Spin stiffness of the anisotropic Heisenberg model on the square lattice and a possible mechanism for pinning of the electronic liquid crystal direction in underdopedYBa2Cu3O6.45

Abstract

Using series expansions and spin-wave theory, we calculate the spin-stiffness anisotropy sx / sy in Heisenberg models on the square lattice with spatially anisotropic couplings J x , J y . We find that for the weakly anisotropic spin-half model ͑J x Ϸ J y ͒, sx / sy deviates substantially from the naive estimate sx / sy Ϸ J x / J y . We argue that this deviation can be responsible for pinning the electronic liquid crystal direction, an effect recently discovered in YBCO. For completeness, we also study the spin stiffness for arbitrary anisotropy J x / J y for spin-half and spin-one models. In the limit of J y / J x → 0, when the model reduces to weakly coupled chains, the two show dramatically different behavior. In the spin-one model, the stiffness along the chains goes to zero, implying the onset of Haldane-gap phase, whereas for the spin-half model, the stiffness along the chains increases monotonically from a value of 0.18J x for J y / J x = 1 toward 0.25J x for J y / J x → 0. In the latter case, spin-wave theory breaks down qualitatively, presumably due to the onset of topological terms with strong anisotropy. This work is motivated by the recent discovery 1,2 of the electronic liquid crystal in underdoped cuprate superconductor YBa 2 Cu 3 O 6.45 . The electronic liquid crystal manifests itself in a strong anisotropy in the low-energy inelastic neutron scattering. The liquid crystal picture implies a spontaneous violation of the directional symmetry: the "crystal" can be oriented either along the ͑1,0͒ or along the ͑0,1͒ axes in the square lattice. The YBa 2 Cu 3 O 6.45 compound has a tetragonal lattice with tiny in-plane lattice anisotropy a ‫ء‬ / b ‫ء‬ Ϸ 0.99. This tiny anisotropy is sufficient to pin the orientation of the electronic liquid crystal along the a ‫ء‬ axis. As a result, the low-energy neutron scattering 1,2 demonstrates a quasi-one-dimensional ͑1D͒ structure along a ‫ء‬ .To understand the pinning mechanism of the electronic crystal, in the present work, we study the anisotropic Heisenberg model. We calculate the in-plane anisotropy of the spin stiffness and demonstrate that this is strongly enhanced by quantum fluctuations. We argue that the enhancement is sufficient to provide a pinning mechanism for the initially spontaneous orientation of the electronic liquid crystal and suggest a specific mechanism for the pinning. The anisotropic Heisenberg model has previously attracted a lot of theoretical interests. 3-9 However, most theoretical studies have focused on the regime of strong anisotropy, where the system reduces to one of weakly coupled spin chains, and the most significant issue there is the dimensional crossover and the onset of long-range antiferromagnetic order. To the best of our knowledge, the anisotropy of spin stiffness has not been studied before. This is an important theoretical problem in itself and therefore, we extend our study to the case of arbitrary strong anisotropy. We consider both the spin-half and spin-one models where in the limit of strong anisotropy, we come to t...

Monte Carlo determination of the low-energy constants for a two-dimensional spin-1 Heisenberg model with spatial anisotropy

“…1 provides a great opportunity to further examine the validity of this method. Indeed as we will demonstrate later, our Monte Carlo results of ρ s1 for the spin-1 model agree quantitatively with those determined by the series expansion method from J 2 /J 1 = 1 down to J 2 /J 1 ∼ 0.08 [32]. Consequently our investigation gives convincing evidence for the correctness of this method.…”

“…In addition, the effective box sizes L shown in Eqs. (6) and (7) ical values of ρ s1 obtained from our Monte Carlo data and the series expansion results determined in [32] are in nice agreement from J 2 /J 1 = 1 down to J 2 /J 1 ≈ 0.08. Although the truncation errors of series expansion results are large for small values of J 2 /J 1 , the outcomes of series expansion without the truncation errors agree very well with those of Monte Carlo for J 2 /J 1 ≥ 0.08.…”

“…For example, NDMAZ, NENP, and PbNi 2 V 2 O 8 are found to be spin-1 quasione-dimensional antiferromagnets. Besides the simplest type of these models which have spatially isotropic couplings, the spatially anisotropic Heisenberg models are also studied thoroughly [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. In particular, these generalized models are frequently used as a route for studying quantum phase transitions.…”

Monte Carlo simulations of an unconventional phase transition for a two-dimensional dimerized quantum Heisenberg model

“…For this purpose, we consider the quantum transition induced by dimerization for the Heisenberg model with a ladder pattern anisotropic couplings (figure 4). For b ∼ 0.95 (22) in Eq. (4), we obtain a good data collapse for the observable (ρ s1 ) in L 2 , here the subscript "in" means the data points are the interpolated one.…”

“…Furthermore, a new proposal of determining the low-energy constant, namely the spinwave velocity c of antiferromagnets with O(2) and O(3) symmetry, through the squares of temporal and spatial winding numbers was verified to be valid and this idea has greatly improved the accuracy of the related low-energy constants 20,21 . On the one hand, Heisenberg-type models on geometrically nonfrustrated lattices are among the best quantitatively understood condensed matter physics systems; on the other hand, despite being well studied, several recent numerical investigation of spatially anisotropic Heisenberg models have led to unexpected results 14,22,23 . In particular, Monte Carlo evidence indicates that the anisotropic Heisenberg model with staggered arrangement of the antiferromagnetic couplings may belong to a new universality class, in contradiction to the theoretical O(3) universality prediction 14 .…”