Spin Chain Model Research Articles

Gustafson integrals for spin magnet

It was observed recently that multidimensional Mellin–Barnes integrals (Gustafson’s integrals) arise naturally in studies of SL(2, R) spin chain models. We extend this analysis to noncompact spin magnets and obtain an integral which generalizes Gustafson’s integrals to the complex case.

Spin chains and Gustafson’s integrals

Gustafson’s integrals are multidimensional generalizations of the classical Mellin–Barnes integrals. We show that some of these integrals arise from relations between matrix elements in Sklyanin’s representation of separated variables in spin chain models. We also present several new integrals.

The gap of the area-weighted Motzkin spin chain is exponentially small

We prove that the energy gap of the model proposed by Zhang et al (2016 arXiv:1606.07795) is exponentially small in the square of the system size. In Movassagh and Shor (2016 Proc. Natl Acad. Sci. USA) a class of exactly solvable quantum spin chain models was proposed that have integer spins (s), with a nearest neighbors Hamiltonian, and a unique ground state. The ground state can be seen as a uniform superposition of all s-colored Motzkin walks. The half-chain entanglement entropy provably violates the area law by a square root factor in the system’s size (∼) for s > 1. For s = 1, the violation is logarithmic (Bravyi et al 2012 Phys. Rev. Lett. 109 207202). Moreover in Movassagh and Shor (2016 Proc. Natl Acad. Sci. USA) it was proved that the gap vanishes polynomially and is O(n−c) with .Recently, a deformation of Movassagh and Shor (2016 Proc. Natl Acad. Sci. USA), which we call ‘weighted Motzkin quantum spin chain’ was proposed Zhang et al (2016 arXiv:1606.07795). This model has a unique ground state that is a superposition of the s-colored Motzkin walks weighted by with t > 1. The most surprising feature of this model is that it violates the area law by a factor of n. Here we prove that the gap of this model is upper bounded by for t > 1 and s > 1.

Spin-chain model for strongly interacting one-dimensional Bose-Fermi mixtures

Strongly interacting one-dimensional (1D) Bose--Fermi mixtures form a tunable $XXZ$ spin chain. Within the spin-chain model developed here, all properties of these systems can be calculated from states representing the ordering of the bosons and fermions within the atom chain and from the ground-state wave function of spinless noninteracting fermions. We validate the model by means of an exact diagonalization of the full few-body Hamiltonian in the strongly interacting regime. Using the model, we explore the phase diagram of the atom chain as a function of the boson-boson (BB) and boson-fermion (BF) interaction strengths and calculate the densities, momentum distributions, and trap-level occupancies for up to 17 particles. In particular, we find antiferromagnetic (AFM) and ferromagnetic (FM) order and a demixing of the bosons and fermions in certain interaction regimes. We find, however, no demixing for equally strong BB and BF interactions, in agreement with earlier calculations that combined the Bethe ansatz with a local-density approximation.

Quantum Coherence and Correlation in Spin Models with Dzyaloshinskii-Moriya Interaction

We study quantum coherence and quantum correlation for detecting quantum phase transition (QPT) by means of quantum renormalization group (QRG) in various spin chain models with Dzyaloshinskii-Moriya (DM) interaction, including XXZ model with DM interaction, Ising model with DM interaction and XY model with DM interaction. It is found that through enough QRG iterations, l 1 norm quantum coherence and one-norm geometric quantum discord can effectively characterize QPT. We also discuss the effect of DM interaction and anisotropy on quantum coherence and quantum correlation.

Series-expansion thermal tensor network approach for quantum lattice models

In this work we propose a series-expansion thermal tensor network (SETTN) approach for efficient simulations of quantum lattice models. This continuous-time SETTN method is based on the numerically exact Taylor series expansion of equilibrium density operator $e^{-\beta H}$ (with $H$ the total Hamiltonian and $\beta$ the imaginary time), and is thus Trotter-error free. We discover, through simulating XXZ spin chain and square-lattice quantum Ising models, that not only the Hamiltonian $H$, but also its powers $H^n$, can be efficiently expressed as matrix product operators, which enables us to calculate with high precision the equilibrium and dynamical properties of quantum lattice models at finite temperatures. Our SETTN method provides an alternative to conventional Trotter-Suzuki renormalization group (RG) approaches, and achieves an unprecedented standard of thermal RG simulations in terms of accuracy and flexibility.

How to fold a spin chain: Integrable boundaries of the Heisenberg XXX and Inozemtsev hyperbolic models

We present a general method of folding an integrable spin chain, defined on a line, to obtain an integrable open spin chain, defined on a half-line. We illustrate our method through two fundamental models with sl2 Lie algebra symmetry: the Heisenberg XXX and the Inozemtsev hyperbolic spin chains. We obtain new long-range boundary Hamiltonians and demonstrate that they exhibit Yangian symmetries, thus ensuring integrability of the models we obtain. The method presented provides a “bottom-up” approach for constructing integrable boundaries and can be applied to any spin chain model.

Total correlations of the diagonal ensemble as a generic indicator for ergodicity breaking in quantum systems

The diagonal ensemble is the infinite time average of a quantum state following unitary dynamics. In analogy to the time average of a classical phase space dynamics, it is intimately related to the ergodic properties of the quantum system giving information on the spreading of the initial state in the eigenstates of the Hamiltonian. In this work we apply a concept from quantum information, known as total correlations, to the diagonal ensemble. Forming an upper-bound on the multipartite entanglement, it quantifies the combination of both classical and quantum correlations in a mixed state. We generalize the total correlations of the diagonal ensemble to more general $\alpha$-Renyi entropies and focus on the the cases $\alpha=1$ and $\alpha=2$ with further numerical extensions in mind. Here we show that the total correlations of the diagonal ensemble is a generic indicator of ergodicity breaking, displaying a sub-extensive behaviour when the system is ergodic. We demonstrate this by investigating its scaling in a range of spin chain models focusing not only on the cases of integrability breaking but also emphasize its role in understanding the transition from an ergodic to a many body localized phase in systems with disorder or quasi-periodicity.

Hamiltonian identifiability assisted by a single-probe measurement

We study the Hamiltonian identifiability of a many-body spin-1/2 system assisted by the measure- ment on a single quantum probe based on the eigensystem realization algorithm (ERA) approach employed in Phys. Rev. Lett. 113, 080401 (2014). We demonstrate a potential application of Gr\"obner basis to the identifiability test of the Hamiltonian, and provide the necessary experimental resources, such as the lower bound in the number of the required measurement points, the upper bound in total required evolution time, and thus the total measurement time. Focusing on the examples of the identifiability in the spin chain model with nearest-neighboring interaction, we classify the spin-chain Hamiltonian based on its identifiability, and provide the control protocols to engineer the nonidentifiable Hamiltonian to be an identifiable Hamiltonian.

Topologically protected localised states in spin chains

We consider spin chain families inspired by the Su, Schrieffer and Hegger (SSH) model. We demonstrate explicitly the topologically induced spatial localisation of quantum states in our systems. We present detailed investigations of the effects of random noise, showing that these topologically protected states are very robust against this type of perturbation. Systems with such topological robustness are clearly good candidates for quantum information tasks and we discuss some potential applications. Thus, we present interesting spin chain models which show promising applications for quantum devices.

Dynamical realization of magnetic states in a strongly interacting Bose mixture

We describe the dynamical preparation of magnetic states in a strongly interacting two-component Bose gas in a harmonic trap. By mapping this system to an effective spin-chain model, we obtain the dynamical spin densities and the fidelities for a few-body system. We show that the spatial profiles transit between ferromagnetic and antiferromagnetic states as the intraspecies interaction parameter is slowly increased.

Out‐of‐time‐order correlations in many‐body localized and thermal phases

We use the out‐of‐time‐order (OTO) correlators to study the slow dynamics in the many‐body localized (MBL) phase. We investigate OTO correlators in the effective (“l‐bit”) model of the MBL phase, and show that their amplitudes after disorder averaging approach their long‐time limits as power‐laws of time. This power‐law dynamics is due to dephasing caused by interactions between the localized operators that fall off exponentially with distance. The long‐time limits of the OTO correlators are determined by the overlaps of the local operators with the conserved l‐bits. We demonstrate numerically our results in the effective model and three other more “realistic” spin chain models. Furthermore, we extend our calculations to the thermal phase and find that for a time‐independent Hamiltonian, the OTO correlators also appear to vanish as a power law at long time, perhaps due to coupling to conserved densities. In contrast, we find that in the thermal phase of a Floquet spin model with no conserved densities the OTO correlator decays exponentially at long times. image

A Floquet model for the many-body localization transition

The nature of the dynamical quantum phase transition between the many-body localized (MBL) phase and the thermal phase remains an open question, and one line of attack on this problem is to explore this transition numerically in finite-size systems. To maximize the contrast between the MBL phase and the thermal phase in such finite-size systems, we argue one should choose a Floquet model with no local conservation laws and rapid thermalization to "infinite temperature" in the thermal phase. Here we introduce and explore such a Floquet spin chain model, and show that standard diagnostics of the MBL-to-thermal transition behave well in this model even at modest sizes. We also introduce a physically motivated spacetime correlation function which peaks at the transition in the Floquet model, but is strongly affected by conservation laws in Hamiltonian models.

Thermodynamic limit and boundary energy of the su(3) spin chain with non-diagonal boundary fields

We investigate the thermodynamic limit of the su(n)-invariant spin chain models with unparallel boundary fields. It is found that the contribution of the inhomogeneous term in the associated T–Q relation to the ground state energy does vanish in the thermodynamic limit. This fact allows us to calculate the boundary energy of the system. Taking the su(2) (or the XXX) spin chain and the su(3) spin chain as concrete examples, we have studied the corresponding boundary energies of the models. The method used in this paper can be generalized to study the thermodynamic properties and boundary energy of other high rank models with non-diagonal boundary fields.

Corrigendum: Exactly solvable spin chain models corresponding to BDI class of topological superconductors.

Scientific Reports 6: Article number: 32720; published online: 06 September 2016; updated: 19 October 2016

Bose-Fermi mapping and a multibranch spin-chain model for strongly interacting quantum gases in one dimension: Dynamics and collective excitations

We show that the wave function of a one dimensional spinor gas with contact $s$-wave interaction, either bosonic or fermionic, can be mapped to the direct product of the wave function of a spinless Fermi gas with short-range $p$-wave interaction and that of a spin system governed by spin parity projection operators. Applying this mapping to strongly interacting spinor gases, we obtain a generalized spin chain model that captures both the static and dynamics properties of the system. Using this spin chain model, we investigate the breathing mode frequency and the quench dynamics of strongly interacting harmonically trapped spinor gases.

Exactly solvable spin chain models corresponding to BDI class of topological superconductors.

We present an exactly solvable extension of the quantum XY chain with longer range multi-spin interactions. Topological phase transitions of the model are classified in terms of the number of Majorana zero modes, nM which are in turn related to an integer winding number, nW. The present class of exactly solvable models belong to the BDI class in the Altland-Zirnbauer classification of topological superconductors. We show that time reversal symmetry of the spin variables translates into a sliding particle-hole (PH) transformation in the language of Jordan-Wigner fermions – a PH transformation followed by a π shift in the wave vector which we call it the πPH. Presence of πPH symmetry restricts the nW (nM) of time-reversal symmetric extensions of XY to odd (even) integers. The πPH operator may serve in further detailed classification of topological superconductors in higher dimensions as well.

Coherence Trade-off relations in Multipartite systems

Quantum coherence is investigated using a new measure with metric properties and entropic nature and decomposed into local and intrinsic contributions. The trade-off relation between these contributions as well as their distribution properties are studied for simple tripartite systems and the more complex spin chain model. We find that the coherence changes its nature with respect to the parameters of the quantum state under investigation.

Momentum distributions and numerical methods for strongly interacting one-dimensional spinor gases

One-dimensional spinor gases with strong $\ensuremath{\delta}$ interaction fermionize and form a spin chain. The spatial degrees of freedom of this atom chain can be described by a mapping to spinless noninteracting fermions and the spin degrees of freedom are described by a spin-chain model with nearest-neighbor interactions. Here, we compute momentum and occupation-number distributions of up to 16 strongly interacting spinor fermions and bosons as a function of their spin imbalance, the strength of an externally applied magnetic field gradient, the length of their spin, and for different excited states of the multiplet. We show that the ground-state momentum distributions resemble those of the corresponding noninteracting systems, apart from flat background distributions, which extend to high momenta. Moreover, we show that the spin order of the spin chain---in particular antiferromagnetic spin order---may be deduced from the momentum and occupation-number distributions of the system. Finally, we present efficient numerical methods for the calculation of the single-particle densities and one-body density matrix elements and of the local exchange coefficients of the spin chain for large systems containing more than 20 strongly interacting particles in arbitrary confining potentials.

An analytic spin chain model with fractional revival

New analytic spin chains with fractional revival are introduced. Their nearest-neighbor couplings and local magnetic fields correspond to the recurrence coefficients of para-Racah polynomials which are orthogonal on quadratic bi-lattices. These models generalize the spin chain associated to the dual-Hahn polynomials. Instances where perfect state transfer also occurs are identified.