Annals of physics

The electronic properties of graphene, a two-dimensional crystal of carbon atoms, are exceptionally novel. For instance, the low-energy quasiparticles in graphene behave as massless chiral Dirac fermions which has led to the experimental observation of many interesting effects similar to those predicted in the relativistic regime. Graphene also has immense potential to be a key ingredient of new devices, such as single molecule gas sensors, ballistic transistors and spintronic devices. Bilayer graphene, which consists of two stacked monolayers and where the quasiparticles are massive chiral fermions, has a quadratic low-energy band structure which generates very different scattering properties from those of the monolayer. It also presents the unique property that a tunable band gap can be opened and controlled easily by a top gate. These properties have made bilayer graphene a subject of intense interest. In this review, we provide an in-depth description of the physics of monolayer and bilayer graphene f...

Properties of graphene: a theoretical perspective

https://archive-ouverte.unige.ch/unige:115288/ATTACHMENT01

Bicrossproduct structure of κ-Poincare group and non-commutative geometry

Topological field theory

We show that the $\kappa$-deformed Poincar\'e quantum algebra proposed for elementary particle physics has the structure of a Hopf agebra bicrossproduct $U(so(1,3))\cobicross T$. The algebra is a semidirect product of the classical Lorentz group $so(1,3)$ acting in a deformed way on the momentum sector $T$. The novel feature is that the coalgebra is also semidirect, with a backreaction of the momentum sector on the Lorentz rotations. Using this, we show that the $\kappa$-Poincar\'e acts covariantly on a $\kappa$-Minkowski space, which we introduce. It turns out necessarily to be deformed and non-commutative. We also connect this algebra with a previous approach to Planck scale physics.

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Bicrossproduct structure of $\kappa$-Poincare group and non-commutative geometry

The path-integral method is used for determination of the quantum corrections to the free energy of nonlinear systems. All quantum effects of the harmonic part of the potential are considered and a variational principle is used to account for the quantum corrections due to the anharmonic part. Correct renormalized frequencies are obtained at any temperature and an effective potential to be inserted in the configurational integral is found. A new general expression for the partition function at any temperature in the low-coupling limit is obtained.

Variational approach to quantum statistical mechanics of nonlinear systems with application to sine-Gordon chains

Contractions of Lie algebras and of their representations are generalized to define new quantum groups. An explicit and complete exposition is made for the one‐dimensional Heisenberg H(1)q and the two‐dimensional Euclidean quantum group E(2)q obtained by contracting SU(2)q.

Three-dimensional quantum groups from contractions of SU(2) q

The structure of the quantum Heisenberg group is studied in the two different frameworks of the Lie algebra deformations and of the quantum matrix pseudogroups. The R‐matrix connecting the two approaches, together with its classical limit r, are explicitly calculated by using the contraction technique and the problems connected with the limiting procedure discussed. Some unusual properties of the quantum enveloping Heisenberg algebra are shown.

The quantum Heisenberg group H(1)q

A contraction procedure starting from SO(4)q is used to determine the quantum analog E(3)q of the three‐dimensional Euclidean group and the structure of its representations. A detailed analysis of the contraction of the R‐matrix is then performed and its explicit expression has been found. The classical limit of R is shown to produce an integrable dynamical system. By means of the R‐matrix the pseudogroup of the noncommutative representative functions is considered. It will finally be shown that a further contraction made on E(3)q produces the two‐dimensional Galilei quantum group and this, in turn, can be used to give a new realization of E(3)q and E(2,1)q.

Riccardo Giachetti

Papers

Variational approach to quantum statistical mechanics of nonlinear systems with application to sine-Gordon chains

Three-dimensional quantum groups from contractions of SU(2) q

The quantum Heisenberg group H(1)q

The three‐dimensional Euclidean quantum group E(3)q and its R‐matrix

Contractions of quantum groups