
Session Chair
Gabriel Lemarié

09:00  09:40

Tomi Ohtsuki
(Sophia University)
Applications of deep 3D convolutional neural network to Anderson and quantum percolation models
The threedimensional Anderson model is a wellstudied model of disordered electron systems that shows the delocalizationlocalization transition. As in the studies on two and threedimensional (2D, 3D) quantum phase transitions [J. Phys. Soc. Jpn. {¥bf 85}, 123706 (2016), {¥bf 86}, 044708 (2017)], we used an image recognition algorithm based on a multilayered convolutional neural network. In contrast to previous studies in which 2D image recognition was used, we applied 3D image recognition to analyze entire 3D wave functions.
We show that a full phase diagram of the disorderenergy plane is obtained once the 3D convolutional neural network has been trained at the band center. We further demonstrate that the full phase diagram for 3D quantum bond and site percolations can be drawn by training the 3D Anderson model at the band center.
References:
T. Mano and T. Ohtsuki: Journal of the Physical Society of Japan 86, 113704 (2017).

09:40  10:10

Xin Wan
(Zhejiang University)
A convolutional neural network study of the quantum Hall plateau transition
Machine learning has been successfully applied to identify phases and phase transitions in condensed matter systems. However, quantitative characterization of the critical fluctuations near phase transitions is lacking. In this study we extract the critical behavior of a quantum Hall plateau transition with a convolutional neural network. We introduce a finitesize scaling approach and show that the localization length critical exponent learned by the neural network is consistent with the value obtained by conventional approaches. We illustrate the physics behind the approach by a crossexamination of the inverse participation ratios.

10:10  10:40

Keith Slevin
(Osaka University)
Critical exponent of the Anderson transition using massively parallel supercomputing
To date the most precise estimations of the critical exponent for the Anderson transition have been made using the transfer matrix method introduced into the field of Anderson localization by Pichard and Sarma [1], and MacKinnon and Kramer [2]. This method involves the simulation of extremely long quasi onedimensional systems. The method is inherently sequential and is not well suited to modern massively parallel supercomputers. The obvious alternative is to simulate a large ensemble of hypercubic systems and average. While this permits taking full advantage of both OpenMP and MPI on massively parallel supercomputers, a straight forward implementation results in data that does not scale. We show that this problem can be avoided by generating random sets of orthogonal starting vectors with an appropriate stationary probability distribution. We have applied this method to the Anderson transition in the threedimensional orthogonal universality class and been able to increase the largest L × L cross section simulated from L=24 in [3] to L=64 here. This permits an estimation of the critical exponent with improved precision and without the necessity of introducing an irrelevant scaling variable. In addition, this approach is better suited to simulations with correlated random potentials such as is needed in quantum Hall or cold atom systems.
1.Pichard, J.L. and G. Sarma, Finite size scaling approach to Anderson localisation. Journal of Physics C: Solid State Physics, 1981. 14(6): p. L127.
2.MacKinnon, A. and B. Kramer, Oneparameter scaling of localization length and conductance in disordered systems. Physical Review Letters, 1981. 47(21): p. 15461549.
3.Slevin, K. and T. Ohtsuki, Critical exponent for the Anderson transition in the threedimensional orthogonal universality class. New Journal of Physics, 2014. 16(1): p. 015012.

10:40  11:10

Coffee break

11:10  11:50

Victor Gurarie
(University of Colorado)
Quantum particles in a random potential in high dimensions
It is well known that quantum particles when moving in a random potential may undergo Anderson localization transition. It has only recently become appreciated that in high enough dimensions a different transition may happen. It manifests itself in the singular disorder averaged density of states as the disorder strength is tuned to the value corresponding to this transition point, and in multifractal wave functions. A particle obeying Schrödinger equation has to be above four dimensions for this to happen, but for Dirac or Weyl particle this happens already in three dimensional space corresponding to the experimentally studied Weyl materials. For certain disordered ion chains this can happen already in one dimensional space. Some evidence in the literature points to this criticality being a sharp crossover instead of a genuine phase transition, but this issue has not yet been unambiguously resolved. I will discuss the theory behind these phenomena and existing open questions.

11:50  12:20

Ivan Khaymovich
(MPI for the Physics of Complex Systems)
Robust multifractal phases and role of correlated hoppings in localization of static and periodicallydriven systems
The standard picture of the Anderson localization in a 3d singleparticle system with shortrange hoppings [1,2]
is represented by the phase transition between ergodic and localized phases at a certain critical disorder strength
with a sharp mobility edge separating ergodic and localized states at the intermediate disorder.
Exactly at the Anderson localization transition (ALT) point nonergodic (multifractal) extended states were proved to appear.
This picture was generalized (see, e.g., a review [2]) to lowdimensional systems with longrange hoppings
by applying the wellknown criterion of the delocalization at the point of the breakdown of the locator expansion [3, 4].
The states with critical statistics emerge typically at ALT in various disordered systems [2].
The examples of the robust multifractality away from phase transitions has been found only recently.
Here I first present such examples.
In lowdimensional systems we have found a class of random matrix models with extremely longrange hoppings, possessing
a whole phase of multifractal extended states, squeezed between Andersonlocalized and ergodic phases [5].
Another example demonstrates a separation of phases with an entire multifractal subband emerging in a singleparticle system
with quasiperiodic "disorder" and shortrange hoppings under a generic periodic drive [5].
In both cases multifractal states are robust to perturbations and are found at a whole range of parameters.
Recently, the standard picture of ALT [3, 4] has been also argued from another perspective.
There have been reported [710] several counterintuitive examples of singleparticle systems with longrange hoppings
where almost all states are (at least powerlaw) localized even in a nominally ergodic regime, where the standard locator expansion breaks down [3, 4].
Some of these "new" models demonstrate critical behavior [7, 8] for disorder strengths below the ones at the ALT [3, 4].
In the other ones [9, 10] wavefunction spatial decay rates obey a "mysterious" duality [10] mapping different powers of powerlaw bending [4].
What is the striking difference between the standard longrange models [35] and the new ones [710]?
The systems [710] belong to a new universality class where correlations of hoppings play a significant role in the localization properties.
In the second part of the talk I address this intriguing question.
I present a general approach applicable to all such models and uncover the role of correlations and the origin of the duality [10].
Based on an example of a system with translationary invariant (random) hoppings,
I demonstrate a simple though powerful method of recovering the standard criterion [3, 4] and giving a clue to the calculation of wavefunction spatial profile.
In the end of my talk I show how uncorrelated random hoppings [35] added to the correlated ones [710] can recover the standard picture of ALT.
[1] P. W. Anderson Phys. Rev. 109, 1492 (1958).
[2] F. Evers and A. D. Mirlin Rev. Mod. Phys. 80, 1355 (2008).
[3] L. S. Levitov, Europhy. Lett. 9, 83 (1989); Phys. Rev. Lett. 64, 547 (1990).
[4] A. D. Mirlin, Y. V. Fyodorov, F.M. Dittes, J. Quezada, and T. H. Seligman, Phys. Rev. E 54, 3221 (1996).
[5] V. E. Kravtsov, I. M. Khaymovich, E. Cuevas, M. Amini, New J. Phys. 17, 122002 (2015).
[6] S. Roy, I. M. Khaymovich, A. Das, R. Moessner, SciPost Phys. 4, 025 (2018).
[7] H. K. Owusu, K. Wagh, and E. A. Yuzbashyan, J. Phys. A: Math. Theor. 42, 035206 (2009).
[8] A. Ossipov, J. Phys. A 46, 105001 (2013).
[9] G. L. Celardo, R. Kaiser, and F. Borgonovi, Phys. Rev. B 94, 144206 (2016).
[10] X. Deng, V. E. Kravtsov, G. V. Shlyapnikov and L. Santos, Phys. Rev. Lett. 120, 110602 (2018).

12:20  13:20

Lunch

13:20  14:00

Discussion


Session Chair
Alexander Altland

14:00  14:40

Siddharth A. Parameswaran
(Oxford University)
Floquet quantum criticality
I will discuss transitions between distinct phases of onedimensional periodically driven (Floquet) systems. I will argue on general grounds [1] that these are controlled by infiniterandomness fixed points of a strongdisorder renormalization group procedure [2] suitably adapted to treat Floquet systems. Working in the fermionic representation of the prototypical Floquet Ising chain, I will leverage infinite randomness physics to provide a simple description of Floquet (multi)criticality in terms of a new type of domain wall associated with timetranslational symmetrybreaking and the formation of `Floquet time crystals’.
Refs:
[1] W. Berdanier, M. Kolodrubetz, SP, R. Vasseur, arXiv:1803.00019,
[2] W. Berdanier, M. Kolodrubetz, SP, R. Vasseur, arXiv:1807.09767.

14:40  15:20

Ilya A. Gruzberg
(The Ohio State University)
Anderson transitions in curved space

15:20  15:30

Closing

15:30  16:00

Coffee break

16:00  18:00

Discussion

18:00  19:00

Dinner

19:30  20:30

Informal discussion
