We present a detailed analysis of the nonanalytic structure of the free energy for the itinerant ferromagnet near the quantum critical point in two and three dimensions. We analyze a model of electrons with an isotropic dispersion interacting through a contact repulsion. A fermionic version of the quantum order-by-disorder mechanism allows us to calculate the free energy as a functional of the dispersion in the presence of homogeneous and spiraling magnetic order. We resum the leading divergent contributions to derive an algebraic expression for the nonanalytic contribution to free energy from quantum fluctuations. Using a recursion which relates subleading divergences to the leading term, we calculate the full T = 0 contribution in d = 3. We propose an interpolating functional form, which allows us to track phase transition lines at temperatures far below the tricritical point and down to T = 0. In d = 2, quantum fluctuations are stronger, and nonanalyticities are more severe. Using a similar resummation approach, we find that despite the different nonanalytic structures, the phase diagrams in two and three dimensions are remarkably similar, exhibiting an incommensurate spiral phase near the avoided quantum critical point.

We explore the geometric phase in N = (2, 2) supersymmetric quantum mechanics. The Witten index ensures the existence of degenerate ground states, resulting in a non-Abelian Berry connection. We exhibit a non-renormalization theorem which prohibits the connection from receiving perturbative corrections. However, we show that it does receive corrections from BPS instantons. We compute the oneinstanton contribution to the Berry connection for the massive CP 1 sigma-model as the potential is varied. This system has two ground states and the associated Berry connection is the smooth SU(2) 't Hooft-Polyakov monopole.1 As pointed out in [4], the first appearance of the Dirac monopole was actually in the context of the Born-Oppenheimer approximation for diatomic systems, in what is now recognised as a Berry connection. This was some two years before Dirac's work and more than fifty years prior to Berry. The monopole tourist can view this connection as the cos θ term in equation (15) of [5].2 The parameters m would not respect Lorentz invariance in four-dimensional theories, which is the reason they are perhaps less familiar than holomorphic parameters which appear in the superpotential. The triplet of masses in quantum mechanics is cousin to the real mass in three dimensions [6] and the complex twisted mass in two dimensions [7].

We investigate the interplay of quantum fluctuations and magnetic anisotropies in metallic ferromagnets. Our central result is that fluctuations close to a quantum critical point can drive the moments to point along a magnetic hard axis. As a proof of concept, we show this behavior explicitly for a generic two-band model with local Coulomb and Hund's interactions, and a spin-orbit-induced easy plane anisotropy. The phase diagram is calculated within the fermionic quantum order-bydisorder approach, which is based on a self-consistent free energy expansion around a magnetically ordered state with unspecified orientation. Quantum fluctuations render the transition of the easyplane ferromagnet first-order below a tricritical point. At even lower temperatures, directionally dependent transverse fluctuations dominate the magnetic anisotropy and the moments flip to lie along the magnetic hard axis. We discuss our findings in the context of recent experiments that show this unusual ordering along the magnetic hard direction.PACS numbers: 74.40. Kb, 75.50.Cc, 75.30.Gw, 75.70.Tj Fluctuations near to quantum critical points in itinerant ferromagnets (FMs) can have drastic and often surprising effects. They generically render a priori continuous phase transitions first order at low temperatures [1][2][3][4][5]. In many systems, quantum phase transitions are preempted by the formation of superconducting [5][6][7], modulated magnetic [8], or unusual spin-glass phases [9,10]. Other metallic FMs show an unexpected ordering along the magnetic hard axis [11,12].It is well understood that the first-order behavior arrises from the coupling of the magnetic order parameter to soft electronic particle-hole fluctuations, giving rise to non-analytic terms in the free energy [13][14][15][16]. Because of this interplay between low-energy quantum fluctuations, metallic FMs are very susceptible towards the formation of incommensurate magnetic [17], spin nematic [14] or modulated superconducting states [18]. This spatial modulation is associated with deformations of the Fermi surface that enhance the phase space for low energy particle-hole fluctuations. The phase reconstruction can therefore be viewed as a fermionic quantum orderby-disorder effect, which can be studied systematically by self-consistently calculating fluctuations around a whole class of possible broken-symmetry states [19][20][21].The coupling to electronic quantum fluctuations can also have counter-intuitive effects upon the direction of the magnetic order parameter. A notable example is the partially ordered phase of the helimagnet MnSi, in which the spiral ordering vector rotates away from the lattice favored directions [22,23]. Similar effects are possible in homogenous itinerant FMs. This is suggested by recent experiments that show unusual ordering of magnetic moments along hard magnetic directions [11,12].The first example is YbRh 2 Si 2 , which is a prototypical system for studying antiferromagnetic quantum criticality, but exhibits strong FM fluctuations [24]. In...