This section gathers some bibliographical references concerning four families of theoretical models, which have been proposed to account for the cascade of phase transitions induced by the magnetic field in "Bechgaard salts", (TMTSF)2X. A short summary of the structure of this molecular compounds family can be found in Appendix 1. For an exhaustive review of organic conductors, the reader can refer to the book by T. Ishiguro, K. Yamaji and G. Saito, Organic Superconductors (Springer-Verlag, Heidelberg, 1998), 2nd ed. More details can be found in line in this site, in the publications of our team devoted to this topic. Particularly, see the introduction of the 1993 habilitation dissertation (Calorimetric studies of the multicritical behaviours of the spin-density-waves phases in a molecular compound).
In (b), the "standard model" groups together many variants of the same theoretical description of the Field-Induced Spin Density Wave phases in terms of SDWs quantisation. This model roughly explains the shape of the experimental phase diagram, and accounts for the existence of a quantised Hall effect in a three-dimensional material. Many theoreticians have given their personal touch to this model, coming from all over the world (Russia, France, USA and Japan, in particular).
In (c), the "interaction" model has been proposed by Victor Yakovenko. It does not call on the particular Fermi surface topography, but rather on the distinctive features of the electronic interactions in a quasi-one-dimensional system.
In (d), a series of models attempt to account for finer aspects of the phase diagram, in particular the arborescence that we have experimentally revealed in (TMTSF)2ClO4 (see section 5). Going beyond the integer quantisation of the main phase transition series, this kind of models proposes to interpret some phenomena in terms of a fractional quantisation.
Finally, in (e), an independent approach by Andreï Lebed' enables to account for another aspect of the phase diagram that our nanocalorimetry investigations have evidenced: the existence of a tetracritical point (see section 6).
The "standard" model of the Field-Induced
Spin Density Wave phases is the first of two families of theoretical models
claiming to explain the cascade of phase transitions between quantised
magnetic states. This model lies on the assumption that the SDW results
from an instability of the quasi-one-dimensional electron gas, because
of particular topological properties of the Fermi surface in these materials
(see the previous section).
The original idea has been stated by Lev Gor'kov and Andreï Lebed' , and independently by Paul Chaikin , who made the remark that the magnetic field tends to localise the electronic trajectories along the TMTSF chains. As the field exceeds some threshold, the electron gas effectively becomes one-dimensional, so that the metallic state becomes unstable against the formation of a density wave. It could be a charge density wave (CDW), but the "standard" model assumes, in an ad hoc way, that the interaction berween electrons is rather favourable to a spin density wave (SDW).
The spin density wave instabilities result from the topography
of the open Fermi surface (see the previous
section), that makes the electronic system extremely unstable to any
perturbation of wave vector Q
coupling a large number of electronic states from one sheet of the Fermi
surface to the other one (due to the "nesting" of the Fermi
surface, Fig. 1).
The instability is related to the divergence of the electronic susceptibility of the 1D electron gas without interaction (the "standard model" assumes the "weak coupling mean field" approximation). The density wave may be considered as resulting from the condensation of the Fermi level states into electron-hole pairs.
Since the nesting is not complete, pockets of non paired
carriers remain (hatching strokes of Fig. 1). Quantisation
in magnetic field arises from the existence of closed electronic trajectories
formed in the pockets, which can thus be quantised in Landau levels. In
order to maintain always completely filled an integer number of these
levels, the wave vector Q(H)
is permanently adjusted to the magnetic field .
The consequence is the experimental observation of the Hall plateaus being
constant (independently, in Orsay and in the USA), in (TMTSF)2ClO4
[5,6] as well as in (TMTSF)2PF6
[7,8] (Fig. 2).
So each plateau is associated to an order parameter, and is characterised by an integer quantum number: the latter is the number of completely filled Landau levels. According to the "quantised nesting" model, the electron-holes pairs act as "carriers reservoir". The vector Q varies with the field, which costs condensation energy because it distorts the spin density wave. When the energetic cost is too important, the nesting vector jumps to a phase characterised by a new quantum number , which yields the phase transitions we have revealed [publis 7, 8, 9], independently of magnetisation experiments in P. Chaikin's team .
Victor Yakovenko's "in interaction" model has been developped by its author after remarking that the spin density wave induced by the field appears only if the zero-field fundamental state is superconducting [17 to 20]. This a "weak coupling" model, like the "standard model", however it does not assume any ad hoc interaction to a priori justify the stability of the spin density wave with respect to the others. On the very contrary, the model attempts to describe the interaction responsible for the condensation into electron-hole pairs.
Yakovenko describes the electron-hole pairing with the help of Landau wave functions, extended over several neighbouring organic chains (Fig. 3). The nesting of the Fermi surface does not play any role in this model.
The quantum states are electron-hole pairs, and the quantum number N of a pair is here the distance (the number of chains) that separate them.
An interesting prediction of the "interaction" model is that it accounts for the reentrance of the metallic state at very high magnetic field (around 27 teslas: Fig. 4 ), which the "standard model" does not.
In order to explain the departure from the "standard model", such as the negative Hall effect, observed in very well ordered (TMTSF)2ClO4  and in (TMTSF)2PF6 under pressure , Héritier proposed a model implying a high order nesting of the Fermi surface, yielding to a fractional quantisation of the SDW wave vector, and not simply an integer quantisation .
The "standard model" accounts for the cascade
of FISDW phases by invoking an interference effect between two periodicities
appearing along the longitudinal direction:
Because of nesting properties, the relevant period is not 2p/Q//, but rather 2p/q// with q//=Q//-2kF. These two periods open a series of band gaps in the quasi particules energy spectrum, the richness of which has not been completely described because a third period of the electronic system has not been incorporated: this is the crystal lattice period along the longitudinal direction, a.
As these three periodicities are explicitely taken into account, the gaps opened by the self-consistent SDW potential are able to stabilise new sub-phases, corresponding to a quantum number n=q//x0, which can take rational values and not anymore simply integer, provided that the interaction between electrons is strong enough [24, 25].
In order to take the periodicity along a into account, the dispersion relation of the electronic energy E=E(k) consists of periodically spaced parts, linearised at the Fermi level (Fig. 5). In the case of a half-filled band (which is the (TMTSF)2ClO4 case if one considers the weak dimerisation along the a axis), the Fermi wave vector is equal to kF=p/2a, and the 4kF vector belongs to the reciprocal lattice.
The 2p/4kF=a period has introduced new commensurate effects with the two periods 2px0 and 2p/q//, but siince these latter two are different from the first one by several orders of magnitude, these effects may be neglected: the lattice parameter a of the Bechgaard salts is about 0.7nm, whereas teh wave number x0 may take values of about300nm, in a field of 3 teslas.
One of the characteristics of this model, unlike the previous two family models, is that it does not assume anymore the weak coupling approximation. On the contrary, it takes into account an intermediate coupling, in a way which is consistent with our nanocalorimetry results (see section 4).
The self-similar structure of this diagram is not without
evoking the "Hofstadter butterfly" ,
Fig. 7, which corresponds to the self-similar structure
of the energetic spectrum of an electron moving within a crystal lattice
and at the same time in a magnetic field [27,26].
The physics looks much like that of the SDW of fractional quantisation: "something happens" as the field takes rational values, that is to say, as a q number of flux quanta, F0=h/e (where h is the Planck constant and e the electron charge), pass through p cells of the lattice. The magnetic flux can thus be written as: F/F0=p/q. The calculation thus shows that q permitted energy sub-bands are expected to open. For example, 2 bands are expected for F/F0=1/2, but 50000 bands for the extremely close value of F/F0=24999/50000!!! This is why the spectrum displays such an irregular shape, with a self-similar nature.
In the case of "Bechgaard salts", the situation is even complicated by the presence of another important period: the wave vecteur of the spin density wave (also the nesting vector of the Fermi surface, in the framework of the fractional nesting model).
Independently of Héritier's fractional nesting model, one may cite other theoretical models that attempt to overcome the limitations of the " standard model". All of them interpret the branching of the phase diagram we have observed as the result of the superposition of a large number of integer phases. The mechanism of this superposition is an "Umklapp" effect, a long range coupling of electronic states in the reciprocal space (for example 4kF) [28-30]. In the next paragraph, we will be more particularily interested in a competing model which exhibits the particularity of having predicted the existence of tetracritical points, even before our experimental discovery was published (see section 6).
Lebed's model attempts to describe the treelike phase
diagram of (TMTSF)2ClO4
(section 5) by assuming that the two order
parameters of two neighbouring FISDW phases, described by quantum numbers
n0 and n0+1,
are coupled through an "Umklapp" process (at 4kF)
. The diagram proposed by this model (Fig.
8) exhibits a series of tetracritical points (in Griffiths' sense
) and "intermediate" phases, corresponding
to the superposition of order parameters (here there is no additional
quantisation, for instance fractional).
A "pure" phase is described by the integer quantum numbers ±n, whereas an "intermediate" phase between the SDW phases n0 and n0-1 is described by four quantum numbers: n0, -n0, n0+1 and -(n0+1) .
This theoretical interpretation is attractive, due to its simplicity, however it imperfectly explains the branching of transition lines. In effect, the model does not describe the arborescence of transition lines that is observed as the temperature is decreased (section 5). Still, it is able to predict how two FISDW sub-phases could exist within an "intermediate" phase, which accounts qualitatively well for our experimental observation of a tetracritical point in the phase diagram of (TMTSF)2ClO4 in magnetic field (section 6).
Lebed' proposed in 2002 a new version of its model for the quantised FISDW phases . The reader may find in it bibliographical references later than 1993.
For more information concerning all these models, by
the way also concerning the spectacular experimental properties of "Bechgaard
salts ", two dissertations may be read on line in this web site:
Organic conductors : theoretical models