Vortex Condensation Research Articles

Statistical analysis of vortex condensate motion in two-dimensional turbulence

An inverse turbulent cascade in a periodic square box produces a coherent system-sized vortex dipole. We study the statistics of its motion by carrying out direct numerical simulations performed for various bottom friction α, pumping intensity ε, and fluid hyperviscosity ν. In the main approximation, coherent vortices can be considered as point vortices, and within this model, they drift at the same dipole velocity, which is determined by their circulation and mutual arrangement. The characteristic value of the dipole velocity is more than an order of magnitude smaller than the polar velocity inside coherent vortices. Turbulent fluctuations give rise to a relative velocity between the vortices, which changes the distance between them. We found that for a strong condensate, the probability density function of the vector ρ, describing the difference in the mutual arrangement of coherent vortices from half the diagonal of the computational domain, has the form of a ring. The radius of the ring weakly depends on control parameters and its width is proportional to parameter δ=ϵ−1/3L2/3α, where ϵ is the inverse energy flux and L is the system size. The random walk around the ring, caused by turbulent fluctuations, has superdiffusion behavior at intermediate times. It results in a finite correlation time of the dipole velocity, which is of the order of turnover time τK=L2/3ϵ−1/3 of system-size eddies produced by an inverse turbulent cascade. The results obtained deepen the understanding of the processes governing the motion of coherent vortices.

Parameterisation of small-scale random forcing in β-plane turbulence

Zonal jet formation in $\beta$ -plane turbulence is investigated with the focus on whether an accurate closure can be developed for the eddy momentum fluxes due to small-scale random forcing. The approach of Srinivasan and Young (J. Atmos. Sci., vol. 71, 2014, pp. 2169–2185) is developed to give a relatively simple expression for the local Reynolds stress, due to a white-in-time random forcing with characteristic length scale much less than the jet spacing. In typical jet flows, however, it is demonstrated that the Srinivasan–Young flux is not the full story because momentum fluxes due to jet-scale waves, present as a result of distinct barotropic instabilities of the eastward and westward jets, respectively, also play a key role in the momentum balance. Numerical simulations that explicitly include the random forcing are then compared with those in which the Srinivasan–Young closure is applied. For typical jet flows, good agreement of the equilibrium zonal flow is found provided that the closure simulation is not truncated to be purely zonal, i.e. jet-scale secondary barotropic instabilities are allowed to develop. Flows in which the geometry or external forcing acts to suppress the development of secondary instabilities are also simulated, and for these flows the Srinivasan–Young closure is shown to be successful as a purely zonal closure. It is argued that vortex condensates in isotropic forced-dissipative 2D turbulence are an example of this latter situation.

Competing orders, the Wess-Zumino-Witten term, and spin liquids

In this paper, we demonstrate that in frustrated magnets when several conventional (i.e., symmetry-breaking) orders compete, and are “intertwined” by a Wess-Zumino-Witten (WZW) term, the possibility of spin liquid arises. The resulting spin liquid could have excitations which carry fractional spins and obey non-trivial self/mutual statistics. As a concrete example, we consider the case where the competing orders are the Néel and valence-bond solid (VBS) order on square lattice. Examining different scenarios of vortex condensation from the VBS side, we show that the intermediate phases, including spin liquids, between the Néel and VBS order always break certain symmetry. Remarkably, our starting theory, without fractionalized particles (partons) and gauge field, predicts results agreeing with those derived from a parton theory. This suggests that the missing link between the Ginzberg-Landau-Wilson action of competing order and the physics of spin liquid is the WZW term.

Discontinuous transitions towards vortex condensates in buoyancy-driven rotating turbulence

Using direct numerical simulations of rotating Rayleigh–Bénard convection, we explore the transitions between turbulent states from a three-dimensional (3-D) flow state towards a quasi-2-D condensate known as the large-scale vortex (LSV). We vary the Rayleigh number $Ra$ as control parameter and study the system response (strength of the LSV) in terms of order parameters, assessing the energetic content in the flow and the upscale energy flux. By sensitively probing the boundaries of the region of existence of the LSV in parameter space, we find discontinuous transitions and we identify the presence of a hysteresis loop as well as a memoryless abrupt growth dynamics. We show furthermore that the creation of the condensate state coincides with a discontinuous transition of the energy transport into the largest mode of the system.

Self-organizing effect of drift term against diffusion term in point vortex system evidenced by numerical simulations on PEZY-SC

We have analytically obtained a kinetic equation for a two-dimensional (2D) point vortex system with a Fokker–Planck type collision term consisting of a diffusion term and a drift term (Yatsuyanagi and Hatori 2015 Fluid Dyn. Res. 47 065506). The equation describes a mechanism of a self-organization in the system. In this paper, analytically anticipated characteristics of the collision term is evidenced by numerical simulations on PEZY-SC supercomputer system. In the previous paper, it is shown that the collision term satisfies following physically good properties: (1) charge separated, self-organized distribution typical for negative absolute temperature is achieved by the drift term, (2) when a system reaches a thermal equilibrium state, stream function ψ and vorticity ω satisfies an inequality which is important for the next property, (3) the obtained kinetic equation conserves total mean field energy and satisfies the H theorem. In this paper we will evidence the above properties numerically. The time-asymptotic distribution of the vortices corresponds to the analytically anticipated local equilibrium state in which many small regions with different local temperatures exist. Especially, the transport process by the drift term is elucidated clearly, which gives a key mechanism of the self-organization, i.e. condensation of the same-sign vortices.

Competition between Ekman Plumes and Vortex Condensates in Rapidly Rotating Thermal Convection.

We perform direct numerical simulations of rotating Rayleigh-Bénard convection (RRBC) of fluids with low (Pr=0.1) and high (Pr≈5) Prandtl numbers in a horizontally periodic layer with no-slip bottom and top boundaries. No-slip boundaries are known to actively promote the formation of plumelike vertical disturbances, through so-called Ekman pumping, that control the ambient flow at sufficiently high rotation rates. At both Prandtl numbers, we demonstrate the presence of competing large-scale vortices (LSVs) in the bulk. Strong buoyant forcing and rotation foster the quasi-two-dimensional turbulent state of the flow that leads to the upscale transfer of kinetic energy that forms the domain-filling LSV condensate. The Ekman plumes from the boundary layers are sheared apart by the large-scale flow, yet we find that their energy feeds the upscale transfer. Our results of RRBC simulations substantiate the emergence of large-scale flows in nature regardless of the specific details of the boundary conditions.

Emergent fractons and algebraic quantum liquid from plaquette melting transitions

Paramagnetic spin systems with spontaneously broken spatial symmetries, such as valence bond solid (VBS) phases, can host topological defects carrying non-trivial quantum numbers, which enables the paradigm of deconfined quantum criticality. In this work, we study the properties of topological defects in valence plaquette solid (VPS) phases on square and cubic lattices. We show that the defects of the VPS order parameter, in addition to possessing non-trivial quantum numbers, have fracton mobility constraints deep in the VPS phase, which has been overlooked previously. The spinon inside a single vortex cannot move freely in any direction, while a dipolar pair of vortices with spinon pairs can only move perpendicular to its dipole moment. These mobility constraints, while they persist, can potentially inhibit the condensation of vortices and preclude a continuous transition from the VPS to the N\'eel antiferromagnet. Instead, the VPS melting transition can be driven by proliferation of spinon dipoles. For example, we argue that a $2d$ VPS can melt into a stable gapless phase in the form of an algebraic bond liquid with algebraic correlations and long range entanglement. Such a bond liquid phase yields a concrete example of the elusive $2d$ Bose metal with symmetry fractionalization. We also study $3d$ valence plaquette and valence cube ordered phase, and demonstrate that the topological defects therein also have fractonic dynamics. Possible nearby phases after melting the valence plaquettes or cubes are also discussed.

Symmetry-Enriched Fracton Phases from Supersolid Duality.

Motivated by the recently established duality between elasticity of crystals and a fracton tensor gauge theory, we combine it with boson-vortex duality, to explicitly account for bosonic statistics of the underlying atoms. We thereby derive a hybrid vector-tensor gauge dual of a supersolid, which features both crystalline and superfluid order. The gauge dual describes a fracton state of matter with full dipole mobility endowed by the superfluid order, as governed by "mutual" axion electrodynamics between the fracton and vortex sectors of the theory, with an associated generalized Witten effect. Vortex condensation restores U(1) symmetry, confines dipoles to be subdimensional (recovering the dislocation glide constraint of a commensurate quantum crystal), and drives a phase transition between two distinct fracton phases. Meanwhile, condensation of elementary fracton dipoles and charges, respectively, provide a gauge dual description of the superhexatic and ordinary superfluid. Consistent with conventional wisdom, in the absence of crystalline order, U(1)-symmetric phases are prohibited at zero temperature via a mechanism akin to deconfined quantum criticality.

Einstein–Bose condensation of Onsager vortices

We have studied statistical mechanics of a gas of vortices in two dimensions. We introduce a new observable—a condensate fraction of Onsager vortices—to quantify the emergence of the vortex condensate. The condensation of Onsager vortices is most transparently observed in a single vortex species system and occurs due to a competition between solid body rotation (see vortex lattice) and potential flow (see multiple quantum vortex state). We propose an experiment to observe the condensation transition of the vortices in such a single vortex species system.

Magnetic monopole versus vortex as gauge-invariant topological objects for quark confinement

First, we give a gauge-independent definition of chromomagnetic monopoles in [Formula: see text] Yang–Mills theory which is derived through a non-Abelian Stokes theorem for the Wilson loop operator. Then we discuss how such magnetic monopoles can give a nontrivial contribution to the Wilson loop operator for understanding the area law of the Wilson loop average. Next, we discuss how the magnetic monopole condensation picture are compatible with the vortex condensation picture as another promising scenario for quark confinement. We analyze the profile function of the magnetic flux tube as the non-Abelian vortex solution of [Formula: see text] gauge-Higgs model, which is to be compared with numerical simulations of the [Formula: see text] Yang–Mills theory on a lattice. This analysis gives an estimate of the string tension based on the vortex condensation picture, and possible interactions between two non-Abelian vortices.

Z3 topological order in the quantum dimer-pentamer model

We introduce the quantum dimer-pentamer model (QDPM) on the square lattice. This model is a generalization of the square lattice quantum dimer model as its configuration space comprises fully-packed hard-core dimer coverings as well as dimer configurations containing pentamers, where four dimers touch a vertex. Thus in the QDPM, the fully-packed, hard-core constraint of the quantum dimer model is relaxed such that the local dimer number at each vertex is fixed modulo 3, resulting in an exact local $Z_3$ gauge symmetry. We construct a local Hamiltonian for which the Rokhsar-Kivelson (RK) equal superposition state is the exact ground state and has a 9-fold topological degeneracy on the torus. Using Monte Carlo calculations, we find no spontaneous symmetry breaking in the RK wavefunction and that its dimer-dimer correlation function decays exponentially. By doping the QDPM RK state with a pair of monomers, we demonstrate that $Z_3$ electric charges are deconfined. Additionally, we introduce a $Z_3$ magnetic string operator that we find decays exponentially and shows no signatures of magnetic vortex condensation and with correlations. These numerical results suggest that the ground state of the QDPM is a dimer liquid with $Z_3$ topological order.

Vortex formation and dynamics in two-dimensional driven-dissipative condensates

We investigate the real-time evolution of lattice bosons in two spatial dimensions whose dynamics is governed by a Markovian quantum master equation. We employ the Wigner-Weyl phase space quantization and derive the functional integral for open quantum many-body systems that governs the time evolution of the Wigner function. Using the truncated Wigner approximation, in which quantum fluctuations are only taken into account in the initial state whereas the dynamics is governed by classical evolution equations, we study the buildup of long-range correlations due to the action of non-Hermitean quantum jump operators that constitute a mechanism for dissipative cooling. Starting from an initially disordered state corresponding to a vortex condensate, the dissipative process results in the annihilation of vortex-antivortex pairs and the establishment of quasi long-range order at late times. We observe that a finite vortex density survives the cooling process which disagrees with the analytically constructed vortex-free Bose-Einstein condensate at asymptotic times. This indicates that quantum fluctuations beyond the truncated Wigner approximation need to be included to fully capture the physics of dissipative Bose-Einstein condensation.

Model of random center vortex lines in continuous 2+1 -dimensional spacetime

A picture of confinement in QCD based on a condensate of thick vortices with fluxes in the center of the gauge group (center vortices) is studied. Previous concrete model realizations of this picture utilized a hypercubic space-time scaffolding, which, together with many advantages, also has some disadvantages, e.g., in the treatment of vortex topological charge. In the present work, we explore a center vortex model which does not rely on such a scaffolding. Vortices are represented by closed random lines in continuous 2+1-dimensional space-time. These random lines are modeled as being piece-wise linear, and an ensemble is generated by Monte Carlo methods. The physical space in which the vortex lines are defined is a torus with periodic boundary conditions. Besides moving, growing and shrinking of the vortex configurations, also reconnections are allowed. Our ensemble therefore contains not a fixed, but a variable number of closed vortex lines. This is expected to be important for realizing the deconfining phase transition. We study both vortex percolation and the potential V (R) between quark and anti-quark as a function of distance R at different vortex densities, vortex segment lengths, reconnection conditions and at different temperatures. We find three deconfinement phase transitions, as a function of density, as a function of vortex segment length, and as a function of temperature.

The effects of Ekman pumping on quasi-geostrophic Rayleigh–Bénard convection

Numerical simulations of three-dimensional rapidly rotating Rayleigh–Bénard convection are performed by employing an asymptotic quasi-geostrophic model that incorporates the effects of no-slip boundaries through (i) parametrized Ekman pumping boundary conditions and (ii) a thermal wind boundary layer that regularizes the enhanced thermal fluctuations induced by pumping. The fidelity of the model, obtained by an asymptotic reduction of the Navier–Stokes equations that implicitly enforces a pointwise geostrophic balance, is explored for the first time by comparisons of simulations against the findings of direct numerical simulations (DNS) and laboratory experiments. Results from these methods have established Ekman pumping as the mechanism responsible for significantly enhancing the vertical heat transport. This asymptotic model demonstrates excellent agreement over a range of thermal forcing for Prandtl number $Pr\approx 1$ when compared with results from experiments and DNS at maximal values of their attainable rotation rates, as measured by the Ekman number ($E\approx 10^{-7}$); good qualitative agreement is achieved for $Pr>1$. Similar to studies with stress-free boundaries, four spatially distinct flow morphologies exists. Despite the presence of frictional drag at the upper and/or lower boundaries, a strong non-local inverse cascade of barotropic (i.e. depth-independent) kinetic energy persists in the final regime of geostrophic turbulence and is dominant at large scales. For mixed no-slip/stress-free and no-slip/no-slip boundaries, Ekman friction is found to attenuate the efficiency of the upscale energy transport and, unlike the case of stress-free boundaries, rapidly saturates the barotropic kinetic energy. For no-slip/no-slip boundaries, Ekman friction is strong enough to prevent the development of a coherent dipole vortex condensate. Instead, vortex pairs are found to be intermittent, varying in both time and strength. For all combinations of boundary conditions, a Nastrom–Gage type of spectrum of kinetic energy is found, where the power-law exponent changes from ${\approx}-3$ to ${\approx}-5/3$, i.e. from steep to shallow, as the spectral wavenumber increases.

Possible Demonstration of a Polaronic Bose-Einstein(-Mott) Condensate in UO2(+x) by Ultrafast THz Spectroscopy and Microwave Dissipation.

Bose-Einstein condensates (BECs) composed of polarons would be an advance because they would combine coherently charge, spin, and a crystal lattice. Following our earlier report of unique structural and spectroscopic properties, we now identify potentially definitive evidence for polaronic BECs in photo- and chemically doped UO2(+x) on the basis of exceptional coherence in the ultrafast time dependent terahertz absorption and microwave spectroscopy results that show collective behavior including dissipation patterns whose precedents are condensate vortex and defect disorder and condensate excitations. That some of these signatures of coherence in an atom-based system extend to ambient temperature suggests a novel mechanism that could be a synchronized, dynamical, disproportionation excitation, possibly via the solid state analog of a Feshbach resonance that promotes the coherence. Such a mechanism would demonstrate that the use of ultra-low temperatures to establish the BEC energy distribution is a convenience rather than a necessity, with the actual requirement for the particles being in the same state that is not necessarily the ground state attainable by other means. A macroscopic quantum object created by chemical doping that can persist to ambient temperature and resides in a bulk solid would be revolutionary in a number of scientific and technological fields.

Chern-Simons-Higgs transitions out of topological superconducting phases

In this study, we examine effective field theories of superconducting phases with topological order, making a connection to proposed realizations of exotic topological phases (including those hosting Ising and Fibonacci anyons) in superconductor-quantum Hall heterostructures. Our effective field theories for the non-Abelian superconducting states are non-Abelian Chern-Simons theories in which the condensation of vortices carrying non-Abelian gauge flux leads to the associated Abelian quantum Hall states. This Chern-Simons-Higgs condensation process is dual to the emergence of superconducting non-Abelian topological phases in coupled chain constructions. In such transitions, the chiral central charge of the system generally changes, so they fall outside the description of bosonic condensation transitions put forth by Bais and Slingerland [F. A. Bais and J. K. Slingerland, Phys. Rev. B 79, 045316 (2009)] (though the two approaches agree when the described transitions coincide). Our condensation process may be generalized to Chern-Simons theories based on arbitrary Lie groups, always describing a transition from a Lie algebra to its Cartan subalgebra. We include several instructive examples of such transitions.

Microscopic realization of two-dimensional bosonic topological insulators.

It is well known that a bosonic Mott insulator can be realized by condensing vortices of a boson condensate. Usually, a vortex becomes an antivortex (and vice versa) under time reversal symmetry, and the condensation of vortices results in a trivial Mott insulator. However, if each vortex or antivortex interacts with a spin trapped at its core, the time reversal transformation of the composite vortex operator will contain an extra minus sign. It turns out that such a composite vortex condensed state is a bosonic topological insulator (BTI) with gapless boundary excitations protected by U(1)⋊Z2(T) symmetry. We point out that in BTI, an external π-flux monodromy defect carries a Kramers doublet. We propose lattice model Hamiltonians to realize the BTI phase, which might be implemented in cold atom systems or spin-1 solid state systems.

Universal profile of the vortex condensate in two-dimensional turbulence.

An inverse turbulent cascade in a restricted two-dimensional periodic domain creates a condensate-a pair of coherent system-size vortices. We perform extensive numerical simulations of this system and carry out theoretical analysis based on momentum and energy exchanges between the turbulence and the vortices. We show that the vortices have a universal internal structure independent of the type of small-scale dissipation, small-scale forcing, and boundary conditions. The theory predicts not only the vortex inner region profile, but also the amplitude, which both perfectly agree with the numerical data.

Critical Capacitance and Charge-Vortex Duality Near the Superfluid-to-Insulator Transition

Using a generalized reciprocity relation between charge and vortex conductivities at complex frequencies in two space dimensions, we identify the capacitance in the insulating phase as a measure of vortex condensate stiffness. We compute the ratio of boson superfluid stiffness to vortex condensate stiffness at mirror points to be 0.21(1) for the relativistic O(2) model. The product of dynamical conductivities at mirror points is used as a quantitative measure of deviations from self-duality between charge and vortex theories. We propose the finite wave vector compressibility as an experimental measure of the vortex condensate stiffness for neutral lattice bosons.

Vortex condensation in [formula omitted] Chern–Simons model with a general Higgs potential on a torus

We consider the self-dual equations in a relativistic U(1)×U(1) Chern–Simons model with a general Higgs potential. We prove the existence of maximal vortex condensates for the system of equations on a torus, and derive some properties of vortex condensates.