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The Quantum Hall Effect of Field Induced Spin Density Wave Phases: the Physics of the Ultra Quantum Crystal
Pascal Lederer
To cite this version:
Pascal Lederer. The Quantum Hall Effect of Field Induced Spin Density Wave Phases: the Physics of the Ultra Quantum Crystal. Journal de Physique I, EDP Sciences, 1996, 6 (12), pp.1899-1916.
�10.1051/jp1:1996196�. �jpa-00247289�
The Quantum Hall Eillect of Field Induced Spin Density Wave Phases: the Physics of trie Ultra Quantum Crystal
Pascal Lederer (*)
Laboratoire de Physique des Solides (**), Bâtiment 41$ Université Paris-Sud, 91405 Orsay Cedex, France
(Receiied 30 May 1996, received in final form 20 June 1996, accepted 1 July 1996)
PACS.72.15.Nj Collective modes le-g-, in one-dimensional conductors) PACS.73.40.Hm Quantum Hall eifect (integer and fractiortal)
PACS.75.30.Fv Spin-density waves
Abstract. The Quantum Hall Eifect of Field Induced Spin Density Waves is accounted for
within a weak coupling theory which assumes that in trie relevant Iow temperature part of the
phase diagram the quasi one-dimensional conductor is weII described by Fermi Iiquid theory.
Recent experimental results show that sign inversion of the Hall plateaux takes place aII the way down from the mstability Iine of the normal state. The Quantum Nesting Model, when ii takes into account small perturbations away from perfect nesting, describes weII, non only
the usual sequence of Hall Plateaux, but also the anomalies connected with sign inversion of the Hall Elfect. Experimental observation of de-doubling of subphase to subphase transition Iines suggests that superposition of SDW order parameters occurs m some parts of the phase diagram. The collective elewentary excitations of the Ultra Quantum Crystal have a specific
magneto-roton structure. The SDW
case exhibits, apart from the usual spm waves, topological
excitations which are either skyrmions or half skyrmions. It is suggested that magneto-rotons may have been observed some years ago in specific heat experiments.
1. Introduction
À step-like Hall voltage behaviour under field was found in the strongly anisotropic quasi one- dimensional compound (TMTSF)2Cl04 Iii very shortly after the first experimental hints of a cascade of phase transitions in quasi-1D conductors under magnetic field were published [2].
It was discussed in the experimental paper in terms of the Quantum Hall Effect [3].
In this review paper, I shall discuss trie present theoretical understanding of this phenomenon,
and of some related aspects -of the physics of Field Induced Spin Density Wave phases. The material indudes known results dating back to 1984, more recent work, some of which still
unpublished at the time I am writing, and new results, not published elsewhere. À review
dealing with trie theory of Field Induced Density Waves up to 1991 cari be found in reference [4j.
Within a weak coupling approach, Gor'kov and Lebed pointed out trie crucial appearance, under an applied magnetic field H~ of a logarithmic divergence in the (one loop) spin staggered
static susceptibility xo(2kF,T, HI because of the open quasi nested Fermi surface [5]. In quasi dassical terms, the electron orbits become one-dimensional under magnetic field and
(* e-mail: pascaltlsolrt.lps.u-psud.fr (**) associé au CNRS
@ Les Éditions de Physique 1996
1900 JOURNAL DE PHYSIQUE I N°12
this restores trie 1-D logarithmic divergence of trie (bare) spin susceptibility. Gor'kov and Lebed discarded an interpretation of trie Hall plateaux in terms of trie Quantum Hall Effect [3], because, at the time, there was no sign of any significant decrease of trie longitudinal resistivity coinciding with trie Hall plateaux.
They pointed out trie thermodynamic nature of trie phenomenon, which they described as
a cascade of phase transitions with periodic re-entrance of the normal phase between two identical Spin Density Wave phases.
However, shortly after trie work by Gor'kov and Lebed [5j, Héritier, Montambaux and Led-
erer [6] suggested that in fact trie step-like Hall voltage was indeed a new form of Quantized
Hall Effect, intimately connected with trie cascade mechanism. Their argument was based on the discovery that trie niost divergent loop, in the presence of trie magnetic field, is obtained for a q~ai~tized, jield depei~dei~t longitudinal wave vector
q = (k~ = 2kF + i~(2~/xo), k~ m ~/b, kz = ~/c). (il
In this expression, trie length xo is trie magnetic length. This length appears naturally if one considers trie area bxo threaded by one flux quantum ç§o between two neighbouring airains at a
distance b under a field H: bxoH = çio " h/(e( (xo
= hi (ebH) is of order 100 nm if H m 10 tesla).
In the following, I shall use trie notation G
= 2~/xo for trie wave vector associated with xo.
There is also an energy scale associated with this length, ltuJc = ltuFG/2
= eufbH/2. Trie
logarithmic growth of trie staggered susceptibility occurs when I~BT < uJc.
Àccording to Héritier, Montambaux and Lederer (6], trie index i~ appearing in trie wave
vector x component of trie Spin Density Wave (SDW) instability labels each SDW subphase
and decreases by one unit from subphase to subphase as H increases. Àt trie same time, this index i~ is the number of exactly filled Landau levels (Landau bands in fact, as will be discussed later on) of unpaired quasiparticles left by the SDW condensation in a situation of imperfect nesting. When H changes, a competition develops between trie condensation energy and trie
diamagnetic energy: trie former is lowered if electrons and holes condense and increase the order parameterj trie latter is lowered if Landau levels are exactly filled and trie Fermi level sits between two Landau le~~els; accordingly, trie SDW wave vector changes, at fixed n, so that trie
pockets of unpaired partides bave exactly trie right area for an mteger number of filled Landau levels below trie Fermi level. Trie SDW wavevector varies smoothly with, say, increasing field until it becomes energetically favourable to jump to trie next quantum number in il. This picture was later on confirmed by trie analytic theory of trie condensed phase, with an order parameter descnbed by a single Fourier component of trie staggered magnetization iii. Trie
general structure of trie phase diagram was also studied independently by Lebed [8j, with similar results. À way of fornmlating this picture is to describe trie quantization condition as
a nesting quantization: trie area between one sheet of trie normal state Fermi surface and trie other sheet translated by q is quantized in terms of trie area quantum eH/h, leading to trie condition 1. Hence trie name "Quantized Nesting Model" (QNM) dubbed by trie authors of reference [6j. Trie Quantized Hall Effect of FISDW phases is thus a special example of a general
result due to Halperm, following whom trie integer quantum Hall effect should be observed in
a bulk system in a magnetic field if trie chemical potential lies m an energy gap [9j.
There is now overwhelmmg evidence for the thermodynamic nature of the cascade of Field Induced SDW (FISDW) phases [loi, and for trie occurrence of a novel type of QHE in those phases Ill]. Bath aspects are intimately connected. Very well defined Hall plateaux with the ratios 1:2:3:4:5 are observed in (TMTSF)2PF6, for example, where trie ratio of trie resistivity
tensor components p~~/p~~ can be as large as 75 within a plateau, and strongly decreases m
trie narrow region between trie plateaux iii]. Figure is a typical QHE curve obtained in
(TàITSF)2PFô.
b (TMTSF)~PF~
00 ~ ? ~ ~
C~ ?5
~
~
' i 2 3 4
fl É 5Ù
p = 9kbar
~ ù 25 ~~ ~~~ ~'~
o oo
0 5 10 15 20
Magnetic field (Lesla)
Fig. 1. Experimental QHE of (TMTSF)2PF6 under
a pressure of 9 kbar. Trie measured resistance
is multiplied by the number of conducting Iayers to obtain p((~ and then renormalized to the value
of the resistance quantum. The msert exhibits trie experimental set-up for the 8 electrical contacts.
(courtesy of Lms Bahcas, thèse, 1995).
In this paper, I discuss trie problems that bave arisen in trie theoretical picture because of progress in experiments over trie last few years; two main phenomena bave led to conflicting
views: trie "Ribault anomaly" and trie fine structure of trie phase diagram. Trie former is trie observation that under certain conditions, for example for certain values of trie applied
pressure, and (in trie case of trie Cl04 compound), for a very slow cooling rate, trie FISDW exhibits a change of sign of trie Hall plateau [12,13j. Trie latter is trie interpretation of specific
heat anomalies within trie domain of existence of trie FISDW in terms of "arborescence" of trie phase diagram [14-16j.
It turns Dut that recent experiments bave helped in clanfying Dur understanding bath of trie
"Ribault anomaly" [13j and of trie nature of trie phase diagram [15,16j. In particular, following
trie important expenments by Balicas, Kriza and Williams [13j, Zanchi and Montambaux bave shown that trie sign reversal of trie QHE can be described within trie Quantized Nesting Model
(trie "Standard Model") ai trie cost of miner conceptual changes (17j. This contrasts with
repeated statements in the literature denying the possibility of accounting for tbis phenomenon
within the "standard model" [18], or with repulsive electron-electron interactions alone [19j.
FISDW are bath a particular example of electron-hale condensate describable as a quantum crystal, and a navet manifestation of quantum orbital resonances. As such, their collective excitations are expected to exhibit specific features. Besides the usual Goldstone basons, pha-
sons and spm waves, which appear as a result of the various broken continuous symmetries in
the FISDW (translation symmetry and spin rotational invariance) I shall discuss trie occur-
rence of the magneto-roton [3j, the existence of which in FISDW was discussed and proved by
Lederer and Poilblanc [20j, and trie skyrmion (and half-skyrmion), discussed in this context by Yakovenko il9]
Trie point of view adopted in this paper is that the Quantum Hall Effect observed in the FISDW phases is reasonably well described within a weak couphng approach I assume that trie normal state iii trie absence of magnetic field is an anisotropic Fermi liquid with quasi 1D
Fermi sheets; trie electronic hopping m all three directions at sufliciently low temperature is coherent. Since aÎÎ the interesting physics occurs at temperatures smaller than tc/kB (where
tc is the smallest interchain hopping, along the field direction) this is a reasonable assumption.
Furthermore, since I consider situations where trie magnetic field is orthogonal to the most con-
ducting plane, hopping along that direction will remam coherent in all cases. Other interesting
and complicated situations may arise in a different field geometry~ if electronic interactions are
sufliciently strong [21].
1902 JOURNAL DE PHYSIQUE I N°12
The specific properties of trie FISDW phases (broken translational symmetry under magnetic field, Quantum Hall Effect. magneto-roton mode and low energy long wavelength Goldstone
modes) are those of a new Mass of electronic Density Wave Phases: trie Ultra Quantum Crystal.
2. The Quantum Hall E~ect of Field Induced SDW Phases
1first recall trie results obtained when trie order parameter is described by a single Fourier component of trie magnetization. For simplicity, I will restrict trie discussion to trie case of a
transverse magnetization, such that trie Zeeman term plays no rote: trie magnetic field is in trie z direction, perpendicular to trie most conducting plane, trie magnetization which appears
as a result of trie orbital effect and of electron-noie pairing is lying in trie ix, y) plane.
2.1. THE QUANTIzED NESTING MODEL. Consider a simple model of an orthorombic, anisotropic quasi two-dimensional conductor. Trie open Fermi surface is described by trie
following dispersion relation, linearized around trie Fermi level in trie longitudinal direction:
e(k) = vF(lk~l kF) + eilk), 12)
fi (k
= -2tb cos kyb 2t[ cos 2kyb 2t~ cos kgc
In trie transverse b direction. a second harmonic is introduced to take into account trie violation of perfect nesting, defined by trie condition e(k)
= -e(k + Q). If t[ vanishes, this equation froids with Q
" (kF,7r/b,7r/c). Trie existence of a non zero t[ anses from hnearization of
trie dispersion relation along trie x direction [22j and for from next nearest neighbour coupling
between chains [23]. Perfect nesting in trie z direction makes trie problem effectively two- dimeusional. Trie magnetic field, parallel to trie c direction is described by trie vector potential
A
= (0, Hz, o). This choice of gauge is crucial to simplify the problem and let the effective
one dimensionality nature of the problem appear in the simplest fashion. The electron transfer
integrals along the three crystal axes hâve the following orders of magnitude:
ta m iifkf Cf 300 mev » tb Cf 30 mev tb > t[ ci t~ cil mev
The equatious là la Gor'kov) which describe the ordered phase are as follows:
~~~" ~ ~~~ Î~~~ ~ ~~ ~~~ ~~~' ~~~
(iwn ivf ~ NVF/xo )f + A*g
= o,
dX
where g and f are diagonal and off-diagonal parts of the Green fuuctiou, the phases of which hâve beeu properly defiued [7, 24].
Â(z)
= AZnlnexp(-mz/zo+1#n), (4)
#n = np z sm p z'sur 2p,
In = ZpJn-2p(z)Jp(='), (5)
z = (4tbxo/vF) cos Qtb/2,
z'
= (2t[xo/vF) cos Qtb,
aud Jp(z) is the pth-order Bessel fuuctiou of argument z. A, the order parameter, aud the wave vector = (Qjj, Qt, 7rIci are determmed self-cousisteutly so as to mmimize the free euergy. we
kuow that Qjj = 2kF + NC. Aix) acts as au effective poteutial which couples electrouic states
uot ouly at k aud k + Q because of SDW orderiug but also k aud k + Q NG(îk~/(k~().
Therefore, the quasi partide spectrum exhibits a series of gaps I?1 /h~ = /hIN opeued at k = ~(1/2)(Qjj NG). The free euergy is minimum wheu the Fermi level lies in the middle of
the largest of these gaps /hN
= /hIN. This occurs wheu QIj(H)
= 2kF + NC. The gaps result from deusity wave orderiug aud orbital quautizatiou: at the level of the oue partide Green fuuctiou. the orbital periodicity acts as a brokeu trauslatioual symmetry, which vauishes at the level of two partide Green fuuctiou, aud deusity correlatiou fuuctiou; this feature, which is a
specific expression of gauge iuvariauce in this problem gives rise to the magueto-rotou minimum,
as will be discussed later. We thus have separate Landau bauds coutaiuiug 1/27rbxo = eH/h
states per unit surface. Trie distance in euergy betweeu trie ceuters of two ueighbouriug Landau bauds is ltw~. Ai zero temperature, each quautized SDW phase bas either completely filled or
completely empty Landau bauds. No FISDW phase cau exist at temperatures T > hw~
If trie Spiu Deusity wave is piuued hi some mechauism or other (say impurities), ouly siugle partide excitations coutribute to trie couductivity. Siuce perfect uestiug aloug trie z direction makes trie problem effectively two-dimeusioual, Laughliu's gauge iuvariauce arguments [3] tell
us that trie surgie atomic layer Hall couductivity is exactly quautized at zero temperature in uuits of e~là, 1.e. we must bave
a~y =
ne~là (6)
The value of n is precisely trie value N which labels trie FISDW subphase where QI -2kF
= NC.
Trie proof was giveu by Poilblauc et ai. [25j usmg au approach due to Stfeda [26j. Followiug
trie latter, at fixed chemical poteutial
~1 aud in a field iudepeudeut poteutial,
a~y = eôp(p,B,/h,Q)/ôB(~,Q (7)
P(li,B,/h.Q) H
"
/
d~Trô(~-H(B,A,Q))where H is trie Hamiltouiau. Takiug iuto accouut trie B depeudeuce of Qii, aud uoticiug that p does uot depeud ou /h aud QL wheu Q
= QN, i-e- wheu trie Fermi level lies m a gap, Stfeda's formula in trie preseut case reads:
~ ~
_~ôojj ôP j~j
~~ ~~ ~Q~
~=~lB,Tj,Q=QN Siuce Q((B,p) = 2kF(p) + NC
= 2kF(p) + N(e(Bb/h, we get ôQ(/ôB
= N(e(b/lt. siuce
ôp/ôQ~
= 1/27rb, we bave eveutually:
a~y =
Ne~ là (9)
This result was rederived by Yakoveuko [19] usiug Kubo formula (which is aise at trie basis of Stfeda's work) [27]. He used trie topological properties of trie wave functious in reciprocal space
which result from trie variation of their phase factor upou trausportiug them aloug closed cou- tours. He argued that in fact trie quantum number N, which results from bis analysis if spiuless
electron-noie pairing ii.e. CDW pairiug) is takeu iuto account should be replaced by 2N in trie
case of SDW pairiug.Trie ratio of QHE plateaux couductivities is trie accurately determmed quautity in experimeuts, siuce trie uumber of layers of a giveu sample is ill determiued.
Thus, trie theory based ou trie notion that trie FISDW order parameter is well described by a siugle harmouic accouuts in a satisfactory way for Hall experimeuts couducted in trie
Bechgaard sait PF6 compouud [III. This is stroug evideuce m favour of trie uew Quantum Hall Effect mechanism described in [6].
1904 JOURNAL DE PHYSIQLiE I N°12
o ~o i/z
° o.z5 ~
Î
o où fl #
~ p=85kbar
~
~ T= 380 mi
_,~~ ~
Po= h/z e? 0 08
~
x ~
x
~
o 04 Q
o
o où
o 5 io 15 zo
Magnetic field (testa)
Fig. 2. The Ribault anomaly observed by Balicas et ai. under a pressure of 8.5 kbar (frein Ref. [13]
The sJgn change is observed ail the way down from the normal phase, aloug a critical fine with about 0.2 T length aloug the magnetic field axJs.
2.2. THE SIGN REVERSALS oF THE QUANTUM HALL EFFECT. Que physical eifect, how-
ever. is conspicuously out of the picture described in the last section: the reversai of the sign
of the Hall effect, which was first discovered by Ribault: under certain conditions of thermal preparation, a few "negative" (by convention) plateaux may appear in (TMTSF)2Cl04 when the field varies. Such sigu reversais have been found to occur under certain circumstances
(pressure, cooling rates, and so on ), m limited field range m trie Cl04, PF6 [28,29] and in trie Re04 compound [30].
A sigmficant progress was achieved recently when Balicas Kriza and Williams [13j reported
a negative Hall plateau (with quantum number -2), inserted between two positive ores (with quantum numbers 3 and 2) which they could follow ail trie way from low temperatures to trie critical fine separating trie FISDW phase from trie normal state. Figure 2 depicts trie experimental results. Trie crucial observation is that trie negative Hall plateau may anse con- tinuously, ma a second order phase transition, from trie normal state, m a limite mterval of
magnetic field. Indeed, this observation means that trie FISDW phase with negative quantum number arises from divergent fluctuations of trie normal phase, at a wave vector Qx
= 2kF G.
It means that no mechanism based on free energy expansions to high order in trie order param-
eter is able to accourt for this phenomenon: trie harmomc term atone already contains trie sign change. which must be a property of trie bare spm susceptibility! This observation was a puz- zle, because analytic and numerical work on trie bare static spin susceptibility xo(q,H) within trie Quantized Nesting Model described in trie previous section showed that trie logarithmic divergences at wave vectors with "negative" quantum uumbers bave smaller amplitude than trie "positive" ores. In other words, within trie standard model, trie normal phase instability
to positive Hall plateaux always overcomes trie transition to negative ores [31,32j.
A very simple due to this puzzle was given very recently by Zanchi and Montambaux [lîj.
They pointed oui that trie competition between logarithmic divergences m io(q, B) at positive and negative quantum numbers depends on small perturbation terms which anse from higher
order harmomcs of trie expansion m Fourier serres of trie dispersion relation m equatioii (2).
They investigated trie following dispersion relation (setting kyb = p)
fi = 2tb cos p + 2t( cos 2p + 2t3 cos 3p + 2t4 cos 4p (la
