The full counting statistics encodes the probability distribution of a dynamical observable and is a dynamical analogue of the thermodynamic partition function. It is naturally discussed within the frameworks of the large deviation theory and the Lee-Yang theory of phase transitions, which we briefly review. By combining the two approaches we point out that, in the presence of dynamical critical point, a rich phenomenology of fluctuations is permissible. As an explicit demonstration we introduce an interacting cellular automaton, where an analytical computation of the full counting statistics is feasible. Asymptotic analysis of the exact solution gives access to the current distribution on all scales and explicit cumulant asymptotics, revealing, among other anomalous features, non-Gaussian typical fluctuations in equilibrium. The scaled cumulant generating function does not generate scaled cumulants. If time permits we will also discuss some recent results on anomalous fluctuations in the (anisotropic) Landau-Lifshitz model, a paradigmatic integrable model of interacting classical spins.

References:
Ž. Krajnik, J. Schmidt, V. Pasquier, E. Ilievski, T. Prosen. Exact anomalous current fluctuations in a deterministic interacting model. arXiv: 2201.05126
Ž. Krajnik, E. Ilievski, T. Prosen. Absence of Normal Fluctuations in an Integrable Magnet, arXiv:2109.13088

The understanding of the relaxation of large quantum systems has received an important boost from the point of view of pure state quantum statistical mechanics, and in particular from the eigenstate thermalization hypothesis. At the same time, the dynamics of quantum systems has been investigated by out of time ordered correlators. Interestingly it was shown, using hydrodynamic theory, that such correlators relax algebraically in the presence of conserved quantities. Here we show how such slow relaxation can be expected from the eigenstate thermalization hypothesis.

At the same time, conserved quantities can lead to the phenomenon of pre-thermalization. Here we show that considering two large non-integrable systems weakly coupled to each other, one can show that in the thermodynamic limit a steady current between the two will emerge, and that this current is typical.

* Dario Poletti, Singapore University for Technology and Design

We study the growth of the number entropy S_N in one-dimensional number-conserving interacting systems with strong disorder, which are believed to display many-body localization. Recently a slow and small growth of S_N has been numerically reported, which, if holding at asymptotically long times in the thermodynamic limit, would imply ergodicity and therefore the absence of true localization. By numerically studying S_N in the disordered isotropic Heisenberg model we first reconfirm that, indeed, there is a small growth of S_N. However, we show that such growth is fully compatible with localization. To be specific, using a simple model that accounts for expected rare resonances we can analytically predict several main features of numerically obtained S_N: trivial initial growth at short times, a slow power-law growth at intermediate times, and the scaling of the saturation value of S_N with the disorder strength. Because resonances crucially depend on individual disorder realizations, the growth of S_N also heavily varies depending on the initial state, and therefore S_N and von Neumann entropy can behave rather differently.

Stark many-body localized (SMBL) systems have been shown both numerically and experimentally to have Bloch many-body oscillations, quantum many-body scars, and fragmentation in the large field tilt limit. Likewise, they are believed to show localization similar to disordered MBL. I will discuss how all of these observations can be analytically understood by rigorously showing the existence of novel algebraic structures that are exponentially stable in time in the large tilt limit. I call this novel operators dynamical l-bits. Moreover, I show that many-body Bloch oscillations persist even at infinite temperature for exponentially long-times. I provide numerical confirmation of these results by studying the prototypical Stark MBL model of a tilted XXZ spin chain. The work explains why thermalization was observed in a recent 2D tilted experiment. As dynamical l-bits are stable, localized and quantum coherent excitations, the work opens new possibilities for quantum information processing in Stark MBL systems.

Reference:
T. Gunawardana, B. Buča. Dynamical l-bits in Stark many-body localization. arXiv:2110.13135 (2021)

* Berislav Buča, University of Oxford

The seminar will be held online via Zoom (ID: 281 621 2459, Password: 667124)

https://uni-lj-si.zoom.us/j/2816212459

We present some solutions of the set-theoretic parametric Yang-Baxter equation. These solutions are birational maps with several invariants and a Lax representation[4]. We show that we can use these maps as building blocks in order to construct higher dimensional birational maps which have nice properties and we prove their integrability in the Liouville sense. These maps can be seen as higher dimensional generalisations of the famous integrable QRT maps [3], known as Adler’s Triad maps[1]. Finally, we discuss some new generalisations [2].

* Georgios Papamikos, University of Essex

The seminar will be held online via Zoom (ID: 281 621 2459, Password: 667124)

https://uni-lj-si.zoom.us/j/2816212459

References
[1]V. E. Adler, On a class of third order mappings with two rational invariants, preprint, arXiv:nlin/0606056v1
[2] S. Konstantinou-Rizos, G.Papamikos, Entwining Yang-Baxter maps related to NLS type equations, Journal of Physics A: Mathematical and Theoretical, 52, 2019
[3] G. R. W. Quispel, J. A. G. Roberts, C. J. Thompson, Integrable mappings and soliton equations II, Physica D: Nonlinear Phenomena34(1989), 183-192.
[4] A. P. Veselov, Yang-Baxter maps and integrable dynamics, Physics Letters A314 (2003), 214 – 221.

Studies of disordered spin chains have recently experienced a renewed interest, inspired by the question to which extent the exact numerical calculations comply with the existence of a many-body localization phase transition. For the paradigmatic random field Heisenberg spin chains, many intriguing features were observed when the disorder is considerable compared to the spin interaction strength. Here, we introduce a phenomenological theory that may explain some of those features [1]. The theory is based on the proximity to the noninteracting limit, in which the system is an Anderson insulator. Taking the spin imbalance as an exemplary observable, we demonstrate that the proximity to the local integrals of motion of the Anderson insulator determines the dynamics of the observable at infinite temperature. In finite interacting systems our theory quantitatively describes its integrated spectral function for a wide range of disorders.

[1] Vidmar, Krajewski, Bonča, Mierzejewski, arXiv:2105.09336 (PRL, in press)

Tuesday 20.7.2021, 14:15h in Kuščerjev seminar, Jadranska 19.

In low dimensional systems heat can propagate faster than diffusion. This leads to a thermal conductivity that diverges with system size pointing to a superdiffusive transport. This leads to the question, if there is an equivalent of the heat equation which can be used to study superdiffusive transport in low dimensional systems. In this talk, I will discuss two simplified models where we establish a Fractional equation description for anomalous heat transport.

* Aritra Kundu, SISSA, Trieste

Tuesday 8.6.2021, 14:15h in Kuščerjev seminar, Jadranska 19.

We compute the Floquet Hamiltonian H_F for weakly interacting fermions subjected to a continuous periodic drive using a Floquet perturbation theory (FPT) with the interaction amplitude being the perturbation parameter. This allows us to address the dynamics of the system at intermediate drive frequencies ~ω_D ≥ V_0  J_0 , where J_0 is the amplitude of the kinetic term, ω_D is the drive frequency, and V_0 is the typical interaction strength between the fermions. We compute, for random initial states, the fidelity F between wavefunctions after a drive cycle obtained using H_F and that obtained using exact diagonalization (ED). We find that FPT yields a substantially larger value of F compared to its Magnus counterpart for V_0 ≤ ~ω_D and V_0  J_0 . We use the H_F obtained to study the nature of the steady state of a weakly interacting fermion chain; we find a wide range of ω_D which leads to subthermal or superthermal steady states for finite chains. The driven fermionic chain displays perfect dynamical localization for V_0 = 0; we address the fate of this dynamical localization in the steady state of a finite interacting chain and show that there is a crossover between localized and delocalized steady states. We discuss the implication of our results for thermodynamically large chains.

Journal Ref:- Phys. Rev. B 102, 235114 – Published 4 December 2020

* Roopayan Ghosh, University of Ljubljana

Tuesday 1.6.2021, 15:15h in Kuščerjev seminar, Jadranska 19.

We apply the inverse scattering method to the sine-Gordon model in discrete space-time.  Building on the results of integrability we formulate two complementary approaches to the thermodynamics of the model based on two distinct sets of canonical variables.

* Žiga Krajnik, University of Ljubljana

Tuesday 18.5.2021, 15:15h in Kuščerjev seminar, Jadranska 19.

We apply the inverse scattering method to the sine-Gordon model in discrete space-time.  Building on the results of integrability we formulate two complementary approaches to the thermodynamics of the model based on two distinct sets of canonical variables.

* Žiga Krajnik, University of Ljubljana