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arXiv:1504.01334 · Europhysics Letters (EPL)

Magnetic quantum phase transitions of the two-dimensional antiferromagnetic J 1 -J 2 Heisenberg model

Tarik P. Cysne, M. B. Silva Neto

Published Nov 1, 2015• 10 citations
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Abstract

We obtain the complete magnetic phase diagram of the two-dimensional antiferromagnetic Heisenberg model, , within the framework of the O ( N ) nonlinear sigma model. We find two magnetically ordered phases, one with Néel order, for , and another with collinear order, for , separated by a nonmagnetic region, for , where a gapped spin liquid is found. The transition at is of the second order while the one at is of the first order and the spin gaps cross at . Our results are exact at N → ∞ and agree with numerical results from different methods.

DOI
10.1209/0295-5075/112/47002

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We obtain the complete magnetic phase diagram of the two-dimensional antiferromagnetic Heisenberg model, , within the framework of the O ( N ) nonlinear sigma model. We find two magnetically ordered phases, one with Néel order, for , and another with collinear order, for , separated by a nonmagnetic region, for , where a gapped spin liquid is found. The transition at is of the second order while the one at is of the first order and the spin gaps cross at . Our results are exact at N → ∞ and agree with numerical results from different methods.
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We study the spin-1/2 Heisenberg model on the square lattice with first- and second-neighbor antiferromagnetic interactions J(1) and J(2), which possesses a nonmagnetic region that has been debated for many years and might realize the interesting Z(2) spin liquid. We use the density matrix renormalization group approach with explicit implementation of SU(2) spin rotation symmetry and study the model accurately on open cylinders with different boundary conditions. With increasing J(2), we find a Néel phase and a plaquette valence-bond (PVB) phase with a finite spin gap. From the finite-size scaling of the magnetic order parameter, we estimate that the Néel order vanishes at J(2)/J(1)≃0.44. For 0.5<J(2)/J(1)<0.61, we find dimer correlations and PVB textures whose decay lengths grow strongly with increasing system width, consistent with a long-range PVB order in the two-dimensional limit. The dimer-dimer correlations reveal the s-wave character of the PVB order. For 0.44<J(2)/J(1)<0.5, spin order, dimer order, and spin gap are small on finite-size systems, which is consistent with a near-critical behavior. The critical exponents obtained from the finite-size spin and dimer correlations could be compatible with the deconfined criticality in this small region. We compare and contrast our results with earlier numerical studies.
Using results for the 4 × 4 and 6 × 6 lattice, we produce the first finite-size scaling analysis of the frustrated Heisenberg model in two dimensions. The results indicate a continuous phase transition from the ordered phase into an intermediate phase without long-range magnetic order, as for the (2 + 1)-dimensional nonlinear sigma-model. The intermediate phase is stable for 0.4 < J2/J1 < 0.65 and exhibits either dimerization or broken chiral symmetry. The transition to the collinear phase at J2/J1 0.65 is apparently of first order.
The two-layer Heisenberg antiferromagnet exhibits a zero temperature quantum phase transition from a disordered dimer phase to a collinear Neel phase, with long range order in the ground state. The spin-wave gap vanishes as ? ? ( J ? – J ? c ) &ngr; approaching the transition point. To account for strong correlations, the S = 1 elementary excitations triplets are described as a dilute Bose gas with infinite on-site repulsion. We apply the Brueckner diagram approach which gives the critical index &ngr; ˜ 0 . 5. We demonstrate also that the linearised in density Brueckner equations give the mean field result &ngr; = 1. Finally, an expansion of the Brueckner equations in powers of the density, combined with the scaling hypothesis, gives &ngr; ˜ 0 . 67. This value agrees reasonably with that of the nonlinear O(3) s model. Our approach demonstrates that for other quantum spin models the critical index can be different from that in the nonlinear s model. We discuss the conditions for this to occur.
We study the zero-temperature phase diagram and the low-lying excitations of a square-lattice spin-half Heisenberg antiferromagnet with two types of regularly distributed nearest-neighbor exchange bonds [J>0[J>0[J>0 (antiferromagnetic) and $\ensuremath{-}\ensuremath{\infty}<{J}^{\ensuremath{'}}<\ensuremath{\infty}]$ using the coupled cluster method (CCM) for high orders of approximation (up to LSUB8). We use a N\'eel model state as well as a helical model state as a starting point for the CCM calculations. We find a second-order transition from a phase with N\'eel order to a finite-gap quantum disordered phase for sufficiently large antiferromagnetic exchange constants ${J}^{\ensuremath{'}}>0.$ For frustrating ferromagnetic couplings ${J}^{\ensuremath{'}}<0$ we find indications that quantum fluctuations favor a first-order phase transition from the N\'eel order to a quantum helical state, by contrast with the corresponding second-order transition in the corresponding classical model. The results are compared to those of exact diagonalizations of finite systems (up to 32 sites) and those of spin-wave and variational calculations. The CCM results agree well with the exact diagonalization data over the whole range of the parameters. The special case of ${J}^{\ensuremath{'}}=0,$ which is equivalent to the honeycomb lattice, is treated more closely.
Strongly correlated systems with geometric frustrations can host the emergent phases of matter with unconventional properties. Here, we study the spin S=1S=1S=1 Heisenberg model on the honeycomb lattice with the antiferromagnetic first- (J1{J}_{1}J1​) and second-neighbor (J2{J}_{2}J2​) interactions ($0.0\ensuremath{\le}{J}_{2}/{J}_{1}\ensuremath{\le}0.5$) by means of density-matrix renormalization group. In the parameter regime ${J}_{2}/{J}_{1}\ensuremath{\lesssim}0.27$, the system sustains a N\'eel antiferromagnetic phase. At the large J2{J}_{2}J2​ side ${J}_{2}/{J}_{1}\ensuremath{\gtrsim}0.32$, a stripe antiferromagnetic phase is found. Between the two magnetic ordered phases $0.27\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.32$, we find a nonmagnetic intermediate region with a plaquette valence-bond order. Although our calculations are limited within a six unit-cell width on cylinder, we present evidence that this plaquette state could be a strong candidate for this nonmagnetic region in the thermodynamic limit. We also briefly discuss the nature of the quantum phase transitions in the system. We gain further insight into the nonmagnetic phases in the spin-1 system by comparing its phase diagram with the spin-1/21/21/2 system.