Three-dimensional Hamiltonian Research Articles

A No-Go Result for Implementing Chiral Symmetries by Locality-Preserving Unitaries in a Three-Dimensional Hamiltonian Lattice Model of Fermions.

We argue that the chiral U(1)_{A} symmetry of a Weyl fermion cannot be implemented by a shallow depth quantum circuit operation in a fermionic lattice Hamiltonian model with finite dimensional onsite Hilbert spaces. We also extend this result to discrete Z_{2N} subgroups of U(1)_{A}, in which case we show that for N_{f} Weyl fermions of the same helicity, this group action cannot be implemented with shallow depth circuits when N_{f} is not an integer multiple of 2N.

Analytical solution for the surface states of the antiferromagnetic topological insulator MnBi2Te4

Recently, the intrinsic magnetic topological insulator MnBi$_2$Te$_4$ has attracted great attention. It has an out-of-plane antiferromagnetic order, which is believed to open a sizable energy gap in the surface states. This gap, however, was not always observable in the latest angle-resolved photoemission spectroscopy (ARPES) experiments. To address this issue, we analytically derive an effective model for the two-dimensional (2D) surface states by starting from a three-dimensional (3D) Hamiltonian for bulk MnBi$_2$Te$_4$ and taking into account the spatial profile of the bulk magnetization. Our calculations suggest that the diminished surface gap may be caused by a much smaller and more localized intralayer ferromagnetic order. In addition, we calculate the spatial distribution and penetration depth of the surface states, which indicates that the surface states are mainly embedded in the first two septuple layers from the terminating surface. From our analytical results, the influence of the bulk parameters on the surface states can be found explicitly. Furthermore, we derive a $\bf{k}\cdot \bf{p}$ model for MnBi$_2$Te$_4$ thin films and show the oscillation of the Chern number between odd and even septuple layers. Our results will be helpful for the ongoing explorations of the MnBi$_x$Te$_y$ family.

Inducing anisotropies in Dirac fermions by periodic driving

We consider the three-dimensional Hamiltonian for Bi2Se3, a second-generation topological insulator, under the effect of a periodic drive for both in-plane and out-of-plane fields. As it will be shown by means of high-frequency expansions up to second order in the Floquet Hamiltonian, the driving induces anisotropies in the Dirac cone and opens up a quasienergy gap for in-plane elliptically polarized fields. Analytic expressions are obtained for the renormalized velocities and the quasienergy gap. These expressions are then compared to numerical calculations performed by discretizing the Hamiltonian in a one-dimensional lattice and following a staggered fermion approach, achieving a remarkable agreement. We believe our work may have an impact on the transport properties of topological insulators.

Fractional charge bound to a vortex in two-dimensional topological crystalline insulators

We establish the correspondence between the fractional charge bound to a vortex in a textured lattice and the relevant bulk band topology in two-dimensional (2D) topological crystalline insulators. As a representative example, we consider the Kekule textured graphene whose bulk band topology is characterized by a 2D $\mathbb{Z}_{2}$ topological invariant $\nu_{\rm 2D}$ protected by inversion symmetry. The fractional charge localized at a vortex in the Kekule texture is shown to be related to the change in the bulk topological invariant $\nu_{\rm 2D}$ around the vortex, as in the case of the Su-Schriefer-Heeger model in which the fractional charge localized at a domain wall is related to the change in the bulk charge polarization between degenerate ground states. We show that the effective three-dimensional (3D) Hamiltonian, where the angle $\theta$ around a vortex in Kekule-textured graphene is a third coordinate, describes a 3D axion insulator with a quantized magnetoelectric polarization. The spectral flow during the adiabatic variation of $\theta$ corresponds to the chiral hinge modes of an axion insulator and determines the accumulated charge localized at the vortex, which is half-quantized when chiral symmetry exists. When chiral symmetry is absent, electric charge localized at the vortex is no longer quantized, but the vortex always carries a half-quantized Wannier charge as long as inversion symmetry exists. For the cases when magnetoelectric polarization is quantized due to the presence of symmetry that reverses the space-time orientation, we classify all possible topological crystalline insulators whose vortex defect carries a fractional charge.

Gauging permutation symmetries as a route to non-Abelian fractons

We discuss the procedure for gauging on-site Z_2Z2 global symmetries of three-dimensional lattice Hamiltonians that permute quasi-particles and provide general arguments demonstrating the non-Abelian character of the resultant gauged theories. We then apply this general procedure to lattice models of several well known fracton phases: two copies of the X-Cube model, two copies of Haah’s cubic code, and the checkerboard model. Where the former two models possess an on-site Z_2Z2 layer exchange symmetry, that of the latter is generated by the Hadamard gate. For each of these models, upon gauging, we find non-Abelian subdimensional excitations, including non-Abelian fractons, as well as non-Abelian looplike excitations and Abelian fully mobile pointlike excitations. By showing that the looplike excitations braid non-trivially with the subdimensional excitations, we thus discover a novel gapped quantum order in 3D, which we term a ‘panoptic" fracton order. This points to the existence of parent states in 3D from which both topological quantum field theories and fracton states may descend via quasi-particle condensation. The gauged cubic code model represents the first example of a gapped 3D phase supporting (inextricably) non-Abelian fractons that are created at the corners of fractal operators.

Hydrogenoid Spectra with Central Perturbations

Through the Kre\u{\i}n-Vi\v{s}ik-Birman extension scheme, unlike the classical analysis based on von Neumann's theory, we reproduce the construction and classification of all self-adjoint realisations of three-dimensional hydrogenoid-like Hamiltonians with singular perturbation supported at the Coulomb centre (the nucleus), as well as of Schr\"{o}dinger operators with Coulomb potentials on the half-line. These two problems are technically equivalent, albeit sometimes treated by their own in the the literature. Based on such scheme, we then recover the formula to determine the eigenvalues of each self-adjoint extension, as corrections of the non-relativistic hydrogenoid energy levels. We discuss in which respect the Kre\u{\i}n-Vi\v{s}ik-Birman scheme is somewhat more natural in yielding the typical boundary condition of self-adjointness at the centre of the perturbation.

Influence of magnetic disorders on quantum anomalous Hall effect in magnetic topological insulator films beyond the two-dimensional limit

Quantum anomalous Hall effect (QAHE) has been experimentally realized in magnetic topological insulator (MTI) thin films fabricated on magnetically doped . In an MTI thin film with the magnetic easy axis along the normal direction (z-direction), orientations of magnetic dopants are randomly distributed around the magnetic easy axis, acting as magnetic disorders. With the aid of the non-equilibrium Green's function and Landauer–Büttiker formalism, we numerically study the influence of magnetic disorders on QAHE in an MTI thin film modeled by a three-dimensional tight-binding Hamiltonian. It is found that, due to the existence of gapless side surface states, QAHE is protected even in the presence of magnetic disorders as long as the z-component of magnetic moment of all magnetic dopants are positive. More importantly, such magnetic disorders also suppress the dissipation of the chiral edge states and enhance the quality of QAHE in MTI films. In addition, the effect of magnetic disorders depends very much on the film thickness, and the optimal influence is achieved at certain thickness. These findings are new features for QAHE in three-dimensional systems, not present in two-dimensional systems.

Quantum regime of a plasma-wave-pumped free-electron laser in the presence of an axial magnetic field

The quantum regime of a plasma-whistler-wave-pumped free-electron laser (FEL) in the presence of an axial-guide magnetic field is presented. By quantizing both the plasma whistler field and axial magnetic field, an N-particle three-dimensional Hamiltonian of quantum-FEL (QFEL) has been derived. Employing Heisenberg evolution equations and introducing a new collective operator which controls the vertical motion of electrons, a quantum dispersion relation of the plasma whistler wiggler has been obtained analytically. Numerical results indicate that, by increasing the intrinsic quantum momentum spread and/or increasing the axial magnetic field strength, the bunching and the radiation fields grow exponentially. In addition, a spiking behavior of the spectrum was observed with increasing cyclotron frequency which provides an enormous improvement in the coherence of QFEL radiation even in a limit close-to-classical regime, where an overlapping of these spikes is observed. Also, an upper limit of the intrinsic quantum momentum spread which depends on the value of the cyclotron frequency was found.

Glassy quantum dynamics in translation invariant fracton models

We investigate relaxation in the recently discovered "fracton" models and discover that these models naturally host glassy quantum dynamics in the absence of quenched disorder. We begin with a discussion of "type I" fracton models, in the taxonomy of Vijay, Haah, and Fu. We demonstrate that in these systems, the mobility of charges is suppressed exponentially in the inverse temperature. We further demonstrate that when a zero temperature type I fracton model is placed in contact with a finite temperature heat bath, the approach to equilibrium is a logarithmic function of time over an exponentially wide window of time scales. Generalizing to the more complex "type II" fracton models, we find that the charges exhibit subdiffusion upto a relaxation time that diverges at low temperatures as a super-exponential function of inverse temperature. This behaviour is reminiscent of "nearly localized" disordered systems, but occurs with a translation invariant three-dimensional Hamiltonian. We also conjecture that fracton models with conserved charge may support a phase which is a thermal metal but a charge insulator.

Matter-waves in Bose–Einstein condensates with spin-orbit and Rabi couplings

We investigate the one-dimensional (1D) and two-dimensional (2D) reduction of a quantum field theory starting from the three-dimensional (3D) many-body Hamiltonian of interacting bosons with spin–orbit (SO) and Rabi couplings. We obtain the effective time-dependent 1D and 2D nonpolynomial Heisenberg equations for both the repulsive and attractive signs of the inter-atomic interaction. Our findings show that in the case in which the many-body state coincides with the Glauber coherent state, the 1D and 2D Heisenberg equations become 1D and 2D nonpolynomial Schrödinger equations (NPSEs). These models were derived in a mean-field approximation from 3D Gross-Pitaevskii equation (GPE), describing a Bose–Einstein condensate (BEC) with SO and Rabi couplings. In the present work self-repulsive and self-attractive localized solutions of the 1D NPSE and the 1D GPE are obtained in a numerical form. The combined action of SO and Rabi couplings produces conspicuous sidelobes on the density profile, for both signs of the interaction. In the case of the attractive nonlinearity, an essential result is the possibility of getting an unstable condensate by the increasing of SO coupling.

Fault-tolerant logical gates in quantum error-correcting codes

Recently, Bravyi and K\"onig have shown that there is a tradeoff between fault-tolerantly implementable logical gates and geometric locality of stabilizer codes. They consider locality-preserving operations which are implemented by a constant depth geometrically local circuit and are thus fault-tolerant by construction. In particular, they shown that, for local stabilizer codes in D spatial dimensions, locality preserving gates are restricted to a set of unitary gates known as the D-th level of the Clifford hierarchy. In this paper, we elaborate this idea and provide several extensions and applications of their characterization in various directions. First, we present a new no-go theorem for self-correcting quantum memory. Namely, we prove that a three-dimensional stabilizer Hamiltonian with a locality-preserving implementation of a non-Clifford gate cannot have a macroscopic energy barrier. Second, we prove that the code distance of a D-dimensional local stabilizer code with non-trivial locality-preserving m-th level Clifford logical gate is upper bounded by $O(L^{D+1-m})$. For codes with non-Clifford gates (m>2), this improves the previous best bound by Bravyi and Terhal. Third we prove that a qubit loss threshold of codes with non-trivial transversal m-th level Clifford logical gate is upper bounded by 1/m. As such, no family of fault-tolerant codes with transversal gates in increasing level of the Clifford hierarchy may exist. This result applies to arbitrary stabilizer and subsystem codes, and is not restricted to geometrically-local codes. Fourth we extend the result of Bravyi and K\"onig to subsystem codes. A technical difficulty is that, unlike stabilizer codes, the so-called union lemma does not apply to subsystem codes. This problem is avoided by assuming the presence of error threshold in a subsystem code, and the same conclusion as Bravyi-K\"onig is recovered.

Control of electron spin–orbit anisotropy in pyramidal InAs quantum dots

We investigate the electron spin–orbit interaction anisotropy of pyramidal InAs quantum dots using a fully three-dimensional Hamiltonian. The dependence of the spin–orbit interaction strength on the orientation of externally applied in-plane magnetic fields is consistent with recent experiments, and it can be explained from the interplay between Rashba and Dresselhaus spin–orbit terms in dots with asymmetric confinement. Based on this, we propose manipulating the dot composition and height as efficient means for controlling the spin–orbit anisotropy.

Is space-time symmetry a suitable generalization of parity-time symmetry?

We discuss space-time symmetric Hamiltonian operators of the form H=H0+igH′, where H0 is Hermitian and g real. H0 is invariant under the unitary operations of a point group G while H′ is invariant under transformation by elements of a subgroup G′ of G. If G exhibits irreducible representations of dimension greater than unity, then it is possible that H has complex eigenvalues for sufficiently small nonzero values of g. In the particular case that H is parity-time symmetric then it appears to exhibit real eigenvalues for all 0<g<gc, where gc is the exceptional point closest to the origin. Point-group symmetry and perturbation theory enable one to predict whether H may exhibit real or complex eigenvalues for g>0. We illustrate the main theoretical results and conclusions of this paper by means of two- and three-dimensional Hamiltonians exhibiting a variety of different point-group symmetries.

Spin–orbit-induced hole spin relaxation in InAs and GaAs quantum dots

We study the effect of valence band spin–orbit interactions (SOI) on the acoustic phonon-assisted spin relaxation of holes confined in quantum dots (QDs). Heavy hole–light hole (hh–lh) mixing and all the spin–orbit terms arising from zinc-blende bulk inversion asymmetry (BIA) are considered on equal footing in a fully three-dimensional Hamiltonian. We show that hh–lh mixing and BIA have comparable contributions to the hole spin relaxation in self-assembled QDs, but BIA becomes dominant in gated QDs. Simultaneously accounting for both mechanisms is necessary for quantitatively correct results in quasi-two-dimensional QDs. The dependence of the hole spin relaxation on the QD geometry and spin splitting energy is drastically different from that of electrons, with a non-monotonic behavior which results from the interplay between SOI terms. Our results reconcile contradictory predictions of previous theoretical works and are consistent with experiments.

Highly correlated ab initio study of the far infrared spectra of methyl acetate

Highly correlated ab initio calculations (CCSD(T)) are used to compute gas phase spectroscopic parameters of three isotopologues of the methyl acetate (CH(3)COOCH(3), CD(3)COOCH(3), and CH(3)COOCD(3)), searching to help experimental assignments and astrophysical detections. The molecule shows two conformers cis and trans separated by a barrier of 4457 cm(-1). The potential energy surface presents 18 minima that intertransform through three internal rotation motions. To analyze the far infrared spectrum at low temperatures, a three-dimensional Hamiltonian is solved variationally. The two methyl torsion barriers are calculated to be 99.2 cm(-1) (C-CH(3)) and 413.1 cm(-1) (O-CH(3)), for the cis-conformer. The three fundamental torsional band centers of CH(3)COOCH(3) are predicted to lie at 63.7 cm(-1) (C-CH(3)), 136.1 cm(-1) (O-CH(3)), and 175.8 cm(-1) (C-O torsion) providing torsional state separations. For the 27 vibrational modes, anharmonic fundamentals and rovibrational parameters are provided. Computed parameters are compared with those fitted using experimental data.

Time reversal, fermion doubling, and the masses of lattice Dirac fermions in three dimensions

Motivated by recent examples of three-dimensional lattice Hamiltonians with massless Dirac fermions in their (bulk) spectrum, I revisit the problem of fermion doubling on bipartite lattices. The number of components of the Dirac fermion in a time-reversal and parity-invariant $d$-dimensional lattice system is determined by the minimal representation of the Clifford algebra of $d+1$ Hermitian Dirac matrices that allows a construction of the time-reversal operator with the square of unity, and it equals ${2}^{d}$ for $d=2$ and 3. Possible mass terms for (spinless) Dirac fermions are listed and discussed. In three dimensions, there are altogether eight independent masses, out of which four are even and four are odd under time reversal. A specific violation of time-reversal symmetry that leads to (minimal) four-component massless Dirac fermion in three dimensions at low energies is constructed.

Quantum dynamics of dissociative chemisorption of CH4 on Ni(111): Influence of the bending vibration

Two-dimensional, three-dimensional, and four-dimensional quantum dynamic calculations are performed on the dissociative chemisorption of CH(4) on Ni(111) using the multiconfiguration time-dependent Hartree (MCTDH) method. The potential energy surface used for these calculations is 15-dimensional (15D) and was obtained with density functional theory for points which are concentrated in the region that is dynamically relevant to reaction. Many reduced dimensionality calculations were already performed on this system, but the molecule was generally treated as pseudodiatomic. The main improvement of our model is that we try to describe CH(4) as a polyatomic molecule by including a degree of freedom describing a bending vibration in our three-dimensional and four-dimensional models. Using a polyspherical coordinate system, a general expression for the 15D kinetic energy operator is derived, which discards all the singularities in the operator and includes rotational and Coriolis coupling. We use seven rigid constraints to fix the CH(3) umbrella of the molecule to its gas phase equilibrium geometry and to derive two-dimensional, three-dimensional, and four-dimensional Hamiltonians, which were used in the MCTDH method. Only four degrees of freedom evolve strongly along the 15D minimum energy path: the distance of the center of mass of the molecule to the surface, the dissociative C[Single Bond]H bond distance, the polar orientation of the molecule, and the bending angle between the dissociative C[Single Bond]H bond and the umbrella. A selection of these coordinates is included in each of our models. The polar rotation is found to be important in determining the mode selective behavior of the reaction. Furthermore, our calculations are in good agreement with the finding of Xiang et al. [J. Chem. Phys. 117, 7698 (2002)] in their reduced dimensional calculation that the helicopter motion of the umbrella symmetry axis is less efficient than its cartwheel motion for promoting the reaction. The effect of pre-exciting the bend modes is qualitatively incorrect at higher energies, suggesting the necessity of including additional rotational and vibrational degrees of freedom in the model.

Quadratic fermionic interactions yield Hamiltonians with large ground-state energy gaps

Polynomially large ground-state energy gaps are rare in many-body quantum systems, but useful in quantum information and an interesting feature of the one-dimensional quantum Ising model. We show analytically that the gap is generically polynomially large not just for the quantum Ising model, but for one-, two-, and three-dimensional interaction lattices and Hamiltonians with certain random interactions. We extend the analysis to Hamiltonian evolutions and we use the Jordan-Wigner transformation and a related transformation for spin-3/2 particles to show that our results can be restated using spin operators in a surprisingly simple manner. These results also yield a new perspective on the one-dimensional cluster state.

Calculated spectroscopic and electric properties of the alkali metal-ammonia complexes from Kn–NH3 to Frn–NH3 (n=,+1)

The newly developed Stuttgart small-core scalar relativistic pseudopotentials for the alkali metals are used to study spectroscopic and electric properties of the heavier alkali metal-ammonia complexes from K(n)-NH(3) to Fr(n)-NH(3) (n=0,+1) at the second-order Moller-Plesset (MP2) and coupled cluster [CCSD(T)] levels of theory. Equilibrium geometries and dissociation energies computed at the MP2 level are in reasonable agreement with their CCSD(T) counterparts, whereas for the dipole polarizabilities MP2 is not performing well overestimating significantly electron correlation effects. The bond distances increase monotonically with increasing mass of the metal atom as relativistic effects are small in these systems. However, the dipole polarizabilities are more sensitive to such effects and we find a decrease in this property from Cs-NH(3) to Fr-NH(3). Combination of CCSD(T) harmonic frequencies and MP2 anharmonic corrections obtained from a perturbative vibrational treatment leads to fundamental frequencies in good agreement with experimental results obtained by Suzer and Andrews [J. Am. Chem. Soc. 109, 300 (1986)]. We also present the results of variational calculations with a three-dimensional vibrational Hamiltonian, making use of CCSD(T) potential energy and electric dipole moment surfaces. Complexation of NH(3) to the metal causes a strong infrared intensification of the symmetric NH(3) stretching mode in the neutral complexes, which is absent in the charged species.

Relativistic extension of the complex scaling method

We construct a tridiagonal matrix representation for the three-dimensional Dirac-Coulomb Hamiltonian that provides for a simple and straightforward relativistic extension of the complex scaling method. Besides the Coulomb interaction, additional vector, scalar, and pseudoscalar coupling to short-range potentials are also included in the same representation. Using that, we are able to obtain highly accurate values for the relativistic bound states and resonance energies. A simple program code is developed to perform the calculation for a given charge, angular momentum, and potential configuration. The resonance structure in the complex relativistic energy plane is also shown graphically. Illustrative examples are given and we verify that in the nonrelativistic limit one obtains known results. As an additional advantage of this tridiagonal representation, we use it to obtain highly accurate evaluation of the relativistic bound state energies for the Woods-Saxon potential (as a model of nuclear interaction) with the nucleus treated as a solid sphere of uniform charge distribution.