Strong and weak thermalization of infinite nonintegrable quantum systems.

When a nonintegrable system evolves out of equilibrium for a long time, local observables are in general expected to attain stationary expectation values, independent of the details of the initial state. But the thermalization of a closed quantum system is not yet well understood. Here we show that it presents indeed a much richer phenomenology than its classical counterpart. Using a new numerical technique, we identify two distinct regimes, strong and weak, occurring for different initial states. Strong thermalization, intrinsically quantum, happens when instantaneous local expectation values converge to the thermal ones. Weak thermalization, well known in classical systems, shows convergence to thermal values only after time averaging. Remarkably, we find a third group of states showing no thermalization, neither strong nor weak, to the time scales one can reliably simulate.

Quantum adiabatic computation with a constant gap is not useful in one dimension.

We show that it is possible to use a classical computer to efficiently simulate the adiabatic evolution of a quantum system in one dimension with a constant spectral gap, starting the adiabatic evolution from a known initial product state. The proof relies on a recently proven area law for such systems, implying the existence of a good matrix product representation of the ground state, combined with an appropriate algorithm to update the matrix product state as the Hamiltonian is changed. This implies that adiabatic evolution with such Hamiltonians is not useful for universal quantum computation. Therefore, adiabatic algorithms which are useful for universal quantum computation either require a spectral gap tending to zero or need to be implemented in more than one dimension (we leave open the question of the computational power of adiabatic simulation with a constant gap in more than one dimension).

Matrix product states for dynamical simulation of infinite chains.

We propose a new method for computing the ground state properties and the time evolution of infinite chains based on a transverse contraction of the tensor network. The method does not require finite size extrapolation and avoids explicit truncation of the bond dimension along the evolution. By folding the network in the time direction prior to contraction, time-dependent expectation values and dynamic correlation functions can be computed after much longer evolution time than with any previous method. Moreover, the algorithm we propose can be used for the study of some noninvariant infinite chains, including impurity models.

Inference from matrix products: a heuristic spin-glass algorithm.

We present an algorithm for finding ground states of two-dimensional spin-glass systems based on ideas from matrix product states in quantum information theory. The algorithm works directly at zero temperature and defines an approximation to the energy whose accuracy depends on a parameter k. We test the algorithm against exact methods on random field and random bond Ising models, and we find that accurate results require a k which scales roughly polynomially with the system size. The algorithm also performs well when tested on small systems with arbitrary interactions, where no fast, exact algorithms exist. The time required is significantly less than Monte Carlo schemes.

Community detection as an inference problem.

We express community detection as an inference problem of determining the most likely arrangement of communities. We then apply belief propagation and mean-field theory to this problem, and show that this leads to fast, accurate algorithms for community detection.

Lieb-Robinson bounds and the generation of correlations and topological quantum order.

The Lieb-Robinson bound states that local Hamiltonian evolution in nonrelativistic quantum mechanical theories gives rise to the notion of an effective light cone with exponentially decaying tails. We discuss several consequences of this result in the context of quantum information theory. First, we show that the information that leaks out to spacelike separated regions is negligible and that there is a finite speed at which correlations and entanglement can be distributed. Second, we discuss how these ideas can be used to prove lower bounds on the time it takes to convert states without topological quantum order to states with that property. Finally, we show that the rate at which entropy can be created in a block of spins scales like the boundary of that block.

Systematic series expansions for processes on networks.

We use series expansions to study dynamics of equilibrium and nonequilibrium systems on networks. This analytical method enables us to include detailed nonuniversal effects of the network structure. We show that even low order calculations produce results which compare accurately to numerical simulation, while the results can be systematically improved. We show that certain commonly accepted analytical results for the critical point on networks with a broad degree distribution need to be modified in certain cases due to disassortativity; the present method is able to take into account the assortativity at sufficiently high order, while previous results correspond to leading and second order approximations in this method. Finally, we apply this method to real-world data.

Universal scaling relations in strongly anisotropic materials.

We consider the critical temperature in strongly anisotropic antiferromagnetic materials, with weak coupling between stacked planes, in order to determine the interplane coupling constant from experimentally measured susceptibilities. We present theoretical arguments for a universal relation between interplane coupling and susceptibility shown numerically by Yasuda et al. [Phys. Rev. Lett. 94, 217201 (2005)10.1103/PhysRevLett.94.217201]. We predict a more general scaling function if the system is close to a quantum critical point, a similar relation for other susceptibilities than considered in Yasuda et al., and the validity of these relations for more general phase transitions.

Conformal invariance and stochastic Loewner evolution processes in two-dimensional Ising spin glasses.

We present numerical evidence that the techniques of conformal field theory might be applicable to two-dimensional Ising spin glasses with Gaussian bond distributions. It is shown that certain domain wall distributions in one geometry can be related to that in a second geometry by a conformal transformation. We also present direct evidence that the domain walls are stochastic Loewner (SLE) processes with kappa approximately 2.1. An argument is given that their fractal dimension d(f) is related to their interface energy exponent theta by d(f) - 1 = 3/[4(3 + theta)], which is consistent with the commonly quoted values d(f) approximately 1.27 and theta approximately -0.28.

Multiscaling at Point J: jamming is a critical phenomenon.

We analyze the jamming transition that occurs as a function of increasing packing density in a disordered two-dimensional assembly of disks at zero temperature for "Point J" of the recently proposed jamming phase diagram. We measure the total number of moving disks and the transverse length of the moving region, and find a power law divergence as the packing density increases toward a critical jamming density. This provides evidence that the T=0 jamming transition as a function of packing density is a second order phase transition. Additionally, we find evidence for multiscaling, indicating the importance of long tails in the velocity fluctuations.

Statistical mechanics of interfering links.

We consider the statistical mechanics of interfering transmissions in a wireless communications protocol. In this case, a connection between two nodes requires all other nodes within communication distance of the given two nodes to remain quiet on the given channel. This leads to an interesting problem of dimers on a lattice, with a restriction that no two dimers can overlap or be nearest neighbors. We consider both an equilibrium and a nonequilibrium, "greedy" dynamics for the links; the equilibrium properties of the model are found to exhibit an interesting spin-glass transition at maximum density on certain lattices, while the greedy construction is related to the problem of random sequential adsorption.

Locality in quantum and Markov dynamics on lattices and networks.

We consider gapped systems governed by either quantum or Markov dynamics, with the low-lying states below the gap being approximately degenerate. For a broad class of dynamics, we prove that ground or stationary state correlation functions can be written as a piece decaying exponentially in space plus a term set by matrix elements between the low-lying states. The key to the proof is a local approximation to the negative energy, or annihilation, part of an operator in a gapped system. Applications to numerical simulation of quantum systems and to networks are discussed.

Decay of correlations in Fermi systems at nonzero temperature.

The locality of correlation functions is considered for Fermi systems at nonzero temperature. We show that for all short-range, lattice Hamiltonians, the correlation function of any two fermionic operators decays exponentially with a correlation length which is of order the inverse temperature for small temperature. We discuss applications to numerical simulation of quantum systems at nonzero temperature.

Nonlinear dynamics, rectification, and phase locking for particles on symmetrical two-dimensional periodic substrates with dc and circular ac drives.

We investigate the dynamical motion of particles on a two-dimensional symmetric periodic substrate in the presence of both a dc drive along a symmetry direction of the periodic substrate and an additional circular ac drive. For large enough ac drives, the particle orbit encircles one or more potential maxima of the periodic substrate. In this case, when an additional increasing dc drive is applied in the longitudinal direction, the longitudinal velocity increases in a series of discrete steps that are integer multiples of a omega/(2 pi), where a is the lattice constant of the substrate. Fractional steps can also occur. These integer and fractional steps correspond to distinct stable dynamical orbits. A number of these phases also show a rectification in the positive or negative transverse direction where a nonzero transverse velocity occurs in the absence of a dc transverse drive. We map out the phase diagrams of the regions of rectification as a function of ac amplitude, and find a series of tongues. Most of the features, including the steps in the longitudinal velocity and the transverse rectification, can be captured with a simple toy model and by arguments from nonlinear maps. We have also investigated the effects of thermal disorder and incommensuration on the rectification phenomena, and find that for increasing disorder, the rectification regions are gradually smeared and the longitudinal velocity steps are no longer flat but show a linearly increasing velocity.

Do vortices entangle?

We propose an experiment for directly constructing and locally probing topologically entangled states of superconducting vortices which can be performed with present-day technology. Calculations using an elastic string vortex model indicate that as the pitch (the winding angle divided by the vertical distance) increases, the vortices approach each other. At values of the pitch higher than a maximum value the entangled state becomes unstable to collapse via a singularity of the model. We provide predicted experimental signatures for both vortex entanglement and vortex cutting. The local probe we propose can also be used to explore a wide range of other quantities.

Roughness scaling for Edwards-Wilkinson relaxation in small-world networks.

Motivated by a fundamental synchronization problem in scalable parallel computing and by a recent criterion for "mean-field" synchronizability in interacting systems, we study the Edwards-Wilkinson model on two variations of a small-world network. In the first version each site has exactly one random link of strength p, while in the second one each site on average has p links of unit strength. We construct a perturbative description for the width of the stationary-state surface (a measure of synchronization), in the weak- and sparse-coupling limits, respectively, and verify the results by performing exact numerical diagonalization. The width remains finite in the limit of infinite system size for both cases, but exhibits anomalous scaling with p in the latter for d< or =2.

Quantum-limited sensitivity of single-electron-transistor-based displacement detectors.

We consider a model of a quantum-mechanical resonator capacitively coupled to a single electron transistor (SET). The tunnel current in the SET is modulated by the vibrations of the resonator, and thus the system operates as a displacement detector. We analyze the effect of the backaction noise of charge fluctuations in the SET onto the dynamics of the resonator and evaluate the displacement sensitivity of the system. The relation between the "classical" and "quantum" parts of the SET charge noise and their effect on the measured system are also discussed.

Mean-field and anomalous behavior on a small-world network.

We consider various equilibrium statistical mechanics models with combined short- and long-range interactions and identify the crossover to mean-field behavior, finding anomalous scaling in the width of the mean-field region, as well as in the mean-field amplitudes. We then show that this model enables us, in many cases, to determine the universal critical properties of systems on a small-world network. Finally, we consider nonequilibrium processes.

Ratchet cellular automata.

In this work we propose a ratchet effect which provides a general means of performing clocked logic operations on discrete particles, such as single electrons or vortices. The states are propagated through the device by the use of an applied ac drive. We numerically demonstrate that a complete logic architecture is realizable using this ratchet. We consider specific nanostructured superconducting geometries using superconducting materials under an applied magnetic field, with the positions of the individual vortices in samples acting as the logic states. These devices can be used as the building blocks for an alternative microelectronic architecture.

Random vibrational networks and the renormalization group.

We consider the properties of vibrational dynamics on random networks, with random masses and spring constants. The localization properties of the eigenstates contrast greatly with the Laplacian case on these networks. We introduce several real-space renormalization techniques which can be used to describe this dynamics on general networks, drawing on strong disorder techniques developed for regular lattices. The renormalization group is capable of elucidating the localization properties, and provides, even for specific network instances, a fast approximation technique for determining the spectra which compares well with exact results.