I outline my past, present and future work, its motivations and strategies.

A common theme in all of the topics that I am involved in is the method of nonperturbative renormalization group. This is a modern incarnation of the original idea by Kenneth Wilson [1], by which one should gradually coarsen the microscopic degrees of freedom, creating in this way a transformation of the scale of the Hamiltonian. The modern implementation of this idea originates from C. Wetterich and is called the effective average action formalism [2] since the starting point of it is an exact equation for the effective average action. This is a starting point for efficient nonperturbative approximations that have shown useful in tackling some of the notoriously difficult questions in physics such as: turbulence [3], transitions in nonequilibrium systems [4], transition in the random field Ising model [5], fermionic systems [6] etc.

This branch of the RG has been in strong rise since 2000ies and the methodology is still being developed. In my work the development of methodology has been a very important aspect and I contributed from purely numerical level in dealing with the solution of the systems of coupled nonlinear partial differential equations to conceptual ones, dealing with the convergence of approximation methods [7] and probabilistic interpretation of the fixed point quantities. I wish to stress the last point as the one which I believe will lead to significant advances in the future.

In parallel I worked on applications of these methods to some notoriously difficult and unresolved questions in statistical physics. One of such problems is the phase transition in the random field Ising model [8]. This model was proposed in the 70ies and has been a treasure trove of analytical results as well as experimental predictions. One of the oldest puzzles related with it is the dimensional reduction puzzle by G. Parisi and N. Sourlas [9] which states that perturbatively the transition in the d dimensional RFIM is in the same universality class as the transition in the pure d-2 dimensional model. This was later found to be invalid [10] in lower dimensions and only NPRG approach gave a physical interpretation of the breaking of the property [11]. RFIM also describes hysteresis phase transition when driven quasistatically out of equilibrium [12]. Here I contributed with several works explaining the similarities and differences between the phase transition in and out of equilibrium. However important questions remain even in these topics.

Lastly, I wish to give a glimpse of an important future development which I am involved in and has to do with the phase transitions dominated by rare events. The most simple case of such systems are systems close to the lower critical dimension, the dimension at which the transition disappears due to excessive fluctuations of the order parameter. It seems that the functional RG is an ideal tool for studying such situations.

 

[1] K. G. Wilson and J. B. Kogut, Phys. Rep. 12, 75 (1974)

[2] J. Berges, N. Tetradis, and C. Wetterich, Phys. Rep. 363, 223 (2002).

[3] L. Canet, B. Delamotte, N. Wschebor, Phys. Rev. E 93 (2016) 063101.

[4] L. Canet, H. Chaté, B. Delamotte, and N. Wschebor, Phys. Rev. Lett. 104, 150601 (2010)

[5] G. Tarjus and M. Tissier, Phys. Rev. Lett. 93, 267008 (2004).

[6] W. Metzner, M. Salmhofer, C. Honerkamp, V. Meden, K. Schönhammer, Rev. Modern Phys. 84 (2012) 299–352.

[7] I. Balog, H. Chaté, B. Delamotte, M. Marohnić, N. Wschebor, Phys. Rev. Lett. 123 (2019) 240604

[8] Y. Imry and S. K. Ma, Phys. Rev. Lett. 35, 1399 (1975)

[9] G. Parisi and N. Sourlas, Phys. Rev. Lett. 43, 744 (1979).

[10] J. Bricmont and A. Kupiainen, Phys. Rev. Lett 59, 1829 (1987).

[11] M. Tissier and G. Tarjus, Phys. Rev. Lett. 107, 041601 (2011)

[12] I. Balog, G. Tarjus, M. Tissier, Phys. Rev. B 97 (2018) 094204

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