Core Structure and Non-Abelian Reconnection of Defects in a Biaxial Nematic Spin-2 Bose-Einstein Condensate

Abstract

We calculate the energetic structure of defect cores and propose controlled methods to imprint a nontrivially entangled vortex pair that undergoes non-Abelian vortex reconnection in a biaxial nematic spin-2 condensate. For a singular vortex, we find three superfluid cores in addition to depletion of the condensate density. These exhibit order parameter symmetries that are different from the discrete symmetry of the biaxial nematic phase, forming an interface between the defect and the bulk superfluid. We provide a detailed analysis of phase mixing in the resulting vortex cores and find an instability dependent upon the orientation of the order parameter. We further show that the spin-2 condensate is a promising system for observing spontaneous deformation of a point defect into an "Alice ring" that has so far avoided experimental detection.Topological defects and textures are ubiquitous across physical systems that seemingly have little in common [1], from liquid crystals [2] and superfluids [3] to cosmic strings [4]. They arise generically from symmetries of a ground state that is described by an order parameter [5]-a function parametrizing the set of physically distinguishable, energetically degenerate states. In the simple example of a scalar superfluid, the order parameter is the phase of the macroscopic wave function. More generally it may be a vector or tensor that is symmetric under particular transformations. In a uniaxial nematic (UN), the order parameter is cylindrically symmetric around a locally defined axis. It also exhibits a twofold discrete symmetry under reversal of the cylinder axis, which leads to half-quantum vortices (HQVs) in atomic spinor BoseEinstein condensates (BECs) [6][7][8][9] and superfluid liquid 3 He [10], and to π disclinations in liquid crystals [1,2]. In a biaxial nematic (BN), also the cylindrical symmetry is broken into the fully discrete symmetry of a rectangular brick, with dramatic consequences: the BN is the simplest order parameter that supports non-Abelian vortices that do not commute [5,11]. As a result, colliding non-Abelian vortices cannot reconnect without leaving traces of the process, but must instead form a connecting rung vortex. Noncommuting defects appear as cosmic strings in theories of the early universe [4], and have been predicted in BN liquid crystals [11].Despite long-standing experimental efforts, BN phases in liquid crystals have experimentally proved more elusive than originally anticipated [12]. In atomic systems, the topological classification and dynamics of non-Abelian vortices have theoretically been studied in the cyclic phase of spin-2 [13-16] and in spin-3 BECs [17], though it remains uncertain whether any alkali-metal atoms exhibit the corresponding ground states. Consequently, any physical system where non-Abelian defects may be reliably studied is still lacking.Spin-2 BECs exhibit-in addition to the ferromagnetic (FM) and cyclic phases-both UN and BN phases [18][19][20], which, however, are degenerate at the mean-field level. Beyond mean-fi...

Introduction to topological quantum computation with non-Abelian anyons

“…Many such high-spin Bose-Einstein condensates, including rubidium (Stamper-Kurn and Ueda, 2013), chromium (Griesmaier et al, 2005), erbium (Aikawa et al, 2012), strontium (Stellmer et al, 2009), ytterbium (Takasu et al, 2003 and dysprosium (Lian et al, 2012;Lu et al, 2011) atoms have already been produced. Such spinor Bose-Einstein condensates may host non-Abelian fractional vortices (Borgh and Ruostekoski, 2016;Huhtamäki et al, 2009;Kobayashi et al, 2009;Kobayashi and Ueda, 2016;Mawson et al, 2017;Semenoff and Zhou, 2007) whose topological invariants (Mermin, 1979;Thouless, 1998) are described by finite non-Abelian symmetry groups. Notwithstanding the finiteness of their underlying symmetry groups, such condensates may possess experimentally realizable ground states with non-Abelian vortex anyons with the capacity to be harnessed for topological quantum computation.…”

Internal structure and stability of vortices in a dipolar spinor Bose-Einstein condensate

“…In a biaxial-nematic spin-2 BEC, the related but simpler dihedral-4 point-group symmetry already leads to highly complex core structures of a half-quantum vortex [32]. As a result of the D 6 symmetry, the spin-3 A-phase vortices are also non-Abelian (i.e., the different topological charges do not all commute), leading to the restricted reconnection dynamics of vortices also predicted in the cyclic and biaxial-nematic spin-2 phases [32,70]. The A-phase D 6 order parameter supports a halfquantum vortex where the π winding of the condensate phase is compensated by a π/3 spin rotation.…”

“…The DI gives rise to a new spin-dependent healing length, adding to the hierarchy of characteristic length scales arising form the contact interaction to determine the structure of singular-vortex cores [27,32,56]. We analyze the interplay of these length scales and demonstrate how the size of a singular-vortex core is determined by the shorter of the spin-dependent healing lengths when the DI and contact interaction both restrict breaking of the ground-state spin condition (the ground-state phase of the bulk superfluid), e.g., in the ferromagnetic (FM) spin-1 BEC.…”

Three-dimensional skyrmions in spin-2 Bose–Einstein condensates

“…Spin-2 BECs exhibit two magnetic phases that have no spin-1 counterparts, namely, the biaxial nematic (BN) and the cyclic (C) phases. To date, research on topological defects in these phases has been restricted to the study of surface solitons [15] and non-Abelian vortices, the latter of which exhibit noncommutative reconnection dynamics [40][41][42] predicted to result in exotic quantum turbulence [43]. However, there have been no detailed studies of three-dimensional skyrmions in these phases.…”