Exact Diagonalization Study Research Articles

Microwave signatures of the Z2 and Z4 fractional Josephson effects

We present a many-body exact diagonalization study of the $\mathbb{Z}_2$ and $\mathbb{Z}_4$ Josephson effects in circuit quantum electrodynamics architectures. Numerical simulations are conducted on Kitaev chain Josephson junctions hosting nearest-neighbor Coulomb interactions. The low-energy effective theory of highly transparent Kitaev chain junctions is shown to be identical to that of junctions created at the edge of a quantum spin-Hall insulator. By capacitively coupling the interacting junction to a microwave resonator, we predict signatures of the fractional Josephson effects on the cavity frequency and on time-resolved reflectivity measurements.

Droplet formation in a one-dimensional system of attractive spinless fermions

A translation invariant one-dimensional system of spinless fermions with a finite-range attraction experiences a quantum phase transition to a phase-separated state. While being a conventional Luttinger liquid for a small interaction strength, spinless fermions form a droplet with the size smaller than the available one-dimensional volume when the interaction strength exceeds some critical value. A particularly remarkable signature of the droplet formation is the change in the lower edge of the many-body excitation spectrum. In the homogeneous phase, it has a Luttinger-liquid shape of periodic arcs on top of the shallow parabolic dispersion of the center-of-mass. When the interaction strength is increased, the arcs disappear completely as soon as the droplet is formed. We perform an exact diagonalization study of this system with the focus on the signatures of the quantum phase transition and the droplet properties. The one-particle and density-density correlation functions, the pressure, the sound velocity, and the droplet density are examined. The value of the critical interaction strength obtained from numerical data reasonably agrees with a simple mean-field analytical estimate. Due to the boson-fermion correspondence valid in one dimension, our results also hold for hard-core bosons with a finite-range attraction.

Interplay of alternation and further neighbor interaction in S=12 spin chains: A case study of Cs2CuAl4O8

Using Wannier function formulation and total energy calculations by first-principles density functional theory (DFT), we derive the underlying spin model of a recently synthesized compound, ${\mathrm{Cs}}_{2}{\mathrm{CuAl}}_{4}{\mathrm{O}}_{8}$, having zeolitelike network structure. The computed magnetic interactions show that the interchain Cu-Cu interactions are negligibly small compared to intrachain Cu-Cu interactions, thus characterizing ${\mathrm{Cs}}_{2}{\mathrm{CuAl}}_{4}{\mathrm{O}}_{8}$ as a prototypical one-dimensional (1D) spin-$\frac{1}{2}$ system. Interestingly, the DFT-derived 1D spin model features a combination of alternating ferromagnetic-antiferromagnetic interactions, together with the presence of both nearest- and next-nearest-neighbor interactions, making it an unprecedented case. The solution of the derived spin model of the compound using the quantum Monte Carlo technique shows reasonably good agreement with the experimental susceptibility data, measured in the presence of the magnetic field. The presence of spin gap is suggested by quantum Monte Carlo simulations in the zero-field condition, which is cross-checked by a more rigorous exact diagonalization study. Motivated by the intricacy of the derived spin model, we further examine the ground-state properties of this model in the parameter space of exchange interactions, which shows the possibility of driving quantum phase transition between gapped and gapless spin excitation. Our study is expected to shed light on the fascinating world of 1D quantum spin systems.

Anomalous and regular transport in spin- 12 chains: ac conductivity

We study magnetization transport in anisotropic spin-$1/2$ chains governed by the integrable XXZ model with and without integrability-breaking perturbations at high temperatures ($T\to \infty$) using a hybrid approach that combines exact sum-rules with judiciously chosen Ans\"atze. In the integrable XXZ model we find (i) super-diffusion at the isotropic (Heisenberg) point, with frequency dependent conductivity $ \sigma'(\omega\to 0) \sim |\omega|^{\alpha}$, where $\alpha=-3/7$ in close numerical agreement with recent $t$-DMRG computations; (ii) a continuously drifting exponent from $\alpha=-1^+$ in the XY limit of the model to $\alpha>0$ within the Ising regime; and (iii) a diffusion constant saturating in the XY coupling deep in the Ising limit. We consider two kinds of integrability breaking perturbations --- a simple next-nearest-neighbor spin-flip term ($J_2$) and a three-spin assisted variant ($t_2$), natural in the fermion particle representation of the spin chain. In the first case we discover a remarkable sensitivity of $\sigma'(\omega)$ to the sign of $J_2$, with enhanced low frequency spectral weight and a pronounced upward shift in the magnitude of $\alpha$ for $J_2>0$. Perhaps even more surprising, we find sub-diffusion ($\alpha>0$) over a range of $J_2<0$. By contrast, the effects of the \enquote{fermionic} three-spin perturbation are sign symmetric; this perturbation produces a clearly observable hydrodynamic relaxation. At large strength of the integrability breaking term $J_2\to \pm \infty$ the problem is effectively non-interacting (fermions hopping on odd and even sublattices) and we find $\alpha\to -1$ behavior reminiscent of the XY limit of the XXZ chain. Exact diagonalization studies largely corroborate these findings at mid-frequencies.

Directional Current-current Correlation Functions in a Two-species Hard-core Bosons in One Dimensional Finite Lattice: An Exact Diagonalization Study

We study a model for two-species hard-core bosons in one dimension. In this model, the same-species bosons have a hard-core condition at the same site, while different-species bosons are allowed to occupy the same site with a local interaction $U$. At half-filling, by Jordan-Wigner transformation, the model can be exactly mapped to a fermionic Hubbard model. Due to this correspondence, the phase transition from superfluid ($U=0$) to Mott insulator ($U>0$) can be explained by simple one-band theory at half-filling. By using an exact diagonalization method adopting a modified Lanczos method, we obtain the ground states as a function of $U$ for the lattice size upto $L=16$. We calculate directional current-current correlation functions in this model, which indicate that there are some remaining counter-flow in the Mott insulating region ($U>0$) and co-flow in the charge-density-wave region ($U<0$) for the finite lattices.

Topological Exciton Fermi Surfaces in Two-Component Fractional Quantized Hall Insulators.

A wide variety of two-dimensional electron systems allow for independent control of the total and relative charge density of two-component fractional quantum Hall (FQH) states. In particular, a recent experiment on bilayer graphene (BLG) observed a continuous transition between a compressible and incompressible phase at total filling ν_{T}=1/2 as charge is transferred between the layers, with the remarkable property that the incompressible phase has a finite interlayer polarizability. We argue that this occurs because the topological order of ν_{T}=1/2 systems supports a novel type of interlayer exciton that carries Fermi statistics. If the fermionic excitons are lower in energy than the conventional bosonic excitons (i.e., electron-hole pairs), they can form an emergent neutral Fermi surface, providing a possible explanation of an incompressible yet polarizable state at ν_{T}=1/2. We perform exact diagonalization studies that demonstrate that fermionic excitons are indeed lower in energy than bosonic excitons. This suggests that a "topological exciton metal" hidden inside a FQH insulator may have been realized experimentally in BLG. We discuss several detection schemes by which the topological exciton metal can be experimentally probed.

Multi-spinon and antiholon excitations probed by resonant inelastic x-ray scattering on doped one-dimensional antiferromagnets

Resonant inelastic x-ray scattering (RIXS) at the oxygen K-edge has recently accessed multi-spinon excitations in the one-dimensional antiferromagnet (1D-AFM) Sr2CuO3, where four-spinon excitations are resolved separately from the two-spinon continuum. This technique, therefore, provides new opportunities to study fractionalized quasiparticle excitations in doped 1D-AFMs. To this end, we carried out exact diagonalization studies of the doped t–J model and provided predictions for oxygen K-edge RIXS experiments on doped 1D-AFMs. We show that the RIXS spectra are rich, containing distinct two- and four-spinon excitations, dispersive antiholon excitations, and combinations thereof. Our results highlight how RIXS complements inelastic neutron scattering experiments by accessing additional charge and spin components of fractionalized quasiparticles.

Does a Single Eigenstate Encode the Full Hamiltonian?

The Eigenstate Thermalization Hypothesis (ETH) posits that the reduced density matrix for a subsystem corresponding to an excited eigenstate is "thermal." Here we expound on this hypothesis by asking: for which class of operators, local or non-local, is ETH satisfied? We show that this question is directly related to a seemingly unrelated question: is the Hamiltonian of a system encoded within a single eigenstate? We formulate a strong form of ETH where in the thermodynamic limit, the reduced density matrix of a subsystem corresponding to a pure, finite energy density eigenstate asymptotically becomes equal to the thermal reduced density matrix, as long as the subsystem size is much less than the total system size, irrespective of how large the subsystem is compared to any intrinsic length scale of the system. This allows one to access the properties of the underlying Hamiltonian at arbitrary energy densities/temperatures using just a $\textit{single}$ eigenstate. We provide support for our conjecture by performing an exact diagonalization study of a non-integrable 1D lattice quantum model with only energy conservation. In addition, we examine the case in which the subsystem size is a finite fraction of the total system size, and find that even in this case, a large class of operators continue to match their canonical expectation values. Specifically, the von Neumann entanglement entropy equals the thermal entropy as long as the subsystem is less than half the total system. We also study, both analytically and numerically, a particle number conserving model at infinite temperature which substantiates our conjectures.

Particle-hole symmetry for composite fermions: An emergent symmetry in the fractional quantum Hall effect

The particle-hole (PH) symmetry of {\em electrons} is an exact symmetry of the electronic Hamiltonian confined to a specific Landau level, and its interplay with the formation of composite fermions has attracted much attention of late. This article investigates an emergent symmetry in the fractional quantum Hall effect, namely the PH symmetry of {\em composite fermions}, which relates states at composite fermion filling factors $\nu^*=n+\bar{\nu}$ and $\nu^*=n+1-\bar{\nu}$, where the integer $n$ is the $\Lambda$ level index and $0\leq \bar{\nu}\leq 1$. Detailed calculations using the microscopic theory of composite fermions demonstrate that for low lying $\Lambda$ levels (small $n$): (i) the 2-body interaction between composite-fermion particles is very similar, apart from a constant additive term and an overall scale factor, to that between composite-fermion holes in the same $\Lambda$ level; and (ii) the 3-body interaction for composite fermions is an order of magnitude smaller than the 2-body interaction. Taken together, these results imply an approximate PH symmetry for composite fermions in low $\Lambda$ levels, which is also supported by exact diagonalization studies and available experiments. This symmetry, which relates states at electron filling factors $\nu={n+\bar{\nu}\over 2(n+\bar{\nu})\pm 1}$ and $\nu={n+1-\bar{\nu}\over 2(n+1-\bar{\nu})\pm 1}$, is not present in the original Hamiltonian and owes its existence entirely to the formation of composite fermions. With increasing $\Lambda$ level index, the 2-body and 3-body pseudopotentials become comparable, but at the same time they both diminish in magnitude, indicating that the interaction between composite fermions becomes weak as we approach $\nu=1/2$.

Robustness of Majorana edge modes and topological order: Exact results for the symmetric interacting Kitaev chain with disorder

We investigate the robustness of Majorana edge modes under disorder and interactions. We exploit a recently found mapping of the interacting Kitaev chain in the symmetric region ($\mu = 0$, $t = \Delta$) to free fermions. Extending the exact solution to the disordered case allows us to calculate analytically the topological phase boundary for all interaction and disorder strengths, which has been thought to be only accessible numerically. We discover a regime in which moderate disorder in the interaction matrix elements enhances topological order well into the strongly interacting regime $U > t$. We also derive the explicit form of the many-body Majorana edge wave function revealing how it is dressed by many-particle fluctuations from interactions. The qualitative features of our analytical results are valid beyond the fine-tuned integrable point as expected from the robustness of topological order and as corroborated here by an exact diagonalization study of small systems.

Ground-state magnetic properties of spin ladder-shaped quantum nanomagnet: Exact diagonalization study

The paper presents a computational study of the ground-state properties of a quantum nanomagnet possessing the shape of a finite two-legged ladder composed of 12 spins S=1/2. The system is described with isotropic quantum Heisenberg model with nearest-neighbour interleg and intraleg interactions supplemented with diagonal interleg coupling between next nearest neighbours. All the couplings can take arbitrary values. The description of the ground state is based on the exact numerical diagonalization of the Hamiltonian. The ground-state phase diagram is constructed and analysed as a function of the interactions and the external magnetic field. The ground-state energy and spin-spin correlations are extensively discussed. The cases of ferro- and antiferromagnetic couplings are compared and contrasted.

Synchronization transition in dipole-coupled two-level systems with positional disorder

We study the decoherence dynamics of dipole-coupled two-level quantum systems in Ramsey-type experiments. We focus on large networks of two-level systems, confined to two spatial dimensions and with positional disorder giving rise to disordered dipolar couplings. This setting is relevant for modeling the decoherence dynamics of the rotational excitations of polar molecules confined to deep optical lattices, where disorder arises from the random filling of lattice sites with occupation probability $p$. We show that the decoherence dynamics exhibits a phase transition at a critical filling $p_c\simeq 0.15$. For $p<p_c$ the dynamics is disorder-dominated and the Ramsey interference signal decays on a timescale $T_2 \propto p^{-3/2}$. For $p>p_c$ the dipolar interactions dominate the disorder, and the system behaves as a collective spin-ordered phase, representing synchronization of the two-level systems and persistent Ramsey oscillations with divergent $T_2$ for large systems. For a finite number of two-level systems, $N$, the spin-ordered phase at $p> p_c$ undergoes a crossover to a collective spin-squeezed state on a timescale $\tau_{\rm sq} \propto \sqrt{N}$. We develop a self-consistent mean-field theory that is capable of capturing the synchronization transition at $p_c$, and provide an intuitive theoretical picture that describes the phase transition in the long-time dynamics. We also show that the decoherence dynamics appear to be ergodic in the vicinity of $p_c$, the long-time behaviour being well described by the predictions of equilibrium thermodynamics. The results are supported by the results of exact diagonalization studies of small systems.

Emergence of chiral spin liquids via quantum melting of noncoplanar magnetic orders

Quantum spin liquids (QSLs) are long-range entangled states of quantum magnets which lie beyond the Landau paradigm of classifying phases of matter via broken symmetries. A physical route to arriving at QSLs is via frustration-induced quantum melting of ordered states such as valence bond crystals or magnetic orders. Here, we show, using extensive exact diagonalization (ED) and density-matrix renormalization group (DMRG) studies of concrete $SU(2)$ invariant spin models on honeycomb, triangular and square lattices, that chiral spin liquids (CSLs) emerge as descendants of triple-$Q$ spin crystals with tetrahedral magnetic order and a large scalar spin chirality. Such ordered-to-CSL melting transitions may yield lattice realizations of effective Chern-Simons-Higgs field theories. Our work provides a distinct unifying perspective on the emergence of CSLs, and suggests that materials with magnetic skyrmion crystal order might provide a good starting point to search for CSLs.

Ground-state properties of Na2IrO3 determined from an ab initio Hamiltonian and its extensions containing Kitaev and extended Heisenberg interactions

We investigate the ground state properties of $\mathrm{Na_2IrO_3}$ based on numerical calculations of the recently proposed {\it ab initio} Hamiltonian represented by Kitaev and extended Heisenberg interactions. To overcome the limitation posed by small tractable system sizes in the exact diagonalization study employed in a previous study (Yamaji $\textit{et al.}$, Phys. Rev. Lett. $\textbf{113}$, 107201 (2014)), we apply two-dimensional density matrix renomalization group, and infinite-size tensor-network method. By calculating at much larger system sizes, we critically test the validity of the exact diagonalization results. The results consistently indicate that the ground state of $\mathrm{Na_2IrO_3}$ is a magnetically ordered state with zigzag configuration in agreement with experimental observations and the previous diagonalization study. Applications of the two independent methods in addition to the exact diagonalization study further uncover a consistent and rich phase diagram near the zigzag phase beyond the accessibility of the exact diagonalization. For example, in the parameter space away from the $\textit{ab initio}$ value of $\mathrm{Na_2IrO_3}$ controlled by the trigonal distortion, we find three phases: (i) an ordered phase with the magnetic moment aligned mutually in $120$ degrees orientation on every third hexagon, (ii) a magnetically ordered phase with a $16$-site unit-cell, and (iii) an ordered phase with presumably incommensurate periodicity of the moment. It suggests that potentially rich magnetic structures may appear in $A_\mathrm{2}\mathrm{IrO_3}$ compounds for $A$ other than Na. The present results also serve to establish the accuracy of the first-principles approach in reproducing the available experimental results thereby further contribute to find a route to realize the Kitaev spin liquid.

Stress Response of Granular Systems

We develop a framework for stress response in two dimensional granular media, with and without friction, that respects vector force balance at the microscopic level. We introduce local gauge degrees of freedom that determine the response of contact forces between constituent grains on a given, disordered, contact network, to external perturbations. By mapping this response to the spectral properties of the graph Laplacian corresponding to the underlying contact network, we show that this naturally leads to spatial localization of forces. We present numerical evidence for localization using exact diagonalization studies of network Laplacians of soft disk packings. Finally, we discuss the role of other constraints, such as torque balance, in determining the stability of a granular packing to external perturbations.

Two Universality Classes for the Many-Body Localization Transition.

We provide a systematic comparison of the many-body localization (MBL) transition in spin chains with nonrandom quasiperiodic versus random fields. We find evidence suggesting that these belong to two separate universality classes: the first dominated by "intrinsic" intrasample randomness, and the second dominated by external intersample quenched randomness. We show that the effects of intersample quenched randomness are strongly growing, but not yet dominant, at the system sizes probed by exact-diagonalization studies on random models. Thus, the observed finite-size critical scaling collapses in such studies appear to be in a preasymptotic regime near the nonrandom universality class, but showing signs of the initial crossover towards the external-randomness-dominated universality class. Our results provide an explanation for why exact-diagonalization studies on random models see an apparent scaling near the transition while also obtaining finite-size scaling exponents that strongly violate Harris-Chayes bounds that apply to disorder-driven transitions. We also show that the MBL phase is more stable for the quasiperiodic model as compared to the random one, and the transition in the quasiperiodic model suffers less from certain finite-size effects.

Unveiling magnetic interactions of ruthenium trichloride via constraining direction of orbital moments: Potential routes to realize a quantum spin liquid

Recent experiments reveal that the honeycomb ruthenium trichloride {\alpha}-RuCl3 is a prime candidate of the Kitaev quantum spin liquid (QSL). However, there is no theoretical model which can properly describe its experimental dynamical response, due to the lack of a full understanding of its magnetic interactions. Here, we propose a general scheme to calculate the magnetic interactions in systems (e.g., {\alpha}-RuCl3) with non-negligible orbital moments by constraining the directions of orbital moments. With this scheme, we put forward a minimal J1-K1-{\Gamma}1-J3-K3 model for {\alpha}-RuCl3 and find that: (I) The third nearest neighbor (NN) antiferromagnetic Heisenberg interaction J3 stabilizes the zigzag antiferromagnetic order; (II) The NN symmetric off-diagonal exchange {\Gamma}1 plays a pivotal role in determining the preferred direction of magnetic moments and generating the spin wave gap. Exact diagonalization study on this model shows that the Kitaev QSL can be realized by suppressing the NN symmetric off-diagonal exchange {\Gamma}1 and the third NN Heisenberg interaction J3. Thus, we not only propose a powerful general scheme for investigating the intriguing magnetism of Jeff=1/2 magnets, but also point out future directions for realizing the Kitaev QSL in the honeycomb ruthenium trichloride {\alpha}-RuCl3.

Exact diagonalization and cluster mean-field study of triangular-lattice XXZ antiferromagnets near saturation

Quantum magnetic phases near the magnetic saturation of triangular-lattice antiferromagnets with XXZ anisotropy have been attracting renewed interest since it has been suggested that a nontrivial coplanar phase, called the $\ensuremath{\pi}$-coplanar or $\mathrm{\ensuremath{\Psi}}$ phase, could be stabilized by quantum effects in a certain range of anisotropy parameter $J/{J}_{z}$ besides the well-known 0-coplanar (known also as $V)$ and umbrella phases. Recently, Sellmann et al. [Phys. Rev. B 91, 081104(R) (2015)] claimed that the $\ensuremath{\pi}$-coplanar phase is absent for $S=1/2$ from an exact-diagonalization analysis in the sector of the Hilbert space with only three down-spins (three magnons). We first reconsider and improve this analysis by taking into account several low-lying eigenvalues and the associated eigenstates as a function of $J/{J}_{z}$ and by sensibly increasing the system sizes (up to 1296 spins). A careful identification analysis shows that the lowest eigenstate is a chirally antisymmetric combination of finite-size umbrella states for $J/{J}_{z}\ensuremath{\gtrsim}2.218$ while it corresponds to a coplanar phase for $J/{J}_{z}\ensuremath{\lesssim}2.218$. However, we demonstrate that the distinction between 0-coplanar and $\ensuremath{\pi}$-coplanar phases in the latter region is fundamentally impossible from the symmetry-preserving finite-size calculations with fixed magnon number. Therefore, we also perform a cluster mean-field plus scaling analysis for small spins $S\ensuremath{\le}3/2$. The obtained results, together with the previous large-$S$ analysis, indicate that the $\ensuremath{\pi}$-coplanar phase exists for any $S$ except for the classical limit $(S\ensuremath{\rightarrow}\ensuremath{\infty})$ and the existence range in $J/{J}_{z}$ is largest in the most quantum case of $S=1/2$.

Exact diagonalization and quantum Monte Carlo study of an ionic Hubbard model in two dimensions

We study quantum phase transitions of an ionic Hubbard model in two dimensions. The ionic Hubbard model explains the quantum states of strongly correlated electrons under the influence of checkerboard-type alternating chemical potentials. For a given amplitude of the alternating potentials Δ, we obtain quantum ground states as we tune the local repulsive energy U between a spin-up electron and a spin-down electron by using an exact diagonalization method of a modified Lanczos algorithm. The system undergoes a quantum phase transition from a band insulator to a Mott insulator as U increases at half-filling. We find the signature of a quantum phase transition by investigating the behavior of ground-state energies and that of double occupancies for the size of L × L = 4 × 4, which was the largest possible lattice in this work. We compare our results with those of quantum Monte Carlo simulations employing the Hirsch-Fye algorithm.

Exact Diagonalization Study of an Extended Hubbard Model for a Cubic Cluster at Quarter Filling

In the paper the thermodynamics of a cubic cluster with 8 sites at quarter filling is characterized by means of exact diagonalization technique. Particular emphasis is put on the behaviour of such response functions as specific heat and magnetic susceptibility. The system is modelled with extended Hubbard model which includes electron hopping between both first and second nearest neighbours as well as coulombic interactions, both on-site and between nearest-neighbour sites. The importance of hopping between second nearest neighbours and coulombic interactions between nearest neighbours for the temperature dependences of thermodynamic response functions is analysed. In particular, the predictions of Schottky model are compared with the calculations based on the full energy spectrum.