Ising Ring Research Articles

Sunburst quantum Ising model under interaction quench: Entanglement and role of initial state coherence.

We study the nonequilibrium dynamics of an isolated bipartite quantum system, the sunburst quantum Ising model, under interaction quench. The prequench limit of this model is two noninteracting integrable systems, namely a transverse Ising chain and finite number of isolated qubits. As a function of interaction strength, the spectral fluctuation property goes from Poisson to Wigner-Dyson statistics. We chose entanglement entropy as a probe to study the approach to thermalization or lack of it in postquench dynamics. In the near-integrable limit, as expected, the linear entropy displays oscillatory behavior, while in the chaotic limit it saturates. Along with the chaotic nature of the time evolution generator, we show the importance of the role played by the coherence of the initial state in deciding the nature of thermalization. We further show that these findings are general by replacing the Ising ring with a disordered XXZ model with disorder strength putting it in the many-body localized phase.

The exact solution of the edge-modified kekulene-like nanostructure Ising-Heisenberg spin system

A ferromagnetic model of kekulene-like nanostructure is established, in which the Ising ring (S = 1) decorated by the edge-spin (S = 3/2) through antiferromagnetic Heisenberg exchange interaction. The thermodynamic properties and magnetization behavior of the system were investigated by use of the exact diagonalization and transfer matrix method . It was found that there are six different phases with different values of magnetic susceptibility multiplied by temperature at low temperature in the parameter space of single-ion anisotropy, but it gets only one value taken in the high-temperature limit. The multi-peak phenomena of the specific heat curve at ultra-low temperature are corresponding to the entropy of the system, which the abundant and dense excited states are near the ground state. It shows many different magnetization plateaus in the magnetization process of the system, but no spontaneous magnetization appears. The number and width of magnetization plateaus and the saturation of the magnetization are affected by the single-ion anisotropy. • The edge-modified kekulene-like nanostructure Ising-Heisenberg spin-(1,3/2) system has been exactly determined. • There are six different phases with different values of the magnetic susceptibility multiplied by temperature at low temperature in the parameter space of single-ion anisotropy, but it gets only one value taken in the high-temperature limit. • The multi-peak phenomena of the specific heat curve at ultra-low temperature are corresponding to the entropy of the system, which the abundant and dense excited states near the ground state. • The magnetization of the system shows many different jumps (magnetic plateaus) with the external magnetic field increasing.

Decoherence and energy flow in the sunburst quantum Ising model

We study the post-quench unitary dynamics of a quantum sunburst spin model, composed of a transverse-field quantum Ising ring which is suddenly coupled to a set of independent external qubits along the longitudinal direction, in a way to respect a residual translation invariance and the Ising symmetry. Starting from the different equilibrium quantum phases of the system, we characterize the decoherence and the energy storage in the external qubits, which may be interpreted as a probing apparatus for the inner Ising ring. Our results show that, in proximity of the quantum transitions of the Ising ring, either first-order or continuous, it is possible to put forward dynamic finite-size scaling frameworks which unveil peculiar scaling regimes, depending on the way in which the large-size limit is taken: either by fixing the number n of probing qubits, or their interspace distance b. In any case, the dependence of the various observables on n can be reabsorbed into a redefinition of the quench parameter by a prefactor. We also address the role of a nearest-neighbor coupling between the external qubits.

Dynamic quantum-enhanced sensing without entanglement in central spin systems

We propose a dynamic quantum sensing scheme by using a quantum many-spin system composed of a central spin interacting with many surrounding spins. Starting from a generalized Ising ring model, we investigate the error propagation formula of the central spin, and it indicates that Heisenberg scaling can be reached while the probe state only needs to be a product state. Particularly, we derive an analytical form of the dynamic quantum Fisher information in a limit case, which explicitly exhibits the Heisenberg scaling. By comparing with numerical results, we demonstrate that the general case can be well approximated by the analytical result when the coupling strength among the surrounding spins is much weaker than the coupling strength between the central and surrounding spins. This analytic result guides us to find the appropriate probe state and the proper measurement time to achieve the Heisenberg scaling in realistic situations. Furthermore, we investigate various effects which are important in practical quantum systems, including the central spin Zeeman term, the anisotropy of the hyperfine interaction and the inhomogeneity of the hyperfine coupling strength. Our result indicates that the dynamic quantum-enhanced sensing scheme seems feasible in realistic quantum central spin systems, like semiconductor quantum dots.

Quantum many-body spin rings coupled to ancillary spins: The sunburst quantum Ising model.

We study the ground-state properties of a quantum sunburst model, composed of a quantum Ising spin ring in a transverse field, symmetrically coupled to a set of ancillary isolated qubits, to maintain a residual translation invariance and also a Z_{2} symmetry. The large-size limit is taken in two different ways: either by keeping the distance between any two neighboring ancillary qubits fixed or by fixing their number while increasing the ring size. Substantially different regimes emerge, depending on the various Hamiltonian parameters: For small energy scale δ of the ancillary subsystem and small ring-qubit interaction κ, we observe rapid and nonanalytic changes in proximity to the quantum transitions of the Ising ring, both first order and continuous, which can be carefully controlled by exploiting renormalization-group and finite-size scaling frameworks. Smoother behaviors are instead observed when keeping δ>0 fixed and in the Ising disordered phase. The effect of an increasing number n of ancillary spins turns out to scale proportionally to sqrt[n] for sufficiently large values of n.

The exact solution of magnetic susceptibility for finite Ising ring with a magnetic impurity

We connected the two ends of a finite spin-1/2 antiferromagnetic Ising chain with a magnetic impurity at one end to form a closed ring, and studied its magnetic susceptibility exactly by using the transfer matrix method. We calculated the magnetic susceptibility in the whole temperature range and gave the phase diagram at ground state of the system about the anisotropy of the impurity and strength of the connection exchange interaction for spin-1 and -3/2 impurities. We also gave the ground state entropy of the system and derived the asymptotic expression of the magnetic susceptibility multiplied by temperature at the zero-temperature limit and the high-temperature limit. It was found that a degenerate phase may exist in some parameter region at zero temperature for the odd spin number system, and the ground state entropy is ln(2) in the nondegenerate phase and is dependent on the spin number in the degenerate phase. The magnetic susceptibility of the system at low temperature exhibits ferromagnetic behavior, and the Curie constant is related to the spin configuration at ground state. When the ground state is nondegenerate, the Curie constant is equal to the square of the net spin, regardless of the parity of the number of the spin. When the spin number is odd and the ground state is degenerate, the Curie constant may be related to the total spin number. At the high-temperature limit, the magnetic susceptibility multiplied by temperature is related to the spin quantum number of the impurity and the spin number in the ring.

Exact one- and two-site reduced dynamics in a finite-size quantum Ising ring after a quench: A semianalytical approach

We study the non-equilibrium dynamics of a homogeneous quantum Ising ring after a quench, in which the transverse field $g$ suddenly changes from zero to a nonzero value. The long-timescale reduced dynamics of a single spin and of two nearest-neighbor spins, which involves the evaluation of expectation values of odd operators that break the fermion parity, is exactly obtained for finite-size but large rings through the use of a recently developed Pfaffian method [N. Wu, Phys. Rev. E 101, 042108 (2020)]. Time dependence of the transverse and longitudinal magnetizations, single-spin purity, expectation value of the string operator $X_j=\prod^{j-1}_{l=1}\sigma^z_l\sigma^x_j$, several equal-time two-site correlators, and pairwise concurrence after quenches to different phases are numerically studied. Our main findings are that (i) The expectation value of a generic odd operator approaches zero in the long-time limit; (ii) $\langle X_j\rangle_t$ exhibits $j$-independent exponential decay for a quench to $g=1$ and the time at which $\langle X_j\rangle_t$ reaches its first maximum scales linearly with $j$; (iii) The single-spin purity dynamics is mainly controlled by $\langle\sigma^x_j\rangle_t$ ($\langle\sigma^z_j\rangle_t$) for a quench to $g<1$ ($g\geq 1$). For quenches to the disordered phase with $g\gg1$, the single-spin tends to be in the maximally mixed state and the transverse and longitudinal correlators $\langle\sigma^z_j\sigma^z_{j+1}\rangle_t$ and $\langle\sigma^x_j\sigma^x_{j+1}\rangle_t$ respectively approaches $-0.25$ and $0.5$ in the thermodynamic limit; (iv) The nearest-neighbor entanglement acquires a finite plateau value that increases with increasing $g$, and approaches a saturated value $\sim0.125$ for $g\gg1$.

Longitudinal magnetization dynamics in the quantum Ising ring: A Pfaffian method based on correspondence between momentum space and real space.

As perhaps the most studied paradigm for a quantum phase transition, the periodic quantum Ising chain is exactly solvable via the Jordan-Wigner transformation followed by a Fourier transform that diagonalizes the model in the momentum space of spinless fermions. Although the above procedures are well-known, there remain some subtle points to be clarified regarding the correspondence between the real-space and momentum-space representations of the finite-size quantum Ising ring, especially those related to fermion parities. In this work, we establish the relationship between the two fully aligned ferromagnetic states in real space and the two degenerate momentum-space ground states of the classical Ising ring, with the former being a special case of the factorized ground states of the more general XYZ model on the frustration-free hypersurface. Based on this observation, we then provide a Pfaffian formula for calculating real-time dynamics of the parity-breaking longitudinal magnetization with the system initially prepared in one of the two ferromagnetic states and under translationally invariant drivings. The formalism is shown to be applicable to large systems with the help of online programs for the numerical computation of the Pfaffian, thus providing an efficient method to numerically study, for example, the emergence of discrete time crystals in related systems.

Dynamical quantum phase transitions and non-Markovian dynamics

In the context of closed quantum systems, when a system prepared in its ground state undergoes a sudden quench, the resulting Loschmidt echo can exhibit zeros, resembling the Fisher zeros in the theory of classical equilibrium phase transitions. These zeros lead to nonanalytical behavior of the corresponding rate function, which is referred to as \textit{dynamical quantum phase transitions} (DQPTs). In this work, we investigate DQPTs in the context of open quantum systems that are coupled to both Markovian and non-Markovian dephasing baths via a conserved quantity. The general framework is corroborated by studying the non-equilibrium dynamics of a transverse-field Ising ring. We show the robustness of DQPT signatures under the action of both engineered dephasing baths, independently on how strongly they couple to the quantum system. Our theory provides insight on the effect of non-Markovian environments on DQPTs.

Ring frustration and factorizable correlation functions of critical spin rings.

Tackling the critical transverse Ising ring with or without ring frustration, we establish the concept of nonlocality in a many-body system in the thermodynamic limit by calculating the nonlocal factors embedded in the factorizable correlation function. Through this exactly solvable prototype, we clearly show the intriguing difference between the periodic chains with odd and even numbers of lattice sites even in the thermodynamic limit. In the context of nonlocality, we also address the important application of finite-size scaling analysis by numerically working out the nonlocal factors of the isotropic XY model and the spin-1/2 Heisenberg model.

Dynamic finite-size scaling after a quench at quantum transitions.

We present a general dynamic finite-size scaling theory for the quantum dynamics after an abrupt quench, at both continuous and first-order quantum transitions. For continuous transitions, the scaling laws are naturally ruled by the critical exponents and the renormalization-group dimension of the perturbation at the transition. In the case of first-order transitions, it is possible to recover a universal scaling behavior, which is controlled by the size behavior of the energy gap between the lowest-energy levels. We discuss these findings in the framework of the paradigmatic quantum Ising ring, and support the dynamic scaling laws by numerical evidence.

Quantum critical environment assisted quantum magnetometer

A central qubit coupled to an Ising ring of N qubits, operating close to a critical point is investigated as a potential precision quantum magnetometer for estimating an applied transverse magnetic field. We compute the quantum Fisher information for the central, probe qubit with the Ising chain initialized in its ground state or in a thermal state. The non-unitary evolution of the central qubit due to its interaction with the surrounding Ising ring enhances the accuracy of the magnetic field measurement. Near the critical point of the ring, Heisenberg-like scaling of the precision in estimating the magnetic field is obtained when the ring is initialized in its ground state. However, for finite temperatures, the Heisenberg scaling is limited to lower ranges of N values.

Out-of-equilibrium dynamics driven by localized time-dependent perturbations at quantum phase transitions

We investigate the quantum dynamics of many-body systems subject to local, i.e. restricted to a limited space region, time-dependent perturbations. If the perturbation drives the system across a quantum transition, an off-equilibrium behavior is observed, even when the perturbation is very slow. We show that, close to the transition, time-dependent quantities obey scaling laws. In first-order quantum transitions, the scaling behavior is universal, and some scaling functions can be exactly computed. For continuous quantum transitions, the scaling laws are controlled by the standard critical exponents and by the renormalization-group dimension of the perturbation at the transition. Our scaling approach is applied to the quantum Ising ring which presents both first-order and continuous quantum transitions.

The a-cycle problem for transverse Ising ring

Traditionally, the transverse Ising model is mapped to the fermionic c-cycle problem, which neglects the boundary effect due to thermodynamic limit. If persisting on a perfect periodic boundary condition, we can get a so-called a-cycle problem that has not been treated seriously so far (Lieb et al 1961 Ann. Phys. 16 407). In this work, we show a little surprising but exact result in this respect. We find the odevity of the number of lattice sites, N, in the a-cycle problem plays an unexpected role even in the thermodynamic limit, , due to the boundary constraint. We pay special attention to the system with , which is in contrast to the one with , because the former suffers a ring frustration. As a new effect, we find the ring frustration induces a low-energy gapless spectrum above the ground state. By proving a theorem for a new type of Toeplitz determinant, we demonstrate that the ground state in the gapless region exhibits a peculiar longitudinal spin–spin correlation. The entangled nature of the ground state is also disclosed by the evaluation of its entanglement entropy. At low temperature, new behavior of specific heat is predicted. We also propose an experimental protocol for observing the new phenomenon due to the ring frustration.

A 16-bit Coherent Ising Machine for One-Dimensional Ring and Cubic Graph Problems

Many tasks in our modern life, such as planning an efficient travel, image processing and optimizing integrated circuit design, are modeled as complex combinatorial optimization problems with binary variables. Such problems can be mapped to finding a ground state of the Ising Hamiltonian, thus various physical systems have been studied to emulate and solve this Ising problem. Recently, networks of mutually injected optical oscillators, called coherent Ising machines, have been developed as promising solvers for the problem, benefiting from programmability, scalability and room temperature operation. Here, we report a 16-bit coherent Ising machine based on a network of time-division-multiplexed femtosecond degenerate optical parametric oscillators. The system experimentally gives more than 99.6% of success rates for one-dimensional Ising ring and nondeterministic polynomial-time (NP) hard instances. The experimental and numerical results indicate that gradual pumping of the network combined with multiple spectral and temporal modes of the femtosecond pulses can improve the computational performance of the Ising machine, offering a new path for tackling larger and more complex instances.

Nonequilibrium statistical mechanics of a two-temperature Ising ring with conserved dynamics.

The statistical mechanics of a one-dimensional Ising model in thermal equilibrium is well-established, textbook material. Yet, when driven far from equilibrium by coupling two sectors to two baths at different temperatures, it exhibits remarkable phenomena, including an unexpected "freezing by heating." These phenomena are explored through systematic numerical simulations. Our study reveals complicated relaxation processes as well as a crossover between two very different steady-state regimes.

Experimental linear-optics simulation of multipartite non-locality in the ground state of a quantum Ising ring.

Critical phenomena involve structural changes in the correlations of its constituents. Such changes can be reproduced and characterized in quantum simulators able to tackle medium-to-large-size systems. We demonstrate these concepts by engineering the ground state of a three-spin Ising ring by using a pair of entangled photons. The effect of a simulated magnetic field, leading to a critical modification of the correlations within the ring, is analysed by studying two- and three-spin entanglement. In particular, we connect the violation of a multipartite Bell inequality with the amount of tripartite entanglement in our ring.

Non-Markovianity, Loschmidt echo, and criticality: A unified picture

A simple relationship between recently proposed measures of non-Markovianity and the Loschmidt echo is established, holding for a two-level system (qubit) undergoing pure dephasing due to a coupling with a many-body environment. We show that the Loschmidt echo is intimately related to the information flowing out from and occasionally back into the system. This, in turn, determines the non-Markovianity of the reduced dynamics. In particular, we consider a central qubit coupled to a quantum Ising ring in the transverse field. In this context, the information flux between system and environment is strongly affected by the environmental criticality; the qubit dynamics is shown to be Markovian exactly and only at the critical point. Therefore non-Markovianity is an indicator of criticality in the model considered here.

GLOBAL QUANTUM CORRELATIONS IN THE ISING MODEL

We study quantum correlations in an isotropic Ising ring under the effects of a transverse magnetic field. After characterizing the behavior of two-spin quantum correlations, we extend our analysis to global properties of the ring, using a figure of merit for quantum correlations that shows enough sensitivity to reveal the drastic changes in the properties of the system at criticality. This opens up the possibility to relate statistical properties of quantum many-body systems to suitably tailored measures of quantum correlations that capture features going far beyond standard quantum entanglement.

Dysprosium-based experimental representatives of an Ising-Heisenberg chain and a decorated Ising ring

It is shown that the bond-decorated Ising model is a realistic model for certain real magnetic compounds containing lanthanide ions. The lanthanide ion plays the role of Ising spin. The required conditions on the crystal-field spectrum of the lanthanide ion for the model to be valid are discussed and found to be in agreement with several recent ab initio calculations on Dy(III) centers. Similarities and differences between the spectra of the simple Ising chain and the decorated Ising chain are discussed and illustrated, with attention to level crossings in a magnetic field. The magnetic properties of two actual examples (a [DyCuMoCu] chain and a Dy4Cr4 ring) are obtained by a transfer-matrix solution of the decorated Ising model. g-factors of the metal ions are directly imported from ab initio results, while exchange coupling constants are fitted to experiment. Agreement with experiment is found to be satisfactory, provided one includes a correction (from ab initio results) for susceptibility and magnetization to account for the presence of excited Kramers doublets on Dy(III).