scholarly journals Universal non-Hermitian skin effect in two and higher dimensions

2021 ◽  
Author(s):  
Kai Zhang ◽  
Zhensen Yang ◽  
Chen Fang
Keyword(s):  
Skin Effect ◽  
Physical Phenomenon ◽  
Higher Dimensions ◽  
Three Dimensions ◽  
One Dimension ◽  
Finite Area ◽  
Open Chain ◽  
Kondo Insulators ◽  
Boundary Spectrum ◽  
Weyl Semimetals

Abstract Skin effect, experimentally discovered in one dimension, describes the physical phenomenon that on an open chain, an extensive number of eigenstates of a non-Hermitian hamiltonian are localized at the end(s) of the chain. Here in two and higher dimensions, we establish a theorem that the skin effect exists, if and only if periodic-boundary spectrum of the hamiltonian covers a finite area on the complex plane. This theorem establishes the universality of the effect, because the above condition is satisfied in almost every generic non-Hermitian hamiltonian, and, unlike in one dimension, is compatible with all spatial symmetries. We propose two new types of skin effect in two and higher dimensions: the corner-skin effect where all eigenstates are localized at one corner of the system, and the geometry-dependent-skin effect where skin modes disappear for systems of a particular shape, but appear on generic polygons. An immediate corollary of our theorem is that any non-Hermitian system having exceptional points (lines) in two (three) dimensions exhibits skin effect, making this phenomenon accessible to experiments in photonic crystals, Weyl semimetals, and Kondo insulators.

Asymptotic properties of the equilibrium probability of identity in a geographically structured population

1977 ◽  
Vol 9 (2) ◽  
pp. 268-282 ◽  
Author(s):  
Stanley Sawyer
Keyword(s):  
Nearest Neighbor ◽  
Asymptotic Properties ◽  
Higher Dimensions ◽  
Two Dimensions ◽  
Three Dimensions ◽  
One Dimension ◽  
Order Zero ◽  
Structured Population ◽  
Equilibrium Probability ◽  
Migration Law

Let I(x, u) be the probability that two genes found a vector distance x apart are the same type in an infinite-allele selectively-neutral migration model with mutation rate u. The creatures involved inhabit an infinite of colonies, are diploid and are held at N per colony. Set in one dimension and in higher dimensions, where σ2 is the covariance matrix of the migration law (which is assumed to have finite fifth moments). Then in one dimension, in two dimensions, and in three dimensions uniformly for Here C0 is a constant depending on the migration law, K0(y) is the Bessel function of the second kind of order zero, and are the eigenvalues of σ2. For symmetric nearest-neighbor migrations, in one dimension and log mi in two. For is known in one dimension and C0 does not appear. In two dimensions, These results extend and make more precise earlier work of Malécot, Weiss and Kimura and Nagylaki.

Asymptotic properties of the equilibrium probability of identity in a geographically structured population

1977 ◽  
Vol 9 (02) ◽  
pp. 268-282 ◽  
Author(s):  
Stanley Sawyer
Keyword(s):  
Nearest Neighbor ◽  
Asymptotic Properties ◽  
Higher Dimensions ◽  
Two Dimensions ◽  
Three Dimensions ◽  
One Dimension ◽  
Order Zero ◽  
Structured Population ◽  
Equilibrium Probability ◽  
Migration Law

Let I(x, u) be the probability that two genes found a vector distance x apart are the same type in an infinite-allele selectively-neutral migration model with mutation rate u. The creatures involved inhabit an infinite of colonies, are diploid and are held at N per colony. Set in one dimension and in higher dimensions, where σ2 is the covariance matrix of the migration law (which is assumed to have finite fifth moments). Then in one dimension, in two dimensions, and in three dimensions uniformly for Here C 0 is a constant depending on the migration law, K0 (y) is the Bessel function of the second kind of order zero, and are the eigenvalues of σ2. For symmetric nearest-neighbor migrations, in one dimension and log m i in two. For is known in one dimension and C 0 does not appear. In two dimensions, These results extend and make more precise earlier work of Malécot, Weiss and Kimura and Nagylaki.

Conclusion

2020 ◽  
pp. 224-232
Author(s):  
Aleida Assmann
Keyword(s):  
Temporal Structure ◽  
Three Dimensions ◽  
One Dimension ◽  
Modern Time ◽  
The Past ◽  
The Future ◽  
Time Regime

This concluding chapter poses the question of whether or not we have too much past and too little future. After all, the notion of the past has dramatically increased in its range of meanings, as has the future. The relation between the past, the present, and the future is a three-fold relationship in which one dimension cannot exist for long without the others. Ordering this three-fold temporal structure anew and bringing the three dimensions into a balanced relation, however, continues to be an open adventure. To be sure, it is also the greatest challenge posed by the demise of the modern time regime.

The probability distribution of the extent of a random chain

1941 ◽  
Vol 37 (3) ◽  
pp. 244-251 ◽  
Author(s):  
H. E. Daniels ◽  
F. Smithies
Keyword(s):  
Probability Distribution ◽  
Three Dimensions ◽  
One Dimension ◽  
Shortest Distance ◽  
A Chain ◽  
Random Configuration

1. Introduction and summary. A chain of N links is allowed to assume a random configuration in space. The extent of the chain in any direction is defined as the shortest distance between a pair of planes perpendicular to that direction, such that the chain is contained entirely between them. In the present paper the probability distribution of the extent is discussed, starting with a chain in one dimension for which formulae are derived for the probability and mean extent for all values of N. The limiting forms for large N are then considered. The results are extended to the case of a chain in three dimensions, and it is shown that the extents in two directions at right angles tend to be independently distributed when N is large. It is assumed that the links are infinitely thin, so that a point in space may be occupied by the chain any number of times.

Reassessing and Reconceptualizing the Multidimensional Nature of Organizational Commitment

1994 ◽  
Vol 75 (3) ◽  
pp. 1379-1390 ◽  
Author(s):  
Syed Akhtar ◽  
Doreen Tan
Keyword(s):  
Organizational Commitment ◽  
Affective Commitment ◽  
Three Dimensional ◽  
Three Dimensions ◽  
One Dimension ◽  
Organizational Goals ◽  
Organizational Membership ◽  
Attitude Theory ◽  
Retail Banks ◽  
Promax Rotation

This study was designed to reassess and reconceptualize the multidimensional nature of organizational commitment. The Organizational Commitment Questionnaire of Porter, Steers, Mowday, and Boulian was administered to 259 employees representing five retail banks. Factor analysis (principal factor, promax rotation) yielded the three dimensions proposed by Porter, et al. in 1974. This conceptualization was inadequate because one dimension, i.e., desire to maintain organizational membership, overlaps the withdrawal construct. A similar criticism has been levelled against Meyer and Allen's 1991 model. Consistent with the three-dimensional attitude theory, organizational commitment was reconceptualized in terms of cognitive, emotive, and conative meanings. The proposed dimensions include normative commitment (amount of cognitive consonance with organizational norms), affective commitment (intensity of emotional attachment to the organization), and volitive commitment (extent of conative orientation towards organizational goals).

scholarly journals CRITICAL LENGTH FOR THE SPREADING–VANISHING DICHOTOMY IN HIGHER DIMENSIONS

2020 ◽  
Vol 62 (1) ◽  
pp. 3-17 ◽  
Author(s):  
MATTHEW J. SIMPSON
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Moving Boundary ◽  
Numerical Solutions ◽  
Critical Length ◽  
Higher Dimensions ◽  
Moving Boundary Problem ◽  
Kolmogorov Equation ◽  
One Dimension ◽  
Spatial Dimensions

We consider an extension of the classical Fisher–Kolmogorov equation, called the “Fisher–Stefan” model, which is a moving boundary problem on $0<x<L(t)$. A key property of the Fisher–Stefan model is the “spreading–vanishing dichotomy”, where solutions with $L(t)>L_{\text{c}}$ will eventually spread as $t\rightarrow \infty$, whereas solutions where $L(t)\ngtr L_{\text{c}}$ will vanish as $t\rightarrow \infty$. In one dimension it is well known that the critical length is $L_{\text{c}}=\unicode[STIX]{x1D70B}/2$. In this work, we re-formulate the Fisher–Stefan model in higher dimensions and calculate $L_{\text{c}}$ as a function of spatial dimensions in a radially symmetric coordinate system. Our results show how $L_{\text{c}}$ depends upon the dimension of the problem, and numerical solutions of the governing partial differential equation are consistent with our calculations.

scholarly journals Averaging over Narain moduli space

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Alexander Maloney ◽  
Edward Witten
Keyword(s):  
Einstein Gravity ◽  
Two Dimensions ◽  
Three Dimensions ◽  
One Dimension ◽  
Simons Theory ◽  
Two Dimensional ◽  
Dual Theory ◽  
Recent Developments ◽  
Chern Simons ◽  
Chern Simons Theory

Abstract Recent developments involving JT gravity in two dimensions indicate that under some conditions, a gravitational path integral is dual to an average over an ensemble of boundary theories, rather than to a specific boundary theory. For an example in one dimension more, one would like to compare a random ensemble of two-dimensional CFT’s to Einstein gravity in three dimensions. But this is difficult. For a simpler problem, here we average over Narain’s family of two-dimensional CFT’s obtained by toroidal compactification. These theories are believed to be the most general ones with their central charges and abelian current algebra symmetries, so averaging over them means picking a random CFT with those properties. The average can be computed using the Siegel-Weil formula of number theory and has some properties suggestive of a bulk dual theory that would be an exotic theory of gravity in three dimensions. The bulk dual theory would be more like U(1)2D Chern-Simons theory than like Einstein gravity.

Hydrothermal synthesis and controlled growth of tungsten disulphide nanostructures from one‐dimension to three‐dimensions

2015 ◽  
Vol 10 (3) ◽  
pp. 183-186 ◽  
Author(s):  
Shixiu Cao ◽  
Tianmo Liu ◽  
Wen Zeng ◽  
Shahid Hussain ◽  
Xianghe Peng
Keyword(s):  
Hydrothermal Synthesis ◽  
Three Dimensions ◽  
One Dimension ◽  
Controlled Growth ◽  
Tungsten Disulphide

Do attractive scattering potentials concentrate particles at the origin in one, two and three dimensions? II. Potentials diverging as — C/r q

1983 ◽  
Vol 388 (1795) ◽  
pp. 419-444 ◽  
Keyword(s):  
Turning Points ◽  
Classical Mechanics ◽  
Preceding Paper ◽  
Incident Beam ◽  
Three Dimensions ◽  
One Dimension ◽  
Zero Energy ◽  
Range Parameter ◽  
Classical Trajectories ◽  
2D And 3D

In classical mechanics (c.m.), and near the semi-classical limit h →0 of quantum mechanics (s.c.l.), the enhancement factors α ≡ ρ 0 /ρ ∞ are found for scattering by attractive central potentials U(r) ; here ρ 0,∞ (and v 0,∞ ) are the particle densities (and speeds) at the origin and far upstream in the incident beam. For finite potentials ( U (0) > — ∞), and when there are no turning points, the preceding paper found both in c.m., and near the s.c.l. (which then covers high v ∞ ), α 1 = v ∞ / v 0 , α 2 = 1, α 3 = v 0 / v ∞ respectively in one dimension (1D), 2D and 3D. The argument is now extended to potentials (still without turning points), where U ( r →0) ~ ─ C/r q , with 0 < q < 1 in ID (where r ≡ | x | ), and 0 < q < 2 in 2D and 3D, since only for such q can classical trajectories and quantum wavefunctions be defined unambiguously. In c.m., α 1 (c.m.) = 0, α 3 (c.m.) = ∞, and α 2 (c.m.) = (1 —½ q ) N , where N = [integer part of (1 ─½ q ) -1 ]is the number of trajectories through any point ( r , θ) in the limit r → 0. All features of U(r) other than q are irrelevant. Near the s.c.l. (which now covers low v ∞ ) a somewhat delicate analysis is needed, matching exact zero-energy solutions at small r to the ordinary W.K.B. approximation at large r ; for small v ∞ / u it yields the leading terms α 1 (s.c.l.) = Λ 1 (q) v ∞ / u , α 2 (s.c.I) = (1 ─½ q ) -1 , α 3 (s.c.l.)= Λ 3 ( q ) u/v ∞ , where u ≡ (C/h q m 1-q ) 1/(2-q) is a generalized Bohr velocity. Here Λ 1,3 are functions of q alone, given in the text; as q →0 the α (s.c.l.) agree with the α quoted above for finite potentials. Even in the limit h = 0, α 2 (s.c.l.) and α 2 (c.m.) differ. This paradox in 2D is interpreted loosely in terms of quantal interference between the amplitudes corresponding to the N classical trajectories. The Coulomb potential ─ C/r is used as an analytically soluble example in 2D as well as in 3D. Finally, if U(r) away from the origin depends on some intrinsic range parameter α(e.g. U = ─ C exp (─r/a)/r q ) , and if, near the s.c.l., v ∞ / u is regarded as a function not of h but more realistically of v ∞ , then the expressions α (s.c.l.) above apply only in an intermediate range 1/ a ≪ mv ∞ / h ≪ ( mC/h 2 ) 1/(2- q ) which exists only if a ≫ ( h 2 / mC ) 1/(2- q ) ).

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