We study the non-equilibrium dynamics due to slowly taking a quasiperiodic Hamiltonian across its quantum critical point. The special quasiperiodic Hamiltonian that we study here has two different types of critical lines belonging to two different universality classes, one of them being the well known quantum Ising universality class. In this paper, we verify the Kibble Zurek scaling which predicts a power law scaling of the density of defects generated as a function of the rate of variation of the Hamiltonian. The exponent of this power law is related to the equilibrium critical exponents associated with the critical point crossed. We show that the power-law behavior is indeed obeyed when the two types of critical lines are crossed, with the exponents that are correctly predicted by Kibble Zurek scaling.
Low-temperature series are calculated for the free energy, magnetisation, susceptibility and field-derivatives of the susceptibility in the Ising model on the quasiperiodic Penrose lattice. The series are computed to order 20 and estimates of the critical exponents alpha, beta and gamma are obtained from Pade approximants.
We present here the non-equilibrium dynamics of the recently studied quasiperiodic Ising model. The zero temperature phase diagram of this model mainly consists of three phases, where each of these three phases can have extended, localized or critically delocalized low energy excited states. We explore the nature of excitations in these different phases by studying the evolution of entanglement entropy after performing quenches of different strengths to different phases. Our results on non-equilibrium dynamics of entanglement entropy are concurrent with the nature of excitations discussed in R…
Quasiperiodic modulation can prevent isolated quantum systems from equilibrating by localizing their degrees of freedom. In this article, we show that such systems can exhibit dynamically stable long-range orders forbidden in equilibrium. Specifically, we show that the interplay of symmetry breaking and localization in the quasiperiodic quantum Ising chain produces a \emph{quasiperiodic Ising glass} stable at all energy densities. The glass order parameter vanishes with an essential singularity at the melting transition with no signatures in the equilibrium properties. The zero temperature pha…
We study the q-dependent susceptibility chi(q) of a series of quasiperiodic Ising models on the square lattice. Several different kinds of aperiodic sequences of couplings are studied, including the Fibonacci and silver-mean sequences. Some identities and theorems are generalized and simpler derivations are presented. We find that the q-dependent susceptibilities are periodic, with the commensurate peaks of chi(q) located at the same positions as for the regular Ising models. Hence, incommensurate everywhere-dense peaks can only occur in cases with mixed ferromagnetic-antiferromagnetic interac…
We investigate zero-field Ising models on periodic approximants of planar quasiperiodic tilings by means of partition function zeros and high-temperature expansions. These are obtained by employing a determinant expression for the partition function. The partition function zeros in the complex temperature plane yield precise estimates of the critical temperature of the quasiperiodic model. Concerning the critical behaviour, our results are compatible with Onsager universality, in agreement with the Harris-Luck criterion based on scaling arguments.
We consider high-temperature expansions for the free energy of zero-field Ising models on planar quasiperiodic graphs. For the Penrose and the octagonal Ammann-Beenker tiling, we compute the expansion coefficients up to 18th order. As a by-product, we obtain exact vertex-averaged numbers of self-avoiding polygons on these quasiperiodic graphs. In addition, we analyze periodic approximants by computing the partition function via the Kac-Ward determinant. For the critical properties, we find complete agreement with the commonly accepted conjecture that the models under consideration belong to th…
The dynamics of a quantum phase transition is inextricably woven with the formation of excitations, as a result of the critical slowing down in the neighborhood of the critical point. We design a transitionless quantum driving through a quantum critical point that allows one to access the ground state of the broken-symmetry phase by a finite-rate quench of the control parameter. The method is illustrated in the one-dimensional quantum Ising model in a transverse field. Driving through the critical point is assisted by an auxiliary Hamiltonian, for which the interplay between the range of the i…
We propose a nonadiabatic approach to quantum annealing, in which we repeat quantum annealing in nonadiabatic time scales, and collect the final states of many realizations to find the ground state among them. In this way, we replace the diffculty of long annealing time in adiabatic quantum annealing by another problem of the number of nonsidabatic (short-time) trials. The one-dimensional transverse-field Ising model is used to test this idea, and it is shown that nonadiabatic quantum annealing has the same computational complexity to find the ground state as the conventional adiabatic anneali…
The strategy of shortcuts to adiabaticity enables us to realize adiabatic dynamics in finite time. In the counter-diabatic driving approach, an auxiliary Hamiltonian which is called the counter-diabatic Hamiltonian is appended to an original Hamiltonian to cancel out diabatic transitions. The counter-diabatic Hamiltonian is constructed by using the eigenstates of the original Hamiltonian. Therefore, it is in general difficult to construct the counter-diabatic Hamiltonian for quantum many-body systems. Even if the counter-diabatic Hamiltonian for quantum many-body systems is obtained, it is gen…
The spin-1/2 Ising octahedron and dodecahedron with a unique antiferromagnetic interaction display an outstanding magnetization jump at zero magnetic field, which consequently leads to a giant magnetocaloric effect during the adiabatic demagnetization. In the present work we report temperature dependences of two basic magnetocaloric response functions: the isothermal entropy change and the adiabatic change of temperature. It is shown that the Ising octahedron and dodecahedron generally exhibit a large negative isothermal entropy change upon increasing of the magnetic field, which serves in evi…
We present a quantitative semi-classical theory for the non-equilibrium dynamics of transverse Ising chains after quantum quenches, in particular sudden changes of the transverse field strength. We obtain accurate predictions for the quench dependent relaxation times and correlation lengths, and also about the recurrence times and quasiperiodicity of time dependent correlations in finite systems with open or periodic boundary conditions. We compare the quantiative predictions of our semi-classical theory (local magnetization, equal time bulk-bulk and surface-to-bulk correlations, and bulk auto…
Quasiperiodic systems are aperiodic but deterministic, so their critical behavior differs from that of clean systems as well as disordered ones. Quasiperiodic criticality was previously understood only in the special limit where the couplings follow discrete quasiperiodic sequences. Here we consider generic quasiperiodic modulations; we find, remarkably, that for a wide class of spin chains, generic quasiperiodic modulations flow to discrete sequences under a real-space renormalization group transformation. These discrete sequences are therefore fixed points of a \emph{functional} renormalizat…
We study the Biswas-Chatterjee-Sen (BCS) model, also known as the KCOD (Kinetic Continuous Opinion Dynamics) model on quasiperiodic lattices by using Kinetic Monte Carlo simulations and Finite Size Scaling technique. Our results are consistent with a continuous phase transition, controlled by an external noise. We obtained the order parameter $M$, defined as the averaged opinion, the fourth-order Binder cumulant $U$, and susceptibility $χ$ as functions of the noise parameter. We estimated the critical noises for Penrose, and Ammann-Beenker lattices. We also considered 7-fold and 9-fold quasipe…
We demonstrate that the invaded cluster algorithm, recently introduced by Machta et al, is a fast and reliable tool for determining the critical temperature and the magnetic critical exponent of periodic and aperiodic ferromagnetic Ising models in two dimensions. The algorithm is shown to reproduce the known values of the critical temperature on various periodic and quasiperiodic graphs with an accuracy of more than three significant digits. On two quasiperiodic graphs which were not investigated in this respect before, the twelvefold symmetric square-triangle tiling and the tenfold symmetric …
We propose a quantum Monte Carlo (QMC) algorithm for non-equilibrium dynamics in a system with a parameter varying as a function of time. The method is based on successive applications of an evolving Hamiltonian to an initial state and delivers results for a whole range of the tuning parameter in a single run, allowing for access to both static and dynamic properties of the system. This approach reduces to the standard Schrödinger dynamics in imaginary time for quasi-adiabatic evolutions, i.e., including the leading non-adiabatic correction to the adiabatic limit. We here demonstrate this quas…
We study the surface magnetization of aperiodic Ising quantum chains. Using fermion techniques, exact results are obtained in the critical region for quasiperiodic sequences generated through an irrational number as well as for the automatic binary Thue-Morse sequence and its generalizations modulo p. The surface magnetization exponent keeps its Ising value, beta_s=1/2, for all the sequences studied. The critical amplitude of the surface magnetization depends on the strength of the modulation and also on the starting point of the chain along the aperiodic sequence.
The system undergoes adiabatic evolution when its population in the instantaneous eigenbasis of its time-dependent Hamiltonian changes only negligibly. Realization of such dynamics requires slow-enough changes of the parameters of the Hamiltonian, a task that can be hard to achieve near quantum critical points. A powerful alternative is provided by the counterdiabatic modification of the Hamiltonian allowing for an arbitrarily quick implementation of the adiabatic dynamics. Such a counterdiabatic driving protocol has been recently proposed for the quantum Ising model [A. del Campo et al., Phys…
We study planar clusters consisting of loops including a Josephson $π$-junction ($π$-rings). Each $π$-ring carries a persistent current and behaves as a classical orbital moment. The type of particular state associated with the orientation of orbital moments at the cluster depends on the interaction between these orbital moments and can be easily controlled, i.e. by a bias current or by other means. We show that these systems can be described by the two-dimensional Ising model with competing nearest-neighbor and diagonal interactions and investigate the phase diagram of this model. The charact…
Magnetization process and adiabatic demagnetization of the antiferromagnetic Ising spin clusters with the shape of regular polyhedra (Platonic solids) are exactly examined within the framework of a simple graph-theoretical approach. While the Ising cube as the only unfrustrated (bipartite) spin cluster shows just one trivial plateau at zero magnetization, the other regular Ising polyhedra (tetrahedron, octahedron, icosahedron and dodecahedron) additionally display either one or two intermediate plateaux at fractional values of the saturation magnetization. The nature of highly degenerate groun…
Energy minimization of Ising spin-glasses has played a central role in statistical and solid-state physics, facilitating studies of phase transitions and magnetism. Recent proposals suggest using Ising spin-glasses for non-traditional computing as a way to harness the nature's ability to find min-energy configurations, and to take advantage of quantum tunneling to boost combinatorial optimization. Laboratory demonstrations have been unconvincing so far and lack a non-quantum baseline for definitive comparisons. In this work we (i) design and evaluate new computational techniques to simulate na…
We study analytically and numerically quench dynamics and defects formation in the quantum Ising model in the presence of a time-dependent transverse magnetic field. We generalize the Landau-Ziner formula to the case of non-adiabatic evolution of the quantum system. For a quasi-static magnetic field, with a slow dependence on time, our outcomes are similar to the results predicted by the Landau-Zener formula. However, a quench dynamics under a shock-wave load is more complicated. The final state of the system depends on the amplitude and pulse velocity, resulting in the mixture of ground and e…
We investigate the entanglement dynamics between two distant qubits by analyzing correlations in the quantum Ising model. Starting from the spin system in a paramagnetic regime enforced by the external magnetic field $B$, we then switch on the ferromagnetic spin-spin coupling $J$. Using the large coordination number expansion, we consider two limiting switching regimes: (1) adiabatic, which monitors the evolution of the ground state through the quantum transition to an ordered state; and (2) instantaneous (quench) which monitors instead the propagation of quantum fluctuations and simulates the…
Ising spin Hamiltonians are often used to encode a computational problem in their ground states. Quantum Annealing (QA) computing searches for such a state by implementing a slow time-dependent evolution from an easy-to-prepare initial state to a low energy state of a target Ising Hamiltonian of quantum spins, $H_I$. Here, we point to the existence of an analytical solution for such a problem for an arbitrary $H_I$ beyond the adiabatic limit for QA. This solution provides insights into the accuracy of nonadiabatic computations. Our QA protocol in the pseudo-adiabatic regime leads to a monotoni…
We study the phase diagram of a class of models in which a generalized cluster interaction can be quenched by Ising exchange interaction and external magnetic field. We characterize the various phases through winding numbers. They may be ordinary phases with local order parameter or exotic ones, known as symmetry protected topologically ordered phases. Quantum phase transitions with dynamical critical exponents z = 1 or z = 2 are found. Quantum phase transitions are analyzed through finite-size scaling of the geometric phase accumulated when the spins of the lattice perform an adiabatic preces…
Motivated by the quantum adiabatic algorithm (QAA), we consider the scaling of the Hamiltonian gap at quantum first order transitions, generally expected to be exponentially small in the size of the system. However, we show that a quantum antiferromagnetic Ising chain in a staggered field can exhibit a first order transition with only an algebraically small gap. In addition, we construct a simple classical translationally invariant one-dimensional Hamiltonian containing nearest-neighbour interactions only, which exhibits an exponential gap at a thermodynamic quantum first-order transition of e…
We investigated the Ising model on a square lattice with ferro and antiferromagnetic interactions modulated by the quasiperiodic Octonacci sequence in both directions of the lattice. We have applied the Replica Exchange Monte Carlo (Parallel Tempering) technique to calculate the thermodynamic quantities of the system. We obtained the order parameter, the associated magnetic susceptibility ($χ$) and the specific heat $(c)$ in order to characterize the universality class of the phase transition. Also, we use the finite size scaling method to obtain the critical temperature of the system and the …
We introduce and study the adiabatic dynamics of free-fermion models subject to a local Lindblad bath and in the presence of a time-dependent Hamiltonian. The merit of these models is that they can be solved exactly, and will help us to study the interplay between non-adiabatic transitions and dissipation in many-body quantum systems. After the adiabatic evolution, we evaluate the excess energy (average value of the Hamiltonian) as a measure of the deviation from reaching the target final ground state. We compute the excess energy in a variety of different situations, where the nature of the b…
We study the nonequilibrium dynamics of a quasiperiodic quantum Ising chain after a sudden change in the strength of the transverse field at zero temperature. In particular we consider the dynamics of the entanglement entropy and the relaxation of the magnetization. The entanglement entropy increases with time as a power-law, and the magnetization is found to exhibit stretched-exponential relaxation. These behaviors are explained in terms of anomalously diffusing quasiparticles, which are studied in a wave packet approach. The nonequilibrium magnetization is shown to have a dynamical phase tra…
Quantum many body systems with long range interactions are known to display many fascinating phenomena experimentally observable in trapped ions, Rydberg atoms and polar molecules. Among these are dynamical phase transitions which occur after an abrupt quench in spin chains with interactions decaying as $r^{-α}$ and whose critical dynamics depend crucially on the power $α$: for systems with $α<1$ the transition is sharp while for $α>1$ it fans out in a chaotic crossover region. In this paper we explore the fate of critical dynamics in Ising chains with long-range interactions when the transver…
We study temporal behavior of a quantum system under a slow external perturbation, which drives the system across a second order quantum phase transition. It is shown that despite the conventional adiabaticity conditions are always violated near the critical point, the number of created excitations still goes to zero in the limit of infinitesimally slow variation of the tuning parameter. It scales with the adiabaticity parameter as a power related to the critical exponents $z$ and $ν$ characterizing the phase transition. We support general arguments by direct calculations for the Boson Hubbard…
We show that the symmetry-breaking gap of the quantum Ising model in the transverse field can be extracted from free evolution of the longitudinal magnetization taking place after a gradual quench of the magnetic field. We perform for this purpose numerical simulations of the Ising chains with either periodic or open boundaries. We also study the condition for adiabaticity of evolution of the longitudinal magnetization finding excellent agreement between our simulations and the prediction based on the Kibble-Zurek theory of non-equilibrium phase transitions. Our results should be relevant for …
In this paper we consider one-dimensional classical and quantum spin-1=2 quasiperiodic Ising chains, with two-valued nearest neighbor interaction modulated by a Fibonacci substitution sequence on two letters. In the quantum case, we investigate the energy spectrum of the Ising Hamiltonian, in presence of constant transverse magnetic field, by employing the techniques that were developed in our previous work. In the classical case, we investigate and prove analyticity of the free energy function when the magnetic field, together with interaction strength couplings, is modulated by the same Fibo…
The Kibble-Zurek mechanism predicts the formation of topological defects and other excitations that quantify how much a quantum system driven across a quantum critical point fails to be adiabatic. We point out that, thanks to the divergent linear susceptibility at the critical point, even a tiny symmetry breaking bias can restore the adiabaticity. The minimal required bias scales like $τ_Q^{-βδ/(1+zν)}$, where $β,δ,z,ν$ are the critical exponents and $τ_Q$ is a quench time. We test this prediction by DMRG simulations of the quantum Ising chain. It is directly applicable to the recent emulation…
A quantum phase transition from paramagnetic to ferromagnetic phase is driven by a time-dependent external magnetic field. For any rate of the transition the evolution is non-adiabatic and finite density of defects is excited in the ferromagnetic state. The density of excitations has only logarithmic dependence on the transition rate. This is much weaker than any usual power law scaling predicted for pure systems by the Kibble-Zurek mechanism.
We investigate the dissipative loss in the $\pm J$ Ising spin glass in three dimensions through the scaling of the hysteresis area, for a maximum magnetic field that is equal to the saturation field. We perform a systematic analysis for the whole range of the bond randomness as a function of the sweep rate, by means of frustration-preserving hard-spin mean field theory. Data collapse within the entirety of the spin-glass phase driven adiabatically (i.e., infinitely-slow field variation) is found, revealing a power-law scaling of the hysteresis area as a function of the antiferromagnetic bond f…
The Kibble-Zurek mechanism demands an initial adiabatic stage before an impulse stage to have a frozen correlation length that generates topological defects in a cooling phase transition. Here we study such a driven critical dynamics but with an initial condition that is near the critical point and that is far away from equilibrium. In this case, there is no initial adiabatic stage at all and thus adiabaticity is broken. However, we show that there again exists a finite length scale arising from the driving that divides the evolution into three stages. A relaxation--finite-time scaling--adiaba…
Let a general quantum many-body system at a low temperature adiabatically cross through the vicinity of the system's quantum critical point. We show that the system's temperature is significantly suppressed due to both the entropy majorization theorem in quantum information science and the entropy conservation law in adiabatic processes. We take the one-dimensional transverse-field Ising model and spinless fermion system as concrete examples to show that the inverse temperature might become divergent around their critical points. Since the temperature is a measurable quantity in experiments, o…
Here we first discuss briefly the quantum annealing technique. We then study the quantum annealing of Sherrington-Kirkpatrick spin glass model with the tuning of both transverse and longitudinal fields. Both the fields are time-dependent and vanish adiabatically at the same time, starting from high values. We solve, for rather small systems, the time-dependent Schrodinger equation of the total Hamiltonian by employing a numerical technique. At the end of annealing we obtain the final state having high overlap with the exact ground state(s) of classical spin glass system (obtained independently…
The purpose of this work is to understand the effect of an external environment on the adiabatic dynamics of a quantum critical system. By means of scaling arguments we derive a general expression for the density of excitations produced in the quench as a function of its velocity and of the temperature of the bath. We corroborate the scaling analysis by explicitly solving the case of a one-dimensional quantum Ising model coupled to an Ohmic bath.
Metastability is a quintessential feature of first order quantum phase transitions, which is lost either by dynamical instability or by nucleating bubbles of a true vacuum through quantum tunneling. By considering a drive across the first order quantum phase transition in the quantum Ising chain in the presence of both transverse and longitudinal fields, we reveal multiple regions in the parameter space where the initial metastable state loses its metastability in successive stages. The mechanism responsible is found to be semi-degenerate resonant tunnelings to states with specific bubble size…
The Kibble-Zurek mechanism captures universality when a system is driven through a continuous phase transition. Here we study the dynamical aspect of quantum phase transitions in the Ising Field Theory where the critical point can be crossed in different directions in the two-dimensional coupling space leading to different scaling laws. Using the Truncated Conformal Space Approach, we investigate the microscopic details of the Kibble-Zurek mechanism in a genuinely interacting field theory. We demonstrate dynamical scaling in the non-adiabatic time window and provide analytic and numerical evid…
By using frustration-preserving hard-spin mean-field theory, we investigated the phase transition dynamics in the three-dimensional field-free $\pm J$ Ising spin glass model. As the temperature $T$ is decreased from paramagnetic phase at high temperatures, with a rate $ω=-dT/dt$ in time $t$, the critical temperature depends on the cooling rate through a clear power-law $ω^a$. With increasing antiferromagnetic bond fraction $p$, the exponent $a$ increases for the transition into the ferromagnetic case for $p<p_\text{c}$, and decreases for the transition into the spin glass phase for $p>p_\text{…
The magnetic and magnetocaloric properties of the mixed spin-(1/2,1) Ising-Heisenberg model on a two-leg ladder with dimer-rung alternation are exactly examined under an adiabatic demagnetization process using the transfer-matrix formalism. We notify that the magnetization curve of the model exhibits plateaux as a function of the applied magnetic field and cyclic four-spin Ising interaction at certain rational fractions of the saturation value. We precisely investigate the ability of cooling/heating of the model nearby the critical points at which discontinuous ground-state phase transition oc…
New effects in the frustrated transverse Ising ring are predicted. The system is solved based on a mapping of Pauli spin operators to the Jordan-Wigner fermions. We group the low-lying energy levels into bands after imposing appropriate parity constraint, which projects out the redundant degrees of freedom brought about by the Jordan-Wigner fermions. In the region of strong antiferromagnetic coupling, we uncover an unusual gapless phase induced by the ring frustration. We demonstrate that its ground state exhibits a strong longitudinal spin-spin correlation and possesses a considerably large e…
