Despite its deceptive simplicity, few concepts have more fundamental implications than chirality, from the therapeutic activity of drugs to the fundamental forces of nature. In magnetic materials, chirality gives rise to unconventional phenomena such as the anomalous Hall effect and multiferroicity, taking an enhanced flavour in the so-called spin-liquid phases where magnetic disorder prevails. Kagome systems sit at the crossroad of these ideas. Motivated by the recent synthesis of rare-earth kagome materials and the progresses in optical-lattice experiments, we bring together an entire network of spin liquids with anisotropic and Dzyaloshinskii–Moriya interactions. This network revolves around the Ising antiferromagnet and ends on (ferromagnetic) chiral spin liquids with spontaneously broken time-reversal symmetry. As for the celebrated Heisenberg antiferromagnet, it now belongs to a triad of equivalently disordered phases. The present work provides a unifying theory of kagome spin liquids with time-reversal invariant nearest-neighbour Hamiltonians.

The kagome lattice is a paragon of geometrical frustration, long-studied for its association with novel ground-states including spin liquids. Many recently synthesized kagome materials feature rare-earth ions, which may be expected to exhibit highly anisotropic exchange interactions. The consequences of this combination of strong exchange anisotropy and extreme geometrical frustration are yet to be fully understood. Here, we establish a general picture of the interactions and resulting ground-states arising from nearest neighbour exchange anisotropy on the kagome lattice. We determine a generic anisotropic exchange Hamiltonian from symmetry arguments. In the high-symmetry case where reflection in the kagome plane is a symmetry of the system, the generic nearest-neighbour Hamiltonian can be locally defined as a XYZ model with out-of-plane Dzyaloshinskii-Moriya interactions. We proceed to study its phase diagram in the classical limit, making use of an exact reformulation of the Hamiltonian in terms of irreducible representations (irreps) of the lattice symmetry group. This reformulation in terms of irreps naturally explains the three-fold mapping between spin liquids recently studied on kagome by the present authors [Nature Communications 7, 10297 (2016)]. In addition, a number of unusual states are stabilised in the regions where different forms of ground-state order compete, including a stripy phase with a local Z8 symmetry and a classical analogue of a chiral spin liquid. This generic Hamiltonian also turns out to be a fruitful hunting ground for coexistence of different forms of magnetic order, or of order and disorder, which we find is a particular property of the kagome lattice arising from the odd number of spins per frustrated unit. These results are compared and contrasted with those obtained on the pyrochlore lattice, and connection is made with recent progress in understanding quantum models with S = 1/2. arXiv:1706.09101v1 [cond-mat.str-el] 28 Jun 2017 where the Greek and Latin indices respectively label the spin components and the sublattices of a triangle [see Figs. 1 and 2 for the labelling convention]. Alternatively, H ∆ can be written in the form of a 9 × 9 coupling matrix J H ∆ = 1 2S   0Ĵ 01Ĵ02 J 10 0Ĵ 12 J 20Ĵ21 0  S = 1 2SĴS ,

The crumpled-to-flat phase transition that occurs in D-dimensional polymerized phantom membranes embedded in a d-dimensional space is investigated nonperturbatively using a field expansion up to order eight in powers of the order parameter. We get the critical dimension dcr(D) that separates a second order region from a first order one everywhere between D = 4 and D = 2. Our approach strongly suggests that the phase transitions that take place in physical membranes are of first order in agreement with most recent numerical simulations.PACS numbers: 87.16. 11.10.Hi, 11.15.Tk Fluctuating or random surfaces are a recurrent concept in physics [1,2]. They occur in soft matter physics or in biology as assemblies of amphiphilic molecules that can form plane or closed structures (vesicles) according to the chemical composition of the membrane itself and its surroundings. Random surfaces also appear in highenergy physics, especially in string theory, as the worldsheet swept out by a string during its spacetime evolution. More recently membranes have received a renewed interest in condensed matter physics where it has been realized that, from the point of view of their mechanical properties, novel materials, like graphene [3], identify with polymerized membranes, providing the first and unique example of genuinely two-dimensional membrane [4,5]. The coexistence of two-dimensional geometry and thermal fluctuations is at the origin of a variety of behaviours depending on the nature of the internal structure of the membrane. Fluid membranes are made of molecules that freely diffuse and re-arrange rapidly when a shear or a stress is performed. This implies that, in absence of an external tension, the dominant energy is the bending energy. It has been shown that, in this case, the height fluctuations are sufficiently strong to prevent the appearance of long-range order; fluid membranes are thus always crumpled [6,7]. Polymerizedor tethered -membranes display a drastically different behaviour. Indeed the existence of an underlying network of linked molecules induces elastic (shearing and stretching) energy contributions that lead to a coupling between height and transverse -phonons -modes. It results from this situation a frustration of the height fluctuations [8] that are strongly reduced at low temperatures giving rise to the appearance of a flat phase with longrange order between the normals [9,10]. The existence of a low-temperature phase accompanied with spontaneous symmetry breaking of rotational invariance is a priori in contradiction with the Mermin-Wagner theorem. However it appears that the effective phonon-mediated interaction between the height fields (more precisely between the Gaussian curvatures) is of long-range kind, allowing to evade the conditions of application of the MerminWagner theorem [9]. Correlatively the low-temperature phase of membranes is characterized by non trivial scaling behaviour in the infrared [11][12][13]:where G hh (q) and G uu (q) are the correlation functions of the out-of-plane and ...