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Showing papers on "Explicit symmetry breaking published in 2015"


Journal ArticleDOI
TL;DR: In this paper, it was shown that inversion symmetry breaking can be used to create acoustic analogues of the topological Haldane model and an effective gauge flux in a reduced two-dimensional system by engineering interlayer couplings.
Abstract: Realizing non-trivial topological effects is challenging in acoustic systems. It is now shown that inversion symmetry breaking can be used to create acoustic analogues of the topological Haldane model. Following the discovery of the quantum Hall effect1,2 and topological insulators3,4, the topological properties of classical waves began to draw attention5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21. Topologically non-trivial bands characterized by non-zero Chern numbers are realized through either the breaking of time-reversal symmetry using an external magnetic field5,6,7,15,16 or dynamic modulation8,17. Owing to the absence of a Faraday-like effect, the breaking of time-reversal symmetry in an acoustic system is commonly realized with moving background fluids20,22, which drastically increases the engineering complexity. Here we show that we can realize effective inversion symmetry breaking and create an effective gauge flux in a reduced two-dimensional system by engineering interlayer couplings, achieving an acoustic analogue of the topological Haldane model2,23. We show that the synthetic gauge flux is closely related to Weyl points24,25,26 in the three-dimensional band structure and the system supports chiral edge states for fixed values of kz.

362 citations


Journal ArticleDOI
TL;DR: In this article, an ultracold atomic cloud bouncing on an oscillating mirror can reveal spontaneous breaking of a discrete time-translation symmetry, which can be induced by atomic losses or by a measurement of particle positions.
Abstract: We show that an ultracold atomic cloud bouncing on an oscillating mirror can reveal spontaneous breaking of a discrete time-translation symmetry. In many-body simulations, we illustrate the process of the symmetry breaking that can be induced by atomic losses or by a measurement of particle positions. The results pave the way for understanding and realization of the time crystal idea where crystalline structures form in the time domain due to spontaneous breaking of continuous time-translation symmetry.

281 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the spontaneous breaking of B − L is accomplished by a singlet scalar field carrying two units of B−L charge, which results in a very natural implementation of the Majorana seesaw mechanism for neutrinos.

188 citations


Journal ArticleDOI
TL;DR: A new mechanism for solving the strong CP problem using a Z_{2} discrete symmetry and an anomalous U(1) symmetry and this model makes the extremely concrete prediction that there exist new colored particles at the TeV scale.
Abstract: We present a new mechanism for solving the strong CP problem using a Z_{2} discrete symmetry and an anomalous U(1) symmetry. A Z_{2} symmetry is used so that two gauge groups have the same theta angle. An anomalous U(1) symmetry makes the difference between the two theta angles physical and the sum unphysical. Two models are presented where the anomalous symmetry manifests itself in the IR in different ways. In the first model, there are massless bifundamental quarks, a solution reminiscent of the massless up quark solution. In the IR of this model, the η^{'} boson relaxes the QCD theta angle to the difference between the two theta angles-in this case zero. In the second model, the anomalous U(1) symmetry is realized in the IR as a dynamically generated mass term that has exactly the phase needed to cancel the theta angle. Both of these models make the extremely concrete prediction that there exist new colored particles at the TeV scale.

159 citations


Journal ArticleDOI
TL;DR: The first transport experiment on Landau level splitting in TDS Cd3As2 single crystals under high magnetic fields is reported, suggesting the removal of spin degeneracy by breaking time reversal symmetry.
Abstract: Three-dimensional topological Dirac semimetals (TDSs) are a new kind of Dirac materials that exhibit linear energy dispersion in the bulk and can be viewed as three-dimensional graphene. It has been proposed that TDSs can be driven to other exotic phases like Weyl semimetals, topological insulators and topological superconductors by breaking certain symmetries. Here we report the first transport experiment on Landau level splitting in TDS Cd3As2 single crystals under high magnetic fields, suggesting the removal of spin degeneracy by breaking time reversal symmetry. The detected Berry phase develops an evident angular dependence and possesses a crossover from non-trivial to trivial state under high magnetic fields, a strong hint for a fierce competition between the orbit-coupled field strength and the field-generated mass term. Our results unveil the important role of symmetry breaking in TDSs and further demonstrate a feasible path to generate a Weyl semimetal phase by breaking time reversal symmetry.

141 citations


Journal ArticleDOI
TL;DR: In this paper, a simple criterion for symmetry enhancement in the ultraviolet was provided, by a direct study of the fermionic zero modes around one-instanton operators, where the fixed points might have larger flavor symmetry or they might even be secretly six-dimensional theories on S^1.
Abstract: Supersymmetric gauge theories in five dimensions often exhibit less symmetry than the ultraviolet fixed points from which they flow. The fixed points might have larger flavor symmetry or they might even be secretly six-dimensional theories on S^1. Here we provide a simple criterion when such symmetry enhancement in the ultraviolet should occur, by a direct study of the fermionic zero modes around one-instanton operators.

121 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that the 5D Nekrasov partition functions enjoy the enhanced global symmetry of the UV fixed point, while the fiber-base duality is responsible for the global symmetry enhancement.
Abstract: We show that the 5D Nekrasov partition functions enjoy the enhanced global symmetry of the UV fixed point. The fiber-base duality is responsible for the global symmetry enhancement. For SU(2) with N f ≤ 7 flavors the fiber-base symmetry together with the manifest flavor SO(2N f ) symmetry generate the $$ {\mathrm{E}}_{N_{f+1}} $$ global symmetry, while in the higher rank case the manifest global symmetry of the two dual theories related by the fiber-base duality map generate the symmetry enhancement. The symmetry enhancement at the level of the partition function is manifest once we chose an appropriate reparametrization for the Coulomb moduli.

109 citations


Journal ArticleDOI
TL;DR: In this article, the relation between discrete gauge symmetries in F-theory compactifications and torsion homology on the associated Calabi-Yau manifold was studied.
Abstract: We study the relation between discrete gauge symmetries in F-theory compactifications and torsion homology on the associated Calabi-Yau manifold. Focusing on the simplest example of a $$ {\mathbb{Z}}_2 $$ symmetry, we show that there are two physically distinct ways that such a discrete gauge symmetry can arise. First, compactifications of M-Theory on Calabi-Yau threefolds which support a genus-one fibration with a bi-section are known to be dual to six-dimensional F-theory vacua with a $$ {\mathbb{Z}}_2 $$ gauge symmetry. We show that the resulting five-dimensional theories do not have a $$ {\mathbb{Z}}_2 $$ symmetry but that the latter emerges only in the F-theory decompactification limit. Accordingly the genus-one fibred Calabi-Yau manifolds do not exhibit torsion in homology. Associated to the bi-section fibration is a Jacobian fibration which does support a section. Compactifying on these related but distinct varieties does lead to a $$ {\mathbb{Z}}_2 $$ symmetry in five dimensions and, accordingly, we find explicitly an associated torsion cycle. We identify the expected particle and membrane system of the discrete symmetry in terms of wrapped M2 and M5 branes and present a field-theory description of the physics for both cases in terms of circle reductions of six-dimensional theories. Our results and methods generalise straightforwardly to larger discrete symmetries and to four-dimensional compactifications.

107 citations


Posted Content
TL;DR: In this paper, the authors characterize condensed matter systems as any state in a Poincare-invariant theory that spontaneously breaks Lorentz boosts while preserving at large distances some form of spatial translations, time-translations, and possibly spatial rotations.
Abstract: We classify condensed matter systems in terms of the spacetime symmetries they spontaneously break. In particular, we characterize condensed matter itself as any state in a Poincare-invariant theory that spontaneously breaks Lorentz boosts while preserving at large distances some form of spatial translations, time-translations, and possibly spatial rotations. Surprisingly, the simplest, most minimal system achieving this symmetry breaking pattern---the "framid"---does not seem to be realized in Nature. Instead, Nature usually adopts a more cumbersome strategy: that of introducing internal translational symmetries---and possibly rotational ones---and of spontaneously breaking them along with their space-time counterparts, while preserving unbroken diagonal subgroups. This symmetry breaking pattern describes the infrared dynamics of ordinary solids, fluids, superfluids, and---if they exist---supersolids. A third, "extra-ordinary", possibility involves replacing these internal symmetries with other symmetries that do not commute with the Poincare group, for instance the galileon symmetry, supersymmetry or gauge symmetries. Among these options, we pick the systems based on the galileon symmetry, the "galileids", for a more detailed study. Despite some similarity, all different patterns produce truly distinct physical systems with different observable properties. For instance, the low-energy $2\to 2$ scattering amplitudes for the Goldstone excitations in the cases of framids, solids and galileids scale respectively as $E^2$, $E^4$, and $E^6$. Similarly the energy momentum tensor in the ground state is "trivial" for framids ($\rho +p=0$), normal for solids ($\rho+p>0$) and even inhomogenous for galileids.

104 citations


Journal ArticleDOI
TL;DR: In this paper, the authors systematically explore symmetry breaking patterns in the scalar sector of the three-Higgs-doublet model (3HDM) and find all possible ways it can break by the Higgs vacuum expectation value alignment.
Abstract: An attractive feature of New Physics models with multiple Higgs fields is that they are equipped with discrete symmetry groups in the Higgs and flavour sectors. These symmetry groups are often broken at the global minimum of the Higgs potential, either completely or to a proper subgroup, with certain phenomenological consequences. Here, we systematically explore these symmetry breaking patterns in the scalar sector of the three-Higgs-doublet model (3HDM). We use the full list of discrete symmetry groups allowed in 3HDM, and for each group we find all possible ways it can break by the Higgs vacuum expectation value alignment. We also discuss the interplay between these symmetry groups and various forms of CP -violation in the scalar sector of 3HDM. Not only do our results solve the problem for 3HDM, but they also hint at several general features in multi-scalar sectors.

92 citations


Journal ArticleDOI
TL;DR: In this paper, the BEH mechanism in Yang-Mills theories was used to reveal the small-distance structure of the gravitational force at the Planck scale, and the symmetry could be as fundamental as Lorentz invariance.
Abstract: Local conformal symmetry is usually considered to be an approximate symmetry of nature, which is explicitly and badly broken. Arguments are brought forward here why it has to be turned into an exact symmetry that is spontaneously broken. As in the BEH mechanism in Yang–Mills theories, we then will have a formalism for disclosing the small-distance structure of the gravitational force. The symmetry could be as fundamental as Lorentz invariance, and guide us towards a complete understanding of physics at the Planck scale.

Journal ArticleDOI
TL;DR: In this paper, a twisted group cohomology description of symmetry fractionalization is proposed, which is encoded in the associativity of fusion rules of the extrinsic ''twist'' defects of the symmetry.
Abstract: Topological order in two dimensions can be described in terms of deconfined quasiparticle excitations - anyons - and their braiding statistics. However, it has recently been realized that this data does not completely describe the situation in the presence of an unbroken global symmetry. In this case, there can be multiple distinct quantum phases with the same anyons and statistics, but with different patterns of symmetry fractionalization - termed symmetry enriched topological (SET) order. When the global symmetry group $G$, which we take to be discrete, does not change topological superselection sectors - i.e. does not change one type of anyon into a different type of anyon - one can imagine a local version of the action of $G$ around each anyon. This leads to projective representations and a group cohomology description of symmetry fractionalization, with $H^2(G,{\cal A})$ being the relevant group. In this paper, we treat the general case of a symmetry group $G$ possibly permuting anyon types. We show that despite the lack of a local action of $G$, one can still make sense of a so-called twisted group cohomology description of symmetry fractionalization, and show how this data is encoded in the associativity of fusion rules of the extrinsic `twist' defects of the symmetry. Furthermore, building on work of Hermele, we construct a wide class of exactly solved models which exhibit this twisted symmetry fractionalization, and connect them to our formal framework.

Journal ArticleDOI
TL;DR: In this article, it was shown that the action of quantum systems evolving in thermal equilibrium is invariant under a symmetry transformation which distinguishes them from generic open systems, and the fluctuation-dissipation relations characterizing the linear response of an equilibrium system to external perturbations can be derived as the Ward-Takahashi identities associated with this symmetry.
Abstract: The time evolution of an extended quantum system can be theoretically described in terms of the Schwinger-Keldysh functional integral formalism, whose action conveniently encodes the information about the dynamics. We show here that the action of quantum systems evolving in thermal equilibrium is invariant under a symmetry transformation which distinguishes them from generic open systems. A unitary or dissipative dynamics having this symmetry naturally leads to the emergence of a Gibbs thermal stationary state. Moreover, the fluctuation-dissipation relations characterizing the linear response of an equilibrium system to external perturbations can be derived as the Ward-Takahashi identities associated with this symmetry. Accordingly, the latter provides an efficient check for the onset of thermodynamic equilibrium and it makes testing the validity of fluctuation-dissipation relations unnecessary. In the classical limit, this symmetry renders the one which is known to characterize equilibrium in the stochastic dynamics of classical systems coupled to thermal baths, described by Langevin equations.

Journal ArticleDOI
Robert Bluhm1
TL;DR: In this paper, the authors considered the gravity sector of the SME for the case of spontaneous symmetry breaking and showed that a useful post-Newtonian limit is only obtained when the symmetry breaking is spontaneous.
Abstract: Gravitational theories with fixed background fields break local Lorentz and diffeomorphism invariance either explicitly or spontaneously. In the case of explicit breaking it is known that conflicts can arise between the dynamics and geometrical constraints, while spontaneous breaking evades this problem. It is for this reason that in the gravity sector of the Standard-Model extension (SME) it is assumed that the background fields (SME coefficients) originate from spontaneous symmetry breaking. However, in other examples, such as Chern-Simons gravity and massive gravity, diffeomorphism invariance is explicitly broken by the background fields, and the potential conflicts between the dynamics and geometry can be avoided in most cases. An analysis of how this occurs is given, and the conditions that are placed on the metric tensor and gravitational structure as a result of the presence of an explicit-breaking background are described. The gravity sector of the SME is then considered for the case of explicit breaking. However, it is found that a useful post-Newtonian limit is only obtained when the symmetry breaking is spontaneous.

Journal ArticleDOI
TL;DR: It is shown that Dirac points can emerge in photonic crystals possessing mirror symmetry when band gap closes, and a photonic analog of Chern insulator can be achieved through time reversal symmetry breaking.
Abstract: We show that Dirac points can emerge in photonic crystals possessing mirror symmetry when band gap closes. The mechanism of generating Dirac points is discussed in a two-dimensional photonic square lattice, in which four Dirac points split out naturally after the touching of two bands with different parity. The emergence of such nodal points, characterized by vortex structure in momentum space, is attributed to the unavoidable band crossing protected by mirror symmetry. The Dirac nodes can be unbuckled through breaking the mirror symmetry and a photonic analog of Chern insulator can be achieved through time reversal symmetry breaking. Breaking time reversal symmetry can lead to unidirectional helical edge states and breaking mirror symmetry can reduce the band gap to amplify the finite size effect, providing ways to engineer helical edge states.

Journal ArticleDOI
TL;DR: Universal properties of the statistics of stochastic efficiency for mesoscopic time-reversal symmetry broken energy transducers are revealed in the Gaussian approximation and the underlying physics is demonstrated through the quantum Hall effect and further elaborated in a triple-quantum-dot three-terminal thermoelectric engine.
Abstract: Universal properties of the statistics of stochastic efficiency for mesoscopic time-reversal symmetry broken energy transducers are revealed in the Gaussian approximation. We also discuss how the second law of thermodynamics restricts the statistics of stochastic efficiency. The tight-coupling limit becomes unfavorable, characterized by an infinitely broad distribution of efficiency at all times, when time-reversal symmetry breaking leads to an asymmetric Onsager response matrix. The underlying physics is demonstrated through the quantum Hall effect and further elaborated in a triple-quantum-dot three-terminal thermoelectric engine.

Journal ArticleDOI
TL;DR: In this article, the authors present an approach allowing perturbatively to keep the dilaton massless, i.e. to preserve conformal symmetry, at any fixed order in perturbation theory.
Abstract: Spontaneous symmetry breaking with necessity leads to the presence of Goldstone field(s). In the case of scale or conformal symmetries the corresponding Goldstone mode is called the dilaton. Consistently coupling a system to the dilaton poses certain difficulties, for the trace of the energy momentum tensor may not necessary be zero, which in turn leads to the dilaton acquiring a mass. In this paper we present the approach allowing perturbatively to keep the dilaton massless, i.e. to preserve conformal symmetry, at any fixed order in perturbation theory.

Journal ArticleDOI
TL;DR: In this article, a new type of symmetry breaking occurs in the steady-state energy distribution of open systems with balanced gain and loss, where the combination of nonlinear saturation effects and the presence of thermal or quantum noise in actual experiments results in unexpected behavior that differs significantly from the usual dynamical picture.
Abstract: The phenomenon of PT (parity- and time-reversal) symmetry breaking is conventionally associated with a change in the complex mode spectrum of a non-Hermitian system that marks a transition from a purely oscillatory to an exponentially amplified dynamical regime. In this work we describe a new type of PT-symmetry breaking, which occurs in the steady-state energy distribution of open systems with balanced gain and loss. In particular, we show that the combination of nonlinear saturation effects and the presence of thermal or quantum noise in actual experiments results in unexpected behavior that differs significantly from the usual dynamical picture. We observe additional phases with preserved or `weakly' broken PT symmetry, and an unconventional transition from a high-noise thermal state to a low-amplitude lasing state with broken symmetry and strongly reduced fluctuations. We illustrate these effects here for the specific example of coupled mechanical resonators with optically-induced loss and gain, but the described mechanisms will be essential for a general understanding of the steady-state properties of actual PT-symmetric systems operated at low amplitudes or close to the quantum regime.

Journal ArticleDOI
TL;DR: It is observed that symmetry breaking leads to increased disorderliness in the dynamical behavior of oscillatory states and consequently results in a rich variety of dynamical states, depending on the strength of the nonisochronicity parameter.
Abstract: We analyze the consequences of symmetry breaking in the coupling in a network of globally coupled identical Stuart-Landau oscillators. We observe that symmetry breaking leads to increased disorderliness in the dynamical behavior of oscillatory states and consequently results in a rich variety of dynamical states. Depending on the strength of the nonisochronicity parameter, we find various dynamical states such as amplitude chimera, amplitude cluster, frequency chimera, and frequency cluster states. In addition we also find disparate transition routes to recently observed chimera death states in the presence of symmetry breaking even with global coupling. We also analytically verify the chimera death region, which corroborates the numerical results. These results are compared with that of the symmetry-preserving case as well.

Journal ArticleDOI
TL;DR: In this article, the first calculations of a symmetry conserving configuration mixing method (SCCM) using time-reversal symmetry breaking Hartree-Fock-Bogoliubov (HFB) states with the Gogny D1S interaction were presented.

Journal ArticleDOI
TL;DR: In this paper, the magnetoconductivity induced by the axial anomaly via the chiral magnetic effect in strongly coupled holographic models was studied and it was shown that axial charge relaxation time grows linearly with magnetic field in the large B regime.
Abstract: We study the magnetoconductivity induced by the axial anomaly via the chiral magnetic effect in strongly coupled holographic models. An important ingredient in our models is that the axial charge is non-conserved beyond the axial anomaly. We achieve this either by explicit symmetry breaking via a non-vanishing non-normalisable mode of an axially charged scalar or using a Stuckelberg field to make the AdS-bulk gauge field massive. The DC magnetoconductivites can be calculated analytically. They take a universal form in terms of gauge field mass at the horizon and quadratic dependence on the magnetic field. The axial charge relaxation time grows linearly with magnetic field in the large B regime. Most strikingly positive magnetoconductivity is still present even when the relaxation times are short τ5 ≈ 1/(πT) and the axial charge can not be thought of as an approximate symmetry. In the U(1) A explicit breaking model, we also observe that the chiral separation conductivity and the axial magnetic conductivity for the consistent axial current vanish in the limit of strong symmetry breaking.

Journal ArticleDOI
TL;DR: In this paper, the authors use a Green's function approach to calculate the tunneling rate between radiatively generated non-degenerate vacua, and apply this to a model that exhibits spontaneous symmetry breaking via the Coleman-Weinberg mechanism.
Abstract: We use a Green’s function approach in order to develop a method for calculating the tunneling rate between radiatively generated nondegenerate vacua. We apply this to a model that exhibits spontaneous symmetry breaking via the Coleman-Weinberg mechanism, where we determine the self-consistent tunneling configuration and illustrate the impact of gradient effects that arise from accounting for the underlying space-time inhomogeneity.

Journal ArticleDOI
TL;DR: In this paper, the authors examined the role of symmetry and effective field theory in inflationary model building, and showed that the only real symmetry in these models is not at all hidden when expressed in the Einstein frame; it is in fact the shift symmetry of a scalar field.

Journal ArticleDOI
TL;DR: In this article, the residual symmetry can be localized to Lie point symmetry by prolonging the original equation to a larger system, which is equivalent to the second type of Darboux transformation.

Journal ArticleDOI
TL;DR: In this paper, the bumblebee model of spontaneous Lorentz symmetry breaking is explored in a cosmological context, considering a single nonzero time component for the vector field.
Abstract: The bumblebee model of spontaneous Lorentz symmetry breaking is explored in a cosmological context, considering a single nonzero time component for the vector field. The relevant dynamic equations for the evolution of the Universe are derived and their properties and physical significance are studied. We conclude that a late-time de Sitter expansion of the Universe can be replicated, and attempt to constrain the parameter of the potential driving the spontaneous symmetry breaking.

Journal ArticleDOI
TL;DR: The observation that radiation from a system of accelerating charges is possible only when there is explicit breaking of symmetry in the electric field in space within the spatial configuration of the radiating system is reported.
Abstract: We report our observation that radiation from a system of accelerating charges is possible only when there is explicit breaking of symmetry in the electric field in space within the spatial configuration of the radiating system Under symmetry breaking, current within an enclosed area around the radiating structure is not conserved at a certain instant of time resulting in radiation in free space Electromagnetic radiation from dielectric and piezoelectric material based resonators are discussed in this context Finally, it is argued that symmetry of a resonator of any form can be explicitly broken to create a radiating antenna

Journal ArticleDOI
TL;DR: In this article, the authors discuss the effective field theory for spacetime symmetry breaking from the local symmetry point of view, and show that the local picture becomes important in particular when taking into account massive modes associated with symmetry breaking, the masses of which are not necessarily high.
Abstract: We discuss the effective field theory for spacetime symmetry breaking from the local symmetry point of view. By gauging spacetime symmetries, the identification of Nambu--Goldstone (NG) fields and the construction of the effective action are performed based on the breaking pattern of diffeomorphism, local Lorentz, and (an)isotropic Weyl symmetries as well as the internal symmetries including possible central extensions in nonrelativistic systems. Such a local picture distinguishes, e.g., whether the symmetry breaking condensations have spins and provides a correct identification of the physical NG fields, while the standard coset construction based on global symmetry breaking does not. We illustrate that the local picture becomes important in particular when we take into account massive modes associated with symmetry breaking, the masses of which are not necessarily high. We also revisit the coset construction for spacetime symmetry breaking. Based on the relation between the Maurer--Cartan one form and connections for spacetime symmetries, we classify the physical meanings of the inverse-Higgs constraints by the coordinate dimension of broken symmetries. Inverse Higgs constraints for spacetime symmetries with a higher dimension remove the redundant NG fields, whereas those for dimensionless symmetries can be further classified by the local symmetry breaking pattern.

Journal ArticleDOI
TL;DR: In this paper, double-trace type deformations of the coset model were studied and compared to changes of boundary conditions for the bulk matter fields in the boundary theory and the bulk theory.
Abstract: Recently, a 2d coset model with $$ \mathcal{N}=3 $$ superconformal symmetry was pro-posed to be holographic dual to a higher spin supergravity on AdS3 and the relation to superstring theory was discussed. However, away from the tensionless limit, there is no higher spin symmetry and the higher spin states are massive. In this paper, we examine the deformations of the coset model which preserve $$ \mathcal{N}=3 $$ superconformal symmetry, but break generic higher spin symmetry. We focus on double-trace type deformations which are dual to changes of boundary conditions for the bulk matter fields. In the bulk theory, the symmetry breaking will generate mass for the higher spin fields. As a concrete example, we compute the Higgs mass of a spin 2 field both from the bulk and the boundary theory.

Posted Content
TL;DR: In this article, the authors considered standing waves in the focusing nonlinear Schrodinger (NLS) equation on a dumbbell graph (two rings attached to a central line segment subject to the Kirchhoff boundary conditions at the junctions).
Abstract: We consider standing waves in the focusing nonlinear Schrodinger (NLS) equation on a dumbbell graph (two rings attached to a central line segment subject to the Kirchhoff boundary conditions at the junctions). In the limit of small $L^2$ norm, the ground state (the orbitally stable standing wave of the smallest energy at a fixed $L^2$ norm) is represented by a constant solution. However, when the $L^2$ norm is increased, this constant solution undertakes two bifurcations, where the first is the pitchfork (symmetry breaking) bifurcation and the second one is the symmetry preserving bifurcation. As a result of the first symmetry breaking bifurcation, the standing wave becomes more localized in one of the two rings. As a result of the second symmetry preserving bifurcation, the standing wave becomes localized in the central line segment. In the limit of large norm solutions, both standing waves are represented by a truncated solitary wave localized in either the ring or the central line segment. Although the asymmetric wave supported in the ring is a ground state near the symmetry breaking bifurcation of the constant solution, it is the symmetric wave supported in the central line segment which becomes the ground state in the limit of large $L^2$ norm. The analytical results are confirmed by numerical approximations of the ground state on the dumbbell graph.

Journal ArticleDOI
TL;DR: In this paper, the authors proposed a superconducting state for strongly hole-doped Ba$ 1-x}$ K$_x$Fe$_2$As$ 2, which breaks time-reversal symmetry (TRS) but does not break any other discrete symmetry.
Abstract: We analyze $s + is$ state proposed as a candidate superconducting state for strongly hole-doped Ba$_{1-x}$K$_x$Fe$_2$As$_2$. Such a state breaks time-reversal symmetry (TRS) but does not break any other discrete symmetry. We address the issue whether TRS breaking alone can generate spontaneous currents near impurity sites, which could be detected in, e.g., muSR experiments. We argue that there are no spontaneous currents if only TRS is broken. However, supercurrents do emerge if the system is put under external strain and C4 lattice rotation symmetry is externally broken.