Arbitrary Spin Quantum Number Research Articles

Projective Spin Adaptation for the Exact Diagonalization of Isotropic Spin Clusters

Spin Hamiltonians, like the Heisenberg model, are used to describe the magnetic properties of exchange-coupled molecules and solids. For finite clusters, physical quantities, such as heat capacities, magnetic susceptibilities or neutron-scattering spectra, can be calculated based on energies and eigenstates obtained by exact diagonalization (ED). Utilizing spin-rotational symmetry SU(2) to factor the Hamiltonian with respect to total spin S facilitates ED, but the conventional approach to spin-adapting the basis is more intricate than selecting states with a given magnetic quantum number M (the spin z-component), as it relies on irreducible tensor-operator techniques and spin-coupling coefficients. Here, we present a simpler technique based on applying a spin projector to uncoupled basis states. As an alternative to Löwdin’s projection operator, we consider a group-theoretical formulation of the projector, which can be evaluated either exactly or approximately using an integration grid. An important aspect is the choice of uncoupled basis states. We present an extension of Löwdin’s theorem for s=12 to arbitrary local spin quantum numbers s, which allows for the direct selection of configurations that span a complete, linearly independent basis in an S sector upon the spin projection. We illustrate the procedure with a few examples.

Designing refrigerators in higher dimensions using quantum spin models

We design quantum refrigerators based on spin-j quantum XYZ and bilinear-biquadratic models with individual spins attached to bosonic thermal baths. By considering both local and global master equations, we illustrate an enhancement in the performance of the refrigerators with an increase in the spin dimension irrespective of the choice of the spin models. To assess the performance of the refrigerators, we introduce a distance-based measure to quantify the local temperature of a particle with arbitrary spin quantum number j. Interestingly, we find that the local temperature quantifier, defined via minimizing the distance between a spin-j thermal state and the evolved state of the spin-j particle in the steady state, coincides with the population-based definition of local temperature known in the literature for spin-1/2 particles. Moreover, we demonstrate that the qualitative behavior of the distance-based local temperature is independent of the choice of the distance measure by comparing the trace distance, Uhlmann's fidelity and relative entropy distance. We further observe by computing local master equation that the quantum refrigerator consisting of a spin-1/2 and a spin-j particle can lead to a lower local temperature compared to a refrigerator with two identical spin-j particles following the XYZ interactions.

Extreme depolarization for any spin

The opportunity to build quantum technologies operating with elementary quantum systems with more than two levels is now increasingly being examined, not least because of the availability of such systems in the laboratory. It is therefore essential to understand how these single systems initially in highly non-classical states decohere on different time scales due to their coupling with the environment. In this work, we consider the depolarization, both isotropic and anisotropic, of a quantum spin of arbitrary spin quantum number $j$ and focus on the study of the most superdecoherent states. We approach this problem from the perspective of the collective dynamics of a system of $N=2j$ constituent spin-$1/2$, initially in a symmetric state, undergoing collective depolarization. This allows us to use the powerful language of quantum information theory to analyze the fading of quantum properties of spin states caused by depolarization. In this framework, we establish a precise link between superdecoherence and entanglement. We present extensive numerical results on the scaling of the entanglement survival time with the Hilbert space dimension for collective depolarization. We also highlight the specific role played by anticoherent spin states and show how their Markovian isotropic depolarization alone can lead to the generation of bound entangled states.

Anisotropy as a diagnostic test for distinct tensor-network wave functions of integer- and half-integer-spin Kitaev quantum spin liquids

Contrasting ground states of quantum magnets with the integer- and half-integer-spin moments are the manifestation of many-body quantum interference effects. In this work, we investigate the distinct nature of the integer- and half-integer-spin quantum spin liquids in the framework of the Kitaev's model on the honeycomb lattice. The models with arbitrary spin quantum numbers are not exactly solvable in contrast to the well-known quantum spin liquid solution of the spin-1/2 system. We use the tensor-network wave functions for the integer-and half-integer-spin quantum spin liquid states to unveil the important difference between these states. We find that the distinct sign structures of the tensor-network wave function for the integer- and half-integer-spin quantum spin liquids are responsible for completely different ground states in the spatially anisotropic limit. Hence the spatial anisotropy would be a useful diagnostic test for distinguishing these quantum spin liquid states, both in the numerical computations and experiments on real materials. We support this discovery via extensive numerics including the tensor-network, DMRG, and exact diagonalization computations.

Generation of spin-adapted and spin-complete substitution operators for (high spin) open-shell coupled cluster of arbitrary order.

A rigorous generation of spin-adapted (spin-free) substitution operators for high spin (S = Sz) references of an arbitrary substitution order and spin quantum number S is presented. The generated operators lead to linearly independent but non-orthogonal configuration state functions (CSFs) when applied to the reference and span the complete spin space. To incorporate spin completeness, spectating substitutions (e.g., Êiv va) are introduced. The presented procedure utilizes Löwdin's projection operator method of spin eigenfunction generation to ensure spin completeness. The generated operators are explicitly checked for (i) their linear independence and (ii) their spin completeness for up to tenfold substitutions and up to a multiplicity of 2S + 1 = 11. A proof of concept implementation utilizing the generated operators in a coupled cluster (CC) calculation was successfully applied to the high spin states of the boron atom. The results show pure spin states and small effects on the correlation energy compared to spin orbital CC. A comparison to spin-adapted but spin-incomplete CC shows a significant spin-incompleteness error.

Onsager's Scars in Disordered Spin Chains.

We propose a class of nonintegrable quantum spin chains that exhibit quantum many-body scars even in the presence of disorder. With the use of the so-called Onsager symmetry, we construct scarred models for arbitrary spin quantum number S. There are two types of scar states, namely, coherent states associated with an Onsager-algebra element and one-magnon scar states. While both of them are highly excited states, they have area-law entanglement and can be written as a matrix product state. Therefore, they explicitly violate the eigenstate thermalization hypothesis. We also investigate the dynamics of the fidelity and entanglement entropy for several initial states. The results clearly show that the scar states are trapped in a perfectly periodic orbit in the Hilbert subspace and never thermalize, whereas other generic states do rapidly. To our knowledge, our model is the first explicit example of disordered quantum many-body scarred models.

Periodic thermodynamics of the Rabi model with circular polarization for arbitrary spin quantum numbers.

We consider a spin s subjected to both a static and an orthogonally applied oscillating, circularly polarized magnetic field while being coupled to a heat bath and analytically determine the quasistationary distribution of its Floquet-state occupation probabilities for arbitrarily strong driving. This distribution is shown to be Boltzmannian with a quasitemperature which is different from the temperature of the bath and independent of the spin quantum number. We discover a remarkable formal analogy between the quasithermal magnetism of the nonequilibrium steady state of a driven ideal paramagnetic material and the usual thermal paramagnetism. Nonetheless, the response of such a material to the combined fields is predicted to show several unexpected features, even allowing one to turn a paramagnet into a diamagnet under strong driving. Thus, we argue that experimental measurements of this response may provide key paradigms for the emerging field of periodic thermodynamics.

Thermodynamics of the pyrochlore-lattice quantum Heisenberg antiferromagnet

We use the rotation-invariant Green's function method (RGM) and the high-temperature expansion to study the thermodynamic properties of the Heisenberg antiferromagnet on the pyrochlore lattice. We discuss the excitation spectra as well as various thermodynamic quantities, such as spin correlations, uniform susceptibility, specific heat, and static and dynamical structure factors. For the ground state we present RGM data for arbitrary spin quantum numbers $S$. At finite temperatures we focus on the extreme quantum cases $S=\frac{1}{2}$ and 1. We do not find indications for magnetic long-range order for any value of $S$. We discuss the width of the pinch point in the static structure factor in dependence on temperature and spin quantum number. We compare our data with experimental results for the pyrochlore compound ${\mathrm{NaCaNi}}_{2}{\mathrm{F}}_{7}$ ($S=1$). Thus, our results for the dynamical structure factor agree well with the experimentally observed features at 3 ...8 meV for ${\mathrm{NaCaNi}}_{2}{\mathrm{F}}_{7}$. We analyze the static structure factor ${S}_{\mathbf{q}}$ to find regions of maximal ${S}_{\mathbf{q}}$. The high-temperature series of the ${S}_{\mathbf{q}}$ provide a fingerprint of weak order by disorder selection of a collinear spin structure, where (classically) the total spin vanishes on each tetrahedron and neighboring tetrahedra are dephased by $\ensuremath{\pi}$.

Phase transition of anisotropic frustrated Heisenberg model on the square lattice.

We have investigated the J_{1}-J_{2} Heisenberg model with exchange anisotropy on a square lattice and focused on possible AF1-AF2 phase transition below the Néel point and its dependence on the exchange anisotropy, where AF1 and AF2 represent Néel state and collinear state, respectively. We use the double-time Green's-function method and adopt the random-phase approximation. The less the exchange anisotropy, the stronger the quantum fluctuation of the system will be. Both the Néel state and collinear state can exist and have the same Néel temperature for arbitrary anisotropy and spin quantum number S when J_{2}/J_{1}=0.5. Under such parameters, the calculated free energies show that there may occur a first-order phase transition between the Néel state and collinear state for an arbitrary S when anisotropy is not strong.

SPIDYAN, a MATLAB library for simulating pulse EPR experiments with arbitrary waveform excitation

Frequency-swept chirp pulses, created with arbitrary waveform generators (AWGs), can achieve inversion over a range of several hundreds of MHz. Such passage pulses provide defined flip angles and increase sensitivity. The fact that spectra are not excited at once, but single transitions are passed one after another, can cause new effects in established pulse EPR sequences. We developed a MATLAB library for simulation of pulse EPR, which is especially suited for modeling spin dynamics in ultra-wideband (UWB) EPR experiments, but can also be used for other experiments and NMR. At present the command line controlled SPin DYnamics ANalysis (SPIDYAN) package supports one-spin and two-spin systems with arbitrary spin quantum numbers. By providing the program with appropriate spin operators and Hamiltonian matrices any spin system is accessible, with limits set only by available memory and computation time. Any pulse sequence using rectangular and linearly or variable-rate frequency-swept chirp pulses, including phase cycling can be quickly created. To keep track of spin evolution the user can choose from a vast variety of detection operators, including transition selective operators. If relaxation effects can be neglected, the program solves the Liouville-von Neumann equation and propagates spin density matrices. In the other cases SPIDYAN uses the quantum mechanical master equation and Liouvillians for propagation. In order to consider the resonator response function, which on the scale of UWB excitation limits bandwidth, the program includes a simple RLC circuit model. Another subroutine can compute waveforms that, for a given resonator, maintain a constant critical adiabaticity factor over the excitation band. Computational efficiency is enhanced by precomputing propagator lookup tables for the whole set of AWG output levels. The features of the software library are discussed and demonstrated with spin-echo and population transfer simulations.

Magnetic susceptibility of frustrated spin-s J1-J2 quantum Heisenberg magnets: High-temperature expansion and exact diagonalization data

Motivated by recent experiments on low-dimensional frustrated quantum magnets with competing nearest-neighbor exchange coupling J1 and next nearest-neighbor exchange coupling J2 we investigate the magnetic susceptibility of two-dimensional J1-J2 Heisenberg models with arbitrary spin quantum number s. We use exact diagonalization and high- temperature expansion up to order 10 to analyze the influence of the frustration strength J2/J1 and the spin quantum number s on the position and the height of the maximum of the susceptibility. The derived theoretical data can be used to get information on the ratio J2/J1 by comparing with susceptibility measurements on corresponding magnetic compounds.

Quantum and Classical Correlations in the Solid-State NMR Free Induction Decay

The free induction decay (FID) of the transverse magnetization in a dipolar-coupled rigid lattice is a fundamental problem in magnetic resonance and in the theory of many-body systems. As it was shown earlier the FID shapes for the systems of classical magnetic moments and for quantum nuclear spin ones coincide if there are many nearly equivalent nearest neighbors n in a solid lattice. In this paper, we reduce a multispin density matrix of above system to a two-spin matrix. Then we obtain analytic expressions for the mutual information and the quantum and classical parts of correlations at the arbitrary spin quantum number S, in the high-temperature approximation. The time dependence of these functions is expressed via the derivative of the FID shape. To extract classical correlations for S > 1/2 we provide generalized POVM measurement (positive-operator-valued measure) using the basis of spin coherent states. We show that in every pair of spins the portion of quantum correlations changes from 1/2 to 1/(S + 1) when S is growing up, and quantum properties disappear completely only if S → ∞.

Tenth-order high-temperature expansion for the susceptibility and the specific heat of spin-sHeisenberg models with arbitrary exchange patterns: Application to pyrochlore and kagome magnets

We present the high-temperature expansion (HTE) up to 10th order of the specific heat C and the uniform susceptibility \chi for Heisenberg models with arbitrary exchange patterns and arbitrary spin quantum number s. We encode the algorithm in a C++ program which allows to get explicitly the HTE series for concrete Heisenberg models. We apply our algorithm to pyrochlore ferromagnets and kagome antiferromagnets using several Pad\'e approximants for the HTE series. For the pyrochlore ferromagnet we use the HTE data for \chi to estimate the Curie temperature T_c as a function of the spin quantum number s. We find that T_c is smaller than that for the simple cubic lattice, although both lattices have the same coordination number. For the kagome antiferromagnet the influence of the spin quantum number s on the susceptibility as a function of renormalized temperature T/s(s+1) is rather weak for temperatures down to T/s(s+1) \sim 0.3. On the other hand, the specific heat as a function of T/s(s+1) noticeably depends on s. The characteristic maximum in C(T) is monotonously shifted to lower values of T/s(s+1) when increasing s.

Quadrupole relaxation enhancement—application to molecular crystals

A general theory of field dependent spin-lattice relaxation for nuclei of the spin quantum number 1/2 ( 1H, 19F, 13C) caused by dipole–dipole interactions with neighboring quadrupolar nuclei (nuclei possessing a quadrupolar moment) is presented. The theory is valid for arbitrary motional conditions and should be treated as a quadrupolar counterpart of the paramagnetic relaxation enhancement theory. When the energy level splitting of the dipolar spin ( I=1/2) matches one of the transition frequencies of the quadrupolar nuclei one can observe a local enhancement of the dipolar spin relaxation (referred to as “quadrupolar peaks”). To see such effects the dynamics modulating the spin interactions has to be relatively slow. This brings the system beyond the validity range of perturbation approaches and requires the stochastic Liouville equation to be applied. The presented theory describes the quadrupolar relaxation enhancement (QRE) for an arbitrary spin quantum number of the quadrupolar nuclei and includes the asymmetry of the quadrupolar coupling. It has been applied to interpret the shape of magnetization curves (amplitude of 1H magnetization versus magnetic field) for the molecular crystal [C 3N 2H 5] 6[Bi 4Br 18] ([C 3N 2H 5]—imidazolium). The magnetization curves show several dips (local minima) attributed to 1H– 14N quadrupolar relaxation enhancement effects. In addition, as a limiting case a perturbation approach to QRE has been presented and its validity conditions have been discussed.

Thermodynamics of layered Heisenberg magnets with arbitrary spin

We present a spin-rotation-invariant Green-function theory of long- and short-range order in the ferro- and antiferromagnetic Heisenberg model with arbitrary spin quantum number $S$ on a stacked square lattice. The thermodynamic quantities (Curie temperature ${T}_{C}$, N\'eel temperature ${T}_{N}$, specific heat ${C}_{V}$, intralayer, and interlayer correlation lengths) are calculated, where the effects of the interlayer coupling and the $S$ dependence are explored. In addition, exact diagonalizations on finite two-dimensional (2D) lattices with $S\ensuremath{\ge}1$ are performed, and a very good agreement between the results of both approaches is found. For the quasi-2D and isotropic three-dimensional magnets, our theory agrees well with available quantum Monte Carlo and high-temperature series-expansion data. Comparing the quasi-2D $S=1/2$ magnets, we obtain the inequalities ${T}_{N}>{T}_{C}$ and, for small enough interlayer couplings, ${T}_{N}<{T}_{C}$. The results for ${C}_{V}$ and the intralayer correlation length are compared to experiments on the quasi-2D antiferromagnets ${\text{Zn}}_{2}\text{VO}{({\text{PO}}_{4})}_{2}$ with $S=1/2$ and ${\text{La}}_{2}{\text{NiO}}_{4}$ with $S=1$, respectively.

Field-dependent nuclear relaxation of spins 1/2 induced by dipole–dipole couplings to quadrupole spins: LaF 3 crystals as an example

A general theory of spin–lattice nuclear relaxation of spins I = 1/2 caused by dipole–dipole couplings to quadrupole spins S ⩾ 1, characterized by a non-zero averaged (static) quadrupole coupling, is presented. In multispin systems containing quadrupolar and dipolar nuclei, transitions of spins 1/2 leading to their relaxation are associated through dipole–dipole couplings with certain transitions of quadrupole spins. The averaged quadrupole coupling attributes to the energy level structure of the quadrupole spin and influences in this manner relaxation processes of the spin 1/2. Typically, quadrupole spins exhibit also a complex multiexponential relaxation sensed by the dipolar spin as an additional modulation of the mutual dipole–dipole coupling. The proposed model includes both effects and is valid for an arbitrary magnetic field and an arbitrary quadrupole spin quantum number. The theory is applied to interpret fluorine relaxation profiles in LaF 3 ionic crystals. The obtained results are compared with predictions of the ‘classical’ Solomon relaxation theory.

Magnetic susceptibilities of diluted magnetic semiconductors and anomalous Hall-voltage noise

The carrier-spin and impurity-spin densities in diluted magnetic semiconductors are considered using a semiclassical approach. Equations of motions for the spin densities and the carrier-spin current density in the paramagnetic phase are derived, exhibiting their coupled diffusive dynamics. The dynamical spin susceptibilities are obtained from these equations. The theory holds for p-type and n-type semiconductors doped with magnetic ions of arbitrary spin quantum number. Spin-orbit coupling in the valence band is shown to lead to anisotropic spin diffusion and to a suppression of the Curie temperature in p-type materials. As an application we derive the Hall-voltage noise in the paramagnetic phase. This quantity is critically enhanced close to the Curie temperature due to the contribution from the anomalous Hall effect.

Linear energy bounds for Heisenberg spin systems

Recently obtained results on linear energy bounds are generalized to arbitrary spin quantum numbers and coupling schemes. Thereby the class of so-called independent magnon states, for which the relative ground-state property can be rigorously established, is considerably enlarged. We still require that the matrix of exchange parameters has constant row sums, but this can be achieved by means of a suitable gauge and need not be considered as a physical restriction.

On the ground state of Ising models with arbitrary spin quantum number

The ground state of a class of Ising models with site dependent arbitrary spin quantum number is shown to be restricted to ± S i MAX state where S i MAX is the spin quantum number at the site i.

Spin parity effect resulting from time-inversion symmetry

The spin parity effect resulting from time-inversion symmetry is derived by generalizing the pure quantum theory of the effect resulting from finite-fold axial-rotation symmetry. The quantum system concerned may consist of either one spin or many spins with arbitrary spin quantum numbers and the states involved may be any pair of eigenstates of spin operator projected on arbitrary axis with their eigenvalues having opposite signs. The two kinds of spin parity effect, resulting from time-inversion symmetry and from axial-rotation symmetry, are in general not equivalent but complementary to each other.