Loop In Parameter Space Research Articles

Topological charge pumping in tetramerized Kitaev chains with nodal loop in parameter space

We investigated the topological pumping charge of a tetramerized Kitaev chain with spatially modulated chemical potential, which hosts nodal loops in parameter space and violates particle number conservation. In the simplest case, with alternatively assigned hopping and pairing terms, we show that the model can be mapped into the Rice–Mele model by a partial particle-hole transformation and subsequently supports topological charge pumping as a demonstration of the Chern number for the ground state. Beyond this special case, analytic analysis shows that the nodal loops are conic curves. Numerical simulation of a finite-size chain indicates that the pumping charge is zero for a quasiadiabatic loop within the nodal loop and is ±1 for a quasiadiabatic passage enclosing the nodal loop. Our findings unveil the topology of Kitaev chains in parameter space.

Topological phase transitions at finite temperature

The ground states of noninteracting fermions in one-dimension with chiral symmetry form a class of topological band insulators, described by a topological invariant that can be related to the Zak phase. Recently, a generalization of this quantity to open systems---known as the ensemble geometric phase (EGP)---has emerged as a robust way to describe topology at nonzero temperature. By using this quantity, we explore the nature of topology allowed for dissipation beyond a Lindblad description, to allow for coupling to external baths at finite temperatures. We introduce two main aspects to the theory of open-system topology. First, we discover topological phase transitions as a function of the temperature $T$, manifesting as changes in differences of the EGP accumulated over a closed loop in parameter space. We characterize the nature of these transitions and reveal that the corresponding nonequilibrium steady state can exhibit a nontrivial structure---contrary to previous studies where it was found to be in a fully mixed state. Second, we demonstrate that the EGP itself becomes quantized when key symmetries are present, allowing it to be viewed as a topological marker which can undergo equilibrium topological transitions at nonzero temperatures.

The deformation of an elastic cell in a circulatory fluid motion

The deformation of a two-dimensional inextensible elastic cell in an inviscid uniform stream with circulation is investigated. An asymptotic expansion based on a conformal mapping is used to obtain equilibria for low far-field flow speeds, and fully nonlinear solutions are obtained numerically. Expanding upon the results of Blyth and Părău (2013) and Yorkston et al. (2020) for an elastic cell in a uniform stream with zero circulation, it is shown that the nature of the cell deformation in response to circulation depends on whether the transmural pressure exceeds a series of critical values. Below the first of these critical values, the deformed cell is elongated vertically against the stream, and the circulation acts to reduce the deformation of the cell from the circular rest-state, while above this critical pressure the deformed cell elongates horizontally parallel to the flow, with stronger circulation resulting in more severe cell deformation until self-intersection. The solution branches which emerge at the second critical transmural pressure are found to form a closed loop in parameter space, which shrinks in size as the circulation is increased to a critical value at which the solution branch vanishes. We also present a set solution branches distinct from those found by Yorkston et al. (2020), which become dominant for large values of circulation.

Fast encirclement of an exceptional point for highly efficient and compact chiral mode converters

Exceptional points (EPs) are degeneracies at which two or more eigenvalues and eigenstates of a physical system coalesce. Dynamically encircling EPs by varying the parameters of a non-Hermitian system enables chiral mode switching, that is, the final state of the system upon a closed loop in parameter space depends on the encircling handedness. In conventional schemes, the parametric evolution during the encircling process has to be sufficiently slow to ensure adiabaticity. Here, we show that fast parametric evolution along the parameter space boundary of the system Hamiltonian can relax this constraint. The proposed scheme enables highly efficient transmission and more compact footprint for asymmetric mode converters. We experimentally demonstrate these principles in a 57 μm-long double-coupled silicon waveguide system, enabling chiral mode switching with near-unity transmission efficiency at 1550 nm. This demonstration paves the way towards high-efficiency and highly integrated chiral mode switching for a wide range of practical applications.

Movable Valley Switch Driven by Berry Phase in Bilayer-Graphene Resonators.

Berry phase, the geometric phase accumulated over a closed loop in parameter space during an adiabatic cyclic evolution, has been demonstrated to play an important role in many quantum systems since its discovery. In gapped Bernal bilayer graphene, the Berry phase can be continuously tuned from zero to 2π, which offers a unique opportunity to explore the tunable Berry phase on physical phenomena. Here, we report experimental observation of Berry-phase-induced valley splitting and crossing in movable bilayer-graphene p-n junction resonators. In our experiment, the resonators are generated by combining the electric field of a scanning tunneling microscope tip with the gap of bilayer graphene. A perpendicular magnetic field changes the Berry phase of the confined bound states in the resonators from zero to 2π continuously and leads to the Berry phase difference for the two inequivalent valleys in the bilayer graphene. As a consequence, we observe giant valley splitting and unusual valley crossing of the lowest bound states. Our results indicate that the bilayer-graphene resonators can be used to manipulate the valley degree of freedom in valleytronics.

Majorana charges, winding numbers and Chern numbers in quantum Ising models

Mapping a many-body state on a loop in parameter space is a simple way to characterize a quantum state. The connections of such a geometrical representation to the concepts of Chern number and Majorana zero mode are investigated based on a generalized quantum spin system with short and long-range interactions. We show that the topological invariants, the Chern numbers of corresponding Bloch band, is equivalent to the winding number in the auxiliary plane, which can be utilized to characterize the phase diagram. We introduce the concept of Majorana charge, the magnitude of which is defined by the distribution of Majorana fermion probability in zero-mode states, and the sign is defined by the type of Majorana fermion. By direct calculations of the Majorana modes we analytically and numerically verify that the Majorana charge is equal to Chern numbers and winding numbers.

Observations of hysteresis in the annual exchange circulation of a large microtidal estuary

AbstractA nonlinear relation between the salinity field and the subtidal exchange circulation in the Tampa Bay estuary is demonstrated using observational data from 1999 to 2011. The data are averaged to form mean monthly climatological values of total freshwater discharge ( ), axial and vertical salinity gradients, and subtidal vertical shear. Well‐known steady state solutions indicate that the exchange circulation is linearly proportional to the horizontal salinity gradient, assuming a constant vertical eddy viscosity ( ). The exchange flow is found to be multivalued with respect to the horizontal salinity gradient, forming a hysteresis loop in parameter space that passes through three dynamical regimes. Regime I is relatively dry with weak salinity gradients and exchange circulation. Regime II is the wet season (June–September) in which all quantities rapidly increase. In regime III, the exchange flow persists even though and the axial salinity gradient are again low. Gradient Richardson numbers and Simpson numbers also form a loop in parameter space with remaining subcritical (turbulent) until the wet season when rises above criticality (weak vertical mixing) where it remains through the end of regime III. The Simpson number is in a narrow range around 0.2, indicating that the horizontal salinity gradient is always a driver of the exchange circulation. The , estimated from a parameterization of the Richardson number, decreases by almost an order of magnitude from regimes I to II. It remains low during III, indicating that the persistent stratification is insulating the exchange flow from destruction by tidal mixing during this time period.

Topological polarization in graphene-like systems

In this paper we investigate the possibility of generating piezoelectric orbital polarization in graphene-like systems which are deformed periodically. We start with discrete two-band models which depend on control parameters; in this setting, time-dependent model Hamiltonians are described by loops in parameter space. Then, the gap structure at a given Fermi energy generates a non-trivial topology on parameter space which then leads to possibly non-trivial polarizations. More precisely, we show the polarization, as given by the King-Smith–Vanderbilt formula, depends only on the homotopy class of the loop; hence, a necessary condition for non-trivial piezo effects is that the fundamental group of the gapped parameter space must not be trivial. The use of the framework of non-commutative geometry implies that our results extend to systems with weak disorder. We then apply this analysis to the uniaxial strain model for graphene which includes nearest-neighbor hopping and a stagger potential, and show that it supports non-trivial piezo effects; this is in agreement with recent physics literature.

Asymmetric effect of slowly varying chirped laser pulses on the adiabatic state exchange of a molecule

Here we show that the adiabatic theorem, originally derived for bound molecular systems, does not necessarily hold for dissociative molecular systems. In particular, we demonstrate here the asymmetric adiabatic evolution of the vibrational states of H+2 under the influence of slowly varying chirped laser pulses. There are some laser parameters for which non-Hermitian degeneracies occur in the spectrum of the problem. These are known as exceptional points. When one slowly cycles the laser intensity and wavelength in a loop in parameter space around an exceptional point, the initial tenth vibrational state of H+2 flips into the ninth vibrational state by the end of the loop. However, within the same excursion around an exceptional point in the laser parameters space, the ninth vibrational state of H+2 does not flip into the tenth vibrational state but winds up unchanged by the end of the loop.

Non-Abelian adiabatic statistics and Hall viscosity in quantum Hall states andpx+ipypaired superfluids

Many trial wavefunctions for fractional quantum Hall states in a single Landau level are given by functions called conformal blocks, taken from some conformal field theory. Also, wavefunctions for certain paired states of fermions in two dimensions, such as p_x+ip_y states, reduce to such a form at long distances. Here we investigate the adiabatic transport of such many-particle trial wavefunctions using methods from two-dimensional field theory. One context for this is to calculate the statistics of widely-separated quasiholes, which has been predicted to be non-Abelian in a variety of cases. The Berry phase or matrix (holonomy) resulting from adiabatic transport around a closed loop in parameter space is the same as the effect of analytic continuation around the same loop with the particle coordinates held fixed (monodromy), provided the trial functions are orthonormal and holomorphic in the parameters so that the Berry vector potential (or connection) vanishes. We show that this is the case (up to a simple area term) for paired states (including the Moore-Read quantum Hall state), and present general conditions for it to hold for other trial states (such as the Read-Rezayi series). We argue that trial states based on a non-unitary conformal field theory do not describe a gapped topological phase, at least in many cases. By considering adiabatic variation of the aspect ratio of the torus, we calculate the Hall viscosity, a non-dissipative viscosity coefficient analogous to Hall conductivity, for paired states, Laughlin states, and more general quantum Hall states. Hall viscosity is an invariant within a topological phase, and is generally proportional to the "conformal spin density" in the ground state.

Adiabatic operation of a molecular machine

Operation of a molecular machine is often thought of as a “far from equilibrium” process in which energy released by some high free energy fuel molecule or by light is used to drive a nonequilibrium “power stroke” to do work on the environment. Here we discuss how a molecular machine can be operated arbitrarily close to chemical equilibrium and still perform significant work at an appreciable rate: micrometer per second velocities against piconewton loads. As a specific example, we focus on a motor based on a three-ring catenane similar to that discussed by Leigh [Leigh DA, Wong JKY, Dehez F, Zerbetto F (2003) Nature 424:174–179]. The machine moves through its working cycle under the influence of external modulation of the energies of the states, where the modulation is carried out slowly enough that the state probabilities obey a Boltzmann equilibrium distribution at every instant. The mechanism can be understood in terms of the geometric phase [Berry MV (1990) Phys Today 43(12):34–40] in which the system moves adiabatically around a closed loop in parameter space, completing, on average, nearly one-half mechanical cycle each time it does so. Because the system is very close to equilibrium at every instant, the efficiency can approach 100%.

Optimal topological test for degeneracies of real Hamiltonians.

We consider adiabatic transport of eigenstates of real Hamiltonians around loops in parameter space. It is demonstrated that loops that map to nontrivial loops in the space of eigenbases must encircle degeneracies. Examples from Jahn-Teller theory are presented to illustrate the test. We show furthermore that the proposed test is optimal.

Arithmetical signatures of the dynamics of the Hénon map.

We report a fourth-degree polynomial that parametrizes analytically all period-4 orbits of the Hénon map and use it to investigate arithmetical signatures of the symbolic coding for this prototypical multidimensional system. A discontinuity in the symbolic dynamics observed by Hansen while following numerically a period-6 orbit along a closed loop in parameter space is shown to exist already for period 4. We obtain an analytical expression for the locus of all such discontinuities in parameter space and explain their origin. Our analytical results allow the accurate location of all discontinuities, in contrast with topological methods based on homoclinic tangencies that exist over continuous intervals.

DESIGN OF ROBUST STABLE MASTER-SLAVE SYSTEMS WITH UNCERTAIN DYNAMICS AND TIME-DELAY

Master-slave and teleoperation systems with kinesthetic feedback enhance the dexterity of an operator performing manipulation tasks in remote, not accessible, or scaled environments. Kinesthetic feedback is achieved by closing bi-directional control loops between the operator and teleoperator site. This article proposes a method for the synthesis of robust bi-directional control loops in parameter space. Using the approach of singular frequencies a method for the synthesis of Hurwitz- and Γ-stable bi-directional controllers is developed and its application is shown by simulation. Robust performance is achieved w.r.t. parameter uncertainties of the operator and teleoperator dynamics and the bi-directional communication time delay.

Charge pumping and photovoltaic effect in open quantum dots

We propose a random matrix theory to describe the influence of a time-dependent external field on electron transport through open quantum dots. We describe the generation of the current by an oscillating field for the dot, connected to two leads with equal chemical potentials. For low frequency fields our results correspond to adiabatic charge pumping. Finite current can be produced if the system goes along a closed loop in parameter space, which covers a finite area. At high frequency finite current is produced even if the loop is a line in parameter space. This result can be explained in the same way as adiabatic pumping but considering the evolution of the system in phase space rather than in parametric space.

Geometric phases and anholonomy for a class of chaotic classical systems.

Berry's phase may be viewed as arising from the parallel transport of a quantal state around a loop in parameter space. In this Letter, the classical limit of this transport is obtained for a particular class of chaotic systems. It is shown that this ``classical parallel transport'' is anholonomic --- transport around a closed curve in parameter space does not bring a point in phase space back to itself --- and is intimately related to the Robbins-Berry classical two-form.

THE GEOMETRIC PHASE AND THE SCHWINGER TERM IN SOME MODELS

We discuss the quantization of fermions interacting with external fields and observe the occurrence of equivalent as well as inequivalent representations of the canonical anticommutation relations. Implementability of gauge and axial gauge transformations leads to generators which fulfil an algebra of current with a Schwinger term. This term can be written as a cocycle and leads to the boson-fermion correspondence. Transport of a quantum-mechanical system along a closed loop of parameter space may yield a geometric phase. We discuss models for which nonintegrable phase factors are obtained from the adiabatic parallel transport. After second quantization, one obtains, in addition, a Schwinger term. Depending on the type of transformation, a subtle relationship between these two obstructions can occur. We indicate finally how we may transport density matrices along closed loops in parameter space.

Remarks on the symbolic dynamics for the Hénon map

We exhibit an unstable periodic orbit in the Hénon map which, under a smooth change of parameters along a loop in parameter space, changes into another orbit with different symbolic dynamics. We show how this is connected with the existence of homoclinic tangencies and discuss the implications of this phenomenon for current proposals for the Hénon map symbolic dynamics.

Phase associated with the single‐particle density of many‐electron systems

AbstractIn excited states of atoms and molecules, as well as in time‐dependent situations, the one‐electron density no longer suffices to completely characterize the electronic state; in addition, one now requires information about the electronic phase or the current density. We show that, for a stationary electronic state, the continuity equation of quantum fluid dynamics represents a differential equation for the electronic phase, which must be solved subject to certain periodicity conditions. These periodicity conditions arise from the nodal topology of the wave function and give rise to quantized vortices of current. The consequences of writing an electronic “wave function” for a many‐electron system directly in terms of the single‐particle density and phase have been investigated. We have shown that such a procedure leads to the appearance of an “internal magnetic vector potential.” We also establish the connection between the electronic phase and the geometrical (“Berry”) phase accompanying the adiabatic transport of a quantal system around a closed loop in parameter space. This leads to a generalization of the current density concept and allows us to discuss the geometrical phase in terms of the circulation of this current in parameter space.

The geometric phase in a simple model

The parallel transport of a quantum mechanical system along closed loops in parameter space may lead to the occurrence of a geometric phase. Despite the simple ideas behind that mechanism not many explicitly solvable models exist. We treat a Schrödinger type system on an interval. All possible self-adjoint operators lead to a four-parameter family of Hamiltonians for which nontrivial phase factors occur.