XX Model Research Articles

Floquet prethermal order by disorder

Frustrated magnets can have accidental ground state degeneracies which may be lifted by various forms of disorder, for example in the form of thermal or quantum fluctuations. This order by disorder (ObD) paradigm is well established in equilibrium and here is generalized to Floquet many-body systems. Investigating a periodically driven XXZ-compass model on the square lattice, we show that in a prethermal regime, dynamical fluctuations induced by high-frequency drives select a discrete set of states out of a degenerate ground state manifold of the lowest order Floquet Hamiltonian chosen as initial states. Remarkably, prior to the ObD selection, an unusual fluctuating regime emerges leading to a prethermalization timescale scaling linearly with the drive frequency. We argue that prethermal ObD with its unusual approach to the selected states is a generic phenomenon of driven frustrated systems and confirm it in the paradigmatic J1−J2 XX model. Published by the American Physical Society 2024

Efficiency of the non-maximally entangled quantum Otto engine

We generalized the maximally entangled Hamiltonian of the isotropic Heisenberg XX model with two spin-1/2 particles to the case of non-maximal entanglement parametrized by the degree of non-maximality 0 < n < 1. The thermal concurrence as the function of the degree of non-maximality in the entanglement at different values of the exchange parameter is calculated. We analyzed the efficiency of the quantum Otto engine for different scenarios of exchange parameter regimes and the degree of maximal entanglement between the spin states. We showed that it is possible to run the quantum Otto engine using non-maximally entangled states and achieve higher efficiencies by controlling the exchange parameter value with respect to a critical value J c and this feature also depends on the degree of non-maximal entanglement n ≠ 1. This result may be useful for harnessing the non-maximal entanglement in quantum heat engine for optimizing the operation of quantum devices interacting with heat bath or environment.

Temporal fluctuations of correlators in integrable and chaotic quantum systems

We provide bounds on temporal fluctuations around the infinite-time average of out-of-time-ordered and time-ordered correlators of many-body quantum systems without energy gap degeneracies. For physical initial states, our bounds predict the exponential decay of the temporal fluctuations as a function of the system size. We numerically verify this prediction for chaotic and interacting integrable spin-1/2 chains, which satisfy the assumption of our bounds. On the other hand, we show analytically and numerically that for the XX model, which is a noninteracting system with gap degeneracies, the temporal fluctuations decay polynomially with system size for operators that are local in the fermion representation and decrease exponentially in the system size for non-local operators. Our results demonstrate that the decay of the long-time temporal fluctuations of correlators cannot be used as a reliable metric of chaos or lack thereof.

Integrability and quench dynamics in the spin-1 central spin XX model

Central spin models provide an idealized description of interactions between a central degree of freedom and a mesoscopic environment of surrounding spins. We show that the family of models with a spin-1 at the center and XX interactions of arbitrary strength with surrounding spins is integrable. Specifically, we derive an extensive set of conserved quantities and obtain the exact eigenstates using the Bethe ansatz. As in the homogenous limit, the states divide into two exponentially large classes: bright states, in which the spin-1 is entangled with its surroundings, and dark states, in which it is not. On resonance, the bright states further break up into two classes depending on their weight on states with central spin polarization zero. These classes are probed in quench dynamics wherein they prevent the central spin from reaching thermal equilibrium. In the single spin-flip sector we explicitly construct the bright states and show that the central spin exhibits oscillatory dynamics as a consequence of the semilocalization of these eigenstates. We relate the integrability to the closely related class of integrable Richardson-Gaudin models, and conjecture that the spin-s central spin XX model is integrable for any s.

Dynamical quantum phase transitions in Stark quantum spin chains

We investigate the nonequilibrium dynamics of one-dimension spin models in the presence of a uniform force. The linear potential induces delocalization-localization transition in the free particles model which is known as the Wannier-Stark effect. We study dynamical quantum phase transition (DQPT) due to sudden global quenches across a quantum critical point when the system undergoes a localization-delocalization transition. In this regard, we consider the XX and XXZ spin chains and explore two types of quenches with and without ramping through the delocalization-localization point. The XX model was mapped to the free fermion particles, so both analytical and numerical results were provided. Results unveil that the dynamical signature of localization-delocalization transition can be characterized by the nonanalyticities in dynamical free energy (corresponds to the zero points in the Loschmit echo). We also explore the interaction effects considering XXZ spin chains, using the time-dependent extension of the numerical DMRG technique. Our results show that depending on the anisotropic parameter $\Delta\lessgtr1.0$, if both the initial and post-quench Hamiltonian are in the same phase or not, DQPTs may happen. Moreover, the interrelation between DQPTs with different correlation measures such as the equilibrium order parameters or entanglement entropy production of the system remains unclear. We provide more analyses on the feature of DQPTs, in both types of quenches, by connecting them to the average local magnetization, entanglement entropy production, and the Schmidt gap.

Nontrivial effect of dephasing: Enhancement of rectification of spin current in graded XX chains.

In order to reveal mechanisms to control and manipulate spin currents, we perform a detailed investigation of the dephasing effects in the open XX model with a Lindblad dynamics involving global dissipators and thermal baths. Specifically, we consider dephasing noise modeled by current-preserving Lindblad dissipators acting on graded versions of these spin systems, that is, systems in which the magnetic field and/or the spin interaction are growing (decreasing) along the chain. In our analysis, we study the nonequilibrium steady state via the covariance matrix using the Jordan-Wigner approach to compute the spin currents. We find that the interplay between dephasing and graded systems gives rise to a nontrivial behavior: When we have homogeneous magnetic field and graded interactions we have rectification enhancement mechanisms, and when we have fully graded systems we can control the spin current in order to keep the direction of the particle and/or spin flow even with inverted baths. We describe our result in detailed numerical analysis and we see that rectification in this simple model indicates that the phenomenon may generally occur in quantum spin systems.

Relaxation of imbalance in a disordered XX model with on-site dephasing

The relaxation of observables to their non-equilibrium steady states in a disordered XX chain subjected to dephasing at every site has been intensely studied in recent years. We comprehensively analyze the relaxation of staggered magnetization, i.e., imbalance, in such a system, starting from the N\'eel initial state. We analytically predict emergence of several timescales in the system and extract results which match with large-system numerics without any extra fitting parameter until a universal timescale. An often reported stretched exponential decay is just one of the regimes which holds in a finite window of time and is therefore in fact not a true stretched exponential decay. Subsequently, the asymptotic decay of imbalance is governed by a power law irrespective of the disorder. We show that this emerges from the continuum limit of the low magnitude eigenspectrum of the Liouvillian. However, for finite systems, due to discreteness of the spectrum, the final phase of relaxation is governed by the relevant smallest Liouvillian gap.

Orbital antiferroelectricity and higher dimensional magnetoelectric order in the spin-1/2 XX chain extended with three-spin interactions

We study the spin-1/2 XX model extended with three-spin interactions of the XZX+YZY and XZY-YZX types. We solve the model exactly and obtain the ground state phase diagram as a function of the two three-spin coupling strengths. We show that even in absence of external electric and magnetic fields there is a phase which exhibits spontaneous magnetoelectric order when both XZX+YZY and XZY-YZX interactions are present. Specifically, in this regime, we show that there exists not only a non-zero magnetization and a scalar chirality but also a vector chiral order. Further, we show the existence of a plaquette vector chirality, or circulating chiral spin current loops, in the plaquettes n, n+1, n+2 with the sense of the current being opposite in adjacent plaquettes. Analogous to charge current loops giving rise to orbital magnetic dipole moments, the circulating spin current loops give rise to orbital electric dipole moments - a novel orbital antiferroelectricity. We characterize this phase by a higher-dimensional scalar and vector toroidal order. Such a novel phase with higher dimensional order arises because of the non-trivial topological connectivity resulting from the presence of both the three-spin interactions. We also study the combined effect of both types of three-spin interactions on the entanglement entropy.

Super-exponential distinguishability of correlated quantum states

In the problem of asymptotic binary i.i.d. state discrimination, the optimal asymptotics of the type I and the type II error probabilities is in general an exponential decrease to zero as a function of the number of samples; the set of achievable exponent pairs is characterized by the quantum Hoeffding bound theorem. A super-exponential decrease for both types of error probabilities is only possible in the trivial case when the two states are orthogonal and hence can be perfectly distinguished using only a single copy of the system. In this paper, we show that a qualitatively different behavior can occur when there is correlation between the samples. Namely, we use gauge-invariant and translation-invariant quasi-free states on the algebra of the canonical anti-commutation relations to exhibit pairs of states on an infinite spin chain with the properties that (a) all finite-size restrictions of the states have invertible density operators and (b) the type I and the type II error probabilities both decrease to zero at least with the speed e^{-nclog n} with some positive constant c, i.e., with a super-exponential speed in the sample size n. Particular examples of such states include the ground states of the XX model corresponding to different transverse magnetic fields. In fact, we prove our result in the setting of binary composite hypothesis testing, and hence, it can be applied to prove super-exponential distinguishability of the hypotheses that the transverse magnetic field is above a certain threshold vs. that it is below a strictly lower value.

Variational quantum simulation of thermal statistical states on a superconducting quantum processer

Quantum computers promise to solve finite-temperature properties of quantum many-body systems, which is generally challenging for classical computers due to high computational complexities. Here, we report experimental preparations of Gibbs states and excited states of Heisenberg XX and XXZ models by using a 5-qubit programmable superconducting processor. In the experiments, we apply a hybrid quantum–classical algorithm to generate finite temperature states with classical probability models and variational quantum circuits. We reveal that the Hamiltonians can be fully diagonalized with optimized quantum circuits, which enable us to prepare excited states at arbitrary energy density. We demonstrate that the approach has a self-verifying feature and can estimate fundamental thermal observables with a small statistical error. Based on numerical results, we further show that the time complexity of our approach scales polynomially in the number of qubits, revealing its potential in solving large-scale problems.

Critical quantum thermometry and its feasibility in spin systems

In this work, we study temperature sensing with finite-sized strongly correlated systems exhibiting quantum phase transitions. We use the quantum Fisher information (QFI) approach to quantify the sensitivity in the temperature estimation, and apply a finite-size scaling framework to link this sensitivity to critical exponents of the system around critical points. We numerically calculate the QFI around the critical points for two experimentally-realizable systems: the spin-1 Bose-Einstein condensate and the spin-chain Heisenberg XX model in the presence of an external magnetic field. Our results confirm finite-size scaling properties of the QFI. Furthermore, we discuss experimentally-accessible observables that (nearly) saturate the QFI at the critical points for these two systems.

Algebraic Bethe Circuits

The Algebraic Bethe Ansatz (ABA) is a highly successful analytical method used to exactly solve several physical models in both statistical mechanics and condensed-matter physics. Here we bring the ABA into unitary form, for its direct implementation on a quantum computer. This is achieved by distilling the non-unitary R matrices that make up the ABA into unitaries using the QR decomposition. Our algorithm is deterministic and works for both real and complex roots of the Bethe equations. We illustrate our method on the spin-12 XX and XXZ models. We show that using this approach one can efficiently prepare eigenstates of the XX model on a quantum computer with quantum resources that match previous state-of-the-art approaches. We run small-scale error-mitigated implementations on the IBM quantum computers, including the preparation of the ground state for the XX and XXZ models on 4 sites. Finally, we derive a new form of the Yang-Baxter equation using unitary matrices, and also verify it on a quantum computer.

Emergent entanglement structures and self-similarity in quantum spin chains.

We introduce an experimentally accessible network representation for many-body quantum states based on entanglement between all pairs of its constituents. We illustrate the power of this representation by applying it to a paradigmatic spin chain model, the XX model, and showing that it brings to light new phenomena. The analysis of these entanglement networks reveals that the gradual establishment of quasi-long range order is accompanied by a symmetry regarding single-spin concurrence distributions, as well as by instabilities in the network topology. Moreover, we identify the existence of emergent entanglement structures, spatially localized communities enforced by the global symmetry of the system that can be revealed by model-agnostic community detection algorithms. The network representation further unveils the existence of structural classes and a cyclic self-similarity in the state, which we conjecture to be intimately linked to the community structure. Our results demonstrate that the use of tools and concepts from complex network theory enables the discovery, understanding and description of new physical phenomena even in models studied for decades. This article is part of the theme issue 'Emergent phenomena in complex physical and socio-technical systems: from cells to societies'.

Scaling of the finite-size effect of the α -Rényi entropy in disjointed intervals under dilation

The $\alpha$-R\'enyi entropy in the gapless models have been obtained by the conformal field theory, which is exact in the thermodynamic limit. However, the calculation of its finite size effect (FSE) is challenging. So far only the FSE in a single interval in the XX model has been understood and the FSE in the other models and in the other conditions are totally unknown. Here we report the FSE of this entropy in disjointed intervals $A = \cup_i A_i$ under a uniform dilation $\lambda A$ in the XY model, showing of a universal scaling law as \begin{equation*} \Delta_{\lambda A}^\alpha = \Delta_A^\alpha \lambda^{-\eta} \mathcal{B}(A, \lambda), \end{equation*} where $|\mathcal{B}(A, \lambda)| \le 1$ is a bounded function and $\eta = \text{min}(2, 2/\alpha)$ when $\alpha < 10$. We verify this relation in the phase boundaries of the XY model, in which the different central charges correspond to the physics of free Fermion and free Boson models. We find that in the disjointed intervals, two FSEs, termed as extrinsic FSE and intrinsic FSE, are required to fully account for the FSE of the entropy. Physically, we find that only the edge modes of the correlation matrix localized at the open ends $\partial A$ have contribution to the total entropy and its FSE. Our results provide some incisive insight into the entanglement entropy in the many-body systems.

Observation of Thermalization and Information Scrambling in a Superconducting Quantum Processor.

Understanding various phenomena in nonequilibrium dynamics of closed quantum many-body systems, such as quantum thermalization, information scrambling, and nonergodic dynamics, is crucial for modern physics. Using a ladder-type superconducting quantum processor, we perform analog quantum simulations of both the XX-ladder model and the one-dimensional XX model. By measuring the dynamics of local observables, entanglement entropy, and tripartite mutual information, we signal quantum thermalization and information scrambling in the XX ladder. In contrast, we show that the XX chain, as free fermions on a one-dimensional lattice, fails to thermalize to the Gibbs ensemble, and local information does not scramble in the integrable channel. Our experiments reveal ergodicity and scrambling in the controllable qubit ladder, and open the door to further investigations on the thermodynamics and chaos in quantum many-body systems.

Unveiling Quantum Entanglement in Many-Body Systems from Partial Information

Quantum entanglement is commonly assumed to be a central resource for quantum computing and quantum simulation. Nonetheless, the capability to detect it in many-body systems is severely limited by the absence of sufficiently scalable and flexible certification tools. This issue is particularly critical in situations where the structure of entanglement is a priori unknown, and where one cannot rely on existing entanglement witnesses. Here, we implement a scheme in which the knowledge of the mean value of arbitrary observables can be used to probe multipartite entanglement in a scalable, certified, and systematic manner. Specifically, we rely on positive semidefinite conditions, independent of partial-transposition-based criteria, necessarily obeyed if the data can be reproduced by a separable state. The violation of any of these conditions yields a specific entanglement witness, tailored to the data of interest, revealing the salient features of the data which are impossible to reproduce without entanglement. We validate this approach by probing theoretical many-body states of several hundreds of qubits relevant to existing experiments: a single-particle quench in a one-dimensional XX chain; a many-body quench in a two-dimensional XX model with 1/r3 interactions; and thermal equilibrium states of Heisenberg and transverse-field Ising chains. In all cases, these investigations have led us to discover new entanglement witnesses, some of which could be characterized analytically, generalizing existing results in the literature. In summary, our paper introduces a flexible data-driven entanglement detection technique for uncharacterized quantum many-body states, of immediate relevance to experiments in a quantum advantage regime.Received 11 August 2021Revised 2 December 2021Accepted 22 February 2022DOI:https://doi.org/10.1103/PRXQuantum.3.010342Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasEntanglement detectionQuantum entanglementPhysical SystemsQuantum spin modelsTechniquesData analysisSemiclassical methodsQuantum InformationAtomic, Molecular & OpticalStatistical Physics

Integrable spin chain with Hilbert space fragmentation and solvable real-time dynamics.

We revisit the so-called folded XXZ model, which was treated earlier by two independent research groups. We argue that this spin-1/2 chain is one of the simplest quantum integrable models, yet it has quite remarkable physical properties. The particles have constant scattering lengths, which leads to a simple treatment of the exact spectrum and the dynamics of the system. The Hilbert space of the model is fragmented, leading to exponentially large degeneracies in the spectrum, such that the exponent depends on the particle content of a given state. We provide an alternative derivation of the Hamiltonian and the conserved charges of the model, including an alternative interpretation of the so-called "dual model" considered earlier. We also construct a nonlocal map that connects the model with the Maassarani-Mathieu spin chain, also known as the SU(3) XX model. We consider the exact solution of the model with periodic and open boundary conditions, and also derive multiple descriptions of the exact thermodynamics of the model. We consider quantum quenches of different types. In one class of problems the dynamics can be treated relatively easily: we compute an example for the real-time dependence of a local observable. In another class of quenches the degeneracies of the model lead to the breakdown of equilibration, and we argue that they can lead to persistent oscillations. We also discuss connections with the TT[over ¯] and hard rod deformations known from quantum field theories.

Conjectures on hidden Onsager algebra symmetries in interacting quantum lattice models

We conjecture the existence of hidden Onsager algebra symmetries in two interacting quantum integrable lattice models, i.e. spin-1/2 XXZ model and spin-1 Zamolodchikov-Fateev model at arbitrary root of unity values of the anisotropy. The conjectures relate the Onsager generators to the conserved charges obtained from semi-cyclic transfer matrices. The conjectures are motivated by two examples which are spin-1/2 XX model and spin-1 U(1)-invariant clock model. A novel construction of the semi-cyclic transfer matrices of spin-1 Zamolodchikov-Fateev model at arbitrary root of unity values of the anisotropy is carried out via the transfer matrix fusion procedure.

Corrections to the singlet-length distribution and the entanglement entropy in random singlet phases

We consider random singlet phases of spin-$\frac{1}{2}$, random, antiferromagnetic spin chains, in which the universal leading-order divergence $\frac{\ln 2}{3}\ln\ell$ of the average entanglement entropy of a block of $\ell$ spins, as well as the closely related leading term $\frac{2}{3}l^{-2}$ in the distribution of singlet lengths are well known by the strong-disorder renormalization group (SDRG) method. Here, we address the question of how large the subleading terms of the above quantities are. By an analytical calculation performed along a special SDRG trajectory of the random XX chain, we identify a series of integer powers of $1/l$ in the singlet-length distribution with the subleading term $\frac{4}{3}l^{-3}$. Our numerical SDRG analysis shows that, for the XX fixed point, the subleading term is generally $O(l^{-3})$ with a non-universal coefficient and also reveals terms with half-integer powers: $l^{-7/2}$ and $l^{-5/2}$ for the XX and XXX fixed points, respectively. We also present how the singlet lengths originating in the SDRG approach can be interpreted and calculated in the XX chain from the one-particle states of the equivalent free-fermion model. These results imply that the subleading term next to the logarithmic one in the entanglement entropy is $O(\ell^{-1})$ for the XX fixed point and $O(\ell^{-1/2})$ for the XXX fixed point with non-universal coefficients. For the XX model, where a comparison with exact diagonalization is possible, the order of the subleading term is confirmed but we find that the SDRG fails to provide the correct non-universal coefficient.

Superstatistics and quantum entanglement in the isotropic spin-1/2 XX dimer from a nonadditive thermodynamics perspective.

In this paper, the impact of temperature fluctuations in the entanglement of two qubits described by a spin-1/2 XX model is studied. To describe the out-of-equilibrium situation, superstatistics is used with fluctuations given by a χ^{2}-distribution function, and its free parameters are chosen in such a way that resembles the nonadditive Tsallis thermodynamics. In order to preserve the Legendre structure of the thermal functions, particular energy constraints are imposed on the density operator and the internal energy. Analytical results are obtained using an additional set of constraints after a parametrization of the physical temperature. We show that the well-known parametrization may lead to undesirable values of the physical temperature so that by analyzing the entropy as a function of energy, the correct values are found. The quantum entanglement is obtained from the concurrence and is compared with the case when the Tsallis restrictions are not imposed on the density operator.