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Theory of spin-anisotropic electron-electron interactions in quasi-one-dimensional metals
T. Giamarchi, H.J. Schulz
To cite this version:
T. Giamarchi, H.J. Schulz. Theory of spin-anisotropic electron-electron interactions in quasi-one-dimensional metals. Journal de Physique, 1988, 49 (5), pp.819-835.
�10.1051/jphys:01988004905081900�. �jpa-00210759�
Theory of spin-anisotropic electron-electron interactions in quasi-one-
dimensional metals
T. Giamarchi and H. J. Schulz
Laboratoire de Physique des Solides, Université Paris-Sud, 91405 Orsay, France
(Requ le 8 décembre 1987, accepté le 3 f6vrier 1988)
Résumé. 2014 Nous construisons un modèle théorique pour des conducteurs unidimensionnels qui prend en compte les interactions dépendantes du spin dues au couplage spin-orbite ou à des interactions dipôle-dipôle
entre électrons. Nous montrons qu’il existe alors un nouveau type d’interaction qui ne conserve pas la composante z du spin, et que, dans le cas où il existe une symétrie par renversement du temps et une symétrie d’espace, quatre constantes différentes sont nécessaires pour décrire le cas le plus général. Ces interactions sont calculées dans le cas d’un couplage spin-orbite et d’un couplage dipolaire. En utilisant la bosonisation et un calcul de renormalisation, nous obtenons les fonctions de corrélation et le diagramme de phase du système à température nulle. Il existe, dans tout le diagramme de phase, un gap dans le spectre des excitations de spin, ce qui conduit à des phases onde de densité de spin totalement anisotropes. Le comportement sous champ magnétique est étudié et l’on trouve une transition de type spin-flop. Afin de décrire des systèmes quasi unidimensionnels, on introduit un couplage effectif inter-chaînes que l’on traite dans l’approximation du champ moyen. Le diagramme de phase est très différent suivant que les anisotropies sont fortes ou faibles.
Nous discutons les implications expérimentales possibles de nos résultats. En particulier le couplage spin-
orbite conduit à un axe facile parallèle à la direction des chaînes, tandis que les interactions dipolaires
conduisent à un axe facile perpendiculaire aux chaînes.
Abstract. 2014 We construct a theoretical model for one-dimensional conductors taking into account the effects of spin-dependent interactions due to spin-orbit or electronic dipole-dipole couplings. We show that a new type of interaction which does not conserve the z-component of the spin is generated, and that in the presence of time-reversal and inversion symmetry four different coupling constants are necessary to describe the most
general case. The spin-dependence of the interactions is computed for spin-orbit and dipolar coupling. The use
of bosonization and a renormalization group calculation allow us to obtain the correlation functions and the zero-temperature phase diagram of a strictly one-dimensional system. There is a gap in the spin excitations in the whole phase diagram which leads to fully anisotropic spin-density wave phases. The behaviour under a
magnetic field is studied and we find a spin-flop transition. In order to describe quasi-one-dimensional systems
we introduce an effective interchain coupling and treat it in the mean-field approximation. The phase diagram
is qualitatively different in the cases of weak and strong anisotropy. We discuss possible experimental implications of our results. In particular, spin-orbit coupling leads to an easy axis parallel to the conducting chains, whereas for dipole-dipole interactions the easy axis is perpendicular to the chains.
Classification
Physics Abstracts
72.15N - 75.30G - 75.90
1. Introduction.
The quasi-one-dimensional compounds with tet- ramethyltetraselenafulvalene (TMTSF) or tet- ramethyltetrathiafulvalene (TMTTF) chains e.g.
(TMTSF)2X and (TMTTF)2X [1], typically exhibit a magnetic phase transition at a temperature of some 10 K. At this transition antiferromagnetic ordering
occurs [2, 3, 4]. The antiferromagnetic phase shows
easy axis anisotropy with the easy axis perpendicular
to the conducting chains, and anisotropy in the plane perpendicular to the easy axis [2, 4]. Under a magnetic field parallel to the easy axis a so-called
spin-flop transition is observed. For He ’" 4.5. kG the antiferromagnetically ordered spins flop to another
axis. The intermediate axis is different for sulphur ( (TMTTF )2X) and selenium compounds ( (TMTSF )2X) [5, 6].
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004905081900
In order to explain such a behaviour one has to
consider spin-anisotropic interactions [4, 7, 8]. In
current theoretical models [9, 10] this is usually crudely achieved by using some spin-dependent
interaction constants g1 =1= g 1 J..’ without specifying
the origin of the anisotropy. In order to give some meaning to this, we here start from a simple
molecular model which takes into account the effects of spin-orbit coupling (SOC) [11] and magnetic dipole-dipole interactions (DD) among the elec- trons, and we deduce from this physical model the spin-dependent interactions between electrons.
The paper is organized as follows : in section 2 we
show that the most general anisotropic process for a one-dimensional electron gas can be described by a
few constants. In addition to the spin-anisotropic
terms previously considered one has to introduce a
spin-nonconserving process. Then we compute these
constants in a simple model which takes into account the effects of SOC and DD interactions. The one-
dimensional Hamiltonian under a magnetic field is
then studied in section 3. The use of bosonisation allows us to obtain renormalisation equations. We study the phase diagram in an arbitrary magnetic
field and find a spin-flop transition. In order to
qualitatively describe real compounds, we introduce
in section 4 a phenomenological interchain coupling
which stabilizes the dominant one-dimensional fluc- tuation. This interaction is treated in the mean-field
approximation, and we study the phase diagram of
the system in the two cases of weak and strong anisotropy. In section 5 we give some semi quantitat-
ive estimate of the anisotropy constants and the spin- flop field. Comparison with the experiments on (TMTTF )2X and (TMTSF )2X compounds is made
and the limitations of this model are pointed out. A
short account of this paper, not including the mag- netic field effects, has been published previously [7].
2. Spin-non-conserving interactions.
2.1 g-ology AND SPIN-DEPENDENT INTERACTIONS.
Let us consider a one dimensional interacting elec-
tron gas. We will use the standard g-ology descrip-
tion [9]. As the important processes are those close to the Fermi surface, the spectrum is linearized around kF and - kF. The interactions between elec- trons are parametrized by constants. If we are not in the case of an half-filled band (no umklapp process),
there are three kinds of interactions denoted by the generic names gl, 92, 94. These processes are shown in figure 1. We won’t consider in the following the
g4-processes, their only effects being a renormalis- ation of the Fermi velocity [9]. Due to the Pauli principle, the gl-processes can be reduced to
g2-processes by permuting the two outgoing electrons
lines (with a minus sign). It is thus sufficient to consider the most general case of a g2-process shown
,
Fig. 1. - Interactions between electrons. Only processes at the Fermi surface are taken into account. Solid lines are
for electrons with momentum + kF and dashed ones for electrons with momentum - kF.
in figure 2. The 16 constants 92 O’l 0"2’ 0’3, 0’4 where
(T i denotes the spin of the electrons, describe all important processes that can occur in a one-dimen- sional electron gas (at the Fermi level). The most general interaction Hamiltonian is therefore :
Fig. 2. - All the interaction processes that occur in a one-
dimensional electron gas are described by the 16 constants
g.11, U2, U3, U4, where a denotes the spin of the electron.
where for simplicity we have omitted the k-sum-
mation. The matrix A is given in the basis T T ) , JIT), ITI, 111) (IT), !> are the two
eigenstates of the a, operator) by :
We have tried to remain as close as possible to the
standard g21, 92 notations in g-ology, and have
introduced the new processes 9 c’ gf. 9 J.. is related to the usual notation gl 1 by g, = - g 1 , . All these
diagrams are given in figure 3.
Fig. 3. - The 10 independent constants sufficient to
parametrize the hermitian matrix gO’I’ 0’2, 0’3, 0’4.
We limit ourselves to time-reversal symmetric
interactions. In this case some degeneracy occurs :
we have 9211T =92111 =9211, , gcl = -gc4, gc2=
- gc3, and gl is real. In order to know the trans- formation properties of A under spin rotations we
will use a more convenient basis :
This corresponds to the singlet-triplet decomposition
of the product of two spinors, and then to the usual
x, y, z orbitals in the subspace of angular momentum
one. In this new basis the matrix A becomes :
The matrix is symmetric and the following notations
have been used :
The S subspace is invariant under spin-rotations and the X, Y, Z) transforms as a vector. We can
.decompose A into irreductible operators for the spin-rotation group :
(1) 1 S) (S] is a scalar. The constant g21. - g 1. is thus invariant under rotations ;
(2) the set IS)XI, IS)YI, S ( Z I is a vector
operator. Under rotations the constants - Re
(gcp), - Im (gcp), gcz are mixed together ;
(3) the restriction of A to the triplet space is a tensor of rank two which can be decomposed to an
invariant under rotations (2 g2 + g21 + g1. ) and a
traceless symmetric tensor of rank two (due to the
time reversal symmetry there is no antisymmetric part). The constants tb Im (gf), Re (gf), Re (gcu), Im (gcu)’ 921 + g1. are mixed together by a
rotation.
By a convenient choice of axes (a spin-rotation)
the restriction of A to the triplet space can be
diagonalized. In this system of coordinates we are
left with the independent constants :
Notice that gcp and gcz describe the interaction between the charge and spin degrees of freedom. If the system is invariant under spatial inversion these constants vanish.
Instead of the S) (S I, IS) Ri , , Ri) (Ri I
(Ri is for X, Y, Z) decomposition of A we can use a more physical basis. Let us introduce the four operators :
where ql t k,, , (Z )is a destruction operator for a left-
(right-) going fermion, (}"(1,2,3) are the three Pauli matrices Grxg (}" Y’ a, respectively and o, (0) is the identity. 00 is the operator for 2 kF charge density
wave fluctuations and Oi 2 3 parametrize the spin- density wave fluctuations [9]. Obviously 00 is a
scalar whereas O1, 2, 3 is a vector operator under spin
rotations. We will thus use the tensors
01 Oj( {i, j} E [0, 3]) as a basis to represent Hint.
We get:
Due to the time reversal symmetry constants such as gzll t - gzlU vanish, and there are no OQ Oi (i E [1, 3]) terms. The OT Oj (i, j E [1, 3]) can be decomposed in one invariant, a vector operator, and
an irreducible tensor operator. One can check that the same invariants and the same transformation
properties as in (2.4) are recovered from equation (2.7).
In the usual g-ology notations our gl is rather
represented by a gl-process between electrons of
opposite spins (g,_L-process). There is also a
gl-process between electrons of the same spins (g, 11 ) which reduces to a g2 -term with a minus sign.
If we choose the axis system where the tensor part is diagonal and if we assume that the system has an inversion center we get for the interaction Hamilto- nian :
Equation (2.8) shows that for a system which is invariant under time reversal and inversion sym- metry all the interaction effects are contained in the constants g2 - g1 , g2 J.. ’ gI J.. ’ gf in properly chosen
axis. The g constant represents the anisotropy in the (rotated) x - y plane. g2 - 91 II - g2 J.. + 91 -L para- metrize the uniaxial anisotropy along the z-direc- tion.
2.2 MICROSCOPIC MODEL. - In this section we will try to estimate the constants of the preceding section, in two cases : for a spin-independent interac-
tion between electrons but in the presence of spin-
orbit coupling with the lattice and for magnetic dipole interactions between electrons.
(TMTSF )2X and (TMTTF )2X compounds are long stacks of molecules in which the conduction band is built out of p-like electrons of the selenium
or sulfur atoms. In order to gain insight into the possible origin of magnetic anisotropy let us consider
a molecule whose levels are, without spin-orbit coupling (SOC), only Kramers degenerate. The
SOC Hamiltonian is :
where V, S, P are respectively the molecular Coulomb potential, the spin and the momentum. As Hso commutes with the time reversal operator K,
each level remains doubly degenerate. The eigenvec-
tors are :
0 is first order in the SOC, K is the time inversion
operator, IT ) , I I are the eigenfunctions of Sz, and for Hso -+ 0 one has S,, I --t ) = ± + ) . We
choose the z-axis in the stack direction and the x-axis in the longitudinal direction of the molecule. In this
simplified model we ignore any difference between the molecular, crystalline and cartesian axis.
We build the tight-binding Bloch function with these two eigenstates. If the cristal has inversion symmetry we obtain :
where a is the distance between molecules along z
and 1 is an integer. ’41,
We suppose that the electrons interact via a spin- independent real potential V (e.g. Coulomb interac-
tion or phonon mediated interaction). The different g constants can be computed using (2.1). Using the
form (2.11) for the Bloch function we see that the vertex of figure 4 corresponds to a factor
and thus even in the presence of SOC the spin is
Fig. 4. - A process which exchanges momentum and spin
is not allowed in the case of spin-independent real
electron-electron potential and time reversal symmetry.
conserved in gl-like process. This is due to time reversal symmetry and the fact that V is spin- independent.
We can compute the contribution of the spin-orbit coupling to the interaction constants and find (for simplicity we take V (r) = V 5 (r)) :
As pointed out the gcu constant can be eliminated by
a rotation. Such a rotation will also slightly redefine
the gl and g constants.
In [7] in order to have a qualitative idea of the SOC effects we use the simplified model of three
nondegenerate p-like orbitals. This allowed us to deduce (g 1 11 - g 1 .l )so = - (gf)so. In fact the point
group of the TMTTF and TMTSF molecules is
D2 h and the determination of the interaction con- stants would need molecular calculations which are not our purpose here. From (2.13) we have the
relation I (g.l )so I > (gf )so . This agrees with the
anisotropy of the Lande g factor [12]. Therefore anisotropy in the plane of the molecule is weaker than the uniaxial anisotropy along z.
We will in the following take the x, y, and z spin
axis along the longitudinal and transverse directions of the molecules and the stacking axis respectively.
The interaction constants will be taken as par- ameters. Fixing the axes this way we expect to have from the SOC interaction (g, )so 0 (for a repulsive
interaction between electrons) and (g.l )so I >>
I (gf)so 1. . In the usual g-ology (gl )so introduces a
difference between g, _L and gl jj processes. We have g1 11 -g1 = (g.l )so. The g f interaction cannot be absorbed into the usual (91, g2 ) description.
The terms g f, 911 - g, 11 are of second order in the
SOC, so to be consistent at the same order of
magnitude we have also to consider the magnetic dipole-dipole interaction among electrons. As we
are mainly concerned with antiferromagnetic instabi- lities, we expect the long-range part of the DD interaction to be canceled by the alternation of
moments. We therefore limit the DD interaction to a single chain and thus consider it as a short-range
but spin-anisotropic interaction [13]. The DD Hamil-
tonian is :
where u is the magnetic moment of the electrons. In units of the Bohr magneton we have tL = cr.
Hdd is a symmetric tensor of rank two in (Ti. The
different g constants generated by such a process can be easily computed by identification with formula
(2.8). We find :
1) g2 and g21 are different (the DD Hamiltonian does not conserve spins) :
with p k = I> Z I> k. There is a negligible contribution to g211 + g2 I j ;
2) a gl -like process :
3) a g process, with :
Note that there are no g,-terms here, and no rotation
of axes is necessary if only the DD interaction is considered.
3. Single chain.
We use the boson representation of fermion operators [14, 15], and define the p and a operators in the standard way [9]. We introduce the phase
fields :
In A, B = ... the upper sign refers to A, and
N,, ± is the number of right- (left-) going fermions
with spin ±. The complete one-dimensional Hamilto- nian is :
where HP and H, are defined by :
where
and 7r]7,, = a, 0,, is the momentum density conjugate
to cp v :
H is invariant under the transformation p v +-+
() v’ gll. H gf, Kv H 1/Kv.
In order to be able to study the spin-flop transition
we also include a magnetic field along the y-direction (i.e. the easy axis direction). The same calculation could also be done with the field along any axis by a simple redefinition of the interaction constants ac-
cording to Section 2.1. The interaction term is :
where h is the magnetic field in units of the Bohr magneton. If we use the bosonized variables the
magnetic term reduces to :
From (3.7) one can see that a study of the Hamilto- nian with the magnetic field in the y-direction would
be difficult. It is more suitable to perform a rotation
in spin space around the x-axis to lave the magnetic
field in the z-direction. The transformation of the interaction constants can be deduced from formula
(2.8). The charge Hamiltonian is invariant and the
spin one keeps the same structure with new constants
G:
In the following we will designate by lowercase
letters the constants in the original system of coordi-
nates (z is the stacking axis) and by capital letters the constants in the rotated system (Y is the chain axis).
With these new axes the magnetic term reduces
to :
Let us consider the free energy F of the system. It is given by :
where /3 = 1 / T. Here we have only to consider the spin part, the charge and spin part of the Hamilto- nian being decoupled. Using F we can compute the magnetisation m per unit length by :
It is more convenient to perform a Legendre trans-
form in order to work with a fixed (externally controlled) magnetisation. All the physical quantities
must be computed with r, the Legendre transform
of F :
where r is function of the variable m. We express r
as :
H’ can easily computed from (3.3) and (3.9). If we write If (Y, r ) = 0 (Y,
7- )7r
2 mY, and if we per-form the transformation (3.12) we get :
where II is now conjugate to qf.
In the rest of this chapter we limit ourselves to zero temperature. From a perturbation expansion of
the correlation functions we obtain the renormali- zation group equations for a change of the length
scale a --+ el « (the details of the calculation can be found in the appendix A) :
We have introduced Y, for G,17Tu,,,. Let us recall
that the equations (3.15) are derived for a magnetic
field along the Z-direction. The Ji are the Bessel
functions, and du is the renormalization of the
velocity of the spin excitations (see appendix A).
The Bessel functions come from the use of a shall’
cutoff in real space. Another cutoff would have
given non-oscillatory functions in the renormalis- ation equations [16]. The renormalization procedure
we use implies dh/dl = h, contrary to reference [16], where one has dm/dl = m and a nontrivial
renormalization of h. Both procedures are of course equivalent.
3.1 ZERO-FIELD CASE. - We will first solve equa- tions (3.15) in the absence of magnetic field (h = 0). The magnetisation and d, remain zero during the renormalisation. The equations simplify (Jo -+ 1) and reduce to the system studied in [7]. Of
course there is no need of a rotation in spin space and the whole calculation can be carried out in the
original axis system (Y, -+ y, in (3.15)). One finds
that there is a critical plane g u - I gl 1. I + I gf I =0.
This plane separates the two behaviours g u -+
± oo. A schematic representation of the renormalis- ation trajectories can be found in figure 5. For
gu -+ - oo the quantum fluctuation term in (3.3) is suppressed and ig, , I increases under renormali- zation. Therefore one has long-range order of the
0,-field, and correlations of (J u decay exponentially
at large distance. The behaviour for g u -+ + 00 is obtained from the transformation 0, *-,, 0,. The
average values of the ordered fields are given by
Away from the critical plane there now is always
one variable scaling to strong coupling, and contrary
to the standard case (gf = 0) there is a gap
aQ in the spin-excitation spectrum in the whole
phase diagram. An explicit solution of the equations
is given in appendix B. Near the critical plane one
has :
Fig. 5. - Different renormalisation trajectories for the system in the absence of magnetic field. The critical plane separates the two behaviours g,,, --+ - oo. The coordinates of points A, B, C, D, E are (- 1, 1, 0), (1, 1, 0), (1, 0, 1), (- 1, 0, 1), and (0, 1, 1), respectively, in arbitrary units of (g u’ g 1 .1 , gf). The substitution g - G in figure 5 gives the
renormalisation trajectories in the rotated axes (X, Y, Z).
The behaviour on the critical plane is easily
studied in the rotated system. It correspond the plane Gf = 0 or G1.L = 0 depending on the signs of
9f, gl.L. The equations reduce to those of a Koster-
litz-Thouless (KT) [17] transition, and lead to the
fixed point Gl 1 = 0, Gl 11 -+ G *. This corresponds
to a fixed line gf = g* and g* = 0. This is also
easily seen directly on the renormalisation equations.
On the critical plane we have g f = g* 11- = -.I/gf 91 -L
To study the influence of the gap on the physical properties of our system we have to consider the one-dimensional fluctuations (2 kF charge density
wave (CDW), spin density wave (SDWI i = x, y, z )
or singlet (SS) and triplet (TSi ) Cooper pairing)
described by the correlation functions [1, 9] :
where T is the usual imaginary time argument, and
the Oi operators are given in the boson represen-
