Finite Critical Value Research Articles

Flux-tunable regimes and supersymmetry in twisted cuprate heterostructures

Van der Waals assembly allows for the creation of Josephson junctions in an atomically sharp interface between two exfoliated Bi2Sr2CaCu2O8+δ (Bi-2212) flakes that are twisted relative to each other. In a narrow range of angles close to 45°, the junction exhibits a regime where time-reversal symmetry can be spontaneously broken, and it can be used to encode an inherently protected qubit called flowermon. In this work, we investigate the physics emerging when two such junctions are integrated in a superconducting quantum interference device circuit threaded by a magnetic flux. We show that the flowermon qubit regime is maintained up to a finite critical value of the magnetic field, and, under appropriate conditions, it is protected against both charge and flux noise. For larger external fluxes, the interplay between the inherent twisted d-wave nature of the order parameter and the external magnetic flux enables the implementation of different artificial atoms, including a flux-biased protected qubit and a supersymmetric quantum circuit.

Spontaneous autophoretic motion of isotropic disks

It is theoretically known that an isotropic chemically active particle in an unbounded solution undergoes symmetry breaking when the intrinsic Péclet number ${{Pe}}$ exceeds a finite critical value (Michelin et al., Phys. Fluids, vol. 25, 2013, 061701). At that value, a transition takes place from a stationary state to spontaneous motion. In two dimensions, where no stationary state is possible in an unbounded domain, a linear stability analysis in a large bounded domain (Hu et al., Phys. Rev. Lett., vol. 123, 2019, 238004) reveals that the critical ${{Pe}}$ value slowly diminishes as the domain size increases. Motivated by these findings, we here consider an unbounded domain from the outset, addressing the two-dimensional problem of steady self-propulsion with a focus on the limit ${{Pe}}\ll 1$ . This singular limit is handled using matched asymptotic expansions, conceptually decomposing the fluid domain into a particle-scale region, where the leading-order solute transport is diffusive, and a remote region, where diffusion and advection are comparable. The expansion parameter is provided by the product of ${{Pe}}$ and $U$ , the unknown particle speed (normalised by the standard autophoretic scale). The problem is unconventional in that the scaling of $U$ with ${{Pe}}$ must be determined in the course of the perturbation analysis. The resulting approximation, $U=4\exp ({-2/{Pe}-\gamma _{E}-1})/{{Pe}}$ ( $\gamma _{E}$ being the Euler–Mascheroni constant), is in remarkable agreement with the numerical predictions of Hu et al. in the common interval of validity.

Saturation phenomena of a nonlocal eigenvalue problem: the Riemannian case

ABSTRACT In this paper we investigate the Riemannian extensibility of saturation phenomena treated first in the Euclidean framework by Brandolini et al. [Sharp estimates and saturation phenomena for a nonlocal eigenvalue problem. Adv Math (N Y). 2011;228(4):2352–2365.]. The saturation problem is formulated in terms of the first eigenvalue of the perturbation of the Laplace-Beltrami operator by the integral of the unknown function: the first eigenvalue increases with the weight affecting the integral up to a finite critical value and then remains constant, i.e. it saturates. Given a Riemannian manifold with certain curvature constraints, by using symmetrization arguments and sharp isoperimetric inequalities, we reduce the general problem to a variational one, formulated on either positively or negatively curved Riemannian model spaces; in addition, the possible scenarios for the optimal domains turn to be either geodesic balls or the union of two disjoint geodesic balls. We then explicitly compute the eigenvalues and eigenfunctions in terms of the radii, curvature and weight. A sufficient condition (incompatibility of a system of nonlinear equations involving special functions) is given that implies similar saturation phenomena to the Euclidean case. Due to its highly nonlinear character of the reduced problem (arising from the presence of curvature and special functions), we provide only partial answers to the original problem. However, both analytical computations and numerical tests suggest that the required incompatibility always persists. In addition, in the limit cases when the curvature tends to zero (for both positive an negative curvature), our results reduce to the Euclidean version.

Resilience of topological superconductivity under particle current

We investigate the robustness of topological superconductors under the perturbing influence of a finite charge current. To this aim, we introduce a modified Kitaev Hamiltonian parametrically dependent on the quasiparticle momentum induced by the current. Using different quantifiers of the topological phase, such as the Majorana polarization and the edge state quantum conditional mutual information, we prove the existence of a finite critical value of the quasiparticle momentum below which edge modes and topological superconductivity survive. We also discuss how a finite current breaks time reversal symmetry and changes the topological class in the Altland-Zirnbauer classification scheme compared to the case of isolated systems. Our findings provide a nontrivial example of the interplay between topology and the nonequilibrium physics of open quantum systems, a relation of crucial importance in the quest to a viable topological quantum electronics.

Fractonic Berezinskii-Kosterlitz-Thouless transition from a renormalization group perspective

Proliferation of defects is a mechanism that allows for topological phase transitions. Such a phase transition is found in two dimensions for the XY model, which lies in the Berezinskii-Kosterlitz-Thouless (BKT) universality class. The transition point can be found using renormalization group analysis. We apply renormalization group arguments to determine the nature of BKT transitions for the three-dimensional plaquette-dimer model, which is a model that exhibits fractonic mobility constraints. We show that an important part of this analysis demands a modified dimensional analysis that changes the interpretation of scaling dimensions upon coarse-graining. Using this modified dimensional analysis, we compute the beta functions of the model and predict a finite critical value above which the fractonic phase melts, proliferating dipoles. Importantly, the transition point is found through a renormalization group analysis that accounts for the phenomenon of UV/IR mixing, characteristic of fractonic models.

Relaxation to a Parity-Time Symmetric Generalized Gibbs Ensemble after a Quantum Quench in a Driven-Dissipative Kitaev Chain.

The construction of the generalized Gibbs ensemble, to which isolated integrable quantum many-body systems relax after a quantum quench, is based upon the principle of maximum entropy. In contrast, there are no universal and model-independent laws that govern the relaxation dynamics and stationary states of open quantum systems, which are subjected to Markovian drive and dissipation. Yet, as we show, relaxation of driven-dissipative systems after a quantum quench can, in fact, be determined by a maximum entropy ensemble, if the Liouvillian that generates the dynamics of the system has parity-time symmetry. Focusing on the specific example of a driven-dissipative Kitaev chain, we show that, similar to isolated integrable systems, the approach to a parity-time symmetric generalized Gibbs ensemble becomes manifest in the relaxation of local observables and the dynamics of subsystem entropies. In contrast, the directional pumping of fermion parity, which is induced by nontrivial non-Hermitian topology of the Kitaev chain, represents a phenomenon that is unique to relaxation dynamics in driven-dissipative systems. Upon increasing the strength of dissipation, parity-time symmetry is broken at a finite critical value, which thus constitutes a sharp dynamical transition that delimits the applicability of the principle of maximum entropy. We show that these results, which we obtain for the specific example of the Kitaev chain, apply to broad classes of noninteracting fermionic models, and we discuss their generalization to a noninteracting bosonic model and an interacting spin chain.

Ultracompact object within a nonlocal Tolman VII model

Tolman VII (TVII) is an analytical model for the nonrotating perfect fluid sphere with a simple quadratic density profile where its value is zero at the surface and a finite critical density value at the center. Therefore, compared to another analytical model like the constant density star, TVII is more realistic. Except for the dominant energy condition (DEC), which is violated in the region near the star's center, the TVII satisfies all energy conditions. However, the causal condition is also violated for $\mathcal{C}>0.26$, and the maximum compactness of the TVII is restricted, i.e., ${\mathcal{C}}_{\mathrm{max}}\ensuremath{\approx}0.38$. Here, we investigate the impacts of nonlocal gravity on the properties of the star of TVII within the range of the compactness of an ultracompact star ($0.33\ensuremath{\le}\mathcal{C}\ensuremath{\le}0.44$). This nonlocal gravity version of TVII (NGTVII) is parametrized by the nonlocal parameters ($\stackrel{\texttildelow{}}{\ensuremath{\beta}}$) and the compactness ($\mathcal{C}$). We have found that NGTVII can reach ${\mathcal{C}}_{\mathrm{max}}=0.43$ with ${\stackrel{\texttildelow{}}{\ensuremath{\beta}}}_{\mathrm{max}}=3$ which is significantly more compact than TVII. The nonlocal density and pressure profile differs from the TVII, depending on the stars' compactness and nonlocal parameter. We have also found that for the relatively small value of $\stackrel{\texttildelow{}}{\ensuremath{\beta}}$ and the compactness, i.e., $\mathcal{C}\ensuremath{\lesssim}0.31$, the causality condition and the DEC are not violated. We have also found that the NGTVII's effective potential in the interior can be larger and deeper than that of the TVII model, indicating the deceleration of the echo time. Moreover, using the effective potential of NGTVII, the quasinormal mode and gravitational echo are calculated using Bohr-Sommerfeld fitting and solving the time-dependent Reggae-Wheeler equation. We can infer that the NGVTII with the maximum compactness and nonlocal parameter values enables the existence of the ultracompact star with more trapped modes.

Dynamics of pearling instability in polymersomes: The role of shear membrane viscosity and spontaneous curvature

The stability of copolymer tethers is investigated theoretically. Self-assembly of diblock or triblock copolymers can lead to tubular polymersomes, which are known experimentally to undergo shape instability under thermal, chemical, and tension stresses. It leads to a periodic modulation of the radius, which evolves to assembly line pearls connected by tiny tethers. We study the contributions of shear surface viscosity and spontaneous curvature and their interplay to understand the pearling instability. The performed linear analysis of stability of this cylinder-to-pearls transition shows that such systems are unstable if the membrane tension is larger than a finite critical value contrary to the Rayleigh–Plateau instability, an already known result, or if the spontaneous curvature is in a specific range, which depends on membrane tension. For the case of spontaneous curvature-induced shape instability, two dynamical modes are identified. The first one is analog to the tension-induced instability with a marginal mode. Its wavenumber associated with the most unstable mode decreases continuously to zero as membrane viscosity increases. The second one has a finite range of unstable wavenumbers. The wavenumber of the most unstable mode tends to be constant as membrane viscosity increases. In this mode, its growth rate becomes independent of the bulk viscosity in the limit of high membrane viscosity and behaves as a pure viscous surface.

Chaitin’s Omega and an algorithmic phase transition

We consider the statistical mechanical ensemble of bit string histories that are computed by a universal Turing machine. The role of the energy is played by the program size. We show that this ensemble has a first-order phase transition at a critical temperature, at which the partition function equals Chaitin’s halting probability Ω. This phase transition has curious properties: the free energy is continuous near the critical temperature, but almost jumps: it converges more slowly to its finite critical value than any computable function. At the critical temperature, the average size of the bit strings diverges. We define a non-universal Turing machine that approximates this behavior of the partition function in a computable way by a super-logarithmic singularity, and discuss its thermodynamic properties. We also discuss analogies and differences between Chaitin’s Omega and the partition function of a quantum mechanical particle, and with quantum Turing machines. For universal Turing machines, we conjecture that the ensemble of bit string histories at the critical temperature has a continuum formulation in terms of a string theory.

Personalized Immunotherapy Treatment Strategies for a Dynamical System of Chronic Myelogenous Leukemia.

Simple SummaryAs computer performance continues to grow at more affordable costs, mathematical modelling and in silico experimentation begin to play a larger role in understanding cancer evolution. The aim of our work is to formulate a control strategy for the Adaptive Cellular Therapy (ACT) that can fully eradicates the Chronic Myelogenous Leukemia (CML) cells population in a mathematical model describing interactions between naive T cells, effector T cells and CML cancer cells in the circulatory blood system. Mathematical analysis and numerical simulations allow us to conclude that it is possible to design a personalized administration protocol for the ACT in the form of a pulse train with asymmetrical waves and a fixed amplitude to achieve complete CML cancer cells eradication. The amplitude of the impulse on which the treatment is applied is given by an arithmetical combination of the parameters of the system with at least a duty cycle of 45 min/day.This paper is devoted to exploring personalized applications of cellular immunotherapy as a control strategy for the treatment of chronic myelogenous leukemia described by a dynamical system of three first-order ordinary differential equations. The latter was achieved by applying both the Localization of Compact Invariant Sets and Lyapunov’s stability theory. Combination of these two approaches allows us to establish sufficient conditions on the immunotherapy treatment parameter to ensure the complete eradication of the leukemia cancer cells. These conditions are given in terms of the system parameters and by performing several in silico experimentations, we formulated a protocol for the therapy application that completely eradicates the leukemia cancer cells population for different initial tumour concentrations. The formulated protocol does not dangerously increase the effector T cells population. Further, complete eradication is considered when solutions go below a finite critical value below which cancer cells cannot longer persist; i.e., one cancer cell. Numerical simulations are consistent with our analytical results.

Phase transitions in a non-Hermitian Aubry-André-Harper model

The Aubry-Andr\'e-Harper model provides a paradigmatic example of aperiodic order in a one-dimensional lattice displaying a delocalization-localization phase transition at a finite critical value ${V}_{c}$ of the quasiperiodic potential amplitude $V$. In terms of the dynamical behavior of the system, the phase transition is discontinuous when one measures the quantum diffusion exponent $\ensuremath{\delta}$ of wave-packet spreading, with $\ensuremath{\delta}=1$ in the delocalized phase $V<{V}_{c}$ (ballistic transport), $\ensuremath{\delta}\ensuremath{\simeq}1/2$ at the critical point $V={V}_{c}$ (diffusive transport), and $\ensuremath{\delta}=0$ in the localized phase $V>{V}_{c}$ (dynamical localization). However, the phase transition turns out to be smooth when one measures, as a dynamical variable, the speed $v(V)$ of excitation transport in the lattice, which is a continuous function of potential amplitude $V$ and vanishes as the localized phase is approached. Here we consider a non-Hermitian extension of the Aubry-Andr\'e-Harper model, in which hopping along the lattice is asymmetric, and show that the dynamical localization-delocalization transition is discontinuous, not only in the diffusion exponent $\ensuremath{\delta}$, but also in the speed $v$ of ballistic transport. This means that even very close to the spectral phase transition point, rather counterintuitively, ballistic transport with a finite speed is allowed in the lattice. Also, we show that the ballistic velocity can increase as $V$ is increased above zero, i.e., surprisingly, disorder in the lattice can result in an enhancement of transport.

Entanglement transitions as a probe of quasiparticles and quantum thermalization

We introduce a diagnostic for quantum thermalization based on mixed-state entanglement. Specifically, given a pure state on a tripartite system $ABC$, we study the scaling of entanglement negativity between $A$ and $B$. For representative states of self-thermalizing systems, either eigenstates or states obtained by a long-time evolution of product states, negativity shows a sharp transition from an area-law scaling to a volume-law scaling when the subsystem volume fraction is tuned across a finite critical value. In contrast, for a system with quasiparticles, it exhibits a volume-law scaling irrespective of the subsystem fraction. For many-body localized systems, the same quantity shows an area-law scaling for eigenstates, and volume-law scaling for long-time evolved product states, irrespective of the subsystem fraction. We provide a combination of numerical observations and analytical arguments in support of our conjecture. Along the way, we prove and utilize a `continuity bound' for negativity: we bound the difference in negativity for two density matrices in terms of the Hilbert-Schmidt norm of their difference.

Fluctuating temperature and baryon chemical potential in heavy-ion collisions and the position of the critical end point in the effective QCD phase diagram

We use the linear sigma model with quarks to locate the critical end point in the effective QCD phase diagram accounting for fluctuations in temperature and quark chemical potential. For this purpose, we use the non-equilibrium formalism provided by the superstatistics framework. We compute the effective potential in the high- and low-temperature approximations up to sixth order and include the contribution of ring diagrams to account for plasma screening effects. We fix the model parameters from relations between the thermal sigma and pion masses imposing a first order phase transition at zero temperature and a finite critical value for the baryon chemical potential that we take of order of the nucleon mass. We find that the CEP displacement due to fluctuations in temperature and/or quark chemical potential is almost negligible.

Renormalization-group study of the many-body localization transition in one dimension

Using a new approximate strong-randomness renormalization group (RG), we study the many-body localized (MBL) phase and phase transition in one-dimensional quantum systems with short-range interactions and quenched disorder. Our RG is built on those of Zhang $\textit{et al.}$ [1] and Goremykina $\textit{et al.}$ [2], which are based on thermal and insulating blocks. Our main addition is to characterize each insulating block with two lengths: a physical length, and an internal decay length $\zeta$ for its effective interactions. In this approach, the MBL phase is governed by a RG fixed line that is parametrized by a global decay length $\tilde{\zeta}$, and the rare large thermal inclusions within the MBL phase have a fractal geometry. As the phase transition is approached from within the MBL phase, $\tilde{\zeta}$ approaches the finite critical value corresponding to the avalanche instability, and the fractal dimension of large thermal inclusions approaches zero. Our analysis is consistent with a Kosterlitz-Thouless-like RG flow, with no intermediate critical MBL phase.

Finite-size effects on pattern selection in immiscible fluids subjected to horizontal vibrations in weightlessness.

A detailed numerical investigation of frozen wave pattern selection in finite rectangular containers under microgravity conditions is presented. The column growth cycle controlling mode transitions is described and the instability diagram showing selected wave number as function of the vibrational velocity is obtained. In contrast to the continuous monotonic dependence of pattern wave number on forcing predicted by linear inviscid theory in the limit of infinitely long containers, the pattern selection process in finite domains is characterized by solution branches that persist over discrete forcing intervals. We describe how properties of the selected frozen wave pattern, including the finite critical forcing value, depend on container length. In long containers, a transition from nearly symmetric to asymmetric columnar development is found at sufficiently high forcing values, with the loss of (approximate) reflection symmetry evident in the mean flow associated with the transient growth of the pattern. The effect of container height is considered separately, in the limit of both shallow and deep layers. Shallow layers suppress the development of long-wave perturbations leading to higher wave-number patterns whose growth may be associated with an asymmetric series of collisions between developing columns and container boundaries. In thick layers, lower pattern wave numbers are observed and, although the column growth is on average more regular, the final state is often asymmetric. Finally, we compare our results with the dependence on container aspect ratio Γ=L/2H predicted by inviscid theory. For Γ≳2, finite-size effects are reasonably weak and the numerically obtained thresholds are similar to but slightly higher than the theoretical values, most likely due to viscous effects. For Γ≲2, finite-size effects come to the fore and onset is significantly delayed.

Weak-strong uniqueness for the compressible Navier-Stokes equations with a hard-sphere pressure law

We consider the Navier-Stokes equations with a pressure function satisfying a hard-sphere law. That means the pressure, as a function of the density, becomes infinite when the density approaches a finite critical value. Under some structural constraints imposed on the pressure law, we show a weak-strong uniqueness principle in periodic spatial domains. The method is based on a modified relative entropy inequality for the system. The main difficulty is that the pressure potential associated with the internal energy of the system is largely dominated by the pressure itself in the area close to the critical density. As a result, several terms appearing in the relative energy inequality cannot be controlled by the total energy.

On gravity’s role in the genesis of rest masses of classical fields

It is shown that in the Einstein-conformally coupled Higgs--Maxwell system with Friedman-Robertson-Walker symmetries the energy density of the Higgs field has stable local minimum only if the mean curvature of the $t={\rm const}$ hypersurfaces is less than a finite critical value $\chi_c$, while for greater mean curvature the energy density is not bounded from below. Therefore, there are extreme gravitational situations in which even quasi-locally defined instantaneous vacuum states of the Higgs sector cannot exist, and hence one cannot at all define the rest mass of all the classical fields. On hypersurfaces with mean curvature less than $\chi_c$ the energy density has the `wine bottle' (rather than the familiar `Mexican hat') shape, and the gauge field can get rest mass via the Brout--Englert--Higgs mechanism. The spacelike hypersurface with the critical mean curvature represents the moment of `genesis' of rest masses.

Effects of slip and rheological parameters on the flow and heat transfer of a Herschel-Bulkley fluid

PurposeThe purpose of this paper is to investigate the combined effects of slip and rheological parameters on the flow and heat transfer of the Herschel-Bulkley fluid.Design/methodology/approachThe combinative dimensionless parameter method is introduced into the equations of the slip flow and heat transfer to make the discussion more comprehensive. More specifically, the slip and rheological parameters are transformed into the dimensionless slip number as well as Herschel-Bulkley number. We solve the dimensionless equations and then focus on the effects of these parameters on the slip flow and heat transfer.FindingsThe results show that, for a given value of Herschel-Bulkley number, there is a finite critical value of slip number at which the pressure gradient reaches the lowest value and both the dimensionless yield radius and slip velocity become 1. Meanwhile, the Nusselt number tends to be infinite at this critical value of slip number. For the case of slip, the Nusselt number also approaches infinity at a finite critical value of Herschel-Bulkley number. Furthermore, the dimensionless velocity as well as temperature of the yield pseudoplastic fluid with higher slip number is lower within a small radius but becomes higher near the wall. Meanwhile, from the velocity and temperature profiles, the effect of Herschel-Bulkley number on these two parameters of the Bingham fluid at the smaller radius is opposite.Originality/valueThese associated expressions can be generalized to the flow and heat transfer of a Herschel-Bulkley fluid under slip boundary condition. It can provide a reference for the engineering application relating to the heat transfer and flow of a Herschel-Bulkley fluid. Meanwhile, it also suggests some revelations for dealing with this similar problem.

Topologically induced swarming phase transition on a 2D percolated lattice

The emergence of collective motion, or swarming, in groups of moving individuals who orient themselves using only information from their neighbors is a very general phenomenon that occurs at multiple spatio-temporal scales. Swarms that occur in natural environments typically have to contend with spatial disorder such as obstacles that can hinder an individual’s motion or can disrupt communication with neighbors. We study swarming agents, possessing both aligning and mutually avoiding repulsive interactions, in a 2D percolated network representing a topologically disordered environment. We numerically find a phase transition from a collectively moving swarm to a disordered gas-like state above a critical value of the topological or environmental disorder. For agents that utilize only alignment interactions, we find that the swarming transition does not exist in the large system size limit, while the addition of a mutually repulsive interaction can restore the existence of the transition at a finite critical value of disorder. We find there is a finite range of topological disorder where swarming can occur and that this range can be maximized by an optimal amount of mutual repulsion.

Single-particle stochastic heat engine.

We have performed an extensive analysis of a single-particle stochastic heat engine constructed by manipulating a Brownian particle in a time-dependent harmonic potential. The cycle consists of two isothermal steps at different temperatures and two adiabatic steps similar to that of a Carnot engine. The engine shows qualitative differences in inertial and overdamped regimes. All the thermodynamic quantities, including efficiency, exhibit strong fluctuations in a time periodic steady state. The fluctuations of stochastic efficiency dominate over the mean values even in the quasistatic regime. Interestingly, our system acts as an engine provided the temperature difference between the two reservoirs is greater than a finite critical value which in turn depends on the cycle time and other system parameters. This is supported by our analytical results carried out in the quasistatic regime. Our system works more reliably as an engine for large cycle times. By studying various model systems, we observe that the operational characteristics are model dependent. Our results clearly rule out any universal relation between efficiency at maximum power and temperature of the baths. We have also verified fluctuation relations for heat engines in time periodic steady state.