Length Exponent Research Articles

Continuum percolation of porous media via random packing of overlapping cube-like particles

The pore configuration in porous medium is assumed to be the randomly distributed cube-like particles which can overlap each other in the periodic cubic domain, and the impact of particle characteristics on the percolation property of these cube-like particle packing systems is analyzed. Firstly, by combining the percolation models and finite-size scaling analysis, three numerical parameters (i.e., percolation transition width ΔL, local percolation threshold ψc(L), and correlation length exponent ν) for the cube-like particle systems with shape parameter s in [1.0, + ∞] are derived successively. Then, based on the relation between the percolation threshold ψc in infinite space and the local percolation threshold ψc(L), the corresponding ψc with s in [1.0, + ∞] are further determined. It is shown from the study that the characteristics of cube-like particles have significant influence on the global percolation threshold ψc of the particle packing systems. As the parameter s increases from 1.0 to = ∞, the percolation threshold ψc will go down persistently. When the surface of cube-like particles is cubical and spherical, respectively, the minimum and maximum thresholds ψc,min and ψc,max are obtained.

Quantum critical behavior of a three-dimensional superfluid-Mott glass transition

The superfluid to insulator quantum phase transition of a three-dimensional particle-hole symmetric system of disordered bosons is studied. To this end, a site-diluted quantum rotor Hamiltonian is mapped onto a classical (3+1)-dimensional XY model with columnar disorder and analyzed by means of large-scale Monte Carlo simulations. The superfluid-Mott insulator transition of the clean, undiluted system is in the 4D XY universality class and shows mean-field critical behavior with logarithmic corrections. The clean correlation length exponent $\nu = 1/2$ violates the Harris criterion, indicating that disorder must be a relevant perturbation. For nonzero dilutions below the lattice percolation threshold of $p_c = 0.688392$, our simulations yield conventional power-law critical behavior with dilution-independent critical exponents $z=1.67(6)$, $\nu = 0.90(5)$, $\beta/\nu = 1.09(3)$, and $\gamma/\nu = 2.50(3)$. The critical behavior of the transition across the lattice percolation threshold is controlled by the classical percolation exponents. Our results are discussed in the context of a classification of disordered quantum phase transitions, as well as experiments in superfluids, superconductors and magnetic systems.

Numerical study of the chiral Z3 quantum phase transition in one spatial dimension

Recent experiments on a one-dimensional chain of trapped alkali atoms [arXiv:1707.04344] have observed a quantum transition associated with the onset of period-3 ordering of pumped Rydberg states. This spontaneous $\mathbb{Z}_3$ symmetry breaking is described by a constrained model of hard-core bosons proposed by Fendley $et\, \,al.$ [arXiv:cond-mat/0309438]. By symmetry arguments, the transition is expected to be in the universality class of the $\mathbb{Z}_3$ chiral clock model with parameters preserving both time-reversal and spatial-inversion symmetries. We study the nature of the order-disorder transition in these models, and numerically calculate its critical exponents with exact diagonalization and density-matrix renormalization group techniques. We use finite-size scaling to determine the dynamical critical exponent $z$ and the correlation length exponent $\nu$. Our analysis presents the only known instance of a strongly-coupled transition between gapped states with $z \ne 1$, implying an underlying nonconformal critical field theory.

Pair-breaking quantum phase transition in superconducting nanowires

Quantum phase transitions (QPT) between distinct ground states of matter are widespread phenomena1–5, yet there are only a few experimentally accessible systems6,7 where the microscopic mechanism of the transition can be tested and understood. These cases are unique and form the experimentally established foundation for our understanding of quantum critical phenomena. Here we report that a magnetic-field-driven QPT in superconducting nanowires—a prototypical one-dimensional system (d=1)—can be fully explained by the critical theory8,9 of pair-breaking transitions characterized by a correlation length exponent v≈1 and dynamic critical exponent z≈2. We find that in the quantum critical regime, the electrical conductivity is in agreement with a theoretically predicted scaling function and, moreover, that the theory quantitatively describes the dependence of conductivity on the critical temperature, field magnitude and orientation, nanowire cross-sectional area, and microscopic parameters of the nanowire material. At the critical field, the conductivity follows a T(d–2)/z dependence predicted by phenomenological scaling theories10,11 and more recently obtained within a holographic framework12. Our work uncovers the microscopic processes governing the transition: the pair-breaking effect of the magnetic field on interacting Cooper pairs overdamped by their coupling to electronic degrees of freedom. It also reveals the universal character of continuous quantum phase transitions. Transport measurements performed on MoGe superconducting nanowires reveal a quantitative agreement with quantum critical behaviour driven by a pair-breaking mechanism.

Sequential superconductor–Bose insulator–Fermi insulator phase transitions in quasi-two-dimensional α -WSi

A zero-temperature magnetic-field-driven superconductor to insulator transition (SIT) in quasi-two-dimensional superconductors is expected to occur when the applied magnetic field crosses a certain critical value [S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar, Rev. Mod. Phys. 69, 315 (1997); A. M. Goldman, Int. J. Mod. Phys. B 24, 4081 (2010)]. A fundamental question is whether this transition is due to the localization of Cooper pairs or due to the destruction of them. Here we address this question by studying the SIT in amorphous WSi. Transport measurements reveal the localization of Cooper pairs at a quantum critical field ${B}_{c}^{1}$ (Bose insulator), with a product of the correlation length and dynamical exponents $z\ensuremath{\nu}\ensuremath{\sim}4/3$ near the quantum critical point. Beyond ${B}_{c}^{1}$, superconducting fluctuations still persist at finite temperatures. Above a second critical field ${B}_{c}^{2}>{B}_{c}^{1}$, the Cooper pairs are destroyed and the film becomes a Fermi insulator. The different phases all merge at a tricritical point at finite temperatures with $z\ensuremath{\nu}=2/3$. Our results suggest a sequential superconductor to Bose insulator to Fermi insulator phase transition, which differs from the conventional scenario involving a single quantum critical point.

Multipartite entanglement, quantum coherence, and quantum criticality in triangular and Sierpiński fractal lattices.

We investigate the quantum phase transitions of the transverse-field quantum Ising model on the triangular lattice and Sierpiński fractal lattices by employing the multipartite entanglement and quantum coherence along with the quantum renormalization group method. It is shown that the quantum criticalities of these high-dimensional models closely relate to the behaviors of the multipartite entanglement and quantum coherence. As the thermodynamic limit is approached, the first derivatives of the multipartite entanglement and quantum coherence exhibit singular behaviors, and the consistent finite-size scaling behaviors for each lattice are also obtained from the first derivatives. The multipartite entanglement and quantum coherence are demonstrated to be good indicators for detecting the quantum phase transitions in the triangular lattice and Sierpiński fractal lattices. Furthermore, the dimensions determine the relations between the critical exponents and the correlation length exponents for these lattices.

Condition factor and length-weight relationship of <i>Bagrus bayad macropterus</i> (Daget, 1954) from Agbura River, Niger Delta, Nigeria

The study on length-weight relationship and condition factor of Bagrus bayad macropterus from Agbura River, a tributary of the Nun River was conducted monthly for eighteen months (April 2012 – December 2013). Three thousand, six hundred and forty (3640) fish samples comprising of one hundred and ninety-three (193) Bagrus bayad macropterus were collected using traditional fishing gear and drag nets. The specimens were preserved in ice-packed box and taken to laboratory for length-weight measurement. The length-weight relationship was estimated using the linear regression model 𝑊=𝑎𝐿𝑏 and condition factor (k) determined using the equation 𝐾=100𝑊𝐿𝑏 .The standard length ranged between 17.6 and 27.2 cm and the weight ranged between 170.56 and 501.30 g in males and 16.8 to 32.2 cm and 190.60 to 510.35 g in females. The value of length exponent was more than 3 showing positive allometric growth pattern. The b-values tested using ANOVA indicated a highly significant regression of weight on standard length (P<0.05). The correlation (r) between increase in length and gain in weight was significant. The condition factor ranged between 1.51 and 3.13 with mean k-value of 2.23. The k-values showed monthly variations with peak values between the months of January and April. The fish species was in good condition in the study area as the K values were above one throughout the period of study.Keywords: Condition factor, length- weight relationship, Bagrus bayad macropterus, Agbura River Niger Delta

Itinerant quantum multicriticality of two-dimensional Dirac fermions

We analyze emergent quantum multi-criticality for strongly interacting, massless Dirac fermions in two spatial dimensions ($d=2$) within the framework of Gross-Neveu-Yukawa models, by considering the competing order parameters that give rise to fully gapped (insulating or superconducting) ground states. We focus only on those competing orders, which can be rotated into each other by generators of an exact or emergent chiral symmetry of massless Dirac fermions, and break $O(S_1)$ and $O(S_2)$ symmetries in the ordered phase. Performing a renormalization group analysis by using the $\epsilon=(3-d)$ expansion scheme, we show that all the coupling constants in the critical hyperplane flow toward a new attractive fixed point, supporting an \emph{enlarged} $O(S_1+S_2)$ chiral symmetry. Such a fixed point acts as an exotic quantum multi-critical point (MCP), governing the \emph{continuous} semimetal-insulator as well as insulator-insulator (for example antiferromagnet to valence bond solid) quantum phase transitions. In comparison with the lower symmetric semimetal-insulator quantum critical points, possessing either $O(S_1)$ or $O(S_2)$ chiral symmetry, the MCP displays enhanced correlation length exponents, and anomalous scaling dimensions for both fermionic and bosonic fields. We discuss the scaling properties of the ratio of bosonic and fermionic masses, and the increased dc resistivity at the MCP. By computing the scaling dimensions of different local fermion bilinears in the particle-hole channel, we establish that most of the four fermion operators or generalized density-density correlation functions display faster power law decays at the MCP compared to the free fermion and lower symmetric itinerant quantum critical points. Possible generalization of this scenario to higher dimensional Dirac fermions is also outlined.

Numerical simulation of fluid flow and sensitivity analysis in rough-wall fractures

Activated natural fractures, poorly-propped and un-propped secondary hydraulic fractures may serve as the connection between reservoir fluids and the main hydraulic fractures (i.e., well-propped fractures). These branched fractures also contribute considerably to well after-fracturing productivity in unconventional tight/shale reservoirs. Rough fracture surface and its effects on fluid flow behavior in such fractures with very limited width is very important in understanding total mass transport in fractured rocks. In this paper, the effect of rough surface properties on fracture conductivity is studied through 3D numerical simulation of fluid flow behavior within the fractures. First, a tensile fracture was created on a core sample retrieved from the Montney formation, and the rough fracture surface topography was scanned via a high-resolution optical profilometer. The surface roughness characterizations were then used as reference to numerically reconstruct multiple three-dimensional fracture models. A finite element method was adopted to simulate fluid flow in the rough-wall fractures, using Navier-Stokes equation, for various fracture surface properties such as mean aperture, root mean square (RMS) asperity height, correlation length, anisotropy and shear displacement distance. Results showed that cubic law tends to overestimate the conductivity of the fractures with rough-wall surfaces, especially when the fracture aperture is small. Other surface properties, including correlation length, anisotropy and Hurst exponent also exert considerable impacts on fracture hydraulic properties. As the shear displacement distance increases, deviation of the fracture conductivity from cubic law first increases and then maintains at a constant level. The results of this paper have advanced the understanding of fracture conductivity and its dependence on fracture surface properties and will aid in improving production prediction and optimization in fractured tight reservoirs.

Dynamics of a quantum phase transition in the Bose-Hubbard model: Kibble-Zurek mechanism and beyond

In this paper, we study the dynamics of the Bose-Hubbard model by using time-dependent Gutzwiller methods. In particular, we vary the parameters in the Hamiltonian as a function of time, and investigate the temporal behavior of the system from the Mott insulator to the superfluid (SF) crossing a second-order phase transition. We first solve a time-dependent Schr\"odinger equation for the experimental setup recently done by Braun et.al. [Proc. Nat. Acad. Sci. 112, 3641 (2015)] and show that the numerical and experimental results are in fairly good agreement. However, these results disagree with the Kibble-Zurek scaling. From our numerical study, we reveal a possible source of the discrepancy. Next, we calculate the critical exponents of the correlation length and vortex density in addition to the SF order parameter for a Kibble-Zurek protocol. We show that beside the "freeze" time $\hat{t}$, there exists another important time, $t_{\rm eq}$, at which an oscillating behavior of the SF amplitude starts. From calculations of the exponents of the correlation length and vortex density with respect to a quench time $\tQ$, we obtain a physical picture of a coarsening process. Finally, we study how the system evolves after the quench. We give a global picture of dynamics of the Bose-Hubbard model.

Saturated imbibition under the influence of gravity and geometry

HypothesisThe effect of gravity was generally neglected in the classical imbibition law for one dimensional geometries. Following researches complemented the classical “Lucas-Washburn law” with consideration of gravity, but no examination of geometries under influence of gravity has been done, while geometry was shown to yield different scaling law for the imbibition process. Hence, it is possible to discover new time exponents for imbibition length in two dimensional and three dimensional imbibition process under gravity. MethodsThrough theoretical analysis and numerical simulations, the size of wetted region under three gravitational scenarios (zero gravity, acceleration and deceleration) in three geometries (one dimensional, two dimensional radial and three dimensional radial) are determined quantitatively. FindingsNew time exponents other than classic 1/2 are identified under different directions of gravity in two dimensional radial and three dimensional radial imbibition, and symmetry of time exponents due to different directions of gravity is discovered. A new time exponent of 1 for the acceleration case in one dimensional imbibition is found. The flow field in the wetted region is also determined from simulations. Discoveries in this paper show that new physical laws for imbibition length exist at the intersection of gravity and geometry.

Non-Abelian fermionization and fractional quantum Hall transitions

There has been a recent surge of interest in dualities relating theories of Chern-Simons gauge fields coupled to either bosons or fermions within the condensed matter community, particularly in the context of topological insulators and the half-filled Landau level. Here, we study the application of one such duality to the long-standing problem of quantum Hall inter-plateaux transitions. The key motivating experimental observations are the anomalously large value of the correlation length exponent $\nu \approx 2.3$ and that $\nu$ is observed to be super-universal, i.e., the same in the vicinity of distinct critical points. Duality motivates effective descriptions for a fractional quantum Hall plateau transition involving a Chern-Simons field with $U(N_c)$ gauge group coupled to $N_f = 1$ fermion. We study one class of theories in a controlled limit where $N_f \gg N_c$ and calculate $\nu$ to leading non-trivial order in the absence of disorder. Although these theories do not yield an anomalously large exponent $\nu$ within the large $N_f \gg N_c$ expansion, they do offer a new parameter space of theories that is apparently different from prior works involving abelian Chern-Simons gauge fields.

Regulator dependence of fixed points in quantum Einstein gravity with R2 truncation

We performed a functional renormalization group analysis for the quantum Einstein gravity including a quadratic term in the curvature. The ultraviolet non-gaussian fixed point and its critical exponent for the correlation length are identified for different forms of regulators in case of dimension 3. We searched for that optimized regulator where the physical quantities show the least regulator parameter dependence. It is shown that the Litim regulator satisfies this condition. The infrared fixed point has also been investigated, it is found that the exponent is insensitive to the third coupling introduced by the R2 term.

Field-tuned superconductor–insulator transitions and Hall resistance in thin polycrystalline MoN films

We report on the superconductor–insulator transitions (SITs) of disordered molybdenum nitride (MoN) thin films on (1 0 0) MgO substrates as a function of the film thickness and magnetic fields. The Tc of the superconducting MoN films, which exhibit a sharp superconducting transition, monotonically decreases as the normal state Rsq increases with a decreasing film thickness. For several films with different thicknesses, we estimate the critical field Hc and the product zν ≃ 0.6 of the dynamical exponent z and the correlation length exponent ν using a finite scaling analysis. The value of this product can be explained by the (2 + 1) XY model. We found that the Hall resistance ΔRxy(H) is maximized when the magnetic field satisfies HHP(T) |1 − T/TC0| in the superconducting state and also in the normal states owning to the superconducting fluctuation corresponding to the ghost critical magnetic field. We measured the Hall conductivity δσxy(H) = σxy(H) − and fit the Gaussian approximation theory for δσxy(H) to the experimental data. Agreement between the data and the theory beyond Hc suggests the survival of the Cooper pair in the insulating region of the SIT.

Influence of Landscape Coverage on Measuring Spatial and Length Properties of Rock Fracture Networks: Insights from Numerical Simulation

Natural fractures are ubiquitous in the Earth’s crust and often deeply buried in the subsurface. Due to the difficulty in accessing to their three-dimensional structures, the study of fracture network geometry is usually achieved by sampling two-dimensional (2D) exposures at the Earth’s surface through outcrop mapping or aerial photograph techniques. However, the measurement results can be considerably affected by the coverage of forests and other plant species over the exposed fracture patterns. We quantitatively study such effects using numerical simulation. We consider the scenario of nominally isotropic natural fracture systems and represent them using 2D discrete fracture network models governed by fractal and length scaling parameters. The groundcover is modelled as random patches superimposing onto the 2D fracture patterns. The effects of localisation and total coverage of landscape patches are further investigated. The fractal dimension and length exponent of the covered fracture networks are measured and compared with those of the original non-covered patterns. The results show that the measured length exponent increases with the reduced localisation and increased coverage of landscape patches, which is more evident for networks dominated by very large fractures (i.e. small underlying length exponent). However, the landscape coverage seems to have a minor impact on the fractal dimension measurement. The research findings of this paper have important implications for field survey and statistical analysis of geological systems.

Effects of surface topography on SERS response: Correlating nanoscopy with spectroscopy

This paper reports for the first time the hidden correlation between the topographical features of the bilayer Langmuir-Blodgett (LB) film substrates of stearic acid (SA) incubated in Au@Ag nanocolloids over various dipping times (DTs) with their corresponding SERS responses. The topographies of the as prepared substrates are investigated from the statistical considerations in terms of lateral correlation length, interface width, Hurst and Lyapnov exponents. The real space of the substrates are mapped directly from the FESEM and AFM images of the bilayer LB film of SA immersed in Au@Ag nanocolloids over various DTs ranging between 6 and 72 h. The SERS spectra of the Rhodamine 6G molecules adsorbed on the as prepared substrates have been reported. The statistical parameters of the substrates that exhibit maximum SERS efficacy have been suggested. The far field distributions in presence and in absence of Raman dipole together with spatial distribution of the near field from the hottest spot of the as prepared substrate have also been reported. To our knowledge, this is the first report that links nanoscopy with SERS spectroscopy from statistical considerations and is expected to open a new window towards the fabrication of more efficient and reproducible SERS active substrates in future endeavours.

Quantum phase transitions to topological Haldane phases in spin-one chains studied by linked-cluster expansions

We use linked-cluster expansions to analyze the quantum phase transitions between symmetry unbroken trivial and topological Haldane phases in two different spin-one chains. The first model is the spin-one Heisenberg chain in the presence of a single-ion anisotropy while the second one is the dimerized spin-one Heisenberg chain. For both models we determine the ground-state energy and the one-particle gap inside the non-topological phase as a high-order series using perturbative continuous unitary transformations. Extrapolations of the gap series are applied to locate the quantum critical point and to extract the associated critical exponent. We find that this approach works unsatisfactory for the anisotropic chain, since the quality of the extrapolation appears insufficient due to the large correlation length exponent. In contrast, extrapolation schemes display very good convergence for the gap closing in the case of the dimerized spin-one Heisenberg chain.

Multiscale Characterization of Joint Surface Roughness

AbstractRecent studies provided detailed characterizations of fault (i.e., shear fracture) roughness at different length scales. Similar investigation for joints (i.e., tensile fractures) are seldom and not as detailed. The present study aims at characterizing joint plumose patterns. We investigated the scale‐dependent surface roughness properties of S‐type plumoses. Joint surface measurements at relatively large scales were carried out with Light Detection and Ranging (LiDAR) technology. Joint surface measurements at the microscopic scale were carried out based on a noncontact optical method, using a Keyence VHX‐2000D microscope. Three parameters were used to characterize fracture surface elevation, standard deviation, Hurst exponent, and correlation length through 3 scale length orders of magnitude. Our study showed that standard deviation and correlation length decrease with scale, similarly to previous findings on faults. In addition, the range of Hurst exponents as a function of scale for the studied joint surface agrees well with those previously found for faults. However, directional analysis showed that correlation length and Hurst exponent of joint surfaces at scales smaller than 1 dm differ significantly from the ones of fault surfaces. In contrast to fault surface ornaments that are mainly characterized by linear structures, plumose structures show marked variability in orientation and anisotropy as a function of position on the joint surface.

Random transverse field spin-glass model on the Cayley tree: phase transition between the two many-body-localized phases

The quantum Ising model with random couplings and random transverse fields on the Cayley tree is studied by real-space-renormalization in order to construct the whole set of eigenstates. The renormalization rules are analyzed via large deviations. The phase transition between the paramagnetic and the spin-glass many-body-localized phases involves the activated exponent and the correlation length exponent . The spin-glass-ordered cluster containing NSG spins is found to be extremely sparse with respect to the total number N of spins: its size grows only logarithmically at the critical point , and it is sub-extensive in the finite region of the spin-glass phase where the continuously varying exponent θ remains in the interval .

On superuniversality in the q-state Potts model with quenched disorder

We obtain the exact scale invariant scattering solutions for two-dimensional field theories with replicated permutational symmetry . After sending to zero the number of replicas they correspond to the renormalization group fixed points of the q-state Potts model with quenched disorder. We find that all solutions with non-zero disorder possess q-independent sectors, pointing to superuniversality (i.e. symmetry independence) of some critical exponents. The solution corresponding to the random bond ferromagnet, for which disorder vanishes as , allows for superuniversality of the correlation length exponent ν (Delfino 2017 Phys. Rev. Lett. 118 250601). Of the two solutions which are strongly disordered for all values of q, one is completely q-independent and accounts for the zero-temperature percolation fixed point of the randomly bond diluted ferromagnet. The other is the main candidate to describe the Nishimori-like fixed point of the Potts model with J disorder, and leaves room for superuniversality of the magnetic exponent η, a possibility not yet excluded by available numerical data.