The recently measured spin susceptibility of the two-dimensional electron gas exhibits a strong dependence on temperature, which is incompatible with the standard Fermi liquid phenomenology. In this article, we show that the observed temperature behavior is inherent to ballistic two-dimensional electrons. Besides the single-particle and collective excitations, the thermodynamics of Fermi liquid systems includes effects of the branch-cut singularities originating from the edges of the continuum of pairs of quasiparticles. As a result of the rescattering induced by interactions, the branch-cut singularities generate nonanalyticities in the thermodynamic potential that reveal themselves in anomalous temperature dependences. Calculation of the spin susceptibility in such a situation requires a nonperturbative treatment of the interactions. As in high-energy physics, a mixture of the collective excitations and pairs of quasiparticles can effectively be described by a pole in the complex momentum plane. This analysis provides a natural explanation for the observed temperature dependence of the spin susceptibility, both in sign and in magnitude.
The recently measured spin susceptibility of the two-dimensional electron gas exhibits a strong dependence on temperature, which is incompatible with the standard Fermi liquid phenomenology. In this article, we show that the observed temperature behavior is inherent to ballistic two-dimensional electrons. Besides the single-particle and collective excitations, the thermodynamics of Fermi liquid systems includes effects of the branch-cut singularities originating from the edges of the continuum of pairs of quasiparticles. As a result of the rescattering induced by interactions, the branch-cut singularities generate nonanalyticities in the thermodynamic potential that reveal themselves in anomalous temperature dependences. Calculation of the spin susceptibility in such a situation requires a nonperturbative treatment of the interactions. As in high-energy physics, a mixture of the collective excitations and pairs of quasiparticles can effectively be described by a pole in the complex momentum plane. This analysis provides a natural explanation for the observed temperature dependence of the spin susceptibility, both in sign and in magnitude.
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 T 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.
If the dynamical quasiparticle spectrum of a Fermi liquid (FL), ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{p}}^{\mathrm{dy}}$, given by the poles of the single-particle propagator, is used in the expression for the entropy of a noninteracting Fermi gas, one finds that the result is different from that calculated directly from the thermodynamic potential. In particular, the coefficient of the T3lnT terms, the leading correction to the linear T dependence for a three-dimensional (3D) FL, is overestimated when calculated from ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{p}}^{\mathrm{dy}}$. The thermodynamic properties may be calculated correctly from a second quasiparticle spectrum, the statistical quasiparticle spectrum, ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{p}}^{\mathrm{st}}$. In this paper I calculate the corrections to leading linear energy and temperature dependences for a 2D FL. I compare the difference between the two spectra in the 2D and 3D cases. Although these differences are qualitatively the same, they are quantitatively larger in the 2D case.
We show that the singularities in the dynamical bosonic response functions of a two-dimensional Fermi liquid give rise to universal nonanalytic corrections to the results of the Fermi-liquid theory. In particular, we find a T2 term in the specific heat, linear-in-T terms in the effective mass and in the uniform spin susceptibility ${\ensuremath{\chi}}_{s}(Q=0,T),$ and ∣Q∣ term in ${\ensuremath{\chi}}_{s}(Q,T=0).$ The existence of these terms has been the subject of recent controversy, which is resolved in this paper. We present exact expressions for all nonanalytic terms to second order in a generic interaction U(Q) and show that the nonanalytic terms originate exclusively from forward- and backward-scattering of particles with zero total momentum.
We show that the singularities in the dynamical bosonic response functions of a two-dimensional Fermi liquid give rise to universal nonanalytic corrections to the results of the Fermi-liquid theory. In particular, we find a T2 term in the specific heat, linear-in-T terms in the effective mass and in the uniform spin susceptibility ${\ensuremath{\chi}}_{s}(Q=0,T),$ and ∣Q∣ term in ${\ensuremath{\chi}}_{s}(Q,T=0).$ The existence of these terms has been the subject of recent controversy, which is resolved in this paper. We present exact expressions for all nonanalytic terms to second order in a generic interaction U(Q) and show that the nonanalytic terms originate exclusively from forward- and backward-scattering of particles with zero total momentum.
We revisit the issue of the leading nonanalytic corrections to the temperature dependence of the specific heat coefficient, $\ensuremath{\gamma}(T)=C(T)∕T$, for a system of interacting fermions in three dimensions. We show that the leading temperature dependence of the specific heat coefficient $\ensuremath{\gamma}(T)\ensuremath{-}\ensuremath{\gamma}(0)\ensuremath{\propto}{T}^{3}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}T$ comes from two physically distinct processes. The first process involves a thermal excitation of a single particle-hole pair, whose components interact via a nonanalytic dynamic vertex. The second process involves an excitation of three particle-hole pairs which interact via the analytic static fixed-point vertex. We show that the single-pair contribution is expressed via the backscattering amplitude of quasiparticles at the Fermi surface. The three-pair contribution does not have a simple expression in terms of scattering in particular directions. We clarify the relation between these results and previous literature on both 3D and 2D systems, and discuss the relation between the nonanalyticities in $\ensuremath{\gamma}$ and those in spin susceptibilities.