Ising Chain Research Articles

Order–disorder transition in a linear system of anti-parallel magnetic dipoles with long-range interactions

In the present study, we investigate the magnetic ordering of a system with a large number of nanomagnetic dipoles, approximated as point dipoles, arranged on a regular ring, and having anti-parallel (anti-ferromagnetic) ordering at zero external magnetic fields. We can impose different restrictions on the degrees of freedom of the dipoles: dipoles along the ring and dipoles perpendicular to the ring. When the dipoles are along the ring, the model is called theϕ-model (the dipoles are along the ϕ-coordinate of the cylindrical coordinate system) or head-to-tail model. When the dipoles are perpendicular to the plane of the ring the model is called the z-model (the dipoles are along the z-axis of the cylindrical coordinate system), and when the dipoles are along the radials of the ring, the model is called the r-model (the radial model). The z-and the r- models are similar to the one-dimensional Ising chain model (spins ‘up’ or ‘down’): the only difference is that in the z-and r-models the interactions are dipolar long-range ones. By a combination of numerical and analytical methods, we have found the exact ground states of the models, which are degenerated. A perfectly ordered system is possible only at zero temperature: by increasing the temperature the perfect order is destroyed. Employing the Monte Carlo simulation, we show that in a system of magnetic dipoles forming a linear chain, similar to the one-dimensional Ising chain model, there are no phase transitions in the thermodynamics limit, when N→∞, except at zero temperature. The system continuously transforms from a perfectly-ordered state at zero temperature to a perfectly-disordered state at very high temperatures. However, the disordered states of the models consist of perfectly ordered domains, where the ordering corresponds to one of the degenerated ground states. With the temperature, the average size of the domains decreases to zero (at very high temperatures), and numerically this size is equal to the correlation length. For finite-sized models the perfectly ordered state is kept to some critical temperature depending on the number of dipoles: T1≈4.8/lnN/2 for the head-to-tail model, Tc≈1.8/lnN/2 for the z- and radial models, Tc is the temperature when the first defect appears in the perfectly-ordered ground state,measured in J=μ0m24πr03 units, where m is the dipole moment of the dipoles, r0 the nearest distance between them. Introducing quasi-dipoles, in the present study we investigate the basic properties of the z-models, such as the internal energy, the heat capacity, the order parameter, the correlation length, etc. Employing Boltzmann’s statistics, we have obtained the exact solution of the z- and 1d Ising models. The zero-temperature state of the z-model is a perfectly ordered configuration of anti-parallel pairs of dipoles, and it is degenerated because inverting the directions of the dipoles does not change the energy of the configuration. When the model consists of N odd number of dipoles, in the ground state there is a natural ‘defect’ – a non-paired dipole, which degenerates additionally N-times the ground state and increases the energy per dipole of the ground state. For any large number of dipoles N∼100 this effect becomes negligible. We have obtained exact analytical expressions for the energy and the magnetic field of the z-model at zero temperature.

Efficient matrix product state methods for extracting spectral information on rings and cylinders

Based on the MPS formalism, we introduce an ansatz for capturing excited states in finite systems with open boundary conditions, providing a very efficient method for computing, e.g., the spectral gap of quantum spin chains. This method can be straightforwardly implemented on top of an existing DMRG or MPS ground-state code. Although this approach is built on open-boundary MPS, we also apply it to systems with periodic boundary conditions. Despite the explicit breaking of translation symmetry by the MPS representation, we show that momentum emerges as a good quantum number, and can be exploited for labeling excitations on top of MPS ground states. We apply our method to the critical Ising chain on a ring and the classical Potts model on a cylinder. Finally, we apply the same idea to compute excitation spectra for 2-D quantum systems on infinite cylinders. Again, despite the explicit breaking of translation symmetry in the periodic direction, we recover momentum as a good quantum number for labeling excitations. We apply this method to the 2-D transverse-field Ising model and the half-filled Hubbard model; for the latter, we obtain accurate results for, e.g., the hole dispersion for cylinder circumferences up to eight sites.

Skeleton of matrix-product-state-solvable models connecting topological phases of matter

Models whose ground states can be written as an exact matrix product state (MPS) provide valuable insights into phases of matter. While MPS-solvable models are typically studied as isolated points in a phase diagram, they can belong to a connected network of MPS-solvable models, which we call the MPS skeleton. As a case study where we can completely unearth this skeleton, we focus on the one-dimensional BDI class -- non-interacting spinless fermions with time-reversal symmetry. This class, labelled by a topological winding number, contains the Kitaev chain and is Jordan-Wigner-dual to various symmetry-breaking and symmetry-protected topological (SPT) spin chains. We show that one can read off from the Hamiltonian whether its ground state is an MPS: defining a polynomial whose coefficients are the Hamiltonian parameters, MPS-solvability corresponds to this polynomial being a perfect square. We provide an explicit construction of the ground state MPS, its bond dimension growing exponentially with the range of the Hamiltonian. This complete characterization of the MPS skeleton in parameter space has three significant consequences: (i) any two topologically distinct phases in this class admit a path of MPS-solvable models between them, including the phase transition which obeys an area law for its entanglement entropy; (ii) we illustrate that the subset of MPS-solvable models is dense in this class by constructing a sequence of MPS-solvable models which converge to the Kitaev chain (equivalently, the quantum Ising chain in a transverse field); (iii) a subset of these MPS states can be particularly efficiently processed on a noisy intermediate-scale quantum computer.

Dynamical phase transitions in quantum spin models with antiferromagnetic long-range interactions

In recent years, dynamical phase transitions and out-of-equilibrium criticality have been at the forefront of ultracold gases and condensed matter research. Whereas universality and scaling are established topics in equilibrium quantum many-body physics, out-of-equilibrium extensions of such concepts still leave much to be desired. Using exact diagonalization and the time-dependent variational principle in uniform martrix product states, we calculate the time evolution of the local order parameter and Loschmidt return rate in transverse-field Ising chains with antiferromagnetic power law-decaying interactions, and map out the corresponding rich dynamical phase diagram. \textit{Anomalous} cusps in the return rate, which are ubiquitous at small quenches within the ordered phase in the case of ferromagnetic long-range interactions, are absent within the accessible timescales of our simulations in the antiferromagnetic case, showing that long-range interactions are not a sufficient condition for their appearance. We attribute this to much weaker domain-wall binding in the antiferromagnetic case. For quenches across the quantum critical point, \textit{regular} cusps appear in the return rate and connect to the local order parameter changing sign, indicating the concurrence of two major concepts of dynamical phase transitions. Our results consolidate conclusions of previous works that a necessary condition for the appearance of anomalous cusps in the return rate after quenches within the ordered phase is for topologically trivial local spin flips to be the energetically dominant excitations in the spectrum of the quench Hamiltonian. Our findings are readily accessible in modern trapped-ion setups, and we outline the associated experimental considerations.

Reinforcement Learning for Digital Quantum Simulation.

Digital quantum simulation on quantum computers provides the potential to simulate the unitary evolution of any many-body Hamiltonian with bounded spectrum by discretizing the time evolution operator through a sequence of elementary quantum gates. A fundamental challenge in this context originates from experimental imperfections, which critically limits the number of attainable gates within a reasonable accuracy and therefore the achievable system sizes and simulation times. In this work, we introduce a reinforcement learning algorithm to systematically build optimized quantum circuits for digital quantum simulation upon imposing a strong constraint on the number of quantum gates. With this we consistently obtain quantum circuits that reproduce physical observables with as little as three entangling gates for long times and large system sizes up to 16 qubits. As concrete examples we apply our formalism to a long-range Ising chain and the lattice Schwinger model. Our method demonstrates that digital quantum simulation on noisy intermediate scale quantum devices can be pushed to much larger scale within the current experimental technology by a suitable engineering of quantum circuits using reinforcement learning.

Lee-Yang theory of criticality in interacting quantum many-body systems

Quantum phase transitions are a ubiquitous many-body phenomenon that occurs in a wide range of physical systems, including superconductors, quantum spin liquids, and topological materials. However, investigations of quantum critical systems also represent one of the most challenging problems in physics, since highly correlated many-body systems rarely allow for an analytic and tractable description. Here we present a Lee-Yang theory of quantum phase transitions including a method to determine quantum critical points which readily can be implemented within the tensor network formalism and even in realistic experimental setups. We apply our method to a quantum Ising chain and the anisotropic quantum Heisenberg model and show how the critical behavior can be predicted by calculating or measuring the high cumulants of properly defined operators. Our approach provides a powerful formalism to analyze quantum phase transitions using tensor networks, and it paves the way for systematic investigations of quantum criticality in two-dimensional systems.

Quantum sampling algorithms, phase transitions, and computational complexity

Drawing independent samples from a probability distribution is an important computational problem with applications in Monte Carlo algorithms, machine learning, and statistical physics. The problem can in principle be solved on a quantum computer by preparing a quantum state that encodes the entire probability distribution followed by a projective measurement. We investigate the complexity of adiabatically preparing such quantum states for the Gibbs distributions of various classical models including the Ising chain, hard-sphere models on different graphs, and a model encoding the unstructured search problem. By constructing a parent Hamiltonian, whose ground state is the desired quantum state, we relate the asymptotic scaling of the state preparation time to the nature of transitions between distinct quantum phases. These insights enable us to identify adiabatic paths that achieve a quantum speedup over classical Markov chain algorithms. In addition, we show that parent Hamiltonians for the problem of sampling from independent sets on certain graphs can be naturally realized with neutral atoms interacting via highly excited Rydberg states.

Magnetotransport properties of Fe substituted Ca3CoMnO6

Ca3CoMnO6, a quasi-1D Ising chain at low temperature offers rich fundamental physics and applications. We have studied the dielectric and magnetoelectric coupling in Fe doped Ca3CoMnO6 (CCMO) bulk ceramics prepared by co-precipitation technique. Single phase hexagonal crystal structure having R-3C space group was confirmed by x-ray diffraction. Extremely low currents were observed up to 20% Fe doping. Doping dependence of magnetoresistance (MR) revealed both positive and negative MR, with anomalously high MR values beyond 3000% in diluted Fe doped CCMO; whereas the higher doping of Fe was found to result in negative MR due to enhanced magnetostriction effects. The dielectric study was carried out for a range of 20 Hz to 10 MHz. The negative value of the colossal magnetodielectric induced in the Fe doped samples can be attributed to the magnetostriction effect along with interfacial Maxwell-Wagner polarization.

Quantum Kibble-Zurek mechanism: Kink correlations after a quench in the quantum Ising chain

The transverse field in the quantum Ising chain is linearly ramped from the para- to the ferromagnetic phase across the quantum critical point at a rate characterized by a quench time $\tau_Q$. We calculate a connected kink-kink correlator in the final state at zero transverse field. The correlator is a sum of two terms: a negative (anti-bunching) Gaussian that depends on the Kibble-Zurek (KZ) correlation length only and a positive term that depends on a second longer scale of length. The second length is made longer by dephasing of the state excited near the critical point during the following ramp across the ferromagnetic phase. This interpretation is corroborated by considering a linear ramp that is halted in the ferromagnetic phase for a finite waiting time and then continued at the same rate as before the halt. The extra time available for dephasing increases the second scale of length that asymptotically grows linearly with the waiting time. The dephasing also suppresses magnitude of the second term making it negligible for waiting times much longer than $\tau_Q$. The same dephasing can be obtained with a smooth ramp that slows down in the ferromagnetic phase. Assuming sufficient dephasing we obtain also higher order kink correlators and the ferromagnetic correlation function.

Localization in the kicked Ising chain

Determining the border between ergodic and localized behavior is of central interest for interacting many-body systems. We consider here the recently quite popular kicked Ising chain. A convenient description of such a many-body system is achieved by the dual operator that evolves the system in contrast to the time-evolution operator not in time but particle direction. We focus in this paper on the largest eigenvalue of a function of the dual operator. It is identified to be a convenient tool to determine if the system shows ergodic or many-body localized features. We analytically explain the eigenvalue structure by perturbation theory in the vicinity of the noninteracting system. This result agrees with the numerics for small times in a previous work [Braun et al., Phys. Rev. E 101, 052201 (2020)]. Given that a direct numerical computation of this largest eigenvalue is only possible for small times, another achievement of this paper is to provide a way to determine this eigenvalue by the ratio of spectral form factors for adjacent particle numbers. By extensive large-time numerical computations of the spectral form factor, we can then distinguish between localized and ergodic system features and compute the Thouless time, i.e., the transition time between these regimes in the thermodynamic limit.

Many-Body Quantum Zeno Effect and Measurement-Induced Subradiance Transition

It is well known that by repeatedly measuring a quantum system it is possible to completely freeze its dynamics into a well defined state, a signature of the quantum Zeno effect. Here we show that for a many-body system evolving under competing unitary evolution and variable-strength measurements the onset of the Zeno effect takes the form of a sharp phase transition. Using the Quantum Ising chain with continuous monitoring of the transverse magnetization as paradigmatic example we show that for weak measurements the entanglement produced by the unitary dynamics remains protected, and actually enhanced by the monitoring, while only above a certain threshold the system is sharply brought into an uncorrelated Zeno state. We show that this transition is invisible to the average dynamics, but encoded in the rare fluctuations of the stochastic measurement process, which we show to be perfectly captured by a non-Hermitian Hamiltonian which takes the form of a Quantum Ising model in an imaginary valued transverse field. We provide analytical results based on the fermionization of the non-Hermitian Hamiltonian in supports of our exact numerical calculations.

Local measures of dynamical quantum phase transitions

In recent years, dynamical quantum phase transitions (DQPTs) have emerged as a useful theoretical concept to characterize nonequilibrium states of quantum matter. DQPTs are marked by singular behavior in an \textit{effective free energy} $\lambda(t)$, which, however, is a global measure, making its experimental or theoretical detection challenging in general. We introduce two local measures for the detection of DQPTs with the advantage of requiring fewer resources than the full effective free energy. The first, called the \textit{real-local} effective free energy $\lambda_M(t)$, is defined in real space and is therefore suitable for systems where locally resolved measurements are directly accessible such as in quantum-simulator experiments involving Rydberg atoms or trapped ions. We test $\lambda_M(t)$ in Ising chains with nearest-neighbor and power-law interactions, and find that this measure allows extraction of the universal critical behavior of DQPTs. The second measure we introduce is the \textit{momentum-local} effective free energy $\lambda_k(t)$, which is targeted at systems where momentum-resolved quantities are more naturally accessible, such as through time-of-flight measurements in ultracold atoms. We benchmark $\lambda_k(t)$ for the Kitaev chain, a paradigmatic system for topological quantum matter, in the presence of weak interactions. Our introduced local measures for effective free energies can further facilitate the detection of DQPTs in modern quantum-simulator experiments.

Extreme statistics of the excitations in the random transverse Ising chain

In random quantum magnets, like the random transverse Ising chain, low energy excitations are localized in rare regions and there are only weak correlations between them. It is an open question whether these correlations are relevant in the sense of the renormalization group. To answer this question, we calculate the distribution of the excitation energy of the random transverse Ising chain in the disordered Griffiths phase with high numerical precision by the strong disorder renormalization group method and---for shorter chains---by free-fermion techniques. Asymptotically, the two methods give identical results, which are well fitted by the Fr\'echet limit law of the extremes of independent and identically distributed random numbers. Considering the finite size corrections, the two numerical methods give very similar results, but these differ from the correction term for uncorrelated random variables, indicating that the weak correlations between low-energy excitations in random quantum magnets are relevant.

Investigating the exchange of Ising chains on a digital quantum computer

The ferromagnetic state of an Ising chain can represent a two-fold degenerate subspace or equivalently a logical qubit which is protected from excitations by an energy gap. We study a a braiding-like exchange operation through the movement of the state in the qubit subspace which resembles that of the localized edge modes in a Kitaev chain. The system consists of two Ising chains in a 1D geometry where the operation is simulated through the adiabatic time evolution of the ground state. The time evolution is implemented via the Suzuki-Trotter expansion on basic single- and two-qubit quantum gates using IBM's Aer QASM simulator. The fidelity of the system is investigated as a function of the evolution and system parameters to obtain optimum efficiency and accuracy for different system sizes. Various aspects of the implementation including the circuit depth, Trotterization error, and quantum gate errors pertaining to the Noisy Intermediate-Scale Quantum (NISQ) hardware are discussed as well. We show that the quantum gate errors, i.e. bit-flip, phase errors, are the dominating factor in determining the fidelity of the system as the Trotter error and the adiabatic condition are less restrictive even for modest values of Trotter time steps. We reach an optimum fidelity $>99\%$ on systems of up to $11$ sites per Ising chain and find that the most efficient implementation of a single braiding-like operation for a fidelity above $90\%$ requires a circuit depth of the order of $\sim 10^{3}$ restricting the individual gate errors to be less than $\sim 10^{-6}$ which is prohibited in current NISQ hardware.

Energy levels and their degeneracies for two-ring Ising chains of spins-$$\frac{1}{2}$$ with NN and NNN couplings: spin frustration of ferromagnetic and antiferromagnetic orders

Energy levels and their degeneracies for two-ring Ising chains of spins-$$\frac{1}{2}$$ with NN and NNN couplings: spin frustration of ferromagnetic and antiferromagnetic orders

Unitary Long-Time Evolution with Quantum Renormalization Groups and Artificial Neural Networks.

In this work, we combine quantum renormalization group approaches with deep artificial neural networks for the description of the real-time evolution in strongly disordered quantum matter. We find that this allows us to accurately compute the long-time coherent dynamics of large many-body localized systems in nonperturbative regimes including the effects of many-body resonances. Concretely, we use this approach to describe the spatiotemporal buildup of many-body localized spin-glass order in random Ising chains. We observe a fundamental difference to a noninteracting Anderson insulating Ising chain, where the order only develops over a finite spatial range. We further apply the approach to strongly disordered two-dimensional Ising models, highlighting that our method can be used also for the description of the real-time dynamics of nonergodic quantum matter in a general context.

Spin dynamics of an Ising chain with bond impurity in a tilt magnetic field

Spin dynamics of quantum Ising system has been the subject of condensed matter physics in the past few decades. In this work, the bond-impurity effects on dynamics of the one-dimensional quantum Ising model with both transverse (Bzi) and longitudinal magnetic field (LMF, Bxi) was studied in the high-temperature limit by the recurrence relations method. Three cases, i.e., the LMF effect on the dynamics of the pure Ising model without impurity (Bxi≠0), the weak- and strong-bond impurity effect on the dynamics of the transverse Ising model without LMF (the impurity strength Jj≠Ji, Bxi=0), and their combining effect (Jj≠Ji, Bxi≠0), were investigated respectively. It is found that the dynamical behaviors of the system depend on both the impurity and the LMF. In general, the central-peak behavior (collective-mode behavior) will be enhanced (depressed) when increasing Jj or Bxi properly. However, when Jj reaches a critical value Jjc , the spin motion will be frozen, i.e., the bond impurity will act as a switch, which may turn off or turn on the spin motion. In contrast, the LMF can active the motion of the frozen spin. Therefore, the dynamical behavior of the Ising system depends on the combining effects of the impurity and the LMF.

Scaling of temporal entanglement in proximity to integrability

Describing dynamics of quantum many-body systems is a formidable challenge due to rapid generation of quantum entanglement between remote degrees of freedom. A promising approach to tackle this challenge, which has been proposed recently, is to characterize the quantum dynamics of a many-body system and its properties as a bath via the Feynman-Vernon influence matrix (IM), which is an operator in the space of time trajectories of local degrees of freedom. Physical understanding of the general scaling of the IM's temporal entanglement and its relation to basic dynamical properties is highly incomplete to present day. In this Article, we analytically compute the exact IM for a family of integrable Floquet models - the transverse-field kicked Ising chain - finding a Bardeen-Cooper-Schrieffer-like "wavefunction" on the Schwinger-Keldysh contour with algebraically decaying correlations. We demonstrate that the IM exhibits area-law temporal entanglement scaling for all parameter values. Furthermore, the entanglement pattern of the IM reveals the system's phase diagram, exhibiting jumps across transitions between distinct Floquet phases. Near criticality, a non-trivial scaling behavior of temporal entanglement is found. The area-law temporal entanglement allows us to efficiently describe the effects of sizeable integrability-breaking perturbations for long evolution times by using matrix product state methods. This work shows that tensor network methods are efficient in describing the effect of non-interacting baths on open quantum systems, and provides a new approach to studying quantum many-body systems with weakly broken integrability.

Exact thermal properties of free-fermionic spin chains

An exact description of integrable spin chains at finite temperature is provided using an elementary algebraic approach in the complete Hilbert space of the system. We focus on spin chain models that admit a description in terms of free fermions, including paradigmatic examples such as the one-dimensional transverse-field quantum Ising and XY models. The exact partition function is derived and compared with the ubiquitous approximation in which only the positive parity sector of the energy spectrum is considered. Errors stemming from this approximation are identified in the neighborhood of the critical point at low temperatures. We further provide the full counting statistics of a wide class of observables at thermal equilibrium and characterize in detail the thermal distribution of the kink number and transverse magnetization in the transverse-field quantum Ising chain.

Dynamics and correlations at a quantum phase transition beyond Kibble-Zurek

Kibble-Zurek theory (KZ) stands out as the most robust theory of defect generation in the dynamics of phase transitions. KZ utilizes the structure of equilibrium states away from the transition point to estimate the excitations due to the transition using adiabatic and impulse approximations. Here we show, the actual nonequilibrium dynamics lead to a qualitatively different scenario from KZ, as far correlations between the defects (rather than their densities) are concerned. For a quantum Ising chain, we show, this gives rise to a Gaussian spatial decay in the domain wall (kinks) correlations, while KZ would predict an exponential fall. We propose a simple but general framework on top of KZ, based on the ``quantum coarsening'' dynamics of local correlators in the supposed impulse regime. We outline how our picture extends to generic interacting situations.