Terms Of Hamiltonian Research Articles

HORRENDOUS NMR: Establishing correlations in solution-state NMR by reinstating non-secular J-coupling terms

Homonuclear isotropic mixing modules allow J-coupled spins to exchange magnetization even when separated by chemical shift offsets that exceed their couplings. This is exploited in TOtal Correlation SpectroscopY (TOCSY) experiments and its variants, which facilitate these homonuclear polarization exchanges by applying broadband RF pulses. These then establish an effective Hamiltonian in which chemical shift offsets are erased, while J-coupling terms –including flip-flop components– remain active. The polarization that these non-secular terms will transfer among systems of chemically inequivalent sites over the course of a mixing period, are widely used modules in 1D and in multidimensional liquid-state NMR. Homonuclear correlation experiments are also common in solids NMR, particularly among X = 13C or 15N nuclei. Solids NMR experiments are often challenged by high-power RF demands which have led to a family of homonuclear solid-state correlation experiments that avoid pulsing on the nuclei of interest, and focus instead on the 1Hs that are bonded to them. These solid experiments usually reintroduce/strengthen 1H-X dipolar couplings; these, in conjunction with assistance from rotational resonance effects, bring back the truncated X-X dipolar interactions and facilitate the generation of cross peaks. The present study explores whether a similar goal can be achieved for solution-state counterparts, based on the reintroduction of truncated flip-flop terms in the J-coupling Hamiltonian via the pulsing on other, heteronuclear species. A proposal to achieve this is derived, and the resulting HOmonucleaR Recoupling by hEteroNuclear DecOUplingS (HORRENDOUS) approach to provide correlations between like nuclei without pulsing on them, is demonstrated.

Indirect NMR detection via proton of nuclei subject to large anisotropic interactions, such as 14N, 195Pt, and 35Cl, using the T-HMQC sequence.

Recently, the T-hetero-nuclear multiple quantum coherence (T-HMQC) sequence using the TRAPDOR (transfer of population in double resonance) recoupling has been introduced for the indirect detection via protons of quadrupolar nuclei with spin I = 1 (14N) or 3/2 (35Cl) in solids at fast magic-angle spinning (MAS). The sequence is simple as it only uses four rectangular pulses and exhibits low t1-noise because the recoupling pulses are applied to the indirectly detected isotope, I. We demonstrate that this sequence is applicable for the detection via protons of spin-1/2 nuclei subject to large chemical shift anisotropy, such as 195Pt. We also report the proton detection of double-quantum (2Q) coherences of 14N nuclei using this sequence. This 2Q version is more robust to the adjustment of the magic angle and the instabilities of the MAS frequencies than its parent single-quantum (1Q) version since the 2Q coherences are not broadened by the first-order quadrupole interaction. In practice, than its 1Q counterpart for the indirect detection of 14N nuclei, the 2Q variant benefits from a slightly higher resolution and comparable sensitivity. In this article, we derive for the first time the Hamiltonian that describes the spin dynamics during the TRAPDOR recoupling. This Hamiltonian demonstrates the importance of the adiabaticity parameter as well as the role of third-order terms in the effective Hamiltonian. The effects of offsets, radio-frequency field, and recoupling time on the efficiency of the T-HMQC sequence are analyzed numerically as well as with experimental detection via protons of 195Pt nuclei in a mixture of cis- and trans-platin and that of 14N and 35Cl isotopes in l-histidine HCl.

Anisotropic BCS-Richardson model and algebraic Bethe ansatz

We consider the limiting j3→∞ case of the elliptic Gaudin-type model in an external magnetic field [13–15,17,27] and of the elliptic BCS-Richardon model with one type of fermions [27] associated with non-skew-symmetric elliptic r-matrix defined on 4:1 unramified covering of the Weierstrass cubic y2=(u+j1)(u+j2)(u+j3). We develop the algebraic Bethe ansatz for the considered elliptic r-matrix in the j3→∞ limit and find the spectrum of its Gaudin-type and Richardson-type Hamiltonians in terms of solutions of the modified Bethe equations.

A new quantum mechanical pseudo material: Application in cryptography

Recently relativistic Dirac material such as graphene and the topological material as topological insulators have played prominent role in providing some of the materials which incorporate more mathematical and relativistic physical nature in general. Here we suggest a material which is quantum mechanical in nature but is a pseudo material. An interacting quantum many-particle system in one dimension representing the prime numbers on the x-axis through the asymptotic form of the prime counting function is formulated. The effective quantum and classical mechanics of a single particle of the system are studied by deriving the potential function that a single particle experiences in this system. We express our classical Hamiltonian in terms of the angle coordinate to illustrate certain interesting mathematical properties related to the prime numbers. Since it is a mathematical material having prime numbers as the atoms interacting with each other like ordinary mater, this material can have real and important application in cryptography. The elliptic curves that appear in the study of this system are related to primality test and are heavily used in communication cryptography. This analysis opens a new way of looking at materials which shows that simulating a material through a potential is a productive one.

Shaken dynamics on the 3d cubic lattice

On the space of ±1 spin configurations on the 3d square lattice, we consider the shaken dynamics, a parallel Markovian dynamics that can be interpreted in terms of Probabilistic Cellular Automata. The transition probabilities are defined in terms of pair ferromagnetic Ising-type Hamiltonians with nearest neighbor interaction J, depending on an additional parameter q, measuring the tendency of the system to remain locally in the same state. Odd times and even times have different transition probabilities. We compute the stationary measure of the shaken dynamics and we investigate its relation with the Gibbs measure for the 3d Ising model. It turns out that the two parameters J and q tune the geometry of the underlying lattice. We conjecture the existence of unique line of critical points in J−q plane. By a judicious use of perturbative methods we delimit the region where such curve must lie and we perform numerical simulation to determine it. Our method allows us to find in a unified way the critical values of J for Ising model with first neighbors interaction, defined on a whole class of lattices, intermediate between the two-dimensional hexagonal and the three-dimensional cubic one, such as, for example, the tetrahedral lattice. Finally we estimate the critical exponents of the magnetic susceptibility and show that our model captures a dimensional transition in the geometry of the system at q=0.

Bounds on spectral norms and barcodes

We investigate the relations between algebraic structures, spectral invariants, and persistence modules, in the context of monotone Lagrangian Floer homology with Hamiltonian term. Firstly, we use the newly introduced method of filtered continuation elements to prove that the Lagrangian spectral norm controls the barcode of the Hamiltonian perturbation of the Lagrangian submanifold, up to shift, in the bottleneck distance. Moreover, we show that it satisfies Chekanov type low-energy intersection phenomena, and non-degeneracy theorems. Secondly, we introduce a new averaging method for bounding the spectral norm from above, and apply it to produce precise uniform bounds on the Lagrangian spectral norm in certain closed symplectic manifolds. Finally, by using the theory of persistence modules, we prove that our bounds are in fact sharp in some cases. Along the way we produce a new calculation of the Lagrangian quantum homology of certain Lagrangian submanifolds, and answer a question of Usher.

Engineering of topological phases in driven thin topological insulators: Structure inversion asymmetry effect

We investigate the effect of a high frequency electromagnetic field with both of circularly and linearly polarization, on the emergence of quantum phases on thin topological insulators. Simultaneously, the influence of the system parameters (such as magnetic impurity, thickness engineering and structural inversion asymmetry of the potential) on emergence of topological phases is studied. We take our attention to the high frequency regime in which it is possible to consider an expansion for the Floquet Hamiltonian in terms of orders of 1/\Omega. The topological invariants are determined and it is demonstrated that some phase transitions between quantum anomalous Hall insulator, quantum pseudospin Hall insulator, quantum spin Hall insulator and normal insulator can be induced by altering the aforementioned parameters of the system. To avoid heating process, tuning of the system parameters gives us the opportunity to observe these phase transitions at small intensities of light.

Deformed explicitly correlated Gaussians.

Deformed explicitly correlated Gaussian (DECG) basis functions are introduced, and their matrix elements are calculated. All matrix elements can be calculated analytically in a closed form, except the Coulomb one, which has to be approximated by a Gaussian expansion. The DECG basis functions can be used to solve problems with nonspherical potentials. One example of such potential is the dipole self-interaction term in the Pauli-Fierz Hamiltonian. Examples are presented showing the accuracy and necessity of deformed Gaussian basis functions to accurately solve light-matter coupled systems in cavity QED.

Quantum thermo-dynamical construction for driven open quantum systems

Quantum dynamics of driven open systems should be compatible with both quantum mechanic and thermodynamic principles. By formulating the thermodynamic principles in terms of a set of postulates we obtain a thermodynamically consistent master equation. Following an axiomatic approach, we base the analysis on an autonomous description, incorporating the drive as a large transient control quantum system. In the appropriate physical limit, we derive the semi-classical description, where the control is incorporated as a time-dependent term in the system Hamiltonian. The transition to the semi-classical description reflects the conservation of global coherence and highlights the crucial role of coherence in the initial control state. We demonstrate the theory by analyzing a qubit controlled by a single bosonic mode in a coherent state.

Cubic-quartic optical soliton perturbation with Fokas–Lenells equation by semi-inverse variation

This paper recovers cubic-quartic bright optical solitons with perturbed Fokas–Lenells equation. The Hamiltonian perturbation terms appear with maximal permissible intensity. The semi-inverse variational principle is employed to retrieve such solitons.

Non-Hermitian time evolution: From static to parametric instability

Eigenmode coalescence imparts remarkable properties to non-hermitian time evolution, culminating in a purely non-hermitian spectral degeneracy known as an exceptional point (EP). Here, we revisit time evolution at the EP and classify two-level non-hermitian Hamiltonians in terms of the M\"obius group. We then leverage that classification to study dynamical EP encircling, by applying it to periodically-modulated (Floquet) Hamiltonians. This reveals that Floquet non-hermitian systems exhibit rich physics whose complexity is not captured by an EP-encircling rule. For example, Floquet EPs can occur without encircling and vice-versa. Instead, we show that the elaborate interplay between non-hermitian and modulation instabilities is better understood through the lens of parametric resonance.

Energy spectrum and structure of confined one-dimensional few-electron systems with and without coupling to light in a cavity

Explicitly Correlated Gaussian basis is used to calculate the energies and wave functions of one dimensional few-electron systems in confinement potentials created by external potentials or coupling to light in cavity. The appearance and properties of Wigner crystal-like structures are discussed. It is shown that one dimensional Wigner crystals can be formed by coupling electrons to light due to the dipole self-interaction term in the light-matter Hamiltonian, provided an additional extremely weak confining potential is present.

On the spectral problem of the quantum KdV hierarchy

The spectral problem for the quantum dispersionless Korteweg–de Vries (KdV) hierarchy, aka the quantum Hopf hierarchy, is solved by Dubrovin. In this article, following Dubrovin, we study Buryak–Rossi’s quantum KdV hierarchy. In particular, we prove a symmetry property and a non-degeneracy property for the quantum KdV Hamiltonians. On the basis of this we construct a complete set of common eigenvectors. The analysis underlying this spectral problem implies certain vanishing identities for combinations of characters of the symmetric group. We also comment on the geometry of the spectral curves of the quantum KdV hierarchy and we give a representation of the quantum dispersionless KdV Hamiltonians in terms of multiplication operators in the class algebra of the symmetric group.

Constructing qudits from infinite-dimensional oscillators by coupling to qubits

An infinite dimensional system such as a quantum harmonic oscillator offers a potentially unbounded Hilbert space for computation, but accessing and manipulating the entire state space requires a physically unrealistic amount of energy. When such a quantum harmonic oscillator is coupled to a qubit, for example via a Jaynes-Cummings interaction, it is well known that the total Hilbert space can be separated into independently accessible subspaces of constant energy, but the number of subspaces is still infinite. Nevertheless, a closed four-dimensional Hilbert space can be analytically constructed from the lowest energy states of the qubit-oscillator system. We extend this idea and show how a $d$-dimensional Hilbert space can be analytically constructed, which is closed under a finite set of unitary operations resulting solely from manipulating standard Jaynes-Cummings Hamiltonian terms. Moreover, we prove that the first-order sideband pulses and carrier pulses comprise a universal set for quantum operations on the qubit-oscillator qudit. This work suggests that the combination of a qubit and a bosonic system may serve as hardware-efficient quantum resources for quantum information processing.

Unusual spin dynamics in the low-temperature magnetically ordered state of Ag3LiIr2O6

Recently, there have been contrary claims of Kitaev spin-liquid behaviour and ordered behavior in the honeycomb compound Ag$_3$LiIr$_2$O$_6$ based on various experimental signatures. Our investigations on this system reveal a low-temperature ordered state with persistent dynamics down to the lowest temperatures. Magnetic order is confirmed by clear oscillations in the muon spin relaxation ($\mu$SR) time spectrum below 9 K till 52 mK. Coincidentally in $^7$Li nuclear magnetic resonance, a wipe-out of the signal is observed below $\sim$ 10 K which again strongly indicates magnetic order in the low temperature regime. This is supported by our density functional theory calculations which show an appreciable Heisenberg exchange term in the spin Hamiltonian that favors magnetic ordering. The $^7$Li shift and spin-lattice relaxation rate also show anomalies at $\sim$ 50 K. They are likely related to the onset of dynamic magnetic correlations, but their origin is not completely clear. Detailed analysis of our $\mu$SR data is consistent with a co-existence of incommensurate N\'eel and striped environments. A significant and undiminished dynamical relaxation rate ($\sim 5$ MHz) as seen in $\mu$SR deep into the ordered phase indicates enhanced quantum fluctuations in the ordered state.

Surface theory of a second-order topological insulator beyond the Dirac approximation

We study the surface states and chiral hinge states of a 3D second-order topological insulator in the presence of an external magnetic gauge field. Surfaces pierced by flux host Landau levels, while surfaces parallel to the applied field are not significantly affected. The chiral hinge modes mediate spectral flow between neighbouring surfaces. As the magnetic field strength is increased, the surface Landau quantization deviates from that of a massive Dirac cone. Quantitatively, the $n = 0$ Landau level falls inside the surface Dirac gap, and not at the gap edge. The $n \ne 0$ levels exhibit a further, qualitative discrepancy: while the massive Dirac cone is expected to produce pairs of levels ($\pm n$) which are symmetric around zero energy, the $n$ and $-n$ levels become asymmetric in our lattice model -- one of the pair may even be absent from the spectrum, or hybridized with the continuum. In order to resolve the issue, we extend the standard 2D massive Dirac surface theory, by including additional Hamiltonian terms at $\mathcal{O} (k^2)$. While these terms do not break particle-hole symmetry in the absence of magnetic field, they lead to the aforementioned Landau level asymmetry once the magnetic field is applied. We argue that similar $\mathcal{O}(k^2)$ correction terms are generically expected in lattice models containing gapped Dirac fermions, using the BHZ model of a 2D topological insulator as an example.

Spin-Pure Stochastic-CASSCF via GUGA-FCIQMC Applied to Iron-Sulfur Clusters.

In this work, we demonstrate how to efficiently compute the one- and two-body reduced density matrices within the spin-adapted full configuration interaction quantum Monte Carlo (FCIQMC) method, which is based on the graphical unitary group approach (GUGA). This allows us to use GUGA-FCIQMC as a spin-pure configuration interaction (CI) eigensolver within the complete active space self-consistent field (CASSCF) procedure and hence to stochastically treat active spaces far larger than conventional CI solvers while variationally relaxing orbitals for specific spin-pure states. We apply the method to investigate the spin ladder in iron–sulfur dimer and tetramer model systems. We demonstrate the importance of the orbital relaxation by comparing the Heisenberg model magnetic coupling parameters from the CASSCF procedure to those from a CI-only (CASCI) procedure based on restricted open-shell Hartree–Fock orbitals. We show that the orbital relaxation differentially stabilizes the lower-spin states, thus enlarging the coupling parameters with respect to the values predicted by ignoring orbital relaxation effects. Moreover, we find that, while CASCI results are well fit by a simple bilinear Heisenberg Hamiltonian, the CASSCF eigenvalues exhibit deviations that necessitate the inclusion of biquadratic terms in the model Hamiltonian.

Theory and simulations of homonuclear three-spin systems in rotating solids.

The homonuclear dipolar coupling is the internal spin interaction that contributes the most to the line shapes in magic-angle-spinning (MAS) 1H NMR spectra of solids, and linewidths typically extend over several hundred Hertz, limiting the 1H resolution. Understanding and reducing this contribution could provide rich structural information for organic solids. Here, we use average Hamiltonian theory to study two- and three-spin systems in the fast MAS regime. Specifically, we develop analytical expressions to third order in the case of two and three inequivalent spins (I = ½). The results show that the full third-order expression of the Hamiltonian, without secular approximations or truncation to second order, is the description that agrees the best, by far, with full numerical calculations. We determine the effect on the NMR spectrum of the different Hamiltonian terms, which are shown to produce both residual shifts and splittings in the three-spin systems. Both the shifts and splittings have a fairly complex dependence on the spinning rate with the eigenstates having a polynomial ωr dependence. The effect on powder line shapes is also shown, and we find that the anisotropic residual shift does not have zero average so that the powder line shape is broadened and shifted from the isotropic position. This suggests that in 1H MAS spectra, even at the fastest MAS rates attainable today, the positions observed are not exactly the isotropic shifts.

Regularity estimates for fully nonlinear elliptic PDEs with general Hamiltonian terms and unbounded ingredients

We develop an optimal regularity theory for $$L^p$$ L p -viscosity solutions of fully nonlinear uniformly elliptic equations in nondivergence form whose gradient growth is described through a Hamiltonian function with measurable and possibly unbounded coefficients. Our approach treats both superlinear and sublinear gradient regimes in a unified way. We show $$C^{0,\alpha }$$ C 0 , α , $$C^{0,\text {Log-Lip}}$$ C 0 , Log-Lip , $$C^{1,\alpha }$$ C 1 , α , $$ C^{1,\text {Log-Lip}}$$ C 1 , Log-Lip and $$C^{2,\alpha }$$ C 2 , α regularity estimates, by displaying the growth allowed to the Hamiltonian in order to deal with an unbounded nonlinear gradient coefficient, whose integrability in turn gets worse as we approach the quadratic regime. These results may be seen as natural extensions of Teixeira (Arch Ration Mech Anal 211(3):911–927, 2014) and Nornberg (J Math Pures Appl 128(9):297–329, 2019). Moreover, we find proper compatibility conditions for which our regularity results depend intrinsically on the integrability of the underlying source term. As a byproduct of our methods, we prove a priori BMO estimates; sharp regularity to associated recession and flat profiles under relaxed convexity assumptions; improved regularity for a class of singular PDEs; and a Perron type result under unbounded ingredients.

Normal-ordering approximations and translational (non)invariance

Normal-ordering provides an approach to approximate three-body forces as effective two-body operators and it is therefore an important tool in many-body calculations with realistic nuclear interactions. The corresponding neglect of certain three-body terms in the normal-ordered Hamiltonian is known to influence translational invariance, although the magnitude of this effect has not yet been systematically quantified. In this work we study in particular the normal-ordering two-body approximation applied to a single harmonic-oscillator reference state. We explicate the breaking of translational invariance and demonstrate the magnitude of the approximation error as a function of model space parameters for $^4\rm{He}$ and $^{16}\rm{O}$ by performing full no-core shell-model calculations with and without three-nucleon forces. We combine two different diagnostics to better monitor the breaking of translational invariance. While the center-of-mass effect is shown to become potentially very large for $^4\rm{He}$, it is also shown to be much smaller for $^{16}\rm{O}$ although full convergence is not reached. These tools can be easily implemented in studies using other many-body frameworks and bases.