One-dimensional Heisenberg Model Research Articles

First-Principle Validation of Fourier's Law: One-Dimensional Classical Inertial Heisenberg Model.

The thermal conductance of a one-dimensional classical inertial Heisenberg model of linear size L is computed, considering the first and last particles in thermal contact with heat baths at higher and lower temperatures, Th and Tl (Th>Tl), respectively. These particles at the extremities of the chain are subjected to standard Langevin dynamics, whereas all remaining rotators (i=2,⋯,L-1) interact by means of nearest-neighbor ferromagnetic couplings and evolve in time following their own equations of motion, being investigated numerically through molecular-dynamics numerical simulations. Fourier's law for the heat flux is verified numerically, with the thermal conductivity becoming independent of the lattice size in the limit L→∞, scaling with the temperature, as κ(T)∼T-2.25, where T=(Th+Tl)/2. Moreover, the thermal conductance, σ(L,T)≡κ(T)/L, is well-fitted by a function, which is typical of nonextensive statistical mechanics, according to σ(L,T)=Aexpq(-Bxη), where A and B are constants, x=L0.475T, q=2.28±0.04, and η=2.88±0.04.

Magnetic properties of a quasi-one-dimensional antiferromagnet Eu2BiS4

We have synthesized c-axis oriented polycrystals of a Eu-based orthorhombic compound Eu2BiS4 and measured transport, magnetic, and thermal properties down to 1.8 K. The crystal structure of Eu2BiS4 was confirmed by powder x-ray diffraction to be orthorhombic CaFe2O4-type with one-dimensional chains of Eu2+ ions along the c-axis. The electrical resistivity ρ increases on cooling from 300 K, which shows that this compound is a semiconductor. Both magnetic susceptibility χ and specific heat C exhibit an obvious peak at TN = 2.5 K, indicating an antiferromagnetic order. Anisotropic behaviors of isothermal magnetization M and χ below TN suggest that the magnetic moments align along the c-axis. Moreover, a tail appears in C(T) in the range of TN < T < 8 K, where χ(T) is reproduced by that calculated with a one-dimensional ferromagnetic Heisenberg model. These behaviors in C(T) and χ(T) are attributed to developed short-range ferromagnetic correlations in the Eu2+ chains.

Optimal Quantum Dataset for Learning a Unitary Transformation

Unitary transformations formulate the time evolution of quantum states. How to learn a unitary transformation efficiently is a fundamental problem in quantum machine learning. The most natural and leading strategy is to train a quantum machine learning model based on a quantum dataset. Although the presence of more training data results in better models, using too much data reduces the efficiency of training. In this work, we solve the problem on the minimum size of sufficient quantum datasets for learning a unitary transformation exactly, which reveals the power and limitation of quantum data. First, we prove that the minimum size of a dataset with pure states is $2^n$ for learning an $n$-qubit unitary transformation. To fully explore the capability of quantum data, we introduce a practical quantum dataset consisting of $n+1$ elementary tensor product states that are sufficient for exact training. The main idea is to simplify the structure utilizing decoupling, which leads to an exponential improvement in the size of the datasets with pure states. Furthermore, we show that the size of the quantum dataset with mixed states can be reduced to a constant, which yields an optimal quantum dataset for learning a unitary. We showcase the applications of our results in oracle compiling and Hamiltonian simulation. Notably, to accurately simulate a 3-qubit one-dimensional nearest-neighbor Heisenberg model, our circuit only uses $96$ elementary quantum gates, which is significantly less than $4080$ gates in the circuit constructed by the Trotter-Suzuki product formula.

Quantum gas microscopy of Kardar-Parisi-Zhang superdiffusion.

The Kardar-Parisi-Zhang (KPZ) universality class describes the coarse-grained behavior of a wealth of classical stochastic models. Surprisingly, KPZ universality was recently conjectured to also describe spin transport in the one-dimensional quantum Heisenberg model. We tested this conjecture by experimentally probing transport in a cold-atom quantum simulator via the relaxation of domain walls in spin chains of up to 50 spins. We found that domain-wall relaxation is indeed governed by the KPZ dynamical exponent z = 3/2 and that the occurrence of KPZ scaling requires both integrability and a nonabelian SU(2) symmetry. Finally, we leveraged the single-spin-sensitive detection enabled by the quantum gas microscope to measure an observable based on spin-transport statistics. Our results yield a clear signature of the nonlinearity that is a hallmark of KPZ universality.

Chebyshev expansion of spectral functions using restricted Boltzmann machines

Calculating the spectral function of two dimensional systems is arguably one of the most pressing challenges in modern computational condensed matter physics. While efficient techniques are available in lower dimensions, two dimensional systems present insurmountable hurdles, ranging from the sign problem in quantum Monte Carlo (MC), to the entanglement area law in tensor network based methods. We hereby present a variational approach based on a Chebyshev expansion of the spectral function and a neural network representation for the wave functions. The Chebyshev moments are obtained by recursively applying the Hamiltonian and projecting on the space of variational states using a modified natural gradient descent method. We compare this approach with a modified approximation of the spectral function which uses a Krylov subspace constructed from the "Chebyshev wave-functions". We present results for the one-dimensional and two-dimensional Heisenberg model on the square lattice, and compare to those obtained by other methods in the literature.

Tangential finite-size scaling at the Gaussian topological transition in the quantum spin-1 anisotropic chain

Scaling aspects of Gaussian topological phase-transitions in quantum spin chains are investigated using the prototypical one-dimensional spin-1 XXZ Heisenberg model with uniaxial single-ion anisotropy $D$. This model presents a critical line separating the gaped Haldane and large-$D$ phases, with the relevant energy gap closing at the transition point. We show that a proper tangential finite-size scaling analysis is able to accurately locate the Gaussian critical line and to probe the continuously varying set of correlation length critical exponents. The specific features of the tangential scaling are highlighted in contrast with the standard scaling holding in the Ising-like transition between the gapless AF-N\'eel and gaped Haldane phases. Our results are compared with field-theoretic predictions and available high-accuracy data for specific points along the Gaussian line.

Quantum Singular Value Decomposition of Spin Correlation Matrix in One-Dimensional Heisenberg Model

We present singular value decomposition of spin correlation matrix defined from the ground state of one-dimensional antiferromagnetic quantum Heisenberg model. The decomposition creates a data set that coincides with various domain excitations from classical antiferromagnetic state. We determine the scaling relation for the singular values as a function of the domain size. The singular values are closely related to the square of weights of the bases in the ground-state wavefunction. The nature of the singular value decomposition is to precisely estimate the appropriate bases and corresponding weights of the ground-state wavefunction.

Numerical Evidence for Many-Body Localization in Two and Three Dimensions.

Disorder and interactions can lead to the breakdown of statistical mechanics in certain quantum systems, a phenomenon known as many-body localization (MBL). Much of the phenomenology of MBL emerges from the existence of ℓ bits, a set of conserved quantities that are quasilocal and binary (i.e., possess only ±1 eigenvalues). While MBL and ℓ bits are known to exist in one-dimensional systems, their existence in dimensions greater than one is a key open question. To tackle this question, we develop an algorithm that can find approximate binary ℓ bits in arbitrary dimensions by adaptively generating a basis of operators in which to represent the ℓ bit. We use the algorithm to study four models: the one-, two-, and three-dimensional disordered Heisenberg models and the two-dimensional disordered hard-core Bose-Hubbard model. For all four of the models studied, our algorithm finds high-quality ℓ bits at large disorder strength and rapid qualitative changes in the distributions of ℓ bits in particular ranges of disorder strengths, suggesting the existence of MBL transitions. These transitions in the one-dimensional Heisenberg model and two-dimensional Bose-Hubbard model coincide well with past estimates of the critical disorder strengths in these models, which further validates the evidence of MBL phenomenology in the other two- and three-dimensional models we examine. In addition to finding MBL behavior in higher dimensions, our algorithm can be used to probe MBL in various geometries and dimensionality.

Probing many-body localization on a noisy quantum computer

A disordered system of interacting particles exhibits localized behavior when the disorder is large compared to the interaction strength. Studying this phenomenon on a quantum computer without error correction is challenging because even weak coupling to a thermal environment destroys most signatures of localization. Fortunately, spectral functions of local operators are known to contain features that can survive the presence of noise. In these spectra, discrete peaks and a soft gap at low frequencies compared to the thermal phase indicate localization. Here, we present the computation of spectral functions on a trapped-ion quantum computer for a one-dimensional Heisenberg model with disorder. Further, we design an error-mitigation technique which is effective at removing the noise from the measurement allowing clear signatures of localization to emerge as the disorder increases. Thus, we show that spectral functions can serve as a robust and scalable diagnostic of many-body localization on the current generation of quantum computers.

Theory of Trotter Error with Commutator Scaling

The Lie-Trotter formula, together with its higher-order generalizations, provides a direct approach to decomposing the exponential of a sum of operators. Despite significant effort, the error scaling of such product formulas remains poorly understood. We develop a theory of Trotter error that overcomes the limitations of prior approaches based on truncating the Baker-Campbell-Hausdorff expansion. Our analysis directly exploits the commutativity of operator summands, producing tighter error bounds for both real- and imaginary-time evolutions. Whereas previous work achieves similar goals for systems with geometric locality or Lie-algebraic structure, our approach holds in general. We give a host of improved algorithms for digital quantum simulation and quantum Monte Carlo methods, including simulations of second-quantized plane-wave electronic structure, $k$-local Hamiltonians, rapidly decaying power-law interactions, clustered Hamiltonians, the transverse field Ising model, and quantum ferromagnets, nearly matching or even outperforming the best previous results. We obtain further speedups using the fact that product formulas can preserve the locality of the simulated system. Specifically, we show that local observables can be simulated with complexity independent of the system size for power-law interacting systems, which implies a Lieb-Robinson bound as a byproduct. Our analysis reproduces known tight bounds for first- and second-order formulas. Our higher-order bound overestimates the complexity of simulating a one-dimensional Heisenberg model with an even-odd ordering of terms by only a factor of $5$, and is close to tight for power-law interactions and other orderings of terms. This suggests that our theory can accurately characterize Trotter error in terms of both asymptotic scaling and constant prefactor.

Structure and magnetic properties of LiMVO4 (M = Mn, Cu)

LiM2+VO4 compounds with spinel-related structures show interesting electrochemical, magnetic and multiferroic properties. Neutron powder diffraction data for LiMVO4 (M ​= ​Cu, Mn) were collected at room temperature and 1.5 ​K. In contrast to single crystal NPD results, no evidence of magnetic ordering was detected for LiCuVO4 at 1.5 ​K due to the incommensurate order and low Cu2+magnetic moment. For LiMnVO4 (space group Cmcm) a clear ordered magnetic state was determined for the first time: the chains of edge-shared MnO6 octahedra have antiferromagnetic (AFM) order within each chain, and also between chains to give C′ centring. The magnetic behaviour of the MnO6 chain can be well described by the classical one-dimensional Heisenberg model with an intra-chain exchange energy of J/k ​= ​−4.32 ​K. It is believed that direct dxy ​− ​dxy AFM interactions are dominant over 90° Mn–O–Mn exchange for determining the magnetic order.

Restricted Boltzmann Machines for Quantum States with Non-Abelian or Anyonic Symmetries.

Although artificial neural networks have recently been proven to provide a promising new framework for constructing quantum many-body wave functions, the parametrization of a quantum wave function with non-abelian symmetries in terms of a Boltzmann machine inherently leads to biased results due to the basis dependence. We demonstrate that this problem can be overcome by sampling in the basis of irreducible representations instead of spins, for which the corresponding ansatz respects the non-abelian symmetries of the system. We apply our methodology to find the ground states of the one-dimensional antiferromagnetic Heisenberg (AFH) model with spin-1/2 and spin-1degrees of freedom, and obtain a substantially higher accuracy than when using the s_{z} basis as an input to the neural network. The proposed ansatz can target excited states, which is illustrated by calculating the energy gap of the AFH model. We also generalize the framework to the case of anyonic spin chains.

Quantum phase transition and topological order scaling in spin-1 bond-alternating Heisenberg model with Dzyaloshinskii-Moriya interaction

&lt;sec&gt;Quantum phase transitions are driven by quantum fluctuations due to the uncertainty principle in many-body physics. In quantum phase transitions, the ground-state changes dramatically. The quantum entanglement, specific heat, magnetization and other physical quantities diverge according to certain functions, and show specific scaling behaviors. In addition, there is a topological quantum phase transition beyond the conventional Landau-Ginzburg-Wilson paradigm, which is relevant to emergent phenomena in strongly correlated electron systems, with topological nonlocal order parameters as a salient feature. Thus, topological order is a new paradigm in the study of topological quantum phase transitions.&lt;/sec&gt;&lt;sec&gt;To investigate competition mechanism of the different quantum spin interactions, in this paper, the one-dimensional spin-1 bond-alternating Heisenberg model with Dzyaloshinskii-Moriya (DM) interaction is considered. The DM interaction drives the quantum fluctuations resulting in a phase transition. By using the one-dimensional infinite matrix product state algorithm in tensor network representation, the quantum entanglement entropy and order parameters are calculated from the ground-state function. The numerical result shows that with the change of bond alternating strength, there is a quantum phase transition from the topological ordered Haldane phase to the local ordered dimer phase. Based on the von Neumann entropy and order parameter, the phase diagram of this model is obtained. There is a critical line that separates the Haldane and the dimer phase. The DM interaction inhibits the dimerization of the quantum spin system and finally breaks the fully dimerization. Due to the fact that the structurally symmetry of system is broken, the local dimer order exists in the whole parameter range when the bond-alternative strength parameter changes. The first derivative of the local dimer order behaves as a peak corresponding to the critical point. Furthermore, from the numerical scaling of the first derivative of dimer order and the non-local string order near the phase transition point, the characteristic critical exponents &lt;i&gt;α&lt;/i&gt; and &lt;i&gt;β&lt;/i&gt; are obtained, respectively. It shows that the characteristic critical exponent &lt;i&gt;α&lt;/i&gt; decreases, and &lt;i&gt;β&lt;/i&gt; increases gradually with the interaction strength of DM increasing. The resulting state i.e. the anti-symmetric anisotropic DM interaction produced by spin-orbit coupling, affects the critical properties of the system in the phase transition. This reveals that the competition mechanism of the quantum spin interaction, also provides some guidance for the future study of the critical behavior in topological quantum phase transition with the DM interaction.&lt;/sec&gt;

Dynamics of the one-dimensional isotropic Heisenberg model with Dzyaloshinskii–Moryia interaction in a random transverse field

The dynamics of the one-dimensional spin-1∕2 isotropic Heisenberg model with Dzyaloshinskii–Moriya (DM) interaction in the presence of a random magnetic field perpendicular to the DM axis (a transverse field) is investigated by using the method of recurrence relations. A bimodal probability distribution function for this transverse field is considered. The first four recurrants, which are used to obtain the short-time expansion for the time-dependent correlation functions, are exactly obtained as function of the probabilities and of the transverse field. A recent proposed extrapolation procedure for the higher-order recurrants is used in order to extend the results to longer times. In the limit of infinite temperature the relaxation and the spectral density functions are computed for several values of the external field and different probability distributions. Depending on the probabilities and on the external fields one has stronger and faster oscillations in the relaxation functions, and a suppression of the central peak in the spectral density.

Measurement Based Quantum Heat Engine with Coupled Working Medium

We consider measurement based single temperature quantum heat engine without feedback control, introduced recently by Yi, Talkner and Kim [Phys. Rev. E 96, 022108 (2017)]. Taking the working medium of the engine to be a one-dimensional Heisenberg model of two spins, we calculate the efficiency of the engine undergoing a cyclic process. Starting with two spin-1/2 particles, we investigate the scenario of higher spins also. We show that, for this model of coupled working medium, efficiency can be higher than that of an uncoupled one. However, the relationship between the coupling constant and the efficiency of the engine is rather involved. We find that in the higher spin scenario efficiency can sometimes be negative (this means work has to be done to run the engine cycle) for certain range of coupling constants, in contrast to the aforesaid work of Yi, Talkner and Kim, where they showed that the extracted work is always positive in the absence of coupling. We provide arguments for this negative efficiency in higher spin scenarios. Interestingly, this happens only in the asymmetric scenarios, where the two spins are different. Given these facts, for judiciously chosen conditions, an engine with coupled working medium gives advantage for the efficiency over the uncoupled one.

Scattering of spinon excitations by potentials in the one-dimensional Heisenberg model

By a semi-analytical Bethe ansatz method and a T-matrix approach we study the scattering of a spinon, the elementary quantum many-body topological excitation in the 1D Heisenberg model, by local and phonon potentials. In particular, we contrast the scattering of a spinon to that of a free spinless fermion in the XY model to highlight the effect of strong correlations. For the one spinon scattering in an odd-site chain, we find a regular behavior of the scattering coefficients. In contrast, in an even-site chain there is a transfer of transmission probability between the two spinon branches that grows exponentially with system size. We link the exponent of the exponential behavior to the dressed charge that characterizes the critical properties of the 1D Heisenberg model, an interplay of topological and critical properties. The aim of this study is a microscopic understanding of spinon scattering by impurities, barriers or phonons, modeled as prototype potentials, an input in the analysis of quantum spin transport experiments.

Quantum disordered vector-spin-chirality state in one dimensional Heisenberg model

We study the existence of a chiral spin ordered state in one-dimensional antiferromagnetic Heisenberg models. We prove that when the coupling constants are short-ranged no quantum vector-spin-chirality order appears at zero temperature for integer spin chains.

Effects of a magnetic field on the dynamics of the one-dimensional Heisenberg model with Dzyaloshinskii-Moriya interactions

We use the method of recurrence relations to investigate the dynamics of the one-dimensional spin-$1/2$ Heisenberg model with Dzyaloshinskii-Moriya (DM) interaction in a magnetic field perpendicular to the DM axis. Our results are valid in the high-temperature limit. We determine exactly the first four recurrants, which produce the time-dependent correlation functions. In order to extend our results to longer times, we introduce a new extrapolation procedure for obtaining higher-order recurrants. Our extrapolation mechanism produces good results when compared to the well-known behavior of the relaxation function in the limit of the $\mathit{XY}$ and isotropic Heisenberg models. The relaxation function and the spectral density function are then analyzed for several values of the external field $B$ for the Heisenberg model with DM interactions in one dimension, in the $T\ensuremath{\rightarrow}\ensuremath{\infty}$ limit. We find that the external field produces stronger and faster oscillations in relaxation functions and a suppression of the central peak in the spectral density. That is accompanied by the appearance of a peak centered at a finite frequency, due to enhancement of the collective mode of spins precessing about the external field.

Time-dependent generalized Gibbs ensembles in open quantum systems

Generalized Gibbs ensembles have been used as powerful tools to describe the steady state of integrable many-particle quantum systems after a sudden change of the Hamiltonian. Here we demonstrate numerically, that they can be used for a much broader class of problems. We consider integrable systems in the presence of weak perturbations which both break integrability and drive the system to a state far from equilibrium. Under these conditions, we show that the steady state and the time-evolution on long time-scales can be accurately described by a (truncated) generalized Gibbs ensemble with time-dependent Lagrange parameters, determined from simple rate equations. We compare the numerically exact time evolutions of density matrices for small systems with a theory based on block-diagonal density matrices (diagonal ensemble) and a time-dependent generalized Gibbs ensemble containing only small number of approximately conserved quantities, using the one-dimensional Heisenberg model with perturbations described by Lindblad operators as an example.

Reorthonormalization of Chebyshev matrix product states for dynamical correlation functions

The Chebyshev expansion offers a numerically efficient and easy-implement algorithm for evaluating dynamic correlation functions using matrix product states (MPS). In this approach, each recursively generated Chebyshev vector is approximately represented by an MPS. However, the recurrence relations of Chebyshev polynomials are broken by the approximation, leading to an error which is accumulated with the increase of the order of expansion. Here we propose a reorthonormalization approach to remove this error introduced in the loss of orthogonality of the Chebyshev polynomials. Our approach, as illustrated by comparison with the exact results for the one-dimensional XY and Heisenberg models, improves significantly the accuracy in the calculation of dynamical correlation functions.