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Add reference to [CTV12].

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Ian Jauslin 2015-07-31 05:18:36 +00:00
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9
Giuliani_Jauslin_2015.tex
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@ -60,8 +60,8 @@ and the dispersion relation is approximately conical around them. This picture i
At lower energies, the effective dispersion relation around the two Fermi points appears to be approximately {\it parabolic}, instead of conical. This implies that the effective mass of the electrons in bilayer graphene does not vanish, unlike those in the monolayer, which has been observed
experimentally~[\cite{novZS}].\par
\indent From an RG point of view, the parabolicity implies that the electron interactions are {\it marginal} in bilayer graphene, thus making the RG analysis non-trivial. The flow of the effective couplings
has been studied by O.~Vafek~[\cite{vafOZ}], who has found that it diverges logarithmically, and has identified the most divergent channels,
thus singling out which of the possible quantum instabilities are dominant.
has been studied by O.~Vafek~[\cite{vafOZ}, \cite{vayOZ}], who has found that it diverges logarithmically, and has identified the most divergent channels,
thus singling out which of the possible quantum instabilities are dominant (see also~\cite{tvOT}).
However, as was mentioned earlier, the assumption of parabolic dispersion relation is only an approximation,
valid in a range of energies between the scale of the transverse hopping and a second threshold, proportional to the cube of the transverse hopping (asymptotically, as this hopping goes to zero).
This range will be called the {\it second regime}.\par
@ -73,7 +73,10 @@ only if the flow of the effective constants has grown significantly in the secon
\indent However, our analysis shows that the flow of the effective couplings in this regime does not grow at all, due to their smallness after integration over the first regime,
which we quantify in terms both of the bare couplings and of the transverse hopping. This puts into question the physical relevance of the ``instabilities'' coming from the logarithmic divergence in the second regime,
at least in the case we are treating, namely small interaction strength and small interlayer hopping.\par
\indent Let us mention that the third regime is not believed to give an adequate description of the system at arbitrarily small energies: at energies smaller than a third threshold (proportional to the fourth power of the transverse hopping) one finds~[\cite{parZS}] that the six extra Fermi points around the two original ones, are actually microscopic ellipses. The analysis of the thermodynamic properties of the system in this regime (to be called the fourth regime) requires new ideas and techniques, due to the extended nature of the singularity, and goes beyond the scope of this paper. It may be possible to adapt the ideas of [\cite{benZS}] to this regime, and we hope to come back to this issue in a future publication.
\indent The transition from a normal phase to one with broken symmetry as the interaction strength is increased from small to intermediate values was studied in~[\cite{ctvOT}] at second order in perturbation theory. Therein, it was found that while at small bare couplings the infrared flow is convergent, at larger couplings it tends to increase, indicating a transition towards an {\it electronic nematic state}.\par
\indent Let us also mention that the third regime is not believed to give an adequate description of the system at arbitrarily small energies: at energies smaller than a third threshold (proportional to the fourth power of the transverse hopping) one finds~\cite{parZS} that the six extra Fermi points around the two original ones, are actually microscopic ellipses. The analysis of the thermodynamic properties of the system in this regime (to be called the fourth regime) requires new ideas and techniques, due to the extended nature of the singularity, and goes beyond the scope of this paper. It may be possible to adapt the ideas of \cite{benZS} to this regime, and we hope to come back to this issue in a future publication.
\par
\bigskip

17
bibliography.BBlog.tex
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@ -38,6 +38,11 @@
\hbox{\parbox[t]{\rw}{[\cite{bkESe}]}\parbox[t]{\colw}{D.~Brydges, T.~Kennedy - {\it Mayer expansions and the Hamilton-Jacobi equation}, Journal of Statistical Physics, Vol.~48, n.~1-2, p.~19-49, 1987.}}\par
\bigskip
\hrefanchor
\outdef{citectvOT}{CTV12}
\hbox{\parbox[t]{\rw}{[\cite{ctvOT}]}\parbox[t]{\colw}{V.~Cvetkovic, R.~Throckmorton, O.~Vafek - {\it Electronic multicriticality in bilayer graphene}, Physical Review B, Vol.~86, n.~075467, 2012.}}\par
\bigskip
\hrefanchor
\outdef{citedoeSeN}{DDe79}
\hbox{\parbox[t]{\rw}{[\cite{doeSeN}]}\parbox[t]{\colw}{R.~Doezema, W.~Datars, H.~Schaber, A.~Van~Schyndel - {\it Far-infrared magnetospectroscopy of the Landau-level structure in graphite}, Physical Review B, Vol.~19, n.~8, p.~4224-4230, 1979.}}\par
@ -115,7 +120,7 @@
\hrefanchor
\outdef{citeluOTh}{Lu13}
\hbox{\parbox[t]{\rw}{[\cite{luOTh}]}\parbox[t]{\colw}{L.~Lu - {\it Constructive analysis of two dimensional Fermi systems at finite temperature}, PhD dissertation, supervised by M.~Salmhofer, Institute for Theoretical Physics, Heidelberg, 2013.}}\par
\hbox{\parbox[t]{\rw}{[\cite{luOTh}]}\parbox[t]{\colw}{L.~Lu - {\it Constructive analysis of two dimensional Fermi systems at finite temperature}, PhD thesis, Institute for Theoretical Physics, Heidelberg, \url{http://www.ub.uni-heidelberg.de/archiv/14947}, 2013.}}\par
\bigskip
\hrefanchor
@ -178,6 +183,11 @@
\hbox{\parbox[t]{\rw}{[\cite{sloFiE}]}\parbox[t]{\colw}{J.~Slonczewski, P.~Weiss - {\it Band structure of graphite}, Physical Review, Vol.~109, p.~272-279, 1958.}}\par
\bigskip
\hrefanchor
\outdef{citetvOT}{TV12}
\hbox{\parbox[t]{\rw}{[\cite{tvOT}]}\parbox[t]{\colw}{R.~Throckmorton, O.~Vafek - {\it Fermions on bilayer graphene: symmetry breaking for $B=0$ and $\nu=0$}, Physical Review B, Vol.~86, 115447, 2012.}}\par
\bigskip
\hrefanchor
\outdef{citetoySeSe}{TDD77}
\hbox{\parbox[t]{\rw}{[\cite{toySeSe}]}\parbox[t]{\colw}{W.~Toy, M.~Dresselhaus, G.~Dresselhaus - {\it Minority carriers in graphite and the H-point magnetoreflection spectra}, Physical Review B, Vol.~15, p.~4077-4090, 1977.}}\par
@ -193,6 +203,11 @@
\hbox{\parbox[t]{\rw}{[\cite{vafOZ}]}\parbox[t]{\colw}{O.~Vafek - {\it Interacting Fermions on the honeycomb bilayer: from weak to strong coupling}, Physical Review B, Vol.~82, 205106, 2010.}}\par
\bigskip
\hrefanchor
\outdef{citevayOZ}{VY10}
\hbox{\parbox[t]{\rw}{[\cite{vayOZ}]}\parbox[t]{\colw}{O.~Vafek, K.~Yang - {\it Many-body instability of Coulomb interacting bilayer graphene: renormalization group approach}, Physical Review B, Vol.~81, 041401, 2010.}}\par
\bigskip
\hrefanchor
\outdef{citewalFSe}{Wa47}
\hbox{\parbox[t]{\rw}{[\cite{walFSe}]}\parbox[t]{\colw}{P.~Wallace - {\it The band theory of graphite}, Physical Review, Vol.~71, n.~9, p.~622-634, 1947.}}\par