Abstract

Screening of vortices and other critical points in a two-dimensional random Gaussian field is studied by using large-scale computer simulations and analytic theory. It is shown that the topological charge imbalance and its variance in a bounded region can be obtained from signed zero crossings on the boundary of the region. A first-principles Gaussian theory of these zero crossings and their correlations is derived for the vortices and shown to be in good agreement with the computer simulation. An exact relationship is obtained between the variance of the charge imbalance and the charge correlation function, and this relationship is verified by comparison with the data. The results obtained are extended to arbitrarily shaped volumes in isotropic spaces of higher dimension.

© 1998 Optical Society of America

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References

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  1. B. W. Roberts, E. Bodenschatz, J. P. Sethna, “A bound on the decay of defect–defect correlation functions in two-dimensional complex order parameter equations,” Physica D 99, 252–268 (1996).
    [CrossRef]
  2. B. I. Halperin, “Statistical mechanics of topological defects,” in Physics of Defects, R. Balian, M. Kleman, J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 814–857.
  3. J. M. Kosterlitz, D. J. Thouless, “Two-dimensional physics,” in Progress in Low Temperature Physics, B. D. F. Brewer, ed. (North-Holland, Amsterdam, 1978), Vol. VIIB, pp. 371–433.
  4. D. J. Thouless, M. Kohmoto, M. P. Nightingale, M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49, 405–408 (1982).
    [CrossRef]
  5. P. N. Walker, M. Wilkinson, “Universal fluctuations of Chern integers,” Phys. Rev. Lett. 74, 4055–4058 (1995).
    [CrossRef] [PubMed]
  6. J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
    [CrossRef]
  7. M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Balian, M. Kleman, J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 453–549.
  8. M. Berry, “Disruption of wave-fronts: statistics of dislocations in incoherent Gaussian random waves,” J. Phys. A 11, 27–37 (1978).
    [CrossRef]
  9. V. I. Arnold, Ordinary Differential Equations (MIT Press, Cambridge, Mass., 1973), Chap. 5, pp. 254–268.
  10. S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, Reading, Mass., 1994).
  11. N. Shvartsman, I. Freund, “Vortices in random wave fields: nearest neighbor anticorrelations,” Phys. Rev. Lett. 72, 1008–1011 (1994).
    [CrossRef] [PubMed]
  12. I. Freund, N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
    [CrossRef] [PubMed]
  13. I. Freund, “Amplitude topological singularities in random electromagnetic wavefields,” Phys. Lett. A 198, 139–144 (1994).
    [CrossRef]
  14. I. Freund, “Saddles, singularities, and extrema in random phase fields,” Phys. Rev. E 52, 2348–2360 (1995).
    [CrossRef]
  15. I. Freund, “‘1001’ correlations in random wave fields,” Waves Random Media 8, 119–158 (1998).
    [CrossRef]
  16. I. Freund, “Vortex derivatives,” Opt. Commun. 137, 118–126 (1997).
    [CrossRef]
  17. R. Loudon, The Quantum Theory of Light (Oxford U. Press, Oxford, 1983), Chap. 6, pp. 226–229.
  18. I. Freund, “Optical vortices in Gaussian random wave fields: statistical probability densities,” J. Opt. Soc. Am. A 11, 1644–1652 (1994).
    [CrossRef]
  19. I. Freund, D. A. Kessler, “Phase autocorrelation of random wave fields,” Opt. Commun. 124, 321–332 (1996).
    [CrossRef]
  20. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 5, pp. 207–222.
  21. D. A. Kessler, I. Freund, “Level-crossing densities in random wave fields,” J. Opt. Soc. Am. A 15, 1608–1618 (1998).
    [CrossRef]
  22. S. O. Rice, “Mathematical analysis of random noise,” in Selected Papers on Noise and Stochastic Processes, N. Wax, ed. (Dover, New York, 1954), pp. 133–294.
  23. S. O. Rice, “Distribution of the duration of fades in radio transmission,” Bell Syst. Tech. J. 37, 581–635 (1958).
    [CrossRef]
  24. M. S. Longuet-Higgins, “The statistical analysis of a random moving surface,” Philos. Trans. R. Soc. London, Ser. A 249, 321–387 (1957).
    [CrossRef]
  25. I. Freund, N. Shvartsman, “Structural correlations in Gaussian random wave fields,” Phys. Rev. E 51, 3770–3773 (1995).
    [CrossRef]
  26. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, San Francisco, 1988), Chap. 4, pp. 73–83.
  27. N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wave-front of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).
  28. A. Weinberg, B. I. Halperin, “Distribution of maxima, minima, and saddle points of the intensity of laser speckle patterns,” Phys. Rev. B 26, 1362–1368 (1982).
    [CrossRef]

1998 (2)

I. Freund, “‘1001’ correlations in random wave fields,” Waves Random Media 8, 119–158 (1998).
[CrossRef]

D. A. Kessler, I. Freund, “Level-crossing densities in random wave fields,” J. Opt. Soc. Am. A 15, 1608–1618 (1998).
[CrossRef]

1997 (1)

I. Freund, “Vortex derivatives,” Opt. Commun. 137, 118–126 (1997).
[CrossRef]

1996 (2)

I. Freund, D. A. Kessler, “Phase autocorrelation of random wave fields,” Opt. Commun. 124, 321–332 (1996).
[CrossRef]

B. W. Roberts, E. Bodenschatz, J. P. Sethna, “A bound on the decay of defect–defect correlation functions in two-dimensional complex order parameter equations,” Physica D 99, 252–268 (1996).
[CrossRef]

1995 (3)

P. N. Walker, M. Wilkinson, “Universal fluctuations of Chern integers,” Phys. Rev. Lett. 74, 4055–4058 (1995).
[CrossRef] [PubMed]

I. Freund, “Saddles, singularities, and extrema in random phase fields,” Phys. Rev. E 52, 2348–2360 (1995).
[CrossRef]

I. Freund, N. Shvartsman, “Structural correlations in Gaussian random wave fields,” Phys. Rev. E 51, 3770–3773 (1995).
[CrossRef]

1994 (4)

I. Freund, “Optical vortices in Gaussian random wave fields: statistical probability densities,” J. Opt. Soc. Am. A 11, 1644–1652 (1994).
[CrossRef]

N. Shvartsman, I. Freund, “Vortices in random wave fields: nearest neighbor anticorrelations,” Phys. Rev. Lett. 72, 1008–1011 (1994).
[CrossRef] [PubMed]

I. Freund, N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[CrossRef] [PubMed]

I. Freund, “Amplitude topological singularities in random electromagnetic wavefields,” Phys. Lett. A 198, 139–144 (1994).
[CrossRef]

1982 (2)

D. J. Thouless, M. Kohmoto, M. P. Nightingale, M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49, 405–408 (1982).
[CrossRef]

A. Weinberg, B. I. Halperin, “Distribution of maxima, minima, and saddle points of the intensity of laser speckle patterns,” Phys. Rev. B 26, 1362–1368 (1982).
[CrossRef]

1981 (1)

N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wave-front of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).

1978 (1)

M. Berry, “Disruption of wave-fronts: statistics of dislocations in incoherent Gaussian random waves,” J. Phys. A 11, 27–37 (1978).
[CrossRef]

1974 (1)

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

1958 (1)

S. O. Rice, “Distribution of the duration of fades in radio transmission,” Bell Syst. Tech. J. 37, 581–635 (1958).
[CrossRef]

1957 (1)

M. S. Longuet-Higgins, “The statistical analysis of a random moving surface,” Philos. Trans. R. Soc. London, Ser. A 249, 321–387 (1957).
[CrossRef]

Arnold, V. I.

V. I. Arnold, Ordinary Differential Equations (MIT Press, Cambridge, Mass., 1973), Chap. 5, pp. 254–268.

Baranova, N. B.

N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wave-front of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).

Berry, M.

M. Berry, “Disruption of wave-fronts: statistics of dislocations in incoherent Gaussian random waves,” J. Phys. A 11, 27–37 (1978).
[CrossRef]

M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Balian, M. Kleman, J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 453–549.

Berry, M. V.

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Bodenschatz, E.

B. W. Roberts, E. Bodenschatz, J. P. Sethna, “A bound on the decay of defect–defect correlation functions in two-dimensional complex order parameter equations,” Physica D 99, 252–268 (1996).
[CrossRef]

den Nijs, M.

D. J. Thouless, M. Kohmoto, M. P. Nightingale, M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49, 405–408 (1982).
[CrossRef]

Freund, I.

I. Freund, “‘1001’ correlations in random wave fields,” Waves Random Media 8, 119–158 (1998).
[CrossRef]

D. A. Kessler, I. Freund, “Level-crossing densities in random wave fields,” J. Opt. Soc. Am. A 15, 1608–1618 (1998).
[CrossRef]

I. Freund, “Vortex derivatives,” Opt. Commun. 137, 118–126 (1997).
[CrossRef]

I. Freund, D. A. Kessler, “Phase autocorrelation of random wave fields,” Opt. Commun. 124, 321–332 (1996).
[CrossRef]

I. Freund, “Saddles, singularities, and extrema in random phase fields,” Phys. Rev. E 52, 2348–2360 (1995).
[CrossRef]

I. Freund, N. Shvartsman, “Structural correlations in Gaussian random wave fields,” Phys. Rev. E 51, 3770–3773 (1995).
[CrossRef]

N. Shvartsman, I. Freund, “Vortices in random wave fields: nearest neighbor anticorrelations,” Phys. Rev. Lett. 72, 1008–1011 (1994).
[CrossRef] [PubMed]

I. Freund, N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[CrossRef] [PubMed]

I. Freund, “Amplitude topological singularities in random electromagnetic wavefields,” Phys. Lett. A 198, 139–144 (1994).
[CrossRef]

I. Freund, “Optical vortices in Gaussian random wave fields: statistical probability densities,” J. Opt. Soc. Am. A 11, 1644–1652 (1994).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 5, pp. 207–222.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, San Francisco, 1988), Chap. 4, pp. 73–83.

Halperin, B. I.

A. Weinberg, B. I. Halperin, “Distribution of maxima, minima, and saddle points of the intensity of laser speckle patterns,” Phys. Rev. B 26, 1362–1368 (1982).
[CrossRef]

B. I. Halperin, “Statistical mechanics of topological defects,” in Physics of Defects, R. Balian, M. Kleman, J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 814–857.

Kessler, D. A.

D. A. Kessler, I. Freund, “Level-crossing densities in random wave fields,” J. Opt. Soc. Am. A 15, 1608–1618 (1998).
[CrossRef]

I. Freund, D. A. Kessler, “Phase autocorrelation of random wave fields,” Opt. Commun. 124, 321–332 (1996).
[CrossRef]

Kohmoto, M.

D. J. Thouless, M. Kohmoto, M. P. Nightingale, M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49, 405–408 (1982).
[CrossRef]

Kosterlitz, J. M.

J. M. Kosterlitz, D. J. Thouless, “Two-dimensional physics,” in Progress in Low Temperature Physics, B. D. F. Brewer, ed. (North-Holland, Amsterdam, 1978), Vol. VIIB, pp. 371–433.

Longuet-Higgins, M. S.

M. S. Longuet-Higgins, “The statistical analysis of a random moving surface,” Philos. Trans. R. Soc. London, Ser. A 249, 321–387 (1957).
[CrossRef]

Loudon, R.

R. Loudon, The Quantum Theory of Light (Oxford U. Press, Oxford, 1983), Chap. 6, pp. 226–229.

Mamaev, A. V.

N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wave-front of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).

Nightingale, M. P.

D. J. Thouless, M. Kohmoto, M. P. Nightingale, M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49, 405–408 (1982).
[CrossRef]

Nye, J. F.

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Pilipetskii, N.

N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wave-front of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).

Rice, S. O.

S. O. Rice, “Distribution of the duration of fades in radio transmission,” Bell Syst. Tech. J. 37, 581–635 (1958).
[CrossRef]

S. O. Rice, “Mathematical analysis of random noise,” in Selected Papers on Noise and Stochastic Processes, N. Wax, ed. (Dover, New York, 1954), pp. 133–294.

Roberts, B. W.

B. W. Roberts, E. Bodenschatz, J. P. Sethna, “A bound on the decay of defect–defect correlation functions in two-dimensional complex order parameter equations,” Physica D 99, 252–268 (1996).
[CrossRef]

Sethna, J. P.

B. W. Roberts, E. Bodenschatz, J. P. Sethna, “A bound on the decay of defect–defect correlation functions in two-dimensional complex order parameter equations,” Physica D 99, 252–268 (1996).
[CrossRef]

Shkukov, V. V.

N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wave-front of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).

Shvartsman, N.

I. Freund, N. Shvartsman, “Structural correlations in Gaussian random wave fields,” Phys. Rev. E 51, 3770–3773 (1995).
[CrossRef]

N. Shvartsman, I. Freund, “Vortices in random wave fields: nearest neighbor anticorrelations,” Phys. Rev. Lett. 72, 1008–1011 (1994).
[CrossRef] [PubMed]

I. Freund, N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[CrossRef] [PubMed]

Strogatz, S. H.

S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, Reading, Mass., 1994).

Thouless, D. J.

D. J. Thouless, M. Kohmoto, M. P. Nightingale, M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49, 405–408 (1982).
[CrossRef]

J. M. Kosterlitz, D. J. Thouless, “Two-dimensional physics,” in Progress in Low Temperature Physics, B. D. F. Brewer, ed. (North-Holland, Amsterdam, 1978), Vol. VIIB, pp. 371–433.

Walker, P. N.

P. N. Walker, M. Wilkinson, “Universal fluctuations of Chern integers,” Phys. Rev. Lett. 74, 4055–4058 (1995).
[CrossRef] [PubMed]

Weinberg, A.

A. Weinberg, B. I. Halperin, “Distribution of maxima, minima, and saddle points of the intensity of laser speckle patterns,” Phys. Rev. B 26, 1362–1368 (1982).
[CrossRef]

Wilkinson, M.

P. N. Walker, M. Wilkinson, “Universal fluctuations of Chern integers,” Phys. Rev. Lett. 74, 4055–4058 (1995).
[CrossRef] [PubMed]

Zel’dovich, B. Y.

N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wave-front of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).

Bell Syst. Tech. J. (1)

S. O. Rice, “Distribution of the duration of fades in radio transmission,” Bell Syst. Tech. J. 37, 581–635 (1958).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Phys. A (1)

M. Berry, “Disruption of wave-fronts: statistics of dislocations in incoherent Gaussian random waves,” J. Phys. A 11, 27–37 (1978).
[CrossRef]

JETP Lett. (1)

N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wave-front of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).

Opt. Commun. (2)

I. Freund, D. A. Kessler, “Phase autocorrelation of random wave fields,” Opt. Commun. 124, 321–332 (1996).
[CrossRef]

I. Freund, “Vortex derivatives,” Opt. Commun. 137, 118–126 (1997).
[CrossRef]

Philos. Trans. R. Soc. London, Ser. A (1)

M. S. Longuet-Higgins, “The statistical analysis of a random moving surface,” Philos. Trans. R. Soc. London, Ser. A 249, 321–387 (1957).
[CrossRef]

Phys. Lett. A (1)

I. Freund, “Amplitude topological singularities in random electromagnetic wavefields,” Phys. Lett. A 198, 139–144 (1994).
[CrossRef]

Phys. Rev. A (1)

I. Freund, N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[CrossRef] [PubMed]

Phys. Rev. B (1)

A. Weinberg, B. I. Halperin, “Distribution of maxima, minima, and saddle points of the intensity of laser speckle patterns,” Phys. Rev. B 26, 1362–1368 (1982).
[CrossRef]

Phys. Rev. E (2)

I. Freund, N. Shvartsman, “Structural correlations in Gaussian random wave fields,” Phys. Rev. E 51, 3770–3773 (1995).
[CrossRef]

I. Freund, “Saddles, singularities, and extrema in random phase fields,” Phys. Rev. E 52, 2348–2360 (1995).
[CrossRef]

Phys. Rev. Lett. (3)

N. Shvartsman, I. Freund, “Vortices in random wave fields: nearest neighbor anticorrelations,” Phys. Rev. Lett. 72, 1008–1011 (1994).
[CrossRef] [PubMed]

D. J. Thouless, M. Kohmoto, M. P. Nightingale, M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49, 405–408 (1982).
[CrossRef]

P. N. Walker, M. Wilkinson, “Universal fluctuations of Chern integers,” Phys. Rev. Lett. 74, 4055–4058 (1995).
[CrossRef] [PubMed]

Physica D (1)

B. W. Roberts, E. Bodenschatz, J. P. Sethna, “A bound on the decay of defect–defect correlation functions in two-dimensional complex order parameter equations,” Physica D 99, 252–268 (1996).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Waves Random Media (1)

I. Freund, “‘1001’ correlations in random wave fields,” Waves Random Media 8, 119–158 (1998).
[CrossRef]

Other (9)

R. Loudon, The Quantum Theory of Light (Oxford U. Press, Oxford, 1983), Chap. 6, pp. 226–229.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 5, pp. 207–222.

M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Balian, M. Kleman, J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 453–549.

B. I. Halperin, “Statistical mechanics of topological defects,” in Physics of Defects, R. Balian, M. Kleman, J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 814–857.

J. M. Kosterlitz, D. J. Thouless, “Two-dimensional physics,” in Progress in Low Temperature Physics, B. D. F. Brewer, ed. (North-Holland, Amsterdam, 1978), Vol. VIIB, pp. 371–433.

V. I. Arnold, Ordinary Differential Equations (MIT Press, Cambridge, Mass., 1973), Chap. 5, pp. 254–268.

S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, Reading, Mass., 1994).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, San Francisco, 1988), Chap. 4, pp. 73–83.

S. O. Rice, “Mathematical analysis of random noise,” in Selected Papers on Noise and Stochastic Processes, N. Wax, ed. (Dover, New York, 1954), pp. 133–294.

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Figures (6)

Fig. 1
Fig. 1

Variance (ΔN)2 of the topological charge imbalance: curve a, vortices; curve b, stationary points of the real part of the wave function; curve c, stationary points of the intensity. In curve a the filled circles are for a square box with perimeter P, and the open circle is for a line of length P. All data points in curves b and c are for lines of length P. For all curves P is measured in units of the transverse coherence Lcoh. The data in curve b are multiplied by 1.5 to prevent overlap with those of curve a. For each data set the line is Eq. (14) with parameters listed in Subsection 2.D.

Fig. 2
Fig. 2

Probability density function (δN) for the vortices, where δN=ΔN/P. The curves are calculated from Eq. (15).

Fig. 3
Fig. 3

Zero-crossing charge correlation function Γ(Δx/Lcoh) defined in Eq. (21) for (a) the vortices, (b) the real part of the wave function, and (c) the intensity. The curve in (a) is calculated from Eq. (29).

Fig. 4
Fig. 4

Critical-point screening. Shown is 2π(Δr/Lcoh)C(Δr/Lcoh) versus Δr/Lcoh for (a) the vortices, (b) the real part of the wave function, and (c) the intensity, where the charge correlation function C(Δr) is defined in Eq. (30). The solid curves are drawn to aid the eye. The dotted curve in (a) is Eq. (32), a first-principles calculation that is due to Halperin (Ref. 2).

Fig. 5
Fig. 5

Screening of extrema. Shown is 2π(Δr/Lcoh)C(Δr/Lcoh) versus Δr/Lcoh for (a) all extrema of the real part R of the wave function and (b) minima of R and maxima of the imaginary part I of the wave function. C(Δr) is defined in Eq. (30). The solid curves are drawn as aids to the eye.

Fig. 6
Fig. 6

Variance (ΔN)2 of topological charge imbalance for small square boxes with perimeter P: (a) the filled circles are measured data, the solid curve is the exact form in Eq. (37), and the dotted curve is the asymptotic form in Eq. (14); (b) percent difference between the asymptotic and exact forms.

Equations (60)

Equations on this page are rendered with MathJax. Learn more.

ϕ=arctan(I/R),
ΔN=12π(ϕ2-ϕ1).
ϴ=arctan(fy/fx).
qv=sgn(RxIy-RyIx),
qs=sgn(fxxfyy-fxyfyx),
ΔN=12π(ϴ2-ϴ1),
σx(fy)=sgn(fxfxy),σx(fx)=sgn(-fyfxx),
σy(fy)=sgn(fxfyy),σy(fx)=sgn(-fyfxy),
ΔN=12Δn,
(ΔN)2=14[(Δn(x))2+(Δn(y))2].
(Δn(x))2=γxn(x),
(Δn(y))2=γyn(y),
n(x)=2n0(x)Lx,
n(y)=2n0(y)Ly,
(ΔN)2=12[γxn0(x)Lx+γyn0(y)Ly].
γxn0(x)=γyn0(y).
(ΔN)2=14γn0P.
(δN)=1[2π(δN)2]1/2exp{-(δN)2/[2(δN)2]}.
ΔN=120Pdx[n+(x)-n-(x)],
(ΔN)2=120Pdx0Pdxn+(x)Δn(x).
(ΔN)2=12-dx-dx uP(x)uP(x)n+(x)Δn(x).
-dx uP(x)uP(x+Δx)=(P-|Δx|)H(P-|Δx|),
(ΔN)2=12-PPd(Δx)(P-|Δx|)n+(0)Δn(Δx).
n+(0)Δn(Δx)=12n0[δ(Δx)+Γ(Δx)],
(ΔN)2=14n0P1+-PPd(Δx)Γ(Δx)-14n0-PPd(Δx)|Δx|Γ(Δx).
γ=1+20d(Δx)Γ(Δx).
Γ(x2-x1)=2n0n+(x1)Δn(x2),
nˆ±=δ(Ri)|Ri|H(IiRi),
(I1, I2; R1, R2)=I(I1, I2)R(R1, R2),
I=KI exp{-[a(I12+I22)+bI1I2]},
R=KR exp{-[c(R12+R22)+dR1R2]}.
KI=(2π)-1DI-1/2,KR=(2π)-2DR-1/2,
DI=1-B2,
DR=(BE-BF+D2+E-F)×(-BE-BF+D2+E+F),
a=(2DI)-1,b=-BDI-1,
c=(B2E-D2-E)(2DR)-1,
d=(BD2-B2F+F)DR-1,
B=W(Δx),D=W(Δx),
E=W(0),F=W(Δx).
Γ(Δx)=-(π2n0)-1[(1-W2)W+WW2]
×(1-W2)-3/2 arcsin(W),
N+(0)ΔN(Δr)=12η[δ(Δr)+C(Δr)],
- d2(Δr)C(Δr)=-1.
C(Δr)
=-W(0)W(Δr){W(Δr)[W(Δr)] 2+W(Δr)[1-W 2(Δr)]}πW(0)Δr[1-W 2(Δr)] 2,
ΔN=A d2r ΔN(r)
(ΔN)2=2-d2(Δr)A(Δr)N+(0)ΔN(Δr),
A(Δr)=-d2r uA(r)uA(r+Δr),
A(Δx, Δy)=(Lx-|Δx|)(Ly-|Δy|)×H(Lx-|Δx|)H(Ly-|Δy|),
(ΔN)2=N+η-d2(Δr)A(Δr)C(Δr),
C(Δr)=-d2k exp(ik·Δr)S(k)
Υ(k)=-d2(Δr)exp(ik·Δr)A(Δr),
(ΔN)2=N+η-d2kΥ(k)S(k).
(ΔN)2=14η[Λs(y)Px+Λs(x)Py],
Λs(x)=-2-d(Δy)-d(Δx)|Δx|C(Δx, Δy),
Λs(y)=-2-d(Δy)-d(Δx)|Δy|C(Δx, Δy),
Λs=-80d(Δr)(Δr)2C(Δr),
(ΔN)2=14ηΛsP.
γ=ηΛsn0.
Λs=-2-dk(Δr)|Δri|C(Δr),

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