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Using the spherically symmetric self-consistent Green's function method, we consider thermodynamic properties of the S = 1 / 2 J 1 – J 2 Heisenberg model on the 2D square lattice. We calculate the temperature dependence of the spin–spin correlation functions c r = 〈 S 0 z S r z 〉 , the gaps in the spin excitation spectrum, the energy E and the heat capacity CV for the whole J1–J2-circle, i.e. for arbitrary φ, J 1 = cos ( φ ) , J 2 = sin ( φ ) . Due to low dimension there is no long-range order at T ≠ 0 , but the short-range holds the memory of the parent zero-temperature ordered phase (antiferromagnetic, stripe or ferromagnetic). E ( φ ) and C V ( φ ) demonstrate extrema “above” the long-range ordered phases and in the regions of rapid short-range rearranging. Tracts of c r ( φ ) lines have several nodes leading to nonmonotonic c r ( T ) dependence. For any fixed φ the heat capacity CV(T) always has maximum, tending to zero at T → 0 , in the narrow vicinity of φ = 155 ° it exhibits an additional frustration-induced low-temperature maximum. We have also found the nonmonotonic behaviour of the spin gaps at φ = 270 ° ± 0 and exponentially small antiferromagnetic gap up to ( T ≲ 0.5 ) for φ ≳ 270 ° .

Frustrated magnetic compounds, in particular low-dimensional, are topical research due to persistent uncover of novel nontrivial quantum states and potential applications. The problem of this field is that many important results are scattered over the localized parameter ranges, while areas in between still contain hidden interesting effects. We consider $J_1-J_2-J_3$ Heisenberg model on the square lattice and use the spherically symmetric self-consistent approach for spin-spin Green's functions in "quasielastic" approximation. We have found a new local order in spin liquids: antiferromagnetic isotropical helices. On the structure factor we see circular concentric dispersionless structures, while on any radial direction the excitation spectrum has "roton" minima. That implies nontrivial magnetic excitations and consequences in magnetic susceptibility and thermodynamics. On the $J_1-J_2-J_3$ exchange parameters globe we discover a crossover between antiferromagnetic-like local order and ferromagnetic-like; we find stripe-like order in the middle. In fact, our "quasielastic" approach allows investigation of the whole $J_1-J_2-J_3$ globe., Comment: 9 pages, 12 figures

A matrix projection formalism is developed for the description of a frustrated two-dimensional antiferromagnet. This approach keeps safe all the advantages of the spherically symmetric self-consistent treatment (it strictly meets the spin constraint and the Mermin—Wagner theorem) but does not have its disadvantages. In particular, even in its simplest implementation, this approach provides a good agreement with the numerical simulations for two best-studied reference points: the zero frustration point and the point of a transition from the antiferromagnetic ordered phase to the disordered phase.

In a spherically symmetric self-consistent approach (SSSA), the spin-1/2 J 1–J 2 Heisenberg model on a two-dimensional square lattice is considered for two-time retarded spin–spin Green’s functions. The spin excitation spectrum, ω(q), and spin gaps at symmetric points are obtained for the entire J 1–J 2 diagram, i.e., for any ϕ, J 1 = cosϕ, and J 2 = sinϕ. The structure factor c q and the correlation length ξ at finite temperature are calculated in the entire range of parameters. A radical difference in the behavior of the system in the upper, frustrated (0 ⩽ ϕ ⩽ π), and the lower, nonfrustrated (π ⩽ ϕ ⩽ 2π), regions of the diagram is demonstrated. In the latter region, there is a first-order phase transition that is unique on the phase diagram. For a weakly frustrated antiferromagnet (J 1 > J 2 > 0), the results obtained are compared with the experimental dependence of ξ on temperature and doping level. A correspondence rule is proposed between frustration in a spin model and the doping of an antiferromagnet with holes.

We analyze the possible types of ordering in a boson--fermion model. The Hamiltonian is inherently related to the Bose--Hubbard model for vector two-species bosons in optical lattices. We show that such model can be reduced to the Kugel--Khomskii type spin--pseudospin model, but in contrast to the usual version of the latter model, we are dealing here with the case of spin $S=1$ and pseudospin $1/2$. We show that the interplay of spin and pseudospin degrees of freedom leads to a rather nontrivial magnetic phase diagram including the spin-nematic configurations. Tuning the spin-channel interaction parameter $U_s$ gives rise to quantum phase transitions. We find that the ground state of the system always has the pseudospin domain structure. On the other hand, the sign change of $U_s$ switches the spin arrangement of the ground state within domains from ferro- to aniferromagnetic one. Finally, we revisit the spin (pseudospin)-1/2 Kugel--Khomskii model and see the inverse picture of phase transitions., Comment: 8 pages, 9 figures (in press), Phys. Rev. B, 2017

The ground state of the J 1-J 2 Heisenberg model with arbitrary signs of exchange is studied for spin S = 1/2 in the case of the two-dimensional (2D) square lattice. The states with different types of spin long-range order (antiferromagnetic checkerboard, stripe, collinear ferromagnetic) as well as the disordered spin liquid states are described in the framework of one analytical approach. In particular, it is shown that the phase transition between the ferromagnetic spin liquid and the ferromagnet with long-range order is of the second order. In the vicinity of such transition, we have found the ferromagnetic state with a rapidly varying condensate function.

For the frustrated two-dimensional $S=1/2$ antiferromagnetic Heisenberg model close to quantum phase transition we consider the singlet ground states retaining both translational and SU(2) symmetry. Besides usually discussed checkerboard, spin-liquid and stripe states an unconventional state with two coexisting long-range orders appears to be possible at sufficiently large damping of spin excitations. The problem is treated in the frames of self-consistent spherically symmetric approach., Comment: 11 pages, 5 figures

The self-consistent treatment of real and imaginary renormalizations in the dynamic spin susceptibility χ ( q , ω ) for the frustrated Heisenberg model reproduces for cuprates at low doping: a spin spectrum ω q , a saddle point for q ≈ ( π / 2 , π / 2 ) , nearly constant q -integrated susceptibility χ 2 D ( ω ) for ω ≲ 150 meV and a scaling law for χ 2 D ( ω ) . Frustration increase (optimally doped case) leads to a stripe scenario with an ω q -saddle point at q ≈ ( π ; π / 2 ) and χ 2 D ( ω ) peak at ω ≈ 30 meV . The obtained χ ( q , ω ) describes neutron scattering results and leads to well-known temperature transport anomalies in doped cuprates.