Numerical Diagonalization Research Articles

Quantum Phase Transition Induced by Geometrical Changes in Spin–Phonon Interaction

Spin–phonon interaction effects in a spin-1/2 antiferromagnetic Heisenberg chain coupled to phonons are investigated in systems with variable crystal structures. By performing a unitary transformation to a Hamiltonian of the system, we derive effective Hamiltonians expanded in powers of spin–phonon and spin–spin couplings. Ground-state phase transitions between a spin-liquid phase and a spin-gapped phase are investigated by a level spectroscopy analysis of numerical diagonalization data of effective systems. Our results depend on whether the ground state of the Heisenberg chain is in the spin-liquid phase or the spin-gapped phase. In the system where the ground state is in the spin-liquid phase, the spin–phonon interaction causes the usual phase transition to the spin-gapped phase. On the other hand, in the system where the ground state is in the spin-gapped phase, the spin–phonon interaction causes the novel phase transition to the spin-liquid phase for some geometrical structures. The phase transition f...

Generalized rotating-wave approximation for the two-qubit quantum Rabi model

The generalized rotating-wave approximation (GRWA) is presented for the two-qubit and cavity coipling system . The analytical expressions in the zeroth order approximation recover the previous adiabatic ones. The counterrotating-wave terms can be eliminated by performing the first order corrections. An effective solvable Hamiltonian with the same form as the ordinary RWA one are then obtained, giving a significantly accurate eigenvalues and eigenstates. Energy levels in the present GRWA are in accordant with the numerical exact diagonalization ones in the a wide range of coupling strength. The atomic population inversion in the GRWA is in quantitative agreement with the numerical results for different detunings in the ultrastrong coupling regime.

Truncated conformal space approach inddimensions: A cheap alternative to lattice field theory?

We show how to perform accurate, nonperturbative and controlled calculations in quantum field theory in d dimensions. We use the Truncated Conformal Space Approach (TCSA), a Hamiltonian method which exploits the conformal structure of the UV fixed point. The theory is regulated in the IR by putting it on a sphere of a large finite radius. The QFT Hamiltonian is expressed as a matrix in the Hilbert space of CFT states. After restricting ourselves to energies below a certain UV cutoff, an approximation to the spectrum is obtained by numerical diagonalization of the resulting finite-dimensional matrix. The cutoff dependence of the results can be computed and efficiently reduced via a renormalization procedure. We work out the details of the method for the phi^4 theory in d dimensions with d not necessarily integer. A numerical analysis is then performed for the specific case d = 2.5, a value chosen in the range where UV divergences are absent. By going from weak to intermediate to strong coupling, we are able to observe the symmetry-preserving, symmetry-breaking, and conformal phases of the theory, and perform rough measurements of masses and critical exponents. As a byproduct of our investigations we find that both the free and the interacting theories in non integral d are not unitary, which however does not seem to cause much effect at low energies.

Novel Quantum Phase Transition in the Frustrated Spin Nanotube

The S=1/2 three-leg quantum spin tube is investigated using the numerical diagonalization. The study indicated a new quantum phase transition between the 1/3 magnetization plateau phase and the plateauless one, with respect to the spin anisotropy. The phase diagram is also presented.

Exotic Field Induced Quantum Phase Transition of the Kagome Lattice Antiferromagnet

The magnetization process of the S=1/2 kagome-lattice antiferromagnet is investigated using the numerical diagonalization up to 42-spin clusters. The critical exponent analysis confirms the unconventional magnetization behavior around the 1/3 magnetization plateau-like anomaly; the field derivative is infinite at the low-field side, while it is zero at the high-field side. We also confirm that a magnetization jump appears at 1/3 of the saturation magnetization in distorted kagome-lattice antiferromagnets.

Dimer-dimer Correlations and Magnetothermodynamics of S=1/2 Spherical Kagome Clusters in W72V30 and Mo72V30

Numerical diagonalization methods are employed to study spin-1/2 spherical kagome clusters realized in W72V30 and Mo72V30, where the former is expected to have the ideal symmetry Ih and the latter is a little distorted. For the ideal model with Ih symmetry we calculate spin correlation functions in the ground state, which resemble the spin-1/2 kagome antiferomagnet closely. We also present the integrated density of states for W72V30 and Mo72V30, and discuss the difference in magnetothermodynamics for both materials.

Möbius molecules and fragile Mott insulators

Motivated by the concept of M\"obius aromatics in organic chemistry, we extend the recently introduced concept of fragile Mott insulators (FMI) to ring-shaped molecules with repulsive Hubbard interactions threaded by a half-quantum of magnetic flux ($hc/2e$). In this context, a FMI is the insulating ground state of a finite-size molecule that cannot be adiabatically connected to a single Slater determinant, i.e., to a band insulator, provided that time-reversal and lattice translation symmetries are preserved. Based on exact numerical diagonalization for finite Hubbard interaction strength $U$ and existing Bethe-ansatz studies of the one-dimensional Hubbard model in the large-$U$ limit, we establish a duality between Hubbard molecules with $4n$ and $4n+2$ sites, with $n$ integer. A molecule with $4n$ sites is an FMI in the absence of flux but becomes a band insulator in the presence of a half-quantum of flux, while a molecule with $4n+2$ sites is a band insulator in the absence of flux but becomes an FMI in the presence of a half-quantum of flux. Including next-nearest-neighbor-hoppings gives rise to new FMI states that belong to multidimensional irreducible representations of the molecular point group, giving rise to a rich phase diagram.

Criteria for first- and second-order vibrational resonances and correct evaluation of the Darling-Dennison resonance coefficients using the canonical Van Vleck perturbation theory.

The second-order vibrational Hamiltonian of a semi-rigid polyatomic molecule when resonances are present can be reduced to a quasi-diagonal form using second-order vibrational perturbation theory. Obtaining exact vibrational energy levels requires subsequent numerical diagonalization of the Hamiltonian matrix including the first- and second-order resonance coupling coefficients. While the first-order Fermi resonance constants can be easily calculated, the evaluation of the second-order Darling-Dennison constants requires more complicated algebra for seven individual cases with different numbers of creation-annihilation vibrational quanta. The difficulty in precise evaluation of the Darling-Dennison coefficients is associated with the previously unrecognized interference with simultaneously present Fermi resonances that affect the form of the canonically transformed Hamiltonian. For the first time, we have presented the correct form of the general expression for the evaluation of the Darling-Dennison constants that accounts for the underlying effect of Fermi resonances. The physically meaningful criteria for selecting both Fermi and Darling-Dennison resonances are discussed and illustrated using numerical examples.

Exotic Quantum Phase Transition of the Spin Nanotube

The S=1/2 three-leg spin tube is investigated using the numerical diagonalization up to the 42-spin clusters. A precise finite-size scaling analysis, so called the level spectroscopy, is carried out. It indicates a quantum phase transition between the spin-gap and gapless phases, with respect to the ratio of the leg and rung exchange interactions. The present analysis gives a precise estimate for the critical value of the ratio.

Novel Field Induced Quantum Phase Transition of the Kagome Lattice Antiferromagnet

The S = 1/2 kagomelattice antiferromagnet is investigated by the numerical diagonalization. We successfully obtained the ground-state full magnetization curve for the 42-spin cluster. Based on the finite-size scaling alnalysis including this new result, we consider the quatnum critical magnetization behavior around the 1/3 of the saturation magnetization. As a result, some unconventional features of the 1/3 magnetization ramp, which was proposed by our previous work, are clarified.

Double semion phase in an exactly solvable quantum dimer model on the kagome lattice

Quantum dimer models typically arise in various low energy theories like those of frustrated antiferromagnets. We introduce a quantum dimer model on the kagome lattice which stabilizes an alternative $\mathbb{Z}_2$ topological order, namely the so-called "double semion" order. For a particular set of parameters, the model is exactly solvable, allowing us to access the ground state as well as the excited states. We show that the double semion phase is stable over a wide range of parameters using numerical exact diagonalization. Furthermore, we propose a simple microscopic spin Hamiltonian for which the low-energy physics is described by the derived quantum dimer model.

Dynamic freezing and defect suppression in the tilted one-dimensional Bose-Hubbard model

We study the dynamics of tilted one-dimensional Bose-Hubbard model for two distinct protocols using numerical diagonalization for finite sized system ($N\le 18$). The first protocol involves periodic variation of the effective electric field $E$ seen by the bosons which takes the system twice (per drive cycle) through the intermediate quantum critical point. We show that such a drive leads to non-monotonic variations of the excitation density $D$ and the wavefunction overlap $F$ at the end of a drive cycle as a function of the drive frequency $\omega_1$, relate this effect to a generalized version of St\"uckelberg interference phenomenon, and identify special frequencies for which $D$ and $1-F$ approach zero leading to near-perfect dynamic freezing phenomenon. The second protocol involves a ramp of both the electric field $E$ (with a rate $\omega_1$) and the boson hopping parameter $J$ (with a rate $\omega_2$) to the quantum critical point. We find that both $D$ and the residual energy $Q$ decrease with increasing $\omega_2$; our results thus demonstrate a method of achieving near-adiabatic protocol in an experimentally realizable quantum critical system. We suggest experiments to test our theory.

Nonempirical anharmonic vibrational perturbation theory applied to biomolecules: free-base porphin.

Anharmonic vibrational frequencies and intensities (infrared and Raman) of an isolated free-base porphin molecule are predicted from the quantum mechanical (QM) geometry, the "semi-diagonal" quartic force field, and dipole moment and polarizability surfaces. The second-order vibrational perturbation theory plus the numerical diagonalization of the Hamiltonian matrix containing off-diagonal Fermi and Darling-Dennison resonance couplings (VPT2+WK) was used. The QM calculations were carried out with the Becke-Lee-Yang-Parr composite exchange-correlation functional (B3LYP) and with the 6-31+G(d,p) basis set. The harmonic force field for the equilibrium configuration was transformed into nonredundant local symmetry internal coordinates, and normal coordinates were defined. The semi-diagonal quartic rectilinear normal coordinate potential energy surface (PES), as well as the cubic surfaces of dipole moment (p) and polarizability (α) components, needed for the VPT2+WK calculation, were constructed by a five-point finite differentiation of Hessians (for PES) and of the values and first derivatives of p and α. They were obtained at the point of equilibrium and for 432 displaced configurations. This theoretical approach provides very good agreement between the predicted and experimental frequencies and intensities. However, the favorable result can be partly attributed to error cancellation within the B3LYP/6-31+G(d,p) QM model, as observed in earlier studies. Reassignments of some observed bands are proposed.

Strain effects on optical properties of tetrapod-shaped CdTe/CdS core–shell nanocrystals

The exciton states of strained CdTe/CdS core–shell tetrapod-shaped nanocrystals were theoretically investigated by the numerical diagonalization of a configuration interaction Hamiltonian based on the single-band effective mass approximation. We found that the inclusion of strain promotes the type-II nature by confining the electrons and holes in nonadjacent regions. This carrier separation is more efficient than that with type-II spherical nanocrystals. The strain leads to a small blue shift of the lowest exciton energy. Its magnitude is smaller than the red shift induced by increasing the shell thickness. Moreover, the larger arm diameter reduces the influence of both shell thickness and strain on the exciton energy. Considering the broken symmetry, a randomness induced carrier separation was revealed which is unique to a branched core–shell structure. The calculated shell thickness dependence of the emission energy agrees well with available experimental data.

Impurities and Landau level mixing in a fractional quantum Hall state in a flat-band lattice model

We study the toplogical checkerboard lattice model around the $\nu=\frac{1}{3}$ fractional quantum Hall phase using numerical exact diagonalization without Landau level projections. We add local perturbations, modified hoppings and on-site potentials, and observe phase transitions from the fractional quantum Hall phase to metallic and insulating phases when the strength and number of impurities is increased. In addition to evaluating the energy spectrum, we identify the phase diagrams by computing the topological Chern number of the many-body ground state manifold, and we show how the ground states lose their correlations due to the impurities by evaluating the spectrum of the one-body reduced density matrix. Our results show that the phase transition from the fractional quantum Hall phase to the metallic phase occurs for both impurity hoppings and potentials. Strong impurity hoppings cause a further transition into the insulating state, regardless of the sign of the hopping, when their density is high enough. In contrast, the same happens only for attractive potentials. Furthermore, the mixing to the higher band in a two-band model, generally denoted as Landau level mixing, is measured concluding that the lowest Landau level projection works well even with remarkably strong interactions and in the presence of impurities.

Phase diagram of the alternating-spin Heisenberg chain with extra isotropic three-body exchange interactions

For the time being isotropic three-body exchange interactions are scarcely explored and mostly used as a tool for constructing various exactly solvable one-dimensional models, although, generally speaking, such competing terms in generic Heisenberg spin systems can be expected to support specific quantum effects and phases. The Heisenberg chain constructed from alternating S=1 and sigma=1/2 site spins defines a realistic prototype model admitting extra three-body exchange terms. Based on numerical density-matrix renormalization group (DMRG) and exact diagonalization (ED) calculations, we demonstrate that the additional isotropic three-body terms stabilize a variety of partially-polarized states as well as two specific non-magnetic states including a critical spin-liquid phase controlled by two Gaussinal conformal theories as well as a critical nematic-like phase characterized by dominant quadrupolar S-spin fluctuations. Most of the established effects are related to some specific features of the three-body interaction such as the promotion of local collinear spin configurations and the enhanced tendency towards nearest-neighbor clustering of the spins. It may be expected that most of the predicted effects of the isotropic three-body interaction persist in higher space dimensions.

Quantum correlations and spatial localization in one-dimensional ultracold bosonic mixtures

We present the complete phase diagram for one-dimensional binary mixtures of bosonic ultracold atomic gases in a harmonic trap. We obtain exact results with direct numerical diagonalization for a small number of atoms, which permits us to quantify quantum many-body correlations. The quantum Monte Carlo method is used to calculate energies and density profiles for larger system sizes. We study the system properties for a wide range of interaction parameters. For the extreme values of these parameters, different correlation limits can be identified, where the correlations are either weak or strong. We investigate in detail how the correlations evolve between the limits. For balanced mixtures in the number of atoms in each species, the transition between the different limits involves sophisticated changes in the one- and two-body correlations. Particularly, we quantify the entanglement between the two components by means of the von Neumann entropy. We show that the limits equally exist when the number of atoms is increased for balanced mixtures. Also, the changes in the correlations along the transitions among these limits are qualitatively similar. We also show that, for imbalanced mixtures, the same limits with similar transitions exist. Finally, for strongly imbalanced systems, only two limits survive, i.e., a miscible limit and a phase-separated one, resembling those expected with a mean-field approach.

Topological states in normal and superconducting p -wave chains

We study a two-band model of fermions in a 1d chain with an antisymmetric hybridization that breaks inversion symmetry. We find that for certain values of its parameters, the sp -chain maps formally into a p -wave superconducting chain, the archetypical 1d system exhibiting Majorana fermions. The eigenspectra, including the existence of zero energy modes in the topological phase, agree for both models. The end states too share several similarities, such as the behavior of the localization length, the non-trivial topological index and robustness to disorder. However, we show that the excitations in the ends of a finite sp chain are conventional fermions though endowed with protected topological properties. Our results are obtained by a scattering approach in a semi-infinite chain with an edge defect treated within the T -matrix approximation. We present exact numerical diagonalization results that extend our analysis to arbitrary parameters and to disordered systems. • Antisymmetric hybridization. • Topological insulator. • Quantum phase transition. • Topological non-trivial phase. • Majorana fermions.

Finite size scaling in crossover among different random matrix ensembles in microscopic lattice models

Using numerical diagonalization we study the crossover among different random matrix ensembles (Poissonian, Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE) and Gaussian symplectic ensemble (GSE)) realized in two different microscopic models. The specific diagnostic tool used to study the crossovers is the level spacing distribution. The first model is a one-dimensional lattice model of interacting hard-core bosons (or equivalently spin 1/2 objects) and the other a higher dimensional model of non-interacting particles with disorder and spin–orbit coupling. We find that the perturbation causing the crossover among the different ensembles scales to zero with system size as a power law with an exponent that depends on the ensembles between which the crossover takes place. This exponent is independent of microscopic details of the perturbation. We also find that the crossover from the Poissonian ensemble to the other three is dominated by the Poissonian to GOE crossover which introduces level repulsion while the crossover from GOE to GUE or GOE to GSE associated with symmetry breaking introduces a subdominant contribution. We also conjecture that the exponent is dependent on whether the system contains interactions among the elementary degrees of freedom or not and is independent of the dimensionality of the system.

Frustration-Induced Magnetic Properties of the Spin-1/2 Heisenberg Antiferromagnet on the Cairo Pentagon Lattice

The magnetic phase diagram under a magnetic field of the spin-1/2 Heisenberg antiferromagnet on the Cairo pentagon lattice has been derived from a magnetization process and correlation functions calculated by numerical diagonalization for small clusters with up to 36 sites. The phase transition between two types of 1/3 magnetization states takes place, which continues to the orthogonal dimer state and ferrimagnetic state in the limits of x = 0 and x = ∞, respectively, where x is the ratio between two types of nearest-neighbor exchange interactions, in the absence of a magnetic field. A peculiar magnetization jump has been found at the lower- or higher-field edge of the 1/3 plateau, when x is larger or smaller than its critical value of ∼0.8, respectively. The temperature dependence of the specific heat shows a double-peak structure.