe-Peierls-Weiss method with a Lanczos exact diagonalization of a cube coupled to its neighbors in $ab$ planes by the mean-field terms --- this approach captures quantum fluctuations on the bonds which decide about the nature of disordered phases in the highly frustrated regime near the orbital degeneracy. We show that the long-range spin order is unstable in a large part of the phase diagram which contains then six phases, including also the valence-bond phase with interlayer spin singlets stabilized by holes in $3z^2-r^2$ orbitals (VB$z$ phase), a disordered plaquette valence-bond (PVB) phase and a crossover phase between the VB$z$ and the $A$-type AF phase. When on-site SO coupling is also included by the $<S_i^z\tau_i^z>$ order parameter, we discover in addition two entangled phases which compete with $A$-type AF phase and another crossover phase in between the $G$-AF phase with occupied $x^2-y^2$ orbitals and the PVB phase. Thus, the present bilayer model provides an example of SO entanglement which generates novel disordered phases. We analyze the order parameters in all phases and identify situations where SO entanglement is crucial and factorization of the spins and orbitals leads to qualitatively incorrect results. We point out that SO entanglement may play a role in a bilayer fluoride K$_3$Cu$_2$F$_7$ which is a realization of the VB$z$ phase.