Pis'ma v ZhETF, vol. 90, iss. 4, pp. 315 - 319 © 2009 August 25
Curvature effects on magnetic susceptibility of ID attractive two
component fermions
T. Veiua*, S.I.Matveenko*+, G. V. Shlyapnikov*v
* Laboratoire de Physique Théorique et Modèles Statistiques, CNRS, Université Paris Sud, 91405 Orsay, France
+ L.D. Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia v Van der Waals-Zeeman Institute, University of Amsterdam, 1018 XE Amsterdam, The Netherlands
Submitted 11 October 2008 Resubmitted 6 July 2009
We develop a bosonization approach for finding magnetic susceptibility of ID attractive two component Fermi gas at the onset of magnetization taking into account the curvature effects. It is shown that the curvature of free dispersion at the Fermi points couples the spin and charge modes and leads to a linear critical behavior and finite susceptibility for a wide range of models. Possible manifestations of spin-charge coupling in cold atomic gases are also briefly discussed.
PACS: 03.75.Ss, 71.10.Pm
Elementary excitations in ID interacting electron systems are not conventional quasiparticles carrying both spin and charge, but rather spin and charge waves that propagate with different velocities [1]. This behavior called spin-charge separation has been addressed in a number of experimental studies and demonstrated, in particular, in experiments with quantum wires in semiconductors [2]. There is a growing interest in revealing effects of spin-charge separation in experiments with cold Fermi gases [3], where the ID regime has been recently achieved [4, 5].
On the other hand, the interaction between spin and charge degrees of freedom can lead to pronounced effects. The spin-charge interaction is seen in exact solutions for integrable systems, for instance in the Fermi Hubbard model for spin-1 /2 fermions [6 - 9]. In this case the spin-charge coupling can also be treated by bosonization accounting for the curvature of the spectrum at the Fermi points [9]. In the presence of two gapless modes, this leads to the phenomenon of charge transfer by spin excitations [9].
In this letter we show that the spin-charge coupling drastically changes the critical behavior at the commensurate-incommensurate (C-IC) phase transition for spin-gapped fermions, ensuring a finite susceptibility. This transition occurs when the gap gets closed by a critical magnetic field and the magnetization emerges in the system. In the absence of spin-charge coupling, in particular at half filling, one has a universal square root dependence of magnetization on the field [10, 11] and an infinite susceptibilty. This is a consequence of the fact that solitons appearing in the spin sector above the crit-
ical field hc behave themselves as free fermions. They have a quadratic dispersion, and the soliton density is proportional to the magnetization to. The kinetic energy of solitons is thus proportional to to3 and minimization of their total energy E ~ [~(h — hc)m + const * to3] in the field h > hc gives the square root dependence to ~ Vh — hc.
Away from half filling the spin-charge interaction, entering the problem through the curvature of the spectrum at the Fermi points, leads to an effective non-local and relatively long-range interaction in the spin sector. The effect of the interaction on the ground state energy is reduced to the change of basic parameters of the spin sector. This provides the appearance of ~ to2 term in the ground state energy, ensuring a linear field dependence of magnetization and a finite susceptibility at the C-IC transition. We develop an effective field theory applicable for a wide range of models. Those include continuum models and extended Fermi Hubbard models with anisotropic interactions [12] and/ or mass (hopping) anisotropy [13]. For the integrable Fermi Hubbard model with only on-site interactions, this type of critical behavior is seen from the Bethe Ansatz solution of Ref. [7]. We give a transparent interpretation of this picture and show how the spin-charge interaction changes the behavior of correlation functions.
We first consider a dilute strong coupling limit for attractively interacting two component fermions and obtain the magnetization across the CIC transition. In the strong coupling limit, spin-f and spin-J. fermions form strongly bound pairs and at low density the interaction between the pairs and uncompensated (for exam-
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T. Véiua, S.I. Matveenko, G. V. Shlyapnikov
pie, spin-up) particles created by the magnetic field can be neglected. Thus in this limit the system represents a mixture of noninteracting hard core bosons (bound pairs) and free fermions (uncompensated spin-f particles) and the density of the thermodynamic potential is:
hdx<f> t 2 A
Ti" cos
•M
> + (dx0p) (dx(fo + 2 dx<j>.
(1)
The fields dx<f>p, dx4>\ and dx6p, dx9^ represent density and current fluctuations for the pairs and uncompensated fermions [1], h is the magnetic field, ¡j, is the chemical potential, and the multiple N = (dx<f>^ + 2dx<f>p)/y/n describes fluctuations of the total number of fermions. The term W cos V^^f provides a gap for spin excitations which in the absence of the field can be described as massive fermions. Indeed, using bosonization ruls, it is easy to see that the spin-up part of Eq.(l) can be rewritten as the Hamiltonian of noninteracting massive gapped fermions, where the term W cos ViTr^f emerges [1]. The gap gets closed by a critical magnetic field hc. At a fixed ¡j, the fields dx(f>p,dx9p and dx(f>f,dx9f are decoupled and one obtains the usual square root dependence of magnetization on the field [10, 11]: to ~ y/h — hc, for h hc + 0. At a constant number of particles we have a constraint (jV) = 0, which provides coupling between the fields of spin-f fermions and pairs and modifies the square root dependence to a linear one. At a critical field hc = 2A, where 2A is equal to the binding energy of the pairs, the low-momentum dispersion relation for spin-f fermions is Ef(k) = ^Jv2k2
A2^A
w|fc2/2A,
with being their velocity. The bound pairs disperse linearly with velocity Ep(k) = vp\k\. Taking into account the constraint (Af) = 0 we have the total energy E = £-Et(fc) + Y,EP(k) ~ m(h - hc) = = vpm2/2ir — (h — hc)m + 0(m3). Minimazing E we obtain linear dépendance m=(h—hc)/nvp for h—¥hc+0 and a finite susceptibility
x = dm/dh\hc = 1/ttvp.
(2)
We now turn to the opposite case of weak coupling and derive in the spin and charge basis an asymptotically exact theory near the critical point. Taking into account the curvature k of the spectrum at the Fermi points [9, 14], the low-energy Euclidean action in the weak coupling limit to the lowest order in k can be written as [9, 15]:
Se
= /dxdr |
J la=c,s
2 K,
■(dT4>a)2]+
9s , x h dx(f>s
— cos(V87r^s)---=
Z7T Vf v27T
¿■k K
KSKC vF
dx(f>sdT(f>sdT(f>c
¿vf
dAc [(dxd>sf + (drfaf/Kl
(3)
where r = ivpt is the Euclidean time, vf is the Fermi velocity, and the subscripts c and s stand for the charge and spin sectors. The field dx<j>c describe fluctuations of the charge (mass) density, while dx<j>s stands for fluctuations of the spin density with <j>CtS = (<f>f±<f>±)/-\/2. The action (3) is applicable for a wide range of models for spin-1/2 fermions, including continuum and extended Hubbard models. The coupling constant gB, Luttinger parameters iTCjS, and spin/charge velocities vCtS = uBtCVF depend on the Fourier transforms of the interaction potential at wavevectors k = 0 and k = 2kp [1]. For spin-gapped fermions which are SU(2) symmetric at h < hc, one has gs < 0, Ks = 1 + gB/2, and the charge sector is gapless. In the weakly interacting regime both Ks and Kc are close to unity. For simplicity, we put vc = vs = vp, which does not affect our main results.
Compared to the standard action which is quadratic in currents and spin-charge separated, Eq.(3) has extra (cubic) terms [14, 9, 15] accounting for the curvature of the free spectrum at the Fermi points. It couples the spin and charge sectors and is proportional to k = d2E(k)/2dk2\hF. Dots in Eq.(3) stand for higher order terms that we neglected and for cubic terms within the charge sector which we omitted as irrelevant modifications of the linearly dispersing charge mode. The cubic terms of Eq.(3) describe a long-range interaction between spin solitons through the charge sector in the second order of perturbation theory. We will show that the effect of this interaction can be reduced to modifications of the basic parameters of the spin sector. As a result the ground state energy shifts proportionally to the soliton density in the square.
For finding the susceptibility at a given number of particles we have to impose a constraint: {dx<f>c) = 0, which allows us to integrate out the charge modes. We calculate the ground state energy at the onset of magnetization, confining ourselves to the terms proportional to to2. For extracting these terms we write: dx<j>s = = : dx(f>s : +V2nto, with the symbol :: standing for the normal ordering with respect to the k/. corresponding to to = 0[1]. This ammounts to separation of dx<j>s into its mean part and fluctuations at h > hcr. Then, after
IlHCbMa b JKST® tom 90 Bbin.3-4 2009
u
u
integrating out charge degrees of freedom, the Euclidean action is Seff = + SK, where:
<?° = _L s 2 K
[ [(: dT<t>s :)2 + (: dx<f>s s J
9sKs
7T
■ cos (V87r : <j)s : +Ai:mx)
2nm
drda;,
(4)
and it does not give rise to an to2 contribution in the ground state energy[10, 11]. Retaining only contributions proportional to to2, the term SB originating from the spin-charge interaction is given by:
2to2K27T2
/«tM
X] [dliViGc(x, y):dXi<j)s(x) ::dyi<j)s(y) :
A — A 1
i,3=0,1
. Gc (x
-Xiyj-
(x,y) ■■dXi^a{x)::dyj^a(y)^ dxdy. (5)
Here x = {x,t} = {îb0,«i}, and y = {y,r'} = {y0,yi}, and the propagator for the charge sector is Gc (x, y) = = —Kc/4irln((a; - y)2/a2 + (r - r')2/a2 + 1), where a is a short distance cut-off.
The ground state energ
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