The Aubry-André model captures the essence of quasiperiodicity.  It consists of a hopping term between lattice sites and an on-site modulation which depends on an irrational parameter.  In numerical calculations this parameter is taken to be the golden ratio and is approximated as a rational number through the ratio of consecutive Fibonacci numbers.  In this talk I will present results of calculations for a many-body non-interacting Aubry-André system with periodic boundary conditions.  We will use the geometric Binder cumulant and other localization sensitive quantities of the modern polarization theory to study the localization transition.  We find that the phase diagram is not a simple function, it can not even be drawn.  It is similar to a Dirichlet indicator function.  At particle densities which extrapolate in the thermodynamic limit to a particular type of irrational number (one rationally well approximated by a ratio of Fibonacci numbers or sums thereof) we find that the system is always localized, otherwise a transition occurs at the potential strength at which single particle states are known to localize.   Light is shed on this state of affairs by the use of the Zeckendorf theorem, which states that all natural numbers can be decomposed (almost) uniquely into a sum of Fibonacci numbers.  The formation of „bands” in such a system, as distinct from systems for which the Bloch theorem is valid, will also be discussed.   The phenomenon is robust: we show that similar behavior is found in an extended Aubry-André model.  Turning on disorder broadens these features, offering hope that it is experimentally measurable. 

 

 

B. Bánfalvi and B. Hetényi, work in progress.

B. Hetényi and I. Balogh, Phys. Rev. B 112 144203 (2025).

B. Hetényi, Phys. Rev. B 110 125124  (2024).

Understanding the structure of entanglement in extended quantum systems is a fundamental problem in theoretical physics. In this framework, a central object is the so-called entanglement (or modular) Hamiltonian (EH), defined as the logarithm of the reduced density matrix, that encodes the full structure of bipartite entanglement. In general the EH  is very hard to compute and is not even expected to be a local operator.  However, in relativistic quantum field theory the locality of the modular Hamiltonian for half-space bipartitions is ensured by the Bisognano–Wichmann theorem, which expresses it as an integral of the energy density with a linear spatial weight. In the presence of conformal symmetry, this result can be extended to other geometries and to some non-equilibrium settings. Since the Bisognano–Wichmann theorem is formulated within relativistic quantum field theory, a natural question concerns its applicability to lattice many-body systems whose low-energy properties are described by a conformal field theory, but which explicitly break Lorentz invariance. Several results addressing this question exist in equilibrium situations, while a lattice realisation of the time-dependent EH in out-of-equilibrium dynamics is still missing.

In this talk, I will present the study of the dynamics of the EH in a system of one-dimensional free fermions, following a local joining quench of two initially disconnected half-chains in their ground states. Applying techniques of conformal field theory, a local expression of the EH is derived, where the left- and right-moving components of the energy density are associated with different weight functions.

The results are then compared to numerical calculations for the hopping chain, which requires to consider a proper continuum limit of the lattice EH, obtaining a good agreement with the field-theory prediction.