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Skyrme Insulators: Insulators at the Brink of Superconductivity BNL-114020-2017-JA Skyrme insulators: insulators at the brink of superconductivity O. Erten, P.-Y. Chang, P. Coleman, and A. M. Tsvelik Submitted to Physics. Rev. Left. June 2017 Condensed Matter Physics and Materials Science Department Brookhaven National Laboratory U.S. Department of Energy USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22) Notice: This manuscript has been authored by employees of Brookhaven Science Associates, LLC under Contract No. DE- SC0012704 with the U.S. Department of Energy. The publisher by accepting the manuscript for publication acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or any third party’s use or the results of such use of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof or its contractors or subcontractors. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. Skyrme insulators: insulators at the brink of superconductivity Onur Erten,1, 2 Po-Yao Chang,1 Piers Coleman,1, 3 and Alexei M. Tsvelik4 1Center for Materials Theory, Rutgers University, Piscataway, New Jersey, 08854, USA 2Max Planck Institute for the Physics of Complex Systems, D-01187 Dresden, Germany 3Department of Physics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK 4Division of Condensed Matter Physics and Material Science, Brookhaven National Laboratory, Upton, NY 11973 Current theories of superfluidity are based on the idea of a coherent quantum state with topologically pro- tected, quantized circulation. When this topological protection is absent, as in the case of 3He-A, the coherent quantum state no longer supports persistent superflow. Here we argue that the loss of topological protection in a superconductor gives rise to an insulating ground state. We specifically introduce the concept of a Skyrme insulator to describe the coherent dielectric state that results from the topological failure of superflow carried by a complex vector order parameter. We apply this idea to the case of SmB6, arguing that the observation of a diamagnetic Fermi surface within an insulating bulk can be understood as a realization of this state. Our theory enables us to understand the linear specific heat of SmB6 in terms of a neutral Majorana Fermi sea and leads us to predict that in low fields of order a Gauss, SmB6 will develop a Meissner effect. While it is widely understood that superfluids and super- conductors carry persistent “supercurrents” associated with the rigidity of the broken symmetry condensate[1], it is less commonly appreciated that the remarkable persistence of su- percurrents has its origins in topology. The order parameter of a conventional superfluid or superconductor lies on a circular manifold (S1) and the topologically stable winding number of the order parameter, like a string wrapped multiple times around a rod, protects a circulating superflow. However, if the order parameter lies on a higher dimensional manifold, such 2 FIG. 1. Illustration of topological stability. The stability of a su- as the surface of a sphere (S ), then the winding has no topo- percurrent is analogous to topological stability of a string wrapped logical protection and putative supercurrents relax their en- around a surface. (a) The winding number of a string wrapped around ergy through a continuous reduction of the winding number, a rod is topologically stable and it can not be unravelled (b) A string leading to dissipation [see Fig.1]. This topological failure of wrapped around the equator of a sphere unravels due to a lack of superfluidity is observed in the A phase of 3He, which exhibits topological stability. dissipation [2–5]. Similar behavior has also been observed in spinor Bose gases, where the decay of Rabi oscillations be- tween two condensates reveals the unravelling superflow[6]. the vector potential should also respond to its time-derivative, the electric field E @A=@t, forming a metal. In other Here we propose an extension of this concept to supercon- ≡ − ductors, arguing that when a charge condensate fails to sup- words, unless the bulk somehow breaks gauge invariance, port a topologically stable circulation, the resulting medium quantized cyclotron motion is incompatible with insulating forms a novel dielectric. Though our arguments enjoy general behavior. This reasoning motivates the hypothesis that SmB6 application, they are specifically motivated by the Kondo in- is a failed superconductor, formed from a topological break- down of an underlying condensate. sulator, SmB6. While transport [7–9] and photoemission [10– 14] measurements demonstrate that SmB6 is an insulator with General arguments tell us that the condition for the stability topological surface states, the observation of bulk quantum of a superfluid is determined by the order parameter manifold oscillations [15, 16], linear specific heat, anomalous thermal G=H formed between the symmetry group G of the Hamilto- and ac optical conductivity[17–20] have raised the fascinat- nian and the invariant subgroup H of the order parameter. The ing possibility of a “neutral” Fermi surface in the bulk, which absence of coherent bulk superflow requires that the first ho- paradoxically, exhibits Landau quantization. Landau quanti- motopy class π1(G=H) = Z is sparse, lacking the infinite set zation is normally understood as a semi-classical quantization of integers which protect6 macroscopic winding of the phase. of cyclotron motion[21]. Rather general arguments tell us that This means that G=H is a higher dimensional non-Abelian gauge invariance makes the Coulomb and Lorentz forces in- manifold, most naturally formed through the condensation of separable: particles interact with the vector potential A via the bosons or Cooper pairs with angular momentum. Thus in gauge invariant kinetic momentum π = (p eA); the corre- spinor Bose gases, an atomic spinor condensate lives on an − sponding equation of motion dπ=dt = q(E + v B) neces- SU(2) manifold with π1(SU(2)) = 0: in this case the ob- sarily contains both E and B as respective temporal× and spa- served decay of vorticity gives rise to Rabi oscillations[6]. tial gradients of the underlying vector potential. Thus quasi- Similarly, in superfluid 3He-A, an SO(3) manifold associ- particles which develop a Landau quantization in response to ated with a dipole-locked triplet paired state[4,5], for which 2 π1(SO(3)) = Z2 allows a single vortex, but no macroscopic In microscopic theory, it is actually more natural to con- circulation in the bulk sider an antiferromagnetic version of composite order, formed In the solid state, the conditions for a topological failure between the staggered magnetization and the pair density, i+j+k of superconductivity are complicated by crystal anisotropy. If Ψ(x) = ( 1) "(x) #(x)S(x) [25, 29–31]. These the condensate carries orbital angular momentum, it will tend details do not− howeverh affect the phenomenology.i to lock to the lattice, collapsing the manifold back to U(1). On We now consider a general Ginzburg Landau free energy the other hand, if the order parameter has s-wave symmetry, for an s-wave triplet condensate. Unlike a p-wave triplet, the its U(1) manifold allows stable vortices. absence of orbital components to the order parameter consid- There are two ways around this no-go argument. The first, erably simpifies the Ginzburg Landau free energy density[32], is if there is an additional “isospin” symmetry of the order 1 2 2 ∗ 2 2 parameter. For example, the half-filled attractive Hubbard f = ( i~ 2eA~)Ψ +a Ψ +b Ψ Ψ +d Ψ Ψ ; 2mj − r− j j j j · j j · j model[22], which forms a “supersolid” ground-state with a (4) perfect spherical (S2) manifold of degenerate charge density where A~ is the vector potential, minimally coupled to the or- and superconducting states, with pure superconductivity along der parameter. Provided d > 0, the condensate energy is min- the equator and a pure density wave at the pole. In this special imized when Ψ Ψ = 0 and the real and imaginary parts of the case, supercurrents can always decay into a density wave. order parameter· are orthogonal Ψ = Ψ (^l + im^ ). The triplet j j A second route is suggested by crystal field theory, which odd-frequency order parameter thus defines a triad (^l; m^ ; n^) allows the restoration of crystalline isotropy for low spin ob- of orthogonal vectors with principal axis n^ = ^l m^ . jects, such as a spin 1/2 ferromagnet in a cubic crystal. Were Eliminating the amplitude degrees of freedom× [25, 32, 33], an analogous s-wave spin-triplet condensate to form, isotropy the long-wavelength action has the following form would be assured. Rather general arguments suggest that the Z " 2 # way to achieve an s-wave spin triplet, is through the devel- 4 ρ? 2 ρs 2 Fµν = d x (@µn^) + (!µ qAµ) + : (5) opment of odd-frequency pairing. The Gorkov function of a F 2 2 − 16π triplet condensate has the form Here q = 2e=~, and we have adopted the relativistic limit d(1 2) = α(1)(iσ2~σ)αβ β(2) : (1) of the action to succinctly include both electric and magnetic − h i 2 2 2 fields[34], using the Minkowski signature (xµ ~x x0 with where i (~x ; t ); (i = 1; 2) are the space-time co-ordinates ≡ − i i c = 1) and denoting A = ( V; A~) as the four-component of the electrons.≡ Exchange statistics enforce the pair wave- µ vector potential.
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