Quantum Ising Model In Transverse Field Research Articles

Learning topological defects formation with neural networks in a quantum phase transition

Neural networks possess formidable representational power, rendering them invaluable in solving complex quantum many-body systems. While they excel at analyzing static solutions, nonequilibrium processes, including critical dynamics during a quantum phase transition, pose a greater challenge for neural networks. To address this, we utilize neural networks and machine learning algorithms to investigate time evolutions, universal statistics, and correlations of topological defects in a one-dimensional transverse-field quantum Ising model. Specifically, our analysis involves computing the energy of the system during a quantum phase transition following a linear quench of the transverse magnetic field strength. The excitation energies satisfy a power-law relation to the quench rate, indicating a proportional relationship between the excitation energy and the kink numbers. Moreover, we establish a universal power-law relationship between the first three cumulants of the kink numbers and the quench rate, indicating a binomial distribution of the kinks. Finally, the normalized kink-kink correlations are also investigated and it is found that the numerical values are consistent with the analytic formula.

Shared purity and concurrence of a mixture of ground and low-lying excited states as indicators of quantum phase transitions

We investigate the efficacy of shared purity, a measure of quantum correlation that is independent of the separability-entanglement paradigm, as a quantum phase transition indicator in comparison with concurrence, a bipartite entanglement measure. The order parameters are investigated for thermal states and pseudo-thermal states, of the systems considered. In the case of the one-dimensional J 1 − J 2 Heisenberg quantum spin model and the one-dimensional transverse-field quantum Ising model, shared purity turns out to be as effective as concurrence in indicating quantum phase transitions. In the two-dimensional J 1 − J 2 Heisenberg quantum spin model, shared purity indicates the two quantum phase transitions present in the model, while concurrence detects only one of them. Moreover, we find diverging finite-size scaling exponents for the order parameters near the transitions in odd- and even-sized systems governed by the one-dimensional J 1 − J 2 model, as had previously been reported for quantum spins on odd- and even-legged ladders. It is plausible that the divergence is related to a Möbius strip-like boundary condition required for odd-sized systems, while for even-sized systems, the usual periodic boundary condition is sufficient.

Large Deviations beyond the Kibble-Zurek Mechanism.

The Kibble-Zurek mechanism (KZM) predicts that the average number of topological defects generated upon crossing a continuous or quantum phase transition obeys a universal scaling law with the quench time. Fluctuations in the defect number near equilibrium are approximately of Gaussian form, in agreement with the central limit theorem. Using large deviations theory, we characterize the universality of fluctuations beyond the KZM and report the exact form of the rate function in the transverse-field quantum Ising model. In addition, we characterize the scaling of large deviations in an arbitrary continuous phase transition, building on recent evidence establishing the universality of the defect number distribution.

The singularities of the rate function of quantum coherent work in one-dimensional transverse field Ising model

Quantum coherence will undoubtedly play a fundamental role in understanding of the dynamics of quantum many-body systems, thereby to reveal its genuine contribution is of great importance. In this paper, we specialize our discussions to the one-dimensional transverse field quantum Ising model initialized in the coherent Gibbs state. After quenching the strength of the transverse field, the effects of quantum coherence are studied by the rate function of quantum work distribution. We find that quantum coherence not only recovers the quantum phase transition destroyed by thermal fluctuations, but also generates some entirely new singularities both in the static state and dynamics. It can be manifested that these singularities are rooted in spin flips causing the sudden change of the domain boundaries of spin polarization. This work sheds new light on the fundamental connection between quantum critical phenomena and quantum coherence.

Fidelity Mechanics: Analogues of the Four Thermodynamic Laws and Landauer's Principle.

Fidelity mechanics is formalized as a framework for investigating critical phenomena in quantum many-body systems. Fidelity temperature is introduced for quantifying quantum fluctuations, which, together with fidelity entropy and fidelity internal energy, constitute three basic state functions in fidelity mechanics, thus enabling us to formulate analogues of the four thermodynamic laws and Landauer’s principle at zero temperature. Fidelity flows, which are irreversible, are defined and may be interpreted as an alternative form of renormalization group flows. Thus, fidelity mechanics offers a means to characterize both stable and unstable fixed points: divergent fidelity temperature for unstable fixed points and zero-fidelity temperature and (locally) maximal fidelity entropy for stable fixed points. In addition, fidelity entropy behaves differently at an unstable fixed point for topological phase transitions and at a stable fixed point for topological quantum states of matter. A detailed analysis of fidelity mechanical-state functions is presented for six fundamental models—the quantum spin- XY model, the transverse-field quantum Ising model in a longitudinal field, the quantum spin- XYZ model, the quantum spin- XXZ model in a magnetic field, the quantum spin-1 XYZ model, and the spin- Kitaev model on a honeycomb lattice for illustrative purposes. We also present an argument to justify why the thermodynamic, psychological/computational, and cosmological arrows of time should align with each other, with the psychological/computational arrow of time being singled out as a master arrow of time.

From Ising Model to Kitaev Chain

In this general article, we map the one-dimensional transverse field quantum Ising model of ferromagnetism to Kitaev's one-dimensional p-wave superconductor, which has its application in fault-tolerant topological quantum computing. Mapping Pauli's spin operators of transverse Ising chain to spinless fermionic creation and annihilation operators by Inverse Jordan-Wigner transformation leads to a Hamiltonian form closely related to Kitaev Chain, which exhibits topological phase transition where phases are characterized by different topological invariant that changes discontinuously at the transition point. Kitaev Chain supports two Majorana zero modes (MZMs) in the non-trivial topological phase, while none is in the trivial phase. The doubly degenerate ground state of the transverse Ising in ferromagnetic phase corresponds to non-local free fermion degree made from MZMs. The quasi-particle excitations of Ising chain, viz., domain wall formation in the ferromagnetic phase and spin-flip in paramagnetic phase maps to Bogoliubon excitations. The mapping suggests that a non-local order parameter can be defined for Kitaev Chain to work with the usual paradigm of Landau's theory.

Quantum many-body scars in transverse field Ising ladders and beyond

We identify quantum many-body scars in the transverse field quantum Ising model on a ladder. We make explicit how the corresponding (mid spectrum, low entanglement) many-body eigenstates can be approximated by injecting quasi-particle excitations into an exact, zero-energy eigenstate, which is of valence bond solid type. Next, we present a systematic construction of product states characterized, in the limit of a weak transverse field, by a sharply peaked local density of states. We describe how the construction of these "peak states" generalizes to arbitrary dimension and show that on the ladder their number scales with system size as the square of the golden ratio.

Loschmidt-amplitude wave function spectroscopy and the physics of dynamically driven phase transitions

We introduce the Loschmidt amplitude as a powerful tool to perform spectroscopy of generic many-body wave functions and use it to interrogate the wave function obtained after ramping the transverse field quantum Ising model through its quantum critical point. Previous results are confirmed and a more complete understanding of the population of defects and of the effects of magnon-magnon interaction or finite-size corrections is obtained. The influence of quantum coherence is clarified.

Dynamics of longitudinal magnetization in transverse-field quantum Ising model: from symmetry-breaking gap to Kibble–Zurek mechanism

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 ongoing cold atom and ion experiments targeting either equilibrium or dynamical aspects of quantum phase transitions. They could be also useful for benchmarking D-Wave machines.

Effect of Fluctuation in the Coupling Strength on Critical Dynamics of 1D Transverse Field Quantum Ising Model

The effect of uniformly distributed fluctuation in the coupling strength of a 100 spins transverse field quantum Ising chain is studied using the Density Matrix Renormalization Group on the Matrix Product States formalism. Uniform noise with mean zero and amplitude η is introduced to uniform nearest neighbour coupling of value unity. Disordered averages of thermodynamic quantities are calculated from 100 disordered realizations for each transverse field reading. We show that the system exhibits distinct behaviour of ordered and disordered state separated at η~1.0. For η ≲ 1.0, thermodynamic behaviour of the system gradually deviates from pure quantum Ising model. For η > 1.0 system exhibits highly fluctuating thermodynamic behaviour. Edward-Anderson order parameters are calculated and shown to be much less fluctuating and suitable as an order parameter for η > 1.0. Qualitative behaviour of the system is not affected by finite-size effect and replacing the fluctuation with normally distributed noise. Finally, for noise level between η = 1.0 and 1.5, the system exhibits a faster phase transition and enhances transverse magnetization before the critical point. For certain ranges of noise amplitude before the critical point, the quantum fluctuations are amplified. This suggests a potential improvement of quantum annealing within that regime.

Critical Dynamics of Transverse-field Quantum Ising Model using Finite-Size Scaling and Matrix Product States

The study of phase transition is usually done by numerical simulation of finite system. Conventional methods such as Monte Carlo simulations and phenomenological renormalization group methods obtain the critical exponents without obtaining the quantum wavefunction of the system. The Matrix Product States formalism allows one to obtain accurate numerical wavefunctions of short ranged interacting quantum many-body systems. In this study we combine the Finite Size Scaling theory and Matrix Product States formalism to study the critical dynamics of one-dimensional quantum Ising model. Finite size simulations of 20, 40, 60, 80, 100 and 120 spins are done using the Density Matrix Renormalization Group to obtain the ground state wavefunction of the system. The thermodynamic quantities such as the magnetization, susceptibility and correlation function are calculated. The critical exponents independently calculated are respectively β/ν = 0.1235(1), γ/ν = 1.7351(2), and η = 0.249(1). They conform with the theoretical values from analytical solution and fulfil the hyperscaling relation. We showed that both methods combined can reliably study the critical dynamics of one-dimensional Ising-like quantum lattice systems. Application of the study on water-ice phase transition of single-file water in nanopores is proposed.

Orthogonality Catastrophe in Dissipative Quantum Many-Body Systems.

We present an analog of the phenomenon of orthogonality catastrophe in quantum many-body systems subject to a local dissipative impurity. We show that the fidelity F(t), giving a measure for distance of the time-evolved state from the initial one, displays a universal scaling form F(t)∝t^{θ}e^{-γt}, when the system supports long-range correlations, in a fashion reminiscent of traditional instances of orthogonality catastrophe in condensed matter. An exponential falloff at rate γ signals the onset of environmental decoherence, which is critically slowed down by the additional algebraic contribution to the fidelity. This picture is derived within a second-order cumulant expansion suited for Liouvillian dynamics, and substantiated for the one-dimensional transverse field quantum Ising model subject to a local dephasing jump operator, as well as for XY and XX quantum spin chains, and for the two-dimensional Bose gas deep in the superfluid phase with local particle heating. Our results hint that local sources of dissipation can be used to inspect real-time correlations and to induce a delay of decoherence in open quantum many-body systems.

Effective metal-insulator nonequilibrium quantum phase transition in the Su-Schrieffer-Heeger model

We consider the steady state behavior of observables in the Su-Schrieffer-Heeger model and in the one dimensional transverse field quantum Ising model after a sudden quantum quench of the parameter controlling the gap. In the thermodynamic limit, and for times $t\rightarrow\infty$, we find non-analyticities even in simple local observables as a function of the quench parameter, that is, a non-equilibrium quantum phase transition. We trace the appearance of this non-equilibrium quantum phase transition to an effective metal-insulator transition which occurs on the level of the generalized Gibbs Ensemble (describing the steady-state of the equilibrated system). Studying whether these transitions are robust, we find, in the paradigmatic case of the SSH model, that they persist for both quantum quench protocols of finite duration in time as well as thermal initial states, while they are washed out in the presence of fermion-fermion interactions and for finite system size.

Quantum neural networks to simulate many-body quantum systems

We conduct experimental simulations of many body quantum systems using a \emph{hybrid} classical-quantum algorithm. In our setup, the wave function of the transverse field quantum Ising model is represented by a restricted Boltzmann machine. This neural network is then trained using variational Monte Carlo assisted by a D-Wave quantum sampler to find the ground state energy. Our results clearly demonstrate that already the first generation of quantum computers can be harnessed to tackle non-trivial problems concerning physics of many body quantum systems.

Universal Statistics of Topological Defects Formed in a Quantum Phase Transition.

When a quantum phase transition is crossed in finite time, critical slowing down leads to the breakdown of adiabatic dynamics and the formation of topological defects. The average density of defects scales with the quench rate following a universal power law predicted by the Kibble-Zurek mechanism. We analyze the full counting statistics of kinks and report the exact kink number distribution in the transverse-field quantum Ising model. Kink statistics is described by the Poisson binomial distribution with all cumulants exhibiting a universal power law scaling with the quench rate. In the absence of finite-size effects, the distribution approaches a normal one, a feature that is expected to apply broadly in systems described by the Kibble-Zurek mechanism.

Multipartite entanglement, quantum coherence, and quantum criticality in triangular and Sierpiński fractal lattices.

We investigate the quantum phase transitions of the transverse-field quantum Ising model on the triangular lattice and Sierpiński fractal lattices by employing the multipartite entanglement and quantum coherence along with the quantum renormalization group method. It is shown that the quantum criticalities of these high-dimensional models closely relate to the behaviors of the multipartite entanglement and quantum coherence. As the thermodynamic limit is approached, the first derivatives of the multipartite entanglement and quantum coherence exhibit singular behaviors, and the consistent finite-size scaling behaviors for each lattice are also obtained from the first derivatives. The multipartite entanglement and quantum coherence are demonstrated to be good indicators for detecting the quantum phase transitions in the triangular lattice and Sierpiński fractal lattices. Furthermore, the dimensions determine the relations between the critical exponents and the correlation length exponents for these lattices.

Eigenstate thermalization in systems with spontaneously broken symmetry.

A strongly nonintegrable system is expected to satisfy the eigenstate thermalization hypothesis, which states that the expectation value of an observable in an energy eigenstate is the same as the thermal value. This must be revised if the observable is an order parameter for a spontaneously broken symmetry, which has multiple thermal values. We propose that in this case the system is unstable towards forming nearby eigenstates which yield each of the allowed thermal values. We provide strong evidence for this from a numerical study of the two-dimensional transverse-field quantum Ising model.

Projected entangled pair states at finite temperature: Iterative self-consistent bond renormalization for exact imaginary time evolution

A projected entangled pair state (PEPS) with ancillas can be evolved in imaginary time to obtain thermal states of a strongly correlated quantum system on a two-dimensional lattice. Every application of a Suzuki-Trotter gate multiplies the PEPS bond dimension $D$ by a factor $k$. It has to be renormalized back to the original $D$. In order to preserve the accuracy of the Suzuki-Trotter (ST) decomposition, the renormalization in principle has to take into account full environment made of the new tensors with the bond dimension $k\ifmmode\times\else\texttimes\fi{}D$. Here, we propose a self-consistent renormalization procedure operating with the original bond dimension $D$, but without compromising the accuracy of the ST decomposition. The iterative procedure renormalizes the bond using full environment made of renormalized tensors with the bond dimension $D$. After every renormalization, the new renormalized tensors are used to update the environment, and then the renormalization is repeated again and again until convergence. As a benchmark application, we obtain thermal states of the transverse field quantum Ising model on a square lattice, both infinite and finite, evolving the system across a second-order phase transition at finite temperature.

Quantum phase transition of the transverse-field quantum Ising model on scale-free networks.

I investigate the quantum phase transition of the transverse-field quantum Ising model in which nearest neighbors are defined according to the connectivity of scale-free networks. Using a continuous-time quantum Monte Carlo simulation method and the finite-size scaling analysis, I identify the quantum critical point and study its scaling characteristics. For the degree exponent λ=6, I obtain results that are consistent with the mean-field theory. For λ=4.5 and 4, however, the results suggest that the quantum critical point belongs to a non-mean-field universality class. Further simulations indicate that the quantum critical point remains mean-field-like if λ>5, but it continuously deviates from the mean-field theory as λ becomes smaller.

Quantum critical behavior of the quantum Ising model on fractal lattices

I study the properties of the quantum critical point of the transverse-field quantum Ising model on various fractal lattices such as the Sierpiński carpet, Sierpiński gasket, and Sierpiński tetrahedron. Using a continuous-time quantum Monte Carlo simulation method and finite-size scaling analysis, I identify the quantum critical point and investigate its scaling properties. Among others, I calculate the dynamic critical exponent and find that it is greater than one for all three structures. The fact that it deviates from one is a direct consequence of the fractal structures not being integer-dimensional regular lattices. Other critical exponents are also calculated. The exponents are different from those of the classical critical point and satisfy the quantum scaling relation, thus confirming that I have indeed found the quantum critical point. I find that the Sierpiński tetrahedron, of which the dimension is exactly 2, belongs to a different universality class than that of the two-dimensional square lattice. I conclude that the critical exponents depend on more details of the structure than just the dimension and the symmetry.