One-dimensional XY Research Articles

The quantum XY chain with boundary fields: finite-size gap and phase behavior

Abstract We present a detailed study of the finite-size one-dimensional quantum XY chain in a transverse field in the presence of boundary fields coupled with the order-parameter spin operator. We consider fields located at the chain boundaries that have the same strength and that are oppositely aligned. We derive exact expressions for the gap Δ as a function of the model parameters for large values of the chain length L. These results allow us to characterize the nature of the ordered phases of the model. We find a magnetic (M) phase ( Δ ∼ e − a L ), a magnetic-incommensurate (MI) phase ( Δ ∼ e − a L f MI ( L ) ), a kink (K) phase ( Δ ∼ L − 2 ) and a kink-incommensurate (KI) phase ( Δ ∼ L − 2 f KI ( L ) ); f MI ( L ) and f KI ( L ) are bounded oscillating functions of L. We also analyze the behavior along the phase boundaries. In particular, we characterize the universal crossover behavior across the K-KI phase boundary. On this boundary, the dynamic critical exponent is z = 4, i.e. Δ ∼ L − 4 for large values of L.

Controlling Energy Storage Crossing Quantum Phase Transitions in an Integrable Spin Quantum Battery.

We investigate the performance of a one-dimensional dimerized XY chain as a spin quantum battery. Such integrable model shows a rich quantum phase diagram that emerges through a mapping of the spins onto auxiliary fermionic degrees of freedom. We consider a charging protocol relying on the double quench of an internal parameter, namely the strength of the dimerization, and address the energy stored in the systems. We observe three distinct regimes, depending on the timescale characterizing the duration of the charging: a short-time regime related to the dynamics of the single dimers, a long-time regime related to the recurrence time of the system at finite size, and a thermodynamic limit time regime. In the latter, the energy stored is almost unaffected by the charging time and the precise values of the charging parameters, provided the quench crosses a quantum phase transition. Finally, we analytically prove that the three-timescale behavior and the strong dependence of the energy stored on the quantum phase diagram also hold in the quantum Ising chain in a transverse field. Our results can play a relevant role in the design of stable solid-state quantum batteries.

Spectral crossovers in non-Hermitian spin chains: Comparison with random matrix theory.

We present a systematic investigation of the short-range spectral fluctuation properties of three non-Hermitian spin-chain Hamiltonians using complex spacing ratios (CSRs). Specifically, we focus on the non-Hermitian variants of the standard one-dimensional anisotropic XY model having intrinsic rotation-time (RT) symmetry that has been explored analytically by Zhang and Song [Phys. Rev. A 87, 012114 (2013)1050-294710.1103/PhysRevA.87.012114]. The corresponding Hermitian counterpart is also exactly solvable and has been widely employed as a toy model in several condensed matter physics problems. We show that the presence of a random field along the x direction together with the one along the z direction facilitates integrability and RT-symmetry breaking, leading to the emergence of quantum chaotic behavior. This is evidenced by a spectral crossover closely resembling the transition from Poissonian to Ginibre unitary ensemble (GinUE) statistics of random matrix theory. Additionally, we consider two phenomenological random matrix models in this paper to examine 1D Poisson to GinUE and 2D Poisson to GinUE crossovers and the associated signatures in CSRs. Here 1D and 2D Poisson correspond to real and complex uncorrelated levels, respectively. These crossovers reasonably capture spectral fluctuations observed in the spin-chain systems within a certain range of parameters.

Low-temperature transport in the XY chain with staggered Dzyaloshinskii–Moriya interaction

The influence of staggered Dzyaloshinskii–Moriya (DM) interaction as well as external magnetic fields on transport properties of the spin-1/2 one-dimensional XY model with spatially modulated DM interaction which has been a complex topic in recent years is investigated. The longitudinal spin conductivity is calculated employing the fermionization approach, where we analyze the effect of the strength of DM interaction on continuum spin conductivity σreg(ω). In addition, the behavior of the Drude’s weight as a function of temperature T is presented, indicating a spin super-current behavior for all T.

Fixed Depth Hamiltonian Simulation via Cartan Decomposition.

Simulating quantum dynamics on classical computers is challenging for large systems due to the significant memory requirements. Simulation on quantum computers is a promising alternative, but fully optimizing quantum circuits to minimize limited quantum resources remains an open problem. We tackle this problem by presenting a constructive algorithm, based on Cartan decomposition of the Lie algebra generated by the Hamiltonian, which generates quantum circuits with time-independent depth. We highlight our algorithm for special classes of models, including Anderson localization in one-dimensional transverse field XY model, where O(n^{2})-gate circuits naturally emerge. Compared to product formulas with significantly larger gate counts, our algorithm drastically improves simulation precision. In addition to providing exact circuits for a broad set of spin and fermionic models, our algorithm provides broad analytic and numerical insight into optimal Hamiltonian simulations.

Many-body localization transition in a frustrated XY chain

We show evidence of many-body localization (MBL) transition in a one-dimensional isotropic XY chain with a weak next-nearest-neighbor frustration in a random magnetic field. We perform finite-size exact diagonalization calculations of level-spacing statistics and fractal dimensions to characterize the MBL transition with increasing the random field amplitude. An equivalent representation of the model in terms of spinless fermions explains the presence of the delocalized phase by the appearance of an effective nonlocal interaction between the fermions. This interaction appears due to frustration provided by the next-nearest-neighbor hopping.

Quantum Coherence and Skew Information with Quantum Phase Transition in One-Dimensional Anisotropic XY Model under Renormalization Group Method

We investigate the behaviors of Wigner–Yanase skew information and the l1 norm, as two different measures of quantum coherence, in a tripartite ground state of the anisotropic spin-1/2 XY model. Fo...

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.

Quantum quench dynamics in XY spin chain with ferromagnetic and antiferromagnetic interactions

In this manuscript we investigate the one-dimensional anisotropic XY model with ferromagnetic and antiferromagnetic interactions, which gives more interesting phase diagram and dynamic critical behaviors. By using quantum renormalization-group method, we find that there are three phases in the system: antiferromagnetic Ising phase ordered in “x direction”, spin-fluid phase and ferromagnetic Ising phase ordered in “y direction”. In order to study the dynamical critical behaviors of the system, two quantum quenching methods are used. In both cases, the concurrence, a measure of entanglement, oscillates periodically over time. It if found that in the both cases the periods are the same and can be used as a order parameter for quantum phase transitions. It is also found that there is a scaling behavior for the evolution period and a reciprocal relation for the evolution period exponent, θ and correlation length exponent, ν.

Floquet Gauge Pumps as Sensors for Spectral Degeneracies Protected by Symmetry or Topology.

We introduce the concept of a Floquet gauge pump whereby a dynamically engineered Floquet Hamiltonian is employed to reveal the inherent degeneracy of the ground state in interacting systems. We demonstrate this concept in a one-dimensional XY model with periodically driven couplings and transverse field. In the high-frequency limit, we obtain the Floquet Hamiltonian consisting of the static XY and dynamically generated Dzyaloshinsky-Moriya interaction (DMI) terms. The dynamically generated magnetization current depends on the phases of complex coupling terms, with the XY interaction as the real and DMI as the imaginary part. As these phases are cycled, the current reveals the ground-state degeneracies that distinguish the ordered and disordered phases. We discuss experimental requirements needed to realize the Floquet gauge pump in a synthetic quantum spin system of interacting trapped ions.

Distribution of RKKY coupling value in 1D crystal with disorder. Specific heat in XY model

The presence of disorder in one-dimensional crystals leads to the localization of all charge carriers and the calculation of the indirect exchange interaction (Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction) cannot be performed perturbatively on disorder. In two and three-dimensional systems it makes sense to calculate the magnitude of RKKY interaction perturbatively treating nearly free carriers scattering on the random potential, and this approach results in a rather high magnitude of the exchange interaction due to interference effects similar to weak localization. We show that in one-dimensional systems the indirect exchange interaction should be described as a random value with heavy-tail distribution function, which is calculated in this work, on scales of carriers localization length. We also demonstrate that heavy tails and the absence of a characteristic value of RKKY interaction magnitude leads to a significant change in observables for these systems. We calculate a specific heat for the one-dimensional XY model taking into account the effect of disorder and assuming that typical distance between impurities exceeds the localization length. In contrast to an ideal system, where specific heat temperature dependence has a peak at a certain temperature proportional to exchange constant describing characteristic energy scale, disorder eliminates the peak as soon as there is no characteristic excitation energy in this case anymore.

Ballistic propagation of a local impact in the one-dimensional XY model

Light-cone-like propagation of information is a universal phenomenon of nonequilibrium dynamics of integrable spin systems. In this paper, we investigate propagation of a local impact in the one-dimensional XY model with the anisotropy γ in a magnetic field h by calculating the magnetization profile. Applying a local and instantaneous unitary operation to the ground state, which we refer to as the local-impact protocol, we numerically observe various types of light-cone-like propagation in the parameter region 0 ⩽ γ ⩽ 1 and 0 ⩽ h ⩽ 2 of the model. By combining numerical integration with an asymptotic analysis, we find the following: (i) for |h| ⩾ |1 − γ 2| except for the case on the line h = 1 with , a wave front propagates with the maximum group velocity of quasiparticles, except for the case γ = 1 and 0 < h < 1, in which there is no clear wave front; (ii) for |h| < |1 − γ 2| as well as on the line h = 1 with , a second wave front appears owing to multiple local extrema of the group velocity; (iii) for |h| = |1 − γ 2|, edges of the second wave front collapses at the origin, and as a result, the magnetization profile exhibits a ridge at the impacted site. Furthermore, we find by an asymptotic analysis that the height of the wave front decays in a power law in time t with various exponents depending on the model parameters: the wave fronts exhibit a power-law decay t −2/3 except for the line h = 1, on which the decay can be given by either ∼t −3/5 or ∼t −1; the ridge at the impacted site for |h| = |1 − γ 2| shows the decay t −1/2 as opposed to the decay t −1 in other cases.

Quantum coherence and criticality in irreversible work

The irreversible work during a driving protocol constitutes one of the most widely studied measures in non-equilibrium thermodynamics, as it constitutes a proxy for entropy production. In quantum systems, it has been shown that the irreversible work has an additional, genuinely quantum mechanical contribution, due to coherence produced by the driving protocol. The goal of this paper is to explore this contribution in systems that undergo a quantum phase transition. Substantial effort has been dedicated in recent years to understand the role of quantum criticality in work protocols. However, practically nothing is known about how coherence contributes to it. To shed light on this issue, we study the entropy production in infinitesimal quenches of the one-dimensional XY model. For quenches in the transverse field, we find that for finite temperatures the contribution from coherence can, in certain cases, account for practically all of the entropy production. At low temperatures, however, the coherence presents a finite cusp at the critical point, whereas the entropy production diverges logarithmically. Alternatively, if the quench is performed in the anisotropy parameter, we find that there are situations where all of the entropy produced is due to quantum coherences.

Out-of-time-order correlators in the one-dimensional XY model

We study the behavior of information spreading in the XY model, using out-of-time-order correlators (OTOCs). The effects of anisotropic parameter γ and external magnetic field λ on OTOCs are studied in detail within thermodynamical limits. The universal form which characterizes the wavefront of information spreading still holds in the XY model. The butterfly speed vB depends on (γ, λ). At a fixed location, the early-time evolution behavior of OTOCs agrees with the results of the Hausdorff–Baker–Campbell expansion. For long-time evolution, OTOCs with local operators decay as for power law t−1, but those with nonlocal operators show different and nontrivial power law behaviors. We also observe temperature dependence for OTOCs when (γ = 0, λ = 1). At low temperature, the OTOCs with nonlocal operators show divergence over time.

Quantum Hopfield Model

We find the free-energy in the thermodynamic limit of a one-dimensional XY model associated to a system of N qubits. The coupling among the σ i z is a long range two-body random interaction. The randomness in the couplings is the typical interaction of the Hopfield model with p patterns ( p &lt; N ), where the patterns are p sequences of N independent identically distributed random variables (i.i.d.r.v.), assuming values ± 1 with probability 1 / 2 . We show also that in the case p ≤ α N , α ≠ 0 , the free-energy is asymptotically independent from the choice of the patterns, i.e., it is self-averaging.

Correlations between Complexity and Entanglement in a One-Dimensional XY Model

Many people believe that the study of complex quantum systems may be simplified by first analyzing the static and dynamic entanglement present in those systems [Phys. Rev. A 66 (2002) 032110]. In this paper, we attempt to complement such notion by adding an order–disorder quantifier called statistical complexity and studying how it is correlated with the degree of entanglement as measured by the concurrence quantifier. We perform such an analysis with reference to a representative system chosen from condensed matter theory, the so-called X Y model. Some interesting insight is obtained as the concurrence and the complexity become correlated in an unexpected fashion.

Repeated Interaction Processes in the Continuous-Time Limit, Applied to Quadratic Fermionic Systems

We study a class of Lindblad equation on finite-dimensional fermionic systems. The model is obtained as the continuous-time limit of a repeated interaction process between fermionic systems with quadratic Hamiltonians, a setup already used by Platini and Karevski for the one-dimensional XY model. We prove a necessary and sufficient condition for the convergence to a unique stationary state, which is similar to the Kalman criterion in control theory. Several examples are treated, including a spin chain with interactions at both ends.

Transition asymptotics of Toeplitz determinants and emergence of Fisher–Hartwig representations

We compute the transition asymptotics (double-scaling limits) of Toeplitz determinants generated by symbols f t possessing Fisher–Hartwig singularities. The symbols f t that we consider depend on a parameter t such that f t has one Fisher–Hartwig singularity when t > 0 and two Fisher–Hartwig singularities when t = 0. Unlike in the other studies of the transition asymptotics of Toeplitz determinants, our setting involves the emergence of Fisher–Hartwig representations as . We use the Riemann–Hilbert problem for orthogonal polynomials and its connection to Painlevé transcendents to obtain the asymptotics. We apply our results to study a special correlator known as the emptiness formation probability (EFP) for the one-dimensional anisotropic XY spin-1/2 chain in a transverse magnetic field, and describe its transition between different regions in the phase diagram across critical lines.

Markovian and Non-Markovian Dynamics in the One-Dimensional Transverse-Field XY Model

We consider an anisotropic spin-1/2 XY Heisenberg chain in the presence of a transverse magnetic field. Selecting the nearest neighbor pair spins as an open quantum system, the rest of the chain plays the role of the structured environment. In fact, the aforementioned system is used as a quantum probe signifying nontrivial features of the environment with which is interacting. We use a general measure that is based on the trace distance for the degree of non-Markovian behavior in open quantum systems. The witness of non-Markovianity takes on nonzero values whenever there is a flow of information from the environment back to the open system. We have shown that the dynamics of the system with isotropic Heisenberg interaction is Markovian. A dynamical transition into the non-Markovian regime is observed as soon as the anisotropy, $$\gamma $$ , is applied. At the Ising value of the anisotropy $$\gamma =1.0$$ , all the information flows back from the environment to the system. The additional dynamical transition from the non-Markovian to the Markovian is obtained by applying the transverse magnetic field. In addition, we have focused on the time evolution of the Loschmidt-echo return rate function. It is found that a non-analyticity can be seen in the time evolution of the Loschmidt-echo return rate function exactly at the critical points where a dynamical transition from the Markovian to the non-Markovian occurs.

Phase Diagrams of One-Dimensional Ising and XY Models with Fully Connected Ferromagnetic and Anti-Ferromagnetic Quantum Fluctuations

We analytically investigated the phase diagrams of one-dimensional Ising and XY models. A second-order phase transition occurs in both models only with the transverse magnetic field. In this study,...