XXZ Hamiltonian Research Articles

Observation of anisotropy-independent magnetization dynamics in spatially disordered Heisenberg spin systems

An important step towards a comprehensive understanding of far-from-equilibrium dynamics of quantum many-body systems is the identification of unifying features that are independent of microscopic details of the system. We experimentally observe such robust features in the magnetization relaxation dynamics of disordered Heisenberg XX, XXZ, and Ising Hamiltonians. We realize these Heisenberg spin models with tunable anisotropy parameter and power-law interactions in an ensemble of Rydberg atoms by encoding the spin in suitable Rydberg state combinations. We consistently observe stretched-exponential relaxation of magnetization for all considered spin models, collapsing onto a single curve after appropriate rescaling of time. This robust short-time relaxation behavior is explained by a perturbative treatment that exploits the strong disorder in pairwise couplings, which leads to a description in terms of approximately independent pairs of spins. In numerical simulations of small systems, we show that these pairs of spins constitute approximate local integrals of motion, which remain at least partially conserved on a timescale exceeding the duration of the relaxation dynamics of the magnetization. Published by the American Physical Society 2024

Quantum correlation of the Heisenberg xxz Hamiltonian with nonuniform external magnetic field and Dzyaloshinskii–Moriya interaction

Quantum correlation of the Heisenberg xxz Hamiltonian with nonuniform external magnetic field and Dzyaloshinskii–Moriya interaction

Entanglement entropy of two disjoint intervals and spin structures in interacting chains in and out of equilibrium

We take the paradigm of interacting spin chains, the Heisenberg spin-12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{1}{2} $$\\end{document} XXZ model, as a reference system and consider interacting models that are related to it by Jordan-Wigner transformations and restrictions to sub-chains. An example is the fermionic analogue of the gapless XXZ Hamiltonian, which, in a continuum scaling limit, is described by the massless Thirring model. We work out the Rényi-α entropies of disjoint blocks in the ground state and extract the universal scaling functions describing the Rényi-α tripartite information in the limit of infinite lengths. We consider also the von Neumann entropy, but only in the limit of large distance. We show how to use the entropies of spin blocks to unveil the spin structures of the underlying massless Thirring model. Finally, we speculate about the tripartite information after global quenches and conjecture its asymptotic behaviour in the limit of infinite time and small quench. The resulting conjecture for the “residual tripartite information”, which corresponds to the limit in which the intervals’ lengths are infinitely larger than their (large) distance, supports the claim of universality recently made studying noninteracting spin chains. Our mild assumptions imply that the residual tripartite information after a small quench of the anisotropy in the gapless phase of XXZ is equal to − log 2.

Restoring permutational invariance in the Jordan–Wigner transformation

The Jordan–Wigner transformation is a powerful tool for converting systems of spins into systems of fermions, or vice versa. While this mapping is exact, the transformation itself depends on the labelling of the spins. One consequence of this dependence is that approximate solutions of a Jordan–Wigner-transformed Hamiltonian may depend on the (physically inconsequential) labelling of the spins. In this work, we turn to an extended Jordan–Wigner transformation which remedies this problem and which may also introduce some correlation atop the Hartree–Fock solution of a transformed spin Hamiltonian. We demonstrate that this extended Jordan–Wigner transformation can be thought of as arising from a unitary version of the Lie algebraic similarity transformation (LAST) theory. We show how these ideas, particularly in combination with the standard (non-unitary) version of LAST, can provide a potentially powerful tool for the treatment of the XXZ and J 1 − J 2 Heisenberg Hamiltonians.

DMRG and Monte Carlo studies of CrI3 magnetic phases and the phase transition

The monolayer of $\textrm{CrI}_3$ has been reported to exhibit the ferromagnetic order, with a Curie temperature of $45 K$ and off-plane easy axis, which has attracted much attention in the community of condensed matter physics. Using the Density Matrix Renormalization Group method, we investigate the role of correlation effects and using classical Monte Carlo simulations analyze the nature of phase transitions in the XXZ Hamiltonian on a honeycomb lattice, which can effectively model $\textrm{CrI}_3$. We show, that the magnetic ordering of the Hamiltonian's ground state can be well approximated by classical models in a wide range of an anisotropy parameter space. Using classical Monte Carlo simulations we estimate the Curie temperature of $49.7 K$ which is in good agreement with experimental result.

Variational Quantum Circuits to Prepare Low Energy Symmetry States

We explore how to build quantum circuits that compute the lowest energy state corresponding to a given Hamiltonian within a symmetry subspace by explicitly encoding it into the circuit. We create an explicit unitary and a variationally trained unitary that maps any vector output by ansatz A(α→) from a defined subspace to a vector in the symmetry space. The parameters are trained varitionally to minimize the energy, thus keeping the output within the labelled symmetry value. The method was tested for a spin XXZ Hamiltonian using rotation and reflection symmetry and H2 Hamiltonian within Sz=0 subspace using S2 symmetry. We have found the variationally trained unitary gives good results with very low depth circuits and can thus be used to prepare symmetry states within near term quantum computers.

Boundary emptiness formation probabilities in the six-vertex model at

We define a new family of overlaps C N,m for the XXZ Hamiltonian on a periodic chain of length N. These are equal to the linear sums of the groundstate components, in the canonical basis, wherein m consecutive spins are fixed to the state ↑. We define the boundary emptiness formation probabilities as the ratios C N,m /C N,0 of these overlaps. In the associated six-vertex model, they correspond to correlation functions on a semi-infinite cylinder of perimeter N. At the combinatorial point , we obtain closed-form expressions in terms of simple products of ratios of integers.

Entanglement and matrix elements of observables in interacting integrable systems.

We study the bipartite von Neumann entanglement entropy and matrix elements of local operators in the eigenstates of an interacting integrable Hamiltonian (the paradigmatic spin-1/2 XXZ chain), and we contrast their behavior with that of quantum chaotic systems. We find that the leading term of the average (over all eigenstates in the zero magnetization sector) eigenstate entanglement entropy has a volume-law coefficient that is smaller than the universal (maximal entanglement) one in quantum chaotic systems. This establishes the entanglement entropy as a powerful measure to distinguish integrable models from generic ones. Remarkably, our numerical results suggest that the volume-law coefficient of the average entanglement entropy of eigenstates of the spin-1/2 XXZ Hamiltonian is very close to, or the same as, the one for translationally invariant quadratic fermionic models. We also study matrix elements of local operators in the eigenstates of the spin-1/2 XXZ Hamiltonian at the center of the spectrum. For the diagonal matrix elements, we show evidence that the support does not vanish with increasing system size, while the average eigenstate-to-eigenstate fluctuations vanish in a power-law fashion. For the off-diagonal matrix elements, we show that they follow a distribution that is close to (but not quite) log-normal, and that their variance is a well-defined function of ω=E_{α}-E_{β} ({E_{α}} are the eigenenergies) proportional to 1/D, where D is the Hilbert space dimension.

Analytic solution of the domain-wall nonequilibrium stationary state

We consider the out-of-equilibrium dynamics generated by joining two domains with arbitrary opposite magnetisations. We study the stationary state which emerges by the unitary evolution via the spin $1/2$ XXZ Hamiltonian, in the gapless regime, where the system develops a stationary spin current. Using the generalized hydrodynamic approach, we present a simple formula for the space-time profile of the spin current and the magnetisation exact in the limit of large times. As a remarkable effect, we show that the stationary state has a strongly discontinuous dependence on the strength of interaction. This feature allows us to give a qualitative estimation for the transient behavior of the current which is compared with numerical simulations. Moreover, we analyse the behavior around the edge of the magnetisation profile and we argue that, unlike the XX free-fermionic point, interactions always prevent the emergence of a Tracy-Widom scaling.

On the origin of magnetic anisotropy in two dimensional CrI3

The observation of ferromagnetic order in a monolayer of CrI3 has been recently reported, with a Curie temperature of 45 K and off-plane easy axis. Here we study the origin of magnetic anisotropy, a necessary ingredient to have magnetic order in two dimensions, combining two levels of modeling, density functional calculations and spin model Hamiltonians. We find two different contributions to the magnetic anisotropy of the material, favoring off-plane magnetization and opening a gap in the spin wave spectrum. First, ferromagnetic super-exchange across the ≃90 degree Cr–I–Cr bonds, are anisotropic, due to the spin–orbit interaction of the ligand I atoms. Second, a much smaller contribution that comes from the single ion anisotropy of the S = 3/2 Cr atom. Our results permit to establish the XXZ Hamiltonian, with a very small single ion anisotropy, as the adequate spin model for this system. Using spin wave theory we estimate the Curie temperature and we highlight the essential role played by the gap that magnetic anisotropy induces on the magnon spectrum.

Quantum Entanglement and Discord Dynamics of two Noninteracting Spin Qubits in Two Independent Spin Baths

We study the dynamics of quantum entanglement and quantum discord between two non-interacting qubits, which couple with two independent spin baths, obeying the XXZ Hamiltonian. After the Holstein-Primakoff transformation, one could reduce the spin bath to a single-mode bosonic bath field. Then we use this model to study the entanglement and discord dynamics of two qubits in their corresponding spin bath. For the initial Werner state, it is indicated that both entanglement and quantum discord exhibit death and revival behavior, while the quantum correlations change more smaller.

On the reality of spectra of Uq(sl2)-invariant XXZ Hamiltonians

.A new inner product is constructed on each standard module over the Temperley–Lieb algebra for and . On these modules, the Hamiltonian is shown to be self-adjoint with respect to this inner product. This implies that its action on these modules is diagonalisable with real eigenvalues. A representation theoretic argument shows that the reality of spectra of the Hamiltonian extends to all other Temperley–Lieb representations. In particular, this result applies to the celebrated -invariant XXZ Hamiltonian, for all .

Quantum state transfer in XXZ spin chains: A measurement induced transport method

We study the information transferring ability of a spin-1∕2 XXZ Hamiltonian for two different proposals of state transfer, namely, the well-studied attaching scenario and the recently proposed measurement induced transport. The latter one has been inspired by recent achievements in optical lattice experiments for local addressability of individual atoms and their time evolution when only local rotations and measurements are available and local control of the Hamiltonian is very limited. We show that while the both scenarios performs with almost similar quality in the case of non-interacting free fermionic XX phase, the difference become more pronounced around the isotropic Heisenberg point. Our study shows that the presence of spin-flip symmetry plays a key point in the quality of state transfer and each scenario which benefits from this symmetry transfers the quantum states with higher fidelity. In fact, for even chains this symmetry exists only for the measurement induced dynamics which then gives higher transport quality and for odd chains the spin-flip symmetry is only valid for the attaching scenarios which become more superior. We also study the effect of thermal fluctuations and environmental interactions on both scenarios.

Times of execution of the quantum NOT gate

It is generally believed that entanglement speeds up Quantum Information Processing (QIP). How- ever, we prove that for a system of two interacting qubits through a XXZ Hamiltonian which are maximally entan- gled it is not possible to execute a quantum NOT gate operating on one of these two qubits. The interaction between the two qubits means presence of noise in one of them. If the two interacting qubits are not entangled, the times of execution of the quantum NOT gate oper- ating on one of the two qubits are not small enough. Since the times of execution of the quantum NOT gate operating on one of the two interacting qubits is extremely large, we conclude that the execution of the quantum NOT gate operating on one of two interacting qubits is not possible.

Transport enhancement from incoherent coupling between one-dimensional quantum conductors

We study the non-equilibrium transport properties of a highly anisotropic two-dimensional lattice of particles governed by a Heisenberg XXZ Hamiltonian. The anisotropy of the lattice allows us to approximate the system at finite temperature as an array of incoherently coupled one-dimensional chains. We show that in the regime of strong intrachain interactions, the weak interchain coupling considerably boosts spin transport in the driven system. Interestingly, we show that this enhancement increases with the length of the chains, which is related to superdiffusive spin transport. We describe the mechanism behind this effect, compare it to a similar phenomenon in single chains induced by dephasing, and explain why the former is much stronger.

The Idempotents of the TL n -Module $${\otimes^{n}\mathbb{C}2}$$ ⊗ n C 2 in Terms of Elements of $${{\rm U}_{q} \mathfrak{sl}_2}$$ U q sl 2

The vector space \({\otimes^{n}\mathbb{C}^2}\) upon which the XXZ Hamiltonian with n spins acts bears the structure of a module over both the Temperley–Lieb algebra \({{\rm TL}_{n}(\beta = q + q^{-1})}\) and the quantum algebra \({{\rm U}_{q} \mathfrak{sl}_2}\). The decomposition of \({\otimes^{n}\mathbb{C}^2}\) as a \({{\rm U}_{q} \mathfrak{sl}_2}\)-module was first described by Rosso (Commun Math Phys 117:581–593, 1988), Lusztig (Cont Math 82:58–77, 1989) and Pasquier and Saleur (Nucl Phys B 330:523–556, 1990) and that as a TL n -module by Martin (Int J Mod Phys A 7:645–673, 1992) (see also Read and Saleur Nucl Phys B 777(3):316–351, 2007; Gainutdinov and Vasseur Nucl Phys B 868:223–270, 2013). For q generic, i.e. not a root of unity, the TL n -module \({\otimes^{n}\mathbb{C}^2}\) is known to be a sum of irreducible modules. We construct the projectors (idempotents of the algebra of endomorphisms of \({\otimes^{n}\mathbb{C}^2}\)) onto each of these irreducible modules as linear combinations of elements of \({{\rm U}_{q} \mathfrak{sl}_2}\). When q = q c is a root of unity, the TL n -module \({\otimes^{n}\mathbb{C}^2}\) (with n large enough) can be written as a direct sum of indecomposable modules that are not all irreducible. We also give the idempotents projecting onto these indecomposable modules. Their expression now involves some new generators, whose action on \({\otimes^{n}\mathbb{C}^2}\) is that of the divided powers \({(S^{\pm})^{(r)} = \lim_{q \rightarrow q_{c}} (S^{\pm})^r/[r]!}\).

Universal quantum computation by the unitary control of ancilla qubits and using a fixed ancilla-register interaction

We characterize a model of universal quantum computation where the register (computational) qubits are controlled by ancillary qubits, using only a single fixed interaction between register and ancillary qubits. No additional access is required to the computational register and the dynamics of both the register and ancilla are unitary. This scheme is inspired by the measurement-based ancilla-driven quantum computation of Anders et al. [Phys. Rev. A 82, 020301(R) (2010)], but does not require measurements of the ancillas, and in this respect is similar to the original gate-based model of quantum computation. We consider what possible forms this ancilla-register interaction can take, with a proof that the interaction is necessarily locally equivalent to swap combined with an entangling controlled gate. We further show which Hamiltonians can create such interactions and discuss two examples; the two-qubit XY Hamiltonian and a particular case of the XXZ Hamiltonian. We then give an example of a simple, finite, and fault-tolerant gate set for universal quantum computation in this model.

The dynamical free energy and the Loschmidt echo for a class of quantum quenches in the Heisenberg spin chain

We consider a class of global quantum quenches in the Heisenberg XXZ spin chain, where the initial states are given by products of local two-site states. The two main examples are the Néel state and the dimer state. We derive an exact analytic result for the ‘Loschmidt echo per site’ at imaginary times and also consider the analytic continuation back to real times. As a by-product we obtain an exact result for the ‘overlap per site’ between the Néel state and the ground state of the XXZ Hamiltonian in the massive regime.

The generalized Gibbs ensemble for Heisenberg spin chains

We consider the generalized Gibbs ensemble (GGE) in the context of global quantum quenches in XXZ Heisenberg spin chains. Embedding the GGE into the quantum transfer matrix formalism, we develop an iterative procedure to fix the Lagrange multipliers and to calculate predictions for the long-time limit of short-range correlators. The main idea is to consider truncated GGEs with only a finite number of charges and to investigate the convergence of the numerical results as the truncation level is increased. As an example we consider a quantum quench situation where the system is initially prepared in the Néel state and then evolves with an XXZ Hamiltonian with anisotropy Δ > 1. We provide predictions for short-range correlators and gather numerical evidence that the iterative procedure indeed converges. The results show that the system retains memory of the initial condition, and there are clear differences between the numerical values of the correlators as calculated from the purely thermal and generalized Gibbs ensembles.

A homomorphism between link and XXZ modules over the periodic Temperley–Lieb algebra

We study finite loop models on a lattice wrapped around a cylinder. A section of the cylinder has N sites. We use a family of link modules over the periodic Temperley–Lieb algebra introduced by Martin and Saleur, and Graham and Lehrer. These are labeled by the numbers of sites N and of defects d, and extend the standard modules of the original Temperley–Lieb algebra. Besides the defining parameters β = u2 + u−2 with u = eiλ/2 (weight of contractible loops) and α (weight of non-contractible loops), this family also depends on a twist parameter v that keeps track of how the defects wind around the cylinder. The transfer matrix TN(λ, ν) depends on the anisotropy ν and the spectral parameter λ that fixes the model. (The thermodynamic limit of TN is believed to describe conformal field theory of central charge c = 1 − 6λ2/(π(λ − π)).) The family of periodic XXZ Hamiltonians is extended to depend on this new parameter v, and the relationship between this family and the loop models is established. The Gram determinant for the natural bilinear form on these link modules is shown to factorize in terms of an intertwiner between these link representations and the eigenspaces of Sz of the XXZ models. This map is shown to be an isomorphism for generic values of u and v, and the critical curves in the plane of these parameters for which fails to be an isomorphism are given.