Nontrivial Ground Research Articles

Localized Majorana-Like Modes in a Number-Conserving Setting: An Exactly Solvable Model

In this Letter we present, in a number conserving framework, a model of interacting fermions in a two-wire geometry supporting nonlocal zero-energy Majorana-like edge excitations. The model has an exactly solvable line, on varying the density of fermions, described by a topologically nontrivial ground state wave function. Away from the exactly solvable line we study the system by means of the numerical density matrix renormalization group. We characterize its topological properties through the explicit calculation of a degenerate entanglement spectrum and of the braiding operators which are exponentially localized at the edges. Furthermore, we establish the presence of a gap in its single particle spectrum while the Hamiltonian is gapless, and compute the correlations between the edge modes as well as the superfluid correlations. The topological phase covers a sizable portion of the phase diagram, the solvable line being one of its boundaries.

Solution of the antiferromagnetic Ising model with multisite interaction on a zigzag ladder.

We consider the antiferromagnetic spin-1/2 Ising model with multisite interaction in an external magnetic field on an infinite zigzag ladder. The model is solved exactly by using the transfer matrix method. Using the exact expression for the total magnetization per site, the magnetic properties of the model are investigated in detail. The model exhibits the formation of magnetization plateaus for low temperatures, and it is shown that their properties depend strongly on the strength of the multisite interaction. All possible ground states of the model are found and discussed. The existence of nontrivial singular ground states is proven and exact explicit expressions for them are found. The macroscopic degeneracy of the ground states is investigated and discussed.

Tensor network implementation of bulk entanglement spectrum

Many topologically nontrivial states of matter possess gapless degrees of freedom on the boundary, and when these boundary states delocalize into the bulk, a phase transition occurs and the system becomes topologically trivial. We show that tensor networks provide a natural framework for analyzing such topological phase transitions in terms of the boundary degrees of freedom which mediate it. To do so, we make use of a correspondence between a topologically nontrivial ground state and its phase transition to a trivial phase established in [1]. This involved computing the bulk entanglement spectrum (BES) of the ground state, upon tracing out an extensive subsystem. This work implements BES via tensor network representations of ground states. In this framework, the universality class of the quantum critical entanglement Hamiltonian in d spatial dimensions is either derived analytically or mapped to a classical statistical model in d + 1 dimensions, which can be studied using Monte Carlo or tensor renormalization group methods. As an example, we analytically derive the universality classes of topological phase transitions from the spin-1 chain Haldane phase and demonstrate that the AKLT wavefunction (and its generalizations) remarkably contains critical six-vertex (and in general eight-vertex) models within it.

Ground state solutions of the periodic discrete coupled nonlinear Schrödinger equations

In this paper, by using the Nehari manifold approach in combination with periodic approximations, we obtain the sufficient conditions on the existence of the nontrivial ground state solutions of the periodic discrete coupled nonlinear Schrödinger equations. Copyright © 2014 John Wiley & Sons, Ltd.

Magnetic short- and long-range order inPbFe0.5Ta0.5O3

PbFe$_{1/2}$Ta$_{1/2}$O$_{3}$ (PFT) belongs to the family of PbB$'_{x}$B$"_{1-x}$O$_{3}$ which have inherent chemical disorder at the B-site. Due to this disorder, a complex magnetic phase diagram is expected in the material. In this paper, we report experimental results of magnetic properties in PFT through macroscopic characterization, neutron scattering and M\"{o}ssbauer spectroscopy techniques. With these results we show for the first time that PbFe$_{1/2}$Ta$_{1/2}$O$_{3}$ behaves very similar to PbFe$_{1/2}$Nb$_{1/2}$O$_{3}$, i.e, it undergoes AF transition at 153 K and has a spinglass transition at 10 K, below which the antiferromagnetism coexists with spinglass. We suggest that the mechanism which is responsible for such a non-trivial ground state can be explained by a speromagnet-like spin arrangement similar to the one proposed for PbFe$_{1/2}$Nb$_{1/2}$O$_{3}$.

Ground state homoclinic orbits of damped vibration problems

In this paper, we consider a class of non-periodic damped vibration problems with superquadratic nonlinearities. We study the existence of nontrivial ground state homoclinic orbits for this class of damped vibration problems under some conditions weaker than those previously assumed. To the best of our knowledge, there has been no work focused on this case. MSC: 49J40, 70H05.

Solution of the antiferromagnetic Ising model on a tetrahedron recursive lattice.

We consider the antiferromagnetic spin-1/2 Ising model on the recursive tetrahedron lattice on which two elementary tetrahedrons are connected at each site. The model represents the simplest approximation of the antiferromagnetic Ising model on the real three-dimensional tetrahedron lattice which takes into account effects of frustration. An exact analytical solution of the model is found and discussed. It is shown that the model exhibits neither the first-order nor the second-order phase transitions. A detailed analysis of the magnetization of the model in the presence of the external magnetic field is performed and the existence of the magnetization plateaus for low temperatures is shown. All possible ground states of the model are found and discussed. The existence of nontrivial singular ground states is proven and exact explicit expressions for them are found.

Ferromagnetic spin-glass behaviour in single-crystalline U2 IrSi3

A single crystal of the U-based ternary silicide U2IrSi3 was investigated by means of magnetic, resistivity and heat-capacity measurements performed in wide ranges of temperature and external magnetic fields. The results hint at the formation of a non-trivial magnetic ground state in which ferromagnetic ordering coexists with spin-glass freezing.

Superquadratic or asymptotically quadratic Hamiltonian systems: ground state homoclinic orbits

Under super or asymptotically quadratic assumptions at infinity, we obtain the existence of nontrivial ground state homoclinic orbits for a class of second-order Hamiltonian systems with general potentials by a variant generalized weak linking theorem. For the asymptotically quadratic case, a necessary and sufficient condition is obtained for the existence of nontrivial homoclinic orbits. For the superquadratic case, we use general superquadratic conditions to replace the Ambrosetti–Rabinowitz condition.

Periodic solutions for second order Hamiltonian systems with general superquadratic potential

In this paper, we study the existence of nontrivial solutions and ground state solutions for the second order Hamiltonian systems: $$ \ddot u(t)+A(t)u(t)+\nabla F(t,u(t))=0\ \ \ \ \ \ \mbox{a.e. } t\in [0,T], $$ where $A(t)$ is a $N\times N$ symmetric matrix, continuous and $T$-periodic in $t$. Replacing the classical Ambrosetti-Rabinowitz superquadratic condition by a general superquadratic condition, we prove some existence theorems, which unify and improve some recent results in the literature.

Symmetry-protected topological phases with charge and spin symmetries: Response theory and dynamical gauge theory in two and three dimensions

A large class of symmetry-protected topological phases (SPT) in boson / spin systems have been recently predicted by the group cohomology theory. In this work, we consider SPT states at least with charge symmetry (U(1) or Z$_N$) or spin $S^z$ rotation symmetry (U(1) or Z$_N$) in 2D, 3D, and the surface of 3D. If both are U(1), we apply external electromagnetic field / `spin gauge field' to study the charge / spin response. For the SPT examples we consider (i.e. U$_c$(1)$\rtimes$Z$^T_2$, U$_s$(1)$\times$Z$^T_2$, U$_c$(1)$\times$[U$_s$(1)$\rtimes$Z$_2$]; subscripts $c$ and $s$ are short for charge and spin; Z$^T_2$ and Z$_2$ are time-reversal symmetry and $\pi$-rotation about $S^y$, respectively), many variants of Witten effect in the 3D SPT bulk and various versions of anomalous surface quantum Hall effect are defined and systematically investigated. If charge or spin symmetry reduces to Z$_N$ by considering charge-$N$ or spin-$N$ condensate, instead of the linear response approach, we gauge the charge/spin symmetry, leading to a dynamical gauge theory with some remaining global symmetry. The 3D dynamical gauge theory describes a symmetry-enriched topological phase (SET), i.e. a topologically ordered state with global symmetry which admits nontrivial ground state degeneracy depending on spatial manifold topology. For the SPT examples we consider, the corresponding SET states are described by dynamical topological gauge theory with topological BF term and axionic $\Theta$-term in 3D bulk. And the surface of SET is described by the chiral boson theory with quantum anomaly.

Improving Point Cloud Accuracy Obtained from a Moving Platform for Consistent Pile Attack Pose Estimation

We present a perception system for enabling automated loading with waist-articulated wheel loaders. To enable autonomous loading of piled materials, using either above-ground wheel loaders or underground load-haul-dump vehicles, 3D data of the pile shape is needed. However, using common 3D scanners, the scan data is distorted while the wheel loader is moving towards the pile. Existing methods that make use of 3D scan data (for autonomous loading as well as tasks such as mapping, localisation, and object detection) typically assume that each 3D scan is accurate. For autonomous robots moving over rough terrain, it is often the case that the vehicle moves a substantial amount during the acquisition of one 3D scan, in which case the scan data will be distorted. We present a study of auto-loading methods, and how to locate piles in real-world scenarios with nontrivial ground geometry. We have compared how consistently each method performs for live scans acquired in motion, and also how the methods perform with different view points and scan configurations. The system described in this paper uses a novel method for improving the quality of distorted 3D scans made from a vehicle moving over uneven terrain. The proposed method for improving scan quality is capable of increasing the accuracy of point clouds without assuming any specific features of the environment (such as planar walls), without resorting to a "stop-scan-go" approach, and without relying on specialised and expensive hardware. Each new 3D scan is registered to the preceding using the normal-distributions transform (NDT). After each registration, a mini-loop closure is performed with a local, per-scan, graph-based SLAM method. To verify the impact of the quality improvement, we present data that shows how auto-loading methods benefit from the corrected scans. The presented methods are validated on data from an autonomous wheel loader, as well as with simulated data. The proposed scan-correction method increases the accuracy of both the vehicle trajectory and the point cloud. We also show that it increases the reliability of pile-shape measures used to plan an efficient attack pose when performing autonomous loading.

Existence and multiplicity of non-trivial solutions for a class of modified Schrödinger–Poisson systems

In the present paper, we study the following modified Schrödinger–Poisson system (P){−△u+V(x)u+ϕu−12u△u2=∣u∣p−1u+h(x),inR3,−△ϕ=u2,inR3, where 4≤p+1<12, h∈L2(R3) and h≠0. This model describes the interaction of a charge particle with an electro-magnetic field. Under some hypotheses on V(x), we prove the existence of a non-trivial ground state solution and two non-trivial ground state solutions for the system (P).

On the ground state solution for a critical fractional Laplacian equation

In this paper, we study the existence of ground state solutions for the following critical fractional Laplacian equation {(−Δ)α2u=|u|2α∗−2u+μuinΩ,u=0on∂Ω, where 0<α<2,Ω⊂RN,N>(1+2)α is a bounded domain. We will prove that the problem possesses a nontrivial ground state solution for any μ>0.

Trivial low energy states for commuting Hamiltonians, and the quantum PCP conjecture

We consider the entanglement properties of ground states of Hamiltonians which are sums of commuting projectors (we call these commuting projector Hamiltonians), in particular whether or not they have ``trivial" ground states, where a state is trivial if it is constructed by a local quantum circuit of bounded depth and range acting on a product state. It is known that Hamiltonians such as the toric code only have nontrivial ground states in two dimensions. Conversely, commuting projector Hamiltonians which are sums of two-body interactions have trivial ground states\cite{bv}. Using a coarse-graining procedure, this implies that any such Hamiltonian with bounded range interactions in one dimension has a trivial ground state. In this paper, we further explore the question of which Hamiltonians have trivial ground states. We define an ``interaction complex" for a Hamiltonian, which generalizes the notion of interaction graph and we show that if the interaction complex can be continuously mapped to a $1$-complex using a map with bounded diameter of pre-images then the Hamiltonian has a trivial ground state assuming one technical condition on the Hamiltonians holds (this condition holds for all stabilizer Hamiltonians, and we additionally prove the result for all Hamiltonians under one assumption on the $1$-complex). While this includes the cases considered by Ref.~\onlinecite{bv}, we show that it also includes a larger class of Hamiltonians whose interaction complexes cannot be coarse-grained into the case of Ref.~\onlinecite{bv} but still can be mapped continuously to a $1$-complex. One motivation for this study is an approach to the quantum PCP conjecture. We note that many commonly studied interaction complexes can be mapped to a $1$-complex after removing a small fraction of sites. For commuting projector Hamiltonians on such complexes, in order to find low energy trivial states for the original Hamiltonian, it would suffice to find trivial ground states for the Hamiltonian with those sites removed. Such trivial states can act as a classical witness to the existence of a low energy state. While this result applies for commuting Hamiltonians and does not necessarily apply to other Hamiltonians, it suggests that to prove a quantum PCP conjecture for commuting Hamiltonians, it is worth investigating interaction complexes which cannot be mapped to $1$-complexes after removing a small fraction of points. We define this more precisely below; in some sense this generalizes the notion of an expander graph. Surprisingly, such complexes do exist as will be shown elsewhere\cite{fh}, and have useful properties in quantum coding theory.

Wigner crystals for a planar, equimolar binary mixture of classical, charged particles

We have investigated the ground state configurations of an equimolar, binary mixture of classical charged particles (with nominal charges $Q_1$ and $Q_2$) that condensate on a neutralizing plane. Using efficient Ewald summation techniques for the calculation of the ground state energies, we have identified the energetically most favourable ordered particle arrangements with the help of a highly reliable optimization tool based on ideas of evolutionary algorithms. Over a large range of charge ratios, $q = Q_2 / Q_1$, we identify six non-trivial ground states, some of which show a remarkable and unexpected structural complexity. For $0.59 \lesssim q < 1$ the system undergoes a phase separation where the two charge species populate in a hexagonal arrangement spatially separated areas.

QUANTUM DISCORD IN THE GROUND STATE OF SPIN CHAINS

The ground state of a quantum spin chain is a natural playground for investigating correlations. Nevertheless, not all correlations are genuinely of quantum nature. Here we review the recent progress to quantify the "quantumness" of the correlations throughout the phase diagram of quantum spin systems. Focusing to one spatial dimension, we discuss the behavior of quantum discord (QD) close to quantum phase transitions (QPT). In contrast to the two-spin entanglement, pairwise discord is effectively long-ranged in critical regimes. Besides the features of QPT, QD is especially feasible to explore the factorization phenomenon, giving rise to nontrivial ground classical states in quantum systems. The effects of spontaneous symmetry breaking are also discussed as well as the identification of quantum critical points through correlation witnesses.

Strong-coupling topological Josephson effect in quantum wires

We investigate the Josephson effect for a setup with two lattice quantum wires featuring Majorana zero energy boundary modes at the tunnel junction. In the weak-coupling regime, the exact solution reproduces the perturbative result for the energy containing a contribution ∼ ± cos(ϕ/2) relative to the tunneling of paired Majorana fermions. As the tunnel amplitude g grows relative to the hopping amplitude w, the gap between the energy levels gradually diminishes until it closes completely at the critical value . At this point the Josephson energies have the principal values , where m =− 1,0,1 and σ =± 1, a result not following from perturbation theory. It represents a transparent regime where three Bogoliubov states merge, leading to additional degeneracies of the topologically nontrivial ground state with an odd number of Majorana fermions at the end of each wire. We also obtain the exact tunnel currents for a fixed parity of the eigenstates. The Josephson current shows the characteristic 4π periodicity expected for a topological Josephson effect. We discuss the additional features of the current associated with a closure of the energy gap between the energy levels.

Theory of topological Kondo insulators

We examine how the properties of the Kondo insulators change when the symmetry of the underlying crystal field multiplets is taken into account. We employ the Anderson lattice model and consider its low-energy physics. We show that in a large class of crystal field configurations, Kondo insulators can develop a topological nontrivial ground state. Such topological Kondo insulators are adiabatically connected to noninteracting insulators with unphysically large spin-orbit coupling, and as such may be regarded as interaction-driven topological insulators. We analyze the entanglement entropy of the Anderson lattice model of Kondo insulators by evaluating its entanglement spectrum. Our results for the entanglement spectrum are consistent with the surface state calculations. Last, we discuss the construction of the maximally localized Wannier wave functions for generic Kondo insulators.

Josephson Junction with a Magnetic-Field Tunable Ground State

We consider an asymmetric 0-π Josephson junction consisting of 0 and π regions of different lengths L(0) and L(π). As predicted earlier this system can be described by an effective sine-Gordon equation for the spatially averaged phase ψ so that the effective current-phase relation of this system includes a negative second harmonic ∝sin(2ψ). If its amplitude is large enough, the ground state of the junction is doubly degenerate ψ=±φ, where φ depends on the amplitudes of the first and second harmonics. We study the behavior of such a junction in an applied magnetic field H and demonstrate that H induces an additional term ∝Hcosψ in the effective current-phase relation. This results in a nontrivial ground state tunable by magnetic field. The dependence of the critical current on H allows for revealing the ground state experimentally.