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arXiv:0811.0996 · Physical Review B

Momentum dependence of the spin susceptibility in two dimensions: Nonanalytic corrections in the Cooper channel

Stefano Chesi, Robert Andrzej Żak, Pascal Simon, Daniel Loss

Published Mar 30, 2009• 13 citations
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Abstract

We consider the effect of rescattering of pairs of quasiparticles in the Cooper channel resulting in the strong renormalization of second-order corrections to the spin susceptibility in a two-dimensional electron system. We use the Fourier expansion of the scattering potential in the vicinity of the Fermi surface to find that each harmonic becomes renormalized independently. Since some of those harmonics are negative, the first derivative of the spin susceptibility is bound to be negative at small momenta, in contrast to the lowest order perturbation theory result, which predicts a positive slope. We present in detail an effective method to calculate diagrammatically corrections to the spin susceptibility to infinite order.

DOI
10.1103/physrevb.79.115445

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We consider the effect of rescattering of pairs of quasiparticles in the Cooper channel resulting in the strong renormalization of second-order corrections to the spin susceptibility in a two-dimensional electron system. We use the Fourier expansion of the scattering potential in the vicinity of the Fermi surface to find that each harmonic becomes renormalized independently. Since some of those harmonics are negative, the first derivative of the spin susceptibility is bound to be negative at small momenta, in contrast to the lowest order perturbation theory result, which predicts a positive slope. We present in detail an effective method to calculate diagrammatically corrections to the spin susceptibility to infinite order.
We consider the nonanalytic terms in the spin susceptibility arising as a result of rescattering of pairs of quasiparticles. We emphasize the importance of rescattering in the Cooper channel for the analysis of the temperature dependences in the two-dimensional electron systems in the ballistic regime. In the calculation of the linear in TTT term we use angular harmonics in the Cooper channel, because for each harmonic the interaction amplitude is renormalized independently. We observe, that as a consequence of strong renormalizations in the Cooper ladder, the temperature derivative of the spin susceptibility may change its sign at low temperatures.
A quantum computer, in contrast to traditional computers based on transistors, is a device that makes direct use of quantum mechanical phenomena, such as superposition and entanglement, to perform computation. One of possible realizations is a so-called spin-qubit quantum computer which uses the intrinsic spin degree of freedom of an electron confined to a quantum dot as a qubit (a unit of quantum information that can be in a linear superposition of the basis states). \n \nElectron spins in semiconductor quantum dots, e.g., in GaAs, are inevitably coupled via hyperfine interaction to the surrounding environment of nuclear spins. This coupling results in decoherence, which is the process leading to the loss of information stored in a qubit. Spontaneous polarization of nuclear spins should suppress decoherence in single-electron spin qubits and ultimately facilitate quantum computing in these systems. \n \nThe main focus of this thesis is to study nonanalytic properties of electron spin susceptibility, which was shown to effectively describe the coupling strength between nuclear spins embedded in a two dimensional electron gas (2DEG), and give detailed insights into the issue of spontaneous polarization of nuclear spins. \n \nIn the first part we consider the effect of rescattering of pairs of quasiparticles in the Cooper channel resulting in the strong renormalization of second-order corrections to the spin susceptibility in a 2DEG. We use the Fourier expansion of the scattering potential in the vicinity of the Fermi surface to find that each harmonic becomes renormalized independently. Since some of those harmonics are negative, the first derivative of the spin susceptibility is bound to be negative at small momenta, in contrast to the lowest order perturbation theory result, which predicts a positive slope. We present in detail an effective method to calculate diagrammatically corrections to the spin susceptibility to infinite order. \n \nThe second part deals with the effect of the Rashba spin-orbit interaction (SOI) on the nonanalytic behavior of the spin susceptibility for a two-dimensional electron liquid. A long-range interaction via virtual particle-hole pairs between Fermi-liquid quasiparticles leads to the nonanalytic behavior of the spin susceptibility as a function of the temperature, magnetic field, and wavenumber . Although the SOI breaks the SU(2) symmetry, it does not eliminate nonanalyticity but rather makes it anisotropic: while the linear scaling of the out-of-plane spin susceptibility with temperature and the magnetic field saturates at the energy scale set by the SOI, that of the in-plane spin susceptibility continues through this energy scale, until renormalization of the electron-electron interaction in the Cooper channel becomes important. We show that the Renormalization Group flow in the Cooper channel has a non-trivial fixed point, and study the consequences of this fixed point for the nonanalytic behavior of the spin susceptibility. \n \nIn the third part we analyze the ordered state of nuclear spins embedded in an interacting 2DEG with Rashba SOI. Stability of the ferromagnetic nuclear-spin phase is governed by nonanalytic dependences of the electron spin susceptibility tensor on the momentum and on the SOI coupling constant. The uniform (zero-momentum) spin susceptibility is anisotropic (with the out-of-plane component being larger than the in-plane one by a term proportional to the SOI coupling to second order in electron-electron interaction). For momenta larger than the SOI coupling, corrections to the leading, linear-in-SOI-coupling term scale linearly with momentum for the in-plane component and are absent for the out-of plane component of the spin susceptibility. This anisotropy has important consequences for the ferromagnetic nuclear-spin phase: (i) the ordered state, if achieved, is of an Ising type and (ii) the spin-wave dispersion is gapped at the vanishing momentum. To second order in electron-electron interaction, the dispersion is a decreasing function of the momentum, and the anisotropy is not sufficient to stabilize long-range order. However, we show that renormalization in the Cooper channel for momenta much larger than the SOI coupling is capable of reversing the sign of the momentum-dependence of the in-plane spin susceptibility and thus stabilizing the ordered state, if the system is sufficiently close to (but not necessarily in the immediate vicinity of) the Kohn-Luttinger instability.
A long-range interaction via virtual particle-hole pairs between Fermi-liquid quasiparticles leads to a nonanalytic behavior of the spin susceptibility $\ensuremath{\chi}$ as a function of the temperature (T)(T)(T), magnetic field (B)(\mathbf{B})(B), and wave number. In this paper, we study the effect of the Rashba spin-orbit interaction (SOI) on the nonanalytic behavior of $\ensuremath{\chi}$ for a two-dimensional electron liquid. Although the SOI breaks the SU(2) symmetry, it does not eliminate nonanalyticity but rather makes it anisotropic: while the linear scaling of ${\ensuremath{\chi}}_{zz}$ with TTT and ∣B∣|\mathbf{B}|∣B∣ saturates at the energy scale set by the SOI, that of ${\ensuremath{\chi}}_{xx}$ $(={\ensuremath{\chi}}_{yy})$ continues through this energy scale until renormalization of the electron-electron interaction in the Cooper channel becomes important. We show that the renormalization group flow in the Cooper channel has a nontrivial fixed point and study the consequences of this fixed point for the nonanalytic behavior of $\ensuremath{\chi}$. An immediate consequence of SOI-induced anisotropy in the nonanalytic behavior of $\ensuremath{\chi}$ is a possible instability of a second-order ferromagnetic quantum phase transition with respect to a first-order transition to an XYXYXY ferromagnetic state.
The wave vector and temperature-dependent static spin susceptibility, ${\mathrm{\ensuremath{\chi}}}_{\mathrm{s}}$(Q,T), of clean interacting Fermi systems is considered in dimensions 1\ensuremath{\leqslant}d\ensuremath{\leqslant}3. We show that at zero temperature ${\mathrm{\ensuremath{\chi}}}_{\mathrm{s}}$ is a nonanalytic function of |Q|, with the leading nonanalyticity being |Q${\mathrm{|}}^{\mathrm{d}\mathrm{\ensuremath{-}}1}$ for 13, and Q2{\mathbf{Q}}^{2}Q2ln |Q| for d=3. For the homogeneous spin susceptibility we find a nonanalytic temperature dependence ${\mathrm{T}}^{\mathrm{d}\mathrm{\ensuremath{-}}1}$ for 13. We give qualitative mode-mode coupling arguments to that effect, and corroborate these arguments by a perturbative calculation to second order in the electron-electron interaction amplitude. The implications of this, in particular for itinerant ferromagnetism, are discussed. We also point out the relation between our findings and established perturbative results for one-dimensional systems, as well as for the temperature dependence of ${\mathrm{\ensuremath{\chi}}}_{\mathrm{s}}$(Q=0) in d=3.
We analyze how the logarithmic renormalizations in the Cooper channel affect the nonanalytic temperature dependence of the specific heat coefficient $\ensuremath{\gamma}(T)\ensuremath{-}\ensuremath{\gamma}(0)=A(T)T$ in a two-dimensional Fermi liquid. We show that A(T)A(T)A(T) is expressed exactly in terms of the fully renormalized backscattering amplitude, which includes the renormalization in the Cooper channel. In contrast to the one-dimensional case, both charge and spin components of the backscattering amplitudes are subject to this renormalization. We show that the logarithmic renormalization of the charge amplitude vanishes for a flat Fermi surface when the system becomes effectively one dimensional.
Spin susceptibility of a clean Fermi gas with repulsion in any dimension is considered using a supersymmetric low energy theory of interacting spin excitations and renormalization scheme recently proposed by Aleiner and Efetov. We generalize this method to include the coupling to the magnetic field. As a result, we obtain for the correction $\ensuremath{\delta}\ensuremath{\chi}$ to the Pauli susceptibility a nonanalytic temperature dependence of the form ${T}^{d\ensuremath{-}1}{\ensuremath{\gamma}}_{b}^{2}(T)$ in dimensions d=2d=2d=2, 3, where ${\ensuremath{\gamma}}_{b}(T)$ is an effective ddd-dependent logarithmically renormalized backscattering amplitude. In one dimension, $\ensuremath{\delta}\ensuremath{\chi}$ is proportional to ${\ensuremath{\gamma}}_{b}(T)$, and we reproduce a well known result obtained long ago by a direct calculation.