3 Commits
v0.0 ... v0.1.2

Author SHA1 Message Date
42f582de1e Fix typo 2015-10-19 14:10:46 +00:00
e20666ffbf Prevent labels from overwriting commands 2015-10-19 14:01:35 +00:00
52f9075093 Add reference to [CTV12]. 2015-07-31 05:18:36 +00:00
3 changed files with 71 additions and 53 deletions

View File

@ -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
@ -230,7 +233,7 @@ in which $\gamma_4$ and $\Delta$ are negligible, and the Fermi surface is approx
\par
\bigskip
{\bf Remark:} If $\gamma_4=\Delta0$, then the error term $O(\epsilon^{4}\|\mathbf k'_{j}\|_{\mathrm{III}}^{-1})$ in (\ref{freeschwinth}) vanishes identically, which allows us to extend the third regime to all momenta satisfying
{\bf Remark:} If $\gamma_4=\Delta=0$, then the error term $O(\epsilon^{4}\|\mathbf k'_{j}\|_{\mathrm{III}}^{-1})$ in (\ref{freeschwinth}) vanishes identically, which allows us to extend the third regime to all momenta satisfying
$$\|\mathbf k'_{j}\|_{\mathrm{III}}\ll\epsilon^3.$$
@ -502,7 +505,7 @@ and complete the proofs of the Main Theorem, as well as of Theorems \ref{theoo},
\section{The model}
\label{themodelsec}
\hfil\framebox{\bf From this point on, we set $\gamma_4=\Delta0$.}
\hfil\framebox{\bf From this point on, we set $\gamma_4=\Delta=0$.}
\bigskip
\indent In this section, we define the model in precise terms, re-express the free energy and two-point Schwinger function in terms of Grassmann integrals and truncated expectations, which we will subsequently explain how to compute, and discuss the symmetries of the model and their representation in this formalism.\par

View File

@ -1,210 +1,225 @@
\hrefanchor
\outdef{citearNE}{AR98}
\outdef{label@citearNE}{AR98}
\hbox{\parbox[t]{\rw}{[\cite{arNE}]}\parbox[t]{\colw}{A.~Abdesselam, V.~Rivasseau - {\it Explicit Fermionic tree expansions}, Letters in Mathematical Physics, Vol.~44, n.~1, p.~77-88, 1998.}}\par
\bigskip
\hrefanchor
\outdef{citebatEF}{BF84}
\outdef{label@citebatEF}{BF84}
\hbox{\parbox[t]{\rw}{[\cite{batEF}]}\parbox[t]{\colw}{G.~Battle, P.~Federbush - {\it A note on cluster expansions, tree graph identities, extra $1/N!$ factors!!!}, Letters in Mathematical Physics, Vol.~8, p.~55-57, 1984.}}\par
\bigskip
\hrefanchor
\outdef{citebenNZ}{BG90}
\outdef{label@citebenNZ}{BG90}
\hbox{\parbox[t]{\rw}{[\cite{benNZ}]}\parbox[t]{\colw}{G.~Benfatto, G.~Gallavotti - {\it Perturbation theory of the Fermi surface in a quantum liquid - a general quasiparticle formalism and one-dimensional systems}, Journal of Statistical Physics, Vol.~59, n.~3-4, p.~541-664, 1990.}}\par
\bigskip
\hrefanchor
\outdef{citebenNFi}{BG95}
\outdef{label@citebenNFi}{BG95}
\hbox{\parbox[t]{\rw}{[\cite{benNFi}]}\parbox[t]{\colw}{G.~Benfatto, G.~Gallavotti - {\it Renormalization Group}, Princeton University Press, 1995.}}\par
\bigskip
\hrefanchor
\outdef{citebmZT}{BM02}
\outdef{label@citebmZT}{BM02}
\hbox{\parbox[t]{\rw}{[\cite{bmZT}]}\parbox[t]{\colw}{G.~Benfatto, V.~Mastropietro - {\it On the density-density critical indices in interacting Fermi systems}, Communications in Mathematical Physics, Vol.~231, n.~1, p.~97-134, 2002.}}\par
\bigskip
\hrefanchor
\outdef{citebenZS}{BGM06}
\outdef{label@citebenZS}{BGM06}
\hbox{\parbox[t]{\rw}{[\cite{benZS}]}\parbox[t]{\colw}{G.~Benfatto, A.~Giuliani, V.~Mastropietro - {\it Fermi liquid behavior in the 2D Hubbard model}, Annales Henri Poincar\'e, Vol.~7, p.~809-898, 2006.}}\par
\bigskip
\hrefanchor
\outdef{citebrySeE}{BF78}
\outdef{label@citebrySeE}{BF78}
\hbox{\parbox[t]{\rw}{[\cite{brySeE}]}\parbox[t]{\colw}{D.~Brydges, P.~Federbush - {\it A new form of the Mayer expansion in classical statistical mechanics}, Journal of Mathematical Physics, Vol.~19, p.~2064, 1978.}}\par
\bigskip
\hrefanchor
\outdef{citebkESe}{BK87}
\outdef{label@citebkESe}{BK87}
\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{citedoeSeN}{DDe79}
\outdef{label@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{label@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
\bigskip
\hrefanchor
\outdef{citedreZT}{DD02}
\outdef{label@citedreZT}{DD02}
\hbox{\parbox[t]{\rw}{[\cite{dreZT}]}\parbox[t]{\colw}{M.~Dresselhaus, G.~Dresselhaus - {\it Intercalation compounds of graphite}, Advances in Physics, Vol.~51, n.~1, p.~1-186, 2002.}}\par
\bigskip
\hrefanchor
\outdef{citefktZFa}{FKT04a}
\outdef{label@citefktZFa}{FKT04a}
\hbox{\parbox[t]{\rw}{[\cite{fktZFa}]}\parbox[t]{\colw}{J.~Feldman, H.~Kn\"orrer, E.~Trubowitz - {\it A two dimensional Fermi liquid. Part~1: Overview}, Communications in Mathematical Physics, Vol.~247, n.~1, p.~1-47, 2004.}}\par
\bigskip
\hrefanchor
\outdef{citefktZFb}{FKT04b}
\outdef{label@citefktZFb}{FKT04b}
\hbox{\parbox[t]{\rw}{[\cite{fktZFb}]}\parbox[t]{\colw}{J.~Feldman, H.~Kn\"orrer, E.~Trubowitz - {\it A two dimensional Fermi liquid. Part~2: Convergence}, Communications in Mathematical Physics, Vol.~247, n.~1, p.~49-111, 2004.}}\par
\bigskip
\hrefanchor
\outdef{citefktZFc}{FKT04c}
\outdef{label@citefktZFc}{FKT04c}
\hbox{\parbox[t]{\rw}{[\cite{fktZFc}]}\parbox[t]{\colw}{J.~Feldman, H.~Kn\"orrer, E.~Trubowitz - {\it A two dimensional Fermi liquid. Part~3: The Fermi surface}, Communications in Mathematical Physics, Vol.~247, n.~1, p.~113-177, 2004.}}\par
\bigskip
\hrefanchor
\outdef{citegalEFi}{GN85}
\outdef{label@citegalEFi}{GN85}
\hbox{\parbox[t]{\rw}{[\cite{galEFi}]}\parbox[t]{\colw}{G.~Gallavotti, F.~Nicol\`o - {\it Renormalization theory for four dimensional scalar fields}, Communications in Mathematical Physics, Vol.~100, p.~545-590 and Vol.~101, p.~247-282, 1985.}}\par
\bigskip
\hrefanchor
\outdef{citegeiZSe}{GN07}
\outdef{label@citegeiZSe}{GN07}
\hbox{\parbox[t]{\rw}{[\cite{geiZSe}]}\parbox[t]{\colw}{A.~Geim, K.~Novoselov - {\it The rise of graphene}, Nature Materials, Vol.~6, p.~183-191, 2007.}}\par
\bigskip
\hrefanchor
\outdef{citegeOZ}{Ge10}
\outdef{label@citegeOZ}{Ge10}
\hbox{\parbox[t]{\rw}{[\cite{geOZ}]}\parbox[t]{\colw}{A.~Geim - {\it Random walk to graphene}, Nobel lecture, 2010.}}\par
\bigskip
\hrefanchor
\outdef{citegenZO}{GM01}
\outdef{label@citegenZO}{GM01}
\hbox{\parbox[t]{\rw}{[\cite{genZO}]}\parbox[t]{\colw}{G.~Gentile, V.~Mastropietro - {\it Renormalization group for one-dimensional fermions - a review on mathematical results}, Physics Reports, Vol.~352, p.~273-437, 2001.}}\par
\bigskip
\hrefanchor
\outdef{citegiuOZ}{GM10}
\outdef{label@citegiuOZ}{GM10}
\hbox{\parbox[t]{\rw}{[\cite{giuOZ}]}\parbox[t]{\colw}{A.~Giuliani, V.~Mastropietro - {\it The two-dimensional Hubbard model on the honeycomb lattice}, Communications in Mathematical Physics, Vol.~293, p.~301-364, 2010.}}\par
\bigskip
\hrefanchor
\outdef{citegiuOZh}{Gi10}
\outdef{label@citegiuOZh}{Gi10}
\hbox{\parbox[t]{\rw}{[\cite{giuOZh}]}\parbox[t]{\colw}{A.~Giuliani - {\it The Ground State Construction of the Two-dimensional Hubbard Model on the Honeycomb Lattice}, Quantum Theory from Small to Large Scales, lecture notes of the Les Houches Summer School, Vol.~95, Oxford University Press, 2010.}}\par
\bigskip
\hrefanchor
\outdef{citegmpOZ}{GMP10}
\outdef{label@citegmpOZ}{GMP10}
\hbox{\parbox[t]{\rw}{[\cite{gmpOZ}]}\parbox[t]{\colw}{A.~Giuliani, V.~Mastropietro, M.~Porta - {\it Lattice gauge theory model for graphene}, Physical Review B, Vol.~82, n.~121418(R), 2010.}}\par
\bigskip
\hrefanchor
\outdef{citegmpOO}{GMP11}
\outdef{label@citegmpOO}{GMP11}
\hbox{\parbox[t]{\rw}{[\cite{gmpOO}]}\parbox[t]{\colw}{A.~Giuliani, V.~Mastropietro, M.~Porta - {\it Absence of interaction corrections in the optical conductivity of graphene}, Physical Review B, Vol.~83, n.~195401, 2011.}}\par
\bigskip
\hrefanchor
\outdef{citegmpOOt}{GMP11b}
\outdef{label@citegmpOOt}{GMP11b}
\hbox{\parbox[t]{\rw}{[\cite{gmpOOt}]}\parbox[t]{\colw}{A.~Giuliani, V.~Mastropietro, M.~Porta - {\it Lattice quantum electrodynamics for graphene}, Annals of Physics, Vol.~327, n.~2, p.~461-511, 2011.}}\par
\bigskip
\hrefanchor
\outdef{citegmpOT}{GMP12}
\outdef{label@citegmpOT}{GMP12}
\hbox{\parbox[t]{\rw}{[\cite{gmpOT}]}\parbox[t]{\colw}{A.~Giuliani, V.~Mastropietro, M.~Porta - {\it Universality of conductivity in interacting graphene}, Communications in Mathematical Physics, Vol.~311, n.~2, p.~317-355, 2012.}}\par
\bigskip
\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
\outdef{label@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 thesis, Institute for Theoretical Physics, Heidelberg, \url{http://www.ub.uni-heidelberg.de/archiv/14947}, 2013.}}\par
\bigskip
\hrefanchor
\outdef{citemalZSe}{MNe07}
\outdef{label@citemalZSe}{MNe07}
\hbox{\parbox[t]{\rw}{[\cite{malZSe}]}\parbox[t]{\colw}{L.~Malard, J.~Nilsson, D.~Elias, J.~Brant, F.~Plentz, E.~Alves, A.~Castro Neto, M.~Pimenta - {\it Probing the electronic structure of bilayer graphene by Raman scattering}, Physical Review B, Vol.~76, n.~201401, 2007.}}\par
\bigskip
\hrefanchor
\outdef{citemasOO}{Ma11}
\outdef{label@citemasOO}{Ma11}
\hbox{\parbox[t]{\rw}{[\cite{masOO}]}\parbox[t]{\colw}{V.~Mastropietro - {\it Conductivity between Luttinger liquids: coupled chains and bilayer graphene}, Physical Review B, Vol.~84, n.~035109, 2011.}}\par
\bigskip
\hrefanchor
\outdef{citemccZS}{MF06}
\outdef{label@citemccZS}{MF06}
\hbox{\parbox[t]{\rw}{[\cite{mccZS}]}\parbox[t]{\colw}{E.~McCann, V.~Fal'ko - {\it Landau-level degeneracy and Quantum Hall Effect in a graphite bilayer}, Physical Review Letters, Vol.~86, 086805, 2006.}}\par
\bigskip
\hrefanchor
\outdef{citemccFiSe}{Mc57}
\outdef{label@citemccFiSe}{Mc57}
\hbox{\parbox[t]{\rw}{[\cite{mccFiSe}]}\parbox[t]{\colw}{J.~McClure - {\it Band structure of graphite and de Haas-van Alphen effect}, Physical review, Vol.~108, p.~612-618, 1957.}}\par
\bigskip
\hrefanchor
\outdef{citemisSeN}{MMD79}
\outdef{label@citemisSeN}{MMD79}
\hbox{\parbox[t]{\rw}{[\cite{misSeN}]}\parbox[t]{\colw}{A.~Misu, E.~Mendez, M.S.~Dresselhaus - {\it Near Infrared Reflectivity of Graphite under Hydrostatic Pressure}, Journal of the Physical Society of Japan, Vol.~47, n.~1, p.~199-207, 1979.}}\par
\bigskip
\hrefanchor
\outdef{citengeZF}{NGe04}
\outdef{label@citengeZF}{NGe04}
\hbox{\parbox[t]{\rw}{[\cite{ngeZF}]}\parbox[t]{\colw}{K.~Novoselov, A.~Geim, S.~Morozov, D.~Jiang, Y.~Zhang, S.~Dubonos, I.~Grigorieva, A.~Firsov - {\it Electric field effect in atomically thin carbon films}, Science, vol.~306, p.~666-669, 2004.}}\par
\bigskip
\hrefanchor
\outdef{citengeZFi}{NGe05}
\outdef{label@citengeZFi}{NGe05}
\hbox{\parbox[t]{\rw}{[\cite{ngeZFi}]}\parbox[t]{\colw}{K.~Novoselov, A.~Geim, S.~Morozov, D.~Jiang, M.~Katsnelson, I.~Grigorieva, S.~Dubonos, A.~Firsov - {\it Two-dimensional gas of massless Dirac fermions in graphene}, Nature, Vol.~438, n.~10, p.~197-200, 2005.}}\par
\bigskip
\hrefanchor
\outdef{citenovZS}{NMe06}
\outdef{label@citenovZS}{NMe06}
\hbox{\parbox[t]{\rw}{[\cite{novZS}]}\parbox[t]{\colw}{K.~Novoselov, E.~McCann, S.~Morozov, V.~Fal'ko, M.~Katsnelson, U.~Zeitler, D.~Jiang, F.~Schedin, A.~Geim - {\it Unconventional quantum Hall effect and Berry's phase of $\pi$ in bilayer graphene}, Nature Physics, Vol.~2, p.~177-180, 2006.}}\par
\bigskip
\hrefanchor
\outdef{citeparZS}{PP06}
\outdef{label@citeparZS}{PP06}
\hbox{\parbox[t]{\rw}{[\cite{parZS}]}\parbox[t]{\colw}{B.~Partoens, F.~Peeters - {\it From graphene to graphite: electronic structure around the $K$ point}, Physical Review B, Vol.~74, n.~075404, 2006.}}\par
\bigskip
\hrefanchor
\outdef{citepsZE}{PS08}
\outdef{label@citepsZE}{PS08}
\hbox{\parbox[t]{\rw}{[\cite{psZE}]}\parbox[t]{\colw}{W.~Pedra, M.~Salmhofer - {\it Determinant bounds and the Matsubara UV problem of many-fermion systems}, Communications in Mathematical Physics, Vol.~282, n.~3, p.~797-818, 2008.}}\par
\bigskip
\hrefanchor
\outdef{citesalOTh}{Sal13}
\outdef{label@citesalOTh}{Sal13}
\hbox{\parbox[t]{\rw}{[\cite{salOTh}]}\parbox[t]{\colw}{M.~Salmhofer - {\it Renormalization: an introduction}, Springer Science \& Business Media, 2013.}}\par
\bigskip
\hrefanchor
\outdef{citesloFiE}{SW58}
\outdef{label@citesloFiE}{SW58}
\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{citetoySeSe}{TDD77}
\outdef{label@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{label@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
\bigskip
\hrefanchor
\outdef{citetriNT}{TMe92}
\outdef{label@citetriNT}{TMe92}
\hbox{\parbox[t]{\rw}{[\cite{triNT}]}\parbox[t]{\colw}{S.~Trickey, F.~M\"uller-Plathe, G.~Diercksen, J.~Boettger - {\it Interplanar binding and lattice relaxation in a graphite dilayer}, Physical Review B, Vol.~45, p.~4460-4468, 1992.}}\par
\bigskip
\hrefanchor
\outdef{citevafOZ}{Va10}
\outdef{label@citevafOZ}{Va10}
\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{citewalFSe}{Wa47}
\outdef{label@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{label@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
\bigskip
\hrefanchor
\outdef{citezteZFi}{ZTe05}
\outdef{label@citezteZFi}{ZTe05}
\hbox{\parbox[t]{\rw}{[\cite{zteZFi}]}\parbox[t]{\colw}{Y.~Zhang, Y.W.~Tan, H.~Stormer, P.~Kim - {\it Experimental observation of the quantum Hall effect and Berry's phase in graphene}, Nature, Vol.~438, n.~10, p.~201-204, 2005.}}\par
\bigskip
\hrefanchor
\outdef{citezhaZE}{ZLe08}
\outdef{label@citezhaZE}{ZLe08}
\hbox{\parbox[t]{\rw}{[\cite{zhaZE}]}\parbox[t]{\colw}{L.~Zhang, Z.~Li, D.~Basov, M.~Fogler, Z.~Hao, M.~Martin - {\it Determination of the electronic structure of bilayer graphene from infrared spectroscopy}, Physical Review B, Vol.~78, n.~235408, 2008.}}\par
\bigskip

View File

@ -74,15 +74,15 @@
%% define a label for the latest tag
%% label defines a command containing the string stored in \tag
\AtBeginDocument{
\def\label#1{\expandafter\outdef{#1}{\safe\tag}}
\def\label#1{\expandafter\outdef{label@#1}{\safe\tag}}
\def\ref#1{%
% check whether the label is defined (hyperlink runs into errors if this check is ommitted)
\ifcsname #1@hl\endcsname%
\hyperlink{ln.\csname #1@hl\endcsname}{\safe\csname #1\endcsname}%
\ifcsname label@#1@hl\endcsname%
\hyperlink{ln.\csname label@#1@hl\endcsname}{\safe\csname label@#1\endcsname}%
\else%
\ifcsname #1\endcsname%
\csname #1\endcsname%
\ifcsname label@#1\endcsname%
\csname label@#1\endcsname%
\else%
{\bf ??}%
\fi%