# Difference between revisions of "Quantum journal club"

Mondays @11:00 am

##### " When the intuition betrays or saves: messages from the 1d world " by Serguei Brazovskii

09/16/19 @LPTMS seminar room
09/23/19 @LPS salle de réunion 2ème étage

Series of two lectures with four excursions.

Theory of (quasi) one-dimensional electronic systems attracts us thanks to its special facilities but also because of some specific curiosities. In these excursions we shall meet some examples of not quite expected relations, difficulties and even mistakes which one can pickup from the half-a-century history of this field.

First excursion will descend to the contradictory story of Luttinger liquid with its curiosities in historical , personal and linguistic aspects. It will also introduce the once unattended role of so called chiral or Schwinger anomalies.

Second excursion will meet the chiral anomalies as they appear already in simplest, MF or BCS types, models particularly at finite temperatures when normal excitations are present. The resulting effective Ginzburg-Landau theory will prove to be quite different from what is commonly expected.

Third excursion will undermine the common belief of the spin-charge separation which is seemingly endorsed by the bosonization and exact solutions. Actually, spin excitations must carry the electric current as it takes place for free fermions. The resolution comes from correct definition of current carrying states and current operators taking into account the band curvature.

Forth excursion will lead to the early days when 1D models were studies to understand the phase diagram of real quasi-1D systems. We will see that, unlike the common beliefs, the 1D (g-ological) phase diagram based upon diverging power-law susceptibilities, does not want at all to reproduce itself when electrons acquire a bandwidth in interchain directions. The system falls to the Fermi-liquid regime unless the “imaginary gaps” appear from external symmetry lowering due to the crystal field of the magnetic field.

##### 11/26/18 : " Correlations of occupation numbers in the canonical ensemble " by Christophe Texier

The connection between the statistical physics of non-interaction indistinguishable particles in quantum mechanics and the theory of symmetric functions will be reviewed.Then, I will study the $p$-point correlation function $\overline {n_{1}\cdots n_{p}}$ of occupation numbers in the canonical ensemble ; in the grand canonical ensemble, they are trivially obtained from the independence of individual quantum states, however the constraint on the number of particles makes the problem non trivial in the canonical ensemble. I will show several representations of these correlation functions. I will illustrate the main formulae by revisiting the problem of Bose-Einstein condensation in a 1D harmonic trap in the canonical ensemble, for which we have obtained several analytical results. In particular, in the temperature regime dominated by quantum correlations, the distribution of the ground state occupancy is shown to be a truncated Gumbel law.

Reference :Olivier Giraud, AurÃ©lien Grabsch & Christophe Texier, Correlations of occupation numbers in the canonical ensemble and application to BEC in a 1D harmonic trap, Phys. Rev. A 97, 053615 (2018).

##### 10/15/18 : " Topological Transition in a Non-Hermitian Quantum Walk " by Leonardo Mazza

References: M. S. Rudner and L. S. Levitov, Phys. Rev. Lett. 102, 065703 (2009)(https://arxiv.org/abs/0807.2048)

##### 07/09/18 : " Out-of-time-order correlators in quantum mechanics" by Bradraj Pandey

References:1. Out-of-time-order correlators in quantum mechanics (https://arxiv.org/abs/1703.09435) 2. Measuring out-of-time-order correlations and multiple quantum spectra in a trapped ion quantum magnet (https://arxiv.org/abs/1608.08938).

##### 06/04/18 : Quantum mechanics in multi-connected space and the origin of new statistics in low dimensional system by Raoul santachiara

We recall how to define the problem of N indinstinguishible quantum particles and argue that the topology of the configuration space plays a crucial role. This observation, that has been put on solid grounds by Leinaas and Mirheim in the 1977, has provided the theoretical framework for the existence of anyonic statistics in two dimensions. Moreover, it inspired the connection between the Conformal field theory and topological phases in two dimensions: via this connection, the occurence of non-Abelian anyons in the fractional quantum Hall effect has been suggested.