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Fisher zeroes and the fluctuations of the spectral form factor of chaotic systems
by Guy Bunin, Laura Foini, Jorge Kurchan
Submission summary
Authors (as registered SciPost users):  Laura Foini · Jorge Kurchan 
Submission information  

Preprint Link:  https://arxiv.org/abs/2207.02473v3 (pdf) 
Date submitted:  20231222 12:01 
Submitted by:  Kurchan, Jorge 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The spectral form factor of quantum chaotic systems has the familiar `ramp $+$ plateau' form. Techniques to determine its form in the semiclassical or the thermodynamic limit have been devised, in both cases based on the average over an energy range or an ensemble of systems. For a single instance, fluctuations are large, do not go away in the limit, and depend on the element of the ensemble itself, thus seeming to question the whole procedure. Considered as the modulus of a partition function in complex inverse temperature $\beta_R+i\beta_I$ ($\beta_I \equiv \tau$ the time), the spectral factor has regions of Fisher zeroes, the analogue of YangLee zeroes for the complex temperature plane. The large spikes in the spectral factor are in fact a consequence of nearmisses of the line parametrized by $\beta_I$ to these zeroes. The largest spikes are indeed extensive and extremely sensitive to details, but we show that they are both exponentially rare and exponentially thin. Motivated by this, and inspired by the work of Derrida on the Random Energy Model, we study here a modified model of random energy levels in which we introduce level repulsion. We also check that the mechanism giving rise to spikes is the same in the SYK model.
List of changes
We have clarified notation and improved the figures as requested by the referees
We have added a phrase :
`` Note that the fact that the larger spikes are exponentially rare and exponentially thin (in $N$) implies that the effect of these fluctuations will be ignored by higher moments of the trace, i.e. by a replica treatment of the problem."
We have now mentioned this:
`` The form factor may be expressed as a sum of the timecorrelations $\sum_n \langle A_n(t) A_n(0)\rangle$ of an exponential set of
operators [.]. It would be interesting to see if the zeroes move chaotically
with the addition of each new term, even for a single sample. ".