Linear Spin Chains Research Articles

Transition metal centers with anharmonic elastic interactions in one-dimensional spin-crossover solids

An exact examination of a one-dimensional (1D) anharmonic elastic model describing the thermodynamic properties of spin-crossover (SCO) solids is presented. This elastic model incorporates both intra-chain and mean field inter-chain interactions, providing a more understanding of 1D SCO materials. Numerical findings for a linear chain of spins unveil intriguing features, such as a two-step-like transition. Extending the model to interacting chains leads to an additional term in the effective ligand field of the system’s Hamiltonian, thereby enhancing its predictive power by accounting for more elastic constants of interactions. This augmentation reinforces the model’s validity in capturing the spin-crossover phenomenon. Numerical results in this extended framework elucidate clearer two-step-like spin crossover transitions, with the HS_LS fraction of spins approaching maximum values at equilibrium temperature Teq.

Coherent two-dimensional THz magnetic resonance spectroscopies for molecular magnets: Analysis of Dzyaloshinskii-Moriya interaction.

To investigate the novel quantum dynamic behaviors of magnetic materials that arise from complex spin-spin interactions, it is necessary to probe the magnetic response at a speed greater than the spin-relaxation and dephasing processes. Recently developed two-dimensional (2D) terahertz magnetic resonance (THz-MR) spectroscopy techniques use the magnetic components of laser pulses, and this allows investigation of the details of the ultrafast dynamics of spin systems. For such investigations, quantum treatment-not only of the spin system itself but also of the environment surrounding the spin system-is important. In our method, based on the theory of multidimensional optical spectroscopy, we formulate nonlinear THz-MR spectra using an approach based on the numerically rigorous hierarchical equations of motion. We conduct numerical calculations of both linear (1D) and 2D THz-MR spectra for a linear chiral spin chain. The pitch and direction of chirality (clockwise or anticlockwise) are determined by the strength and signof the Dzyaloshinskii-Moriya interaction (DMI). We show that not only the strength but also the signof the DMI can be evaluated through the use of 2D THz-MR spectroscopic measurements, while 1D measurements allow us to determine only the strength.

From classical to quantum stochastic processes.

In this paper, we construct quantum analogs starting from classical stochastic processes, by replacing random "which path" decisions with superpositions of all paths. This procedure typically leads to nonunitary quantum evolution, where coherences are continuously generated and destroyed. In spite of their transient nature, these coherences can change the scaling behavior of classical observables. Using the zero temperature Glauber dynamics in a linear Ising spin chain, we find quantum analogs with different domain growth exponents. In some cases, this exponent is even smaller than for the original classical process, which means that coherence can play an important role to speed up the relaxation process.

Thermal decay of magnons in MnF2

It is shown that for nearly all magnets only two types of magnon dispersions occur. For the first group of magnets, magnon dispersions are as for the linear spin chain (a sine-function of wave-vector for antiferromagnets but a sine-function squared for ferromagnets), quite independent of the lattice structure. Since the observed magnon dispersions are specific to the individual domain, the identical magnon dispersions show that the domains of all magnets are very similar in that all spins are perfectly aligned along the domain axis. The linear chain dispersion is explained by a strong coupling of the spins to the one-dimensionally ordered boson field in each domain. The bosons were identified as essentially magnetic dipole radiation generated essentially through stimulated emission by the precessing spins. In the ordered state of the boson field, the bosons are in a coherent and one-dimensional state, like the radiation field of a LASER. For the second group of magnets, magnon dispersions exhibit an analytical crossover from a power function of wave vector (∼qx) plus a gap to the dispersion of the linear spin chain. Occasionally it turns out to be necessary to add a phase shift in the argument of the sine function. The functionality of the local exchange interactions is restricted to determine the relative orientation of neighboring spins, i.e. to decide between the sine-function and the sine-function squared and to define the absolute values of the magnon energies. As a consequence, instead by exchange interactions, magnon dispersions have to be characterized by four parameters: the gap, the pre-factor of the ∼qx power function, the amplitude of the sine-function and the phase-shift in the argument of the sine function. For MnF2 the temperature dependence of these four parameters is evaluated, as far as experimental data are available. The exponent x is the only parameter that is characteristic of the dynamic dimensionality of the system and is temperature-independent. For MnF2 with a half-integer spin of S = 5/2 and a one-dimensional dynamic symmetry x = 3/2.

1H NMR in a quasi-one-dimensional zig-zag spin chain of hambergite, Be2BO3(OH)

One-dimensional spin chains are important testbeds for the investigation of quantum many-body systems. Here, we introduce a new quasi-one-dimensional chain of spin-1/2 1H nuclei in hambergite (Be2BO3(OH)) crystal. The distinctive feature of hambergite is the zig-zag arrangement of the spins in the chains compared to the systems reported earlier. A single crystal of hambergite was investigated experimentally using 1H nuclear magnetic resonance at different orientations in the external magnetic field. The experimental spectra were compared with the Van Vleck’s second moments calculated from the structural data. For the orientation of the chain along the external magnetic field the resonance resembles that of the linear spin chains and the calculated contributions to the second moment from spins within one chain of hambergite amount to 96%. For other orientations the spectra demonstrate features specific to the zig-zag chain. Our results show that 1H nuclei in hambergite crystals could serve a good model of a well-isolated zig-zag spin chain.

Universality in the temperature dependence of the heat capacity of magnetic solids

It is shown that the temperature dependence of the heat capacity of the magnetic solids can completely be described by a sequence of universal power functions of temperature. Characteristic for universality is that each power function holds over a finite temperature range and has a rational exponent. The analytical change from one to the adjacent power function is a typical crossover event. Universality reveals that the temperature dependence of the heat capacity is determined by a boson field whereas the absolute values are given by all magnetic and non-magnetic inter-atomic interactions. Universality for temperatures outside the critical range at Tc, i.e. for temperatures for which the phonons dominate the heat capacity has to be characterized as non-intrinsic, arising from interactions of the phonons with the bosons of the continuous magnetic medium. As we have shown earlier, the bosons of the continuous magnetic solid are essentially magnetic dipole radiation generated via stimulated emission by the precessing spins. We have called them Goldstone bosons. The interactions of the Goldstone bosons with the magnons modify the wave-vector dependence of the magnons. For cubic crystals the dispersions along [ζ, 0, 0] direction are essentially as for the linear spin chain, i.e. one-dimensional. As the different rational exponent values in the temperature power function of the heat capacity show, there exists a number of distinct modes of interaction between the Goldstone boson field and the phonons. The actual exponent depends additionally on the proportion between the magnetic and the non-magnetic energy contributions and therefore changes with temperature. The observed exponents are, however, difficult to interpret.

Bosonic and magnonic magnon dispersions

The analytical crossover events in the magnon dispersion relations are discussed. As the available experimental data show, only a few types of magnon dispersion relations are observed for all spin- and lattice-structures. When the magnon spectrum exhibits a finite energy gap at q = 0 the dispersion consists of two q-sections with different functions of wave vector. Such analytical changes indicate that two mechanisms dominate the dispersion relation alternately. For magnon energies of larger than the gap, the dispersion exhibits over a finite q-range a single qx power function of wave vector. Within the experimental error limits, the exponent x is a rational number and is independent of the spin structure, i.e. universal, but depends on whether the spin quantum number is integer or half-integer. As we now know, this is the typical indication that the qx function is determined by the bosons of the continuous magnetic medium. The qx function holds up to the crossover to a sine-function of wave vector for antiferromagnets but to a sine-function squared for ferromagnets. One therefore has to distinguish between the bosonic part of the magnon dispersion relation, at small q-values, and the magnonic part at large q-values. As is well known, sine functions of wave vector are the dispersion of the linear spin chain. Since the linear spin chain is not ordered at any finite temperature it follows that the observed magnons are not indicative of a long-range magnetic order. From the fact that no principal change of the magnon dispersions occurs upon crossing the magnetic ordering temperature it follows that the exchange interactions are not involved directly in the magnetic ordering process. The long-range ordered system is the boson field. At the critical temperature, the boson field orders. Ordering of the boson field is associated with the formation of domains and with the emergence of a magnon gap. In the ordered state, propagation of the bosons is restricted to the few different domain axes. The dimensionality of the ordered boson field can be recognized from the number of the in-equivalent domain orientations. For all lattice structures the boson field within each magnetic domain is perfectly one-dimensional, and aligns all spins parallel. This is the origin of the linear chain dispersion. The mass-less bosons are, however, not visible for neutrons. If the magnon-boson interaction is strong, the q-range of the sine functions is small. It then proves necessary to add a phenomenological phase shift in the argument of the sine function. As a consequence, magnon dispersions cannot be understood considering exchange interactions alone.

Linear spin chains in paramagnetic and in ordered bulk magnets

Published magnon dispersion data are discussed, showing that magnon dispersions of bulk magnets are essentially as for linear spin chains, i.e. one-dimensional. This applies to all propagation directions in cubic magnets, in layered two-dimensional magnets and in axial magnets. Propagation of magnons seems to be restricted to the individual magnetic domain. Within each domain the spin structure is collinear (one-dimensional). Perfect collinear spin structures up to ordering temperature, in spite of local exchange anisotropies, cannot be understood on the basis of atomistic models. As we have argued earlier, collinear spin structures are caused by a one-dimensional boson field. The bosons are essentially magnetic dipole radiation emitted by the precessing spins. Since emission is by stimulated emission, perfect one-dimensional boson fields result. The one-dimensional boson fields are considered as responsible for the magnon dispersions of linear chain type. However, magnon dispersions follow sine function of wave vector (for antiferromagnets) and sine function squared (for ferromagnets) not perfectly. At low q-values, magnon dispersions often deflect from sine function (or from sine function squared) towards a magnon gap at q≡0. This part of the magnon dispersion is excellently described by gap energy plus qx power function of wave vector. The two analytically different sections of wave vector have different temperature dependencies. The gap closes for T→Tc, while the sine function persists nearly unperturbed into the paramagnetic phase. In contrast to the sine functions, the qx term is universal that is, independent of spin structure but depends on whether the spin is integer or half-integer. It is concluded, that for all q-values, the functional wave-vector dependence of the magnon dispersions (not their absolute energies) is determined by boson fields with well defined dimensionalities.

Langevin dynamics simulation of a one-dimensional linear spin chain with long-range interactions

In this paper we study the critical behavior of a simple one-dimensional rotor spin in the form of a linear chain with long-range interactions, using the mean field Langevin dynamics approach and in the presence of fluctuations added by a heat bath. We have computed the specific heat, the magnetic susceptibility, the Binder fourth-order cumulant, and the magnetization, and then we have calculated the critical exponents using finite-size scaling. In addition, we provide a relation between the thermal bath temperature and the temperature of the system. Our results confirm the existence of a second-order critical temperature in the one-dimensional chain of spins with long-range interaction.

Thermal stability in exchange-spring chains of spins

Thermal stability and switching behaviour have been compared in pure-hard and soft–hard Heisenberg linear spin chains of the same total length and equal magnetic parameters (except for magnetic anisotropy) with the anisotropy axis and external magnetic field parallel to the chain direction. The zero-temperature energy barriers and finite-temperature transition rates between remanent equilibrium states have been calculated by utilizing the string method and the forward flux sampling (FFS) method, respectively. Depending on the assumed interfaces, the FFS method could in fact fail to correctly sample the characteristic transition paths at interfaces at which these paths have probabilities much lower than those associated with other non-characteristic transition paths. This can especially occur in the case of the asymmetric energy landscapes and multiple asymmetric minimum energy paths (MEPs) of soft–hard systems. Therefore, a proper interface definition is needed in order to deduce the correct transition rates. In particular, we show that the thermal switching of soft–hard chains starting in the soft or in the hard part turns out to occur with an equal rate provided that the interfaces of the FFS method are defined on the basis of the corresponding zero-temperature MEPs. The thermal stability of a soft–hard chain in the remanent equilibrium state could be to some extent lower with respect to that of a pure-hard chain, due to the shorter hard-part length crossed by the domain wall formed in the chain and also to the related slightly smaller energy barrier. However, its switching field at zero temperature is verified to be widely lower than that of the pure-hard chain. Analytical expressions of switching fields and energy barriers have been deduced in various cases.

Time-optimized quantum QFT gate in an Ising coupling system

Quantum Fourier transform (QFT) is a quantum analogue of the classical discrete Fourier transform. It is a fundamental quantum gate in quantum algorithms which has an exponential advantage over the classical computation and has been excessively studied. Normally, an n-qubit quantum Fourier transform could be resolved into the tensor product of n single-qubit operations, and each operation could be implemented by a Hadamard gate and a controlled phase gate. Then the complexity of an n-qubit QFT is of order O(n2). To reduce the complexity of quantum operations, optimal control (OC) method has recently been used successfully to find the minimum time for implementing a quantum operation. Up to now, two types of quantum optimal control methods have been presented, i.e. analytical and numerical methods. The analytical approach is to change the problem of efficient synthesis of unitary transformations into the geometrical one of finding the shortest paths. Numerical optimal control procedures are based on the gradient methods (GRAPE, Gradient Ascent Pulse Engineering) and Krotov methods. Notable application mainly focus on nuclear magnetic resonance fields, including imaging, liquid-state NMR, solid-state NMR, and NMR quantum computation. One obvious advantage of optimal control NMR quantum computation is that the OC unitary evolution transformation pulse sequences are normally shorter than the conventional corresponding ones. Here we use the optimal control method to find the minimum duration for implementing QFT quantum gate. A linear spin chain with nearest-neighbor Ising interaction is used to find the optimization. And the optimized pulse sequence is experimentally demonstrated on an NMR quantum information processor. By using optimal control method with numerical calculation, a three-qubit QFT in an indirect-linear-coupling chain system is optimized. The duration of the OC QFT is obviously shorter than that of conventional approaches. The OC pulse sequence has been experimentally implemented on a liquid-state NMR spectrometer. To verify the optimally controlled pulse sequence for the three-qubit QFT, different initial states are assumed. The accuracy of the OC pulse sequence could be demonstrated by the consistency of theoretical simulation spectra with the experimental results. The good consistency between the simulation and the experimental spectra demonstrates that the OC QFT is of high fidelity.

Mixing time and local exponential ergodicity of the East-like process in ℤ d

The East process, a well known reversible linear chain of spins, represents the prototype of a general class of interacting particle systems with constraints modeling the dynamics of real glasses. In this paper we consider a generalization of the East process living in the d-dimensional lattice and we establish new progresses on the out-of-equilibrium behavior. Despite the fact that convergence to the stationary reversible measure in the uniform norm cannot hold because of the presence of blocked configurations, we prove a form of (local) exponential ergodicity when the initial distribution is different from the stationary one. We also establish that the mixing time in a finite box grows linearly in the side of the box.

Time-optimized quantum gates on linear three-qubit systems with indirect Ising coupling

Optimal control based numerical algorithms were used to explore the minimum duration of a variety of quantum gates between the first and the last spins in a linear spin chain with identical next neighbor Ising couplings. The true SWAP gate, the planar SWAP gate and the generation of an effective direct Ising coupling between the first and the third spin were studied. Pulse sequences with shortest durations were experimentally implemented in a three-qubit NMR system. The experimental results show the accuracy and feasibility of the optimized quantum gates.

Diamond as a Solid State Quantum Computer with a Linear Chain of Nuclear Spins System

By removing a 12C atom from the tetrahedral configuration of the diamond, replacing it by a 13C atom, and repeating this in a linear direction, it is possible to have a linear chain of nuclear spins one half and to build a solid state quantum computer. One qubit rotation, controlled-not (CNOT) and controlled-controlled-not (CCNOT) quantum gates are obtained immediately from this configuration. CNOT and CCNOT quantum gates are used to determined the design parameters of this quantum computer.

Decay of spin coherences in one-dimensional spin systems

Strategies to protect multi-qubit states against decoherence are difficult to formulate because of their complex many-body dynamics. A better knowledge of the decay dynamics would help in the construction of dynamical decoupling control schemes. Here we use solid-state nuclear magnetic resonance techniques to experimentally investigate decay of coherent multi-spin states in linear spin chains. Leveraging on the quasi-one-dimension geometry of fluorapatite crystal spin systems, we can gain a deeper insight on the multi-spin states created by the coherent evolution, and their subsequent decay, than it is possible in three-dimensional (3D) systems. We are then able to formulate an analytical model that captures the key features of the decay. We can thus compare the decoherence behavior for different initial states of the spin chain and link their decay rate to the state characteristics, in particular their coherence and long-range correlation among spins. Our experimental and theoretical study shows that the spin chains undergo a rich dynamics, with a slower decay rate than for the 3D case, and thus might be more amenable to decoupling techniques.

Rates for irreversible Gibbsian Ising models

Dynamics under which a system of Ising spins relaxes to a stationary state with Boltzmann–Gibbs measure and which do not fulfil the condition of detailed balance are irreversible and asymmetric. We revisit the problem of the determination of rates yielding such a stationary state for models with single-spin flip dynamics. We add some supplementary material to this study and confirm that Gibbsian irreversible Ising models exist for one- and two-dimensional lattices, but not for the three-dimensional cubic lattice. We also analyse asymmetric Gibbsian dynamics in the limit of infinite temperature. We finally revisit the case of a linear chain of spins under asymmetric conserved dynamics.

Multiple-spin coherence transfer in linear Ising spin chains and beyond: Numerically optimized pulses and experiments

We study multiple-spin coherence transfers in linear Ising spin chains with nearest neighbor couplings. These constitute a model for efficient information transfers in future quantum computing devices and for many multi-dimensional experiments for the assignment of complex spectra in nuclear magnetic resonance spectroscopy. We complement prior analytic techniques for multiple-spin coherence transfers with a systematic numerical study where we obtain strong evidence that a certain analytically-motivated family of restricted controls is sufficient for time-optimality. In the case of a linear three-spin system, additional evidence suggests that prior analytic pulse sequences using this family of restricted controls are time-optimal even for arbitrary local controls. In addition, we compare the pulse sequences for linear Ising spin chains to pulse sequences for more realistic spin systems with additional long-range couplings between non-adjacent spins. We experimentally implement the derived pulse sequences in three and four spin systems and demonstrate that they are applicable in realistic settings under relaxation and experimental imperfections-in particular-by deriving broadband pulse sequences which are robust with respect to frequency offsets.

An Exactly Solvable Spin Chain Related to Hahn Polynomials

We study a linear spin chain which was originally introduced by Shi et al. [Phys. Rev. A 71 (2005), 032309, 5 pages], for which the coupling strength contains a parameter $\alpha$ and depends on the parity of the chain site. Extending the model by a second parameter $\beta$, it is shown that the single fermion eigenstates of the Hamiltonian can be computed in explicit form. The components of these eigenvectors turn out to be Hahn polynomials with parameters $(\alpha,\beta)$ and $(\alpha+1,\beta-1)$. The construction of the eigenvectors relies on two new difference equations for Hahn polynomials. The explicit knowledge of the eigenstates leads to a closed form expression for the correlation function of the spin chain. We also discuss some aspects of a $q$-extension of this model.

Time-optimal CNOT between indirectly coupled qubits in a linear Ising chain**Dedicated to Alberto Carlini's mother, Giovanna Novaro Mascarello.

We give analytical solutions for the time-optimal synthesis of entangling gates between indirectly coupled qubits 1 and 3 in a linear spin chain of three qubits subject to an Ising Hamiltonian interaction with a symmetric coupling J plus a local magnetic field acting on the intermediate qubit. The energy available is fixed, but we relax the standard assumption of instantaneous unitary operations acting on single qubits. The time required for performing an entangling gate which is equivalent, modulo local unitary operations, to the CNOT(1, 3) between the indirectly coupled qubits 1 and 3 is , i.e. faster than a previous estimate based on a similar Hamiltonian and the assumption of local unitaries with zero time cost. Furthermore, performing a simple Walsh–Hadamard rotation in the Hilbert space of qubit 3 shows that the time-optimal synthesis of the CNOT±(1, 3) (which acts as the identity when the control qubit 1 is in the state |0⟩, while if the control qubit is in the state |1⟩, the target qubit 3 is flipped as | ± ⟩ → |∓⟩) also requires the same time T.

Quantum communication and state transfer in spin chains

We investigate the time evolution of a single spin excitation state in certain linear spin chains, as a model for quantum communication. We consider first the simplest possible spin chain, where the spin chain data (the nearest neighbour interaction strengths and the magnetic field strengths) are constant throughout the chain. The time evolution of a single spin state is determined, and this time evolution is illustrated by means of an animation. Some years ago it was discovered that when the spin chain data are of a special form so-called perfect state transfer takes place. These special spin chain data can be linked to the Jacobi matrix entries of Krawtchouk polynomials or dual Hahn polynomials. We discuss here the case related to Krawtchouk polynomials, and illustrate the possibility of perfect state transfer by an animation showing the time evolution of the spin chain from an initial single spin state. Very recently, these ideas were extended to discrete orthogonal polynomials of q-hypergeometric type. Here, a remarkable result is a new analytic model where perfect state transfer is achieved: this is when the spin chain data are related to the Jacobi matrix of q-Krawtchouk polynomials. This case is discussed here, and again illustrated by means of an animation.