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arXiv:1508.07189 · Physical review. B./Physical review. B

Spin- S kagome quantum antiferromagnets in a field with tensor networks

Thibaut Picot, Marc Ziegler, Román Orús, Didier Poilblanc

Published Feb 19, 2016• 109 citations
arXivHTMLPDF

Abstract

Spin-$S$ Heisenberg quantum antiferromagnets on the kagome lattice offer, when placed in a magnetic field, a fantastic playground to observe exotic phases of matter with (magnetic analogs of) superfluid, charge, bond, or nematic orders, or a coexistence of several of the latter. In this context, we have obtained the (zero-temperature) phase diagrams up to $S=2$ directly in the thermodynamic limit owing to infinite projected entangled pair states, a tensor network numerical tool. We find incompressible phases characterized by a magnetization plateau versus field and stabilized by spontaneous breaking of point group or lattice translation symmetry(ies). The nature of such phases may be semiclassical, as the plateaus at the $\frac{1}{3}\mathrm{th},\left(1\ensuremath{-}\frac{2}{9S}\right)\mathrm{th}$, and $\left(1\ensuremath{-}\frac{1}{9S}\right)\mathrm{th}$ of the saturated magnetization (the latter followed by a macroscopic magnetization jump), or fully quantum as the spin-$\frac{1}{2}$ $\frac{1}{9}$ plateau exhibiting a coexistence of charge and bond orders. Upon restoration of the spin rotation $U(1)$ symmetry, a finite compressibility appears, although lattice symmetry breaking persists. For integer spin values we also identify spin gapped phases at low enough fields, such as the $S=2$ (topologically trivial) spin liquid with no symmetry breaking, neither spin nor lattice.

DOI
10.1103/physrevb.93.060407

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Spin-SSS Heisenberg quantum antiferromagnets on the kagome lattice offer, when placed in a magnetic field, a fantastic playground to observe exotic phases of matter with (magnetic analogs of) superfluid, charge, bond, or nematic orders, or a coexistence of several of the latter. In this context, we have obtained the (zero-temperature) phase diagrams up to S=2S=2S=2 directly in the thermodynamic limit owing to infinite projected entangled pair states, a tensor network numerical tool. We find incompressible phases characterized by a magnetization plateau versus field and stabilized by spontaneous breaking of point group or lattice translation symmetry(ies). The nature of such phases may be semiclassical, as the plateaus at the 13th,(1\ensuremath−29S)th\frac{1}{3}\mathrm{th},\left(1\ensuremath{-}\frac{2}{9S}\right)\mathrm{th}31​th,(1\ensuremath−9S2​)th, and (1\ensuremath−19S)th\left(1\ensuremath{-}\frac{1}{9S}\right)\mathrm{th}(1\ensuremath−9S1​)th of the saturated magnetization (the latter followed by a macroscopic magnetization jump), or fully quantum as the spin-12\frac{1}{2}21​ 19\frac{1}{9}91​ plateau exhibiting a coexistence of charge and bond orders. Upon restoration of the spin rotation U(1)U(1)U(1) symmetry, a finite compressibility appears, although lattice symmetry breaking persists. For integer spin values we also identify spin gapped phases at low enough fields, such as the S=2S=2S=2 (topologically trivial) spin liquid with no symmetry breaking, neither spin nor lattice.
Optimizing translationally invariant infinite-projected entangled pair states, we investigate the spin-2 Affleck-Kennedy-Lieb-Tasaki and spin-1 Heisenberg models on the kagome lattice as a function of magnetic field. We found that the magnetization curves offer a wide variety of compressible and incompressible phases. Incompressible nematic phases breaking the lattice C3{C}_{3}C3​ rotation, for which we propose simple qualitative pictures, give rise to magnetization plateaus at reduced magnetization mz=56{m}_{z}=\frac{5}{6}mz​=65​ and 13\frac{1}{3}31​ for spin-2 and -1, respectively, in addition to the mz=0{m}_{z}=0mz​=0 plateaus characteristic of zero-field gapped spin liquids. Moving away from the plateaus, we observe a rich variety of compressible superfluid nematic, named ``supernematic,'' phases breaking spontaneously both point group and spin-U(1) symmetries, as well as a superfluid phase preserving lattice symmetries. We also identify the nature, continuous or first order, of the various phase transitions. Possible connections to experimental spin-1 systems are discussed.
We study the ground-state phase diagram of the spin-1/21/21/2 antiferromagnetic Heisenberg model on the square-kagome lattice using infinite projected entangled-pair states (iPEPS). By systematically varying the ratio of exchange couplings on triangular and square plaquettes, we establish a complete quantum phase diagram in the thermodynamic limit. In the intermediate-coupling regime, we identify four distinct nonmagnetic phases that are unambiguously characterized as valence-bond crystals (VBCs) by their symmetry-inequivalent patterns of strong and weak spin-spin correlations. These include a plaquette crossed-dimer VBC, a loop-six VBC stabilized around the isotropic point, a generalized pinwheel VBC with reduced rotational symmetry, and a decorated loop-six VBC proximate to ferrimagnetic order. We determine the phase boundaries using a combination of bond-resolved correlation functions, entanglement entropy, and magnetization. For transitions not accompanied by sharp entanglement signatures, we extract the spin gap from finite-field simulations, allowing us to distinguish gapped and gapless VBC phases. At larger coupling ratios, the system undergoes transitions into imperfect and perfect ferrimagnetic states. Our results resolve long-standing ambiguities in the square-kagome Heisenberg model and provide a quantitatively reliable reference for future theoretical and experimental studies of frustrated quantum magnets.
We perform a systematic study of the antiferromagnetic Heisenberg model on the Husimi lattice using numerical tensor-network methods based on projected entangled simplex states. The nature of the ground state varies strongly with the spin quantum number SSS. For S=12S=\frac{1}{2}S=21​, it is an algebraic (gapless) quantum spin liquid. For S=1S=1S=1, it is a gapped, nonmagnetic state with spontaneous breaking of triangle symmetry (a trimerized simplex-solid state). For S=2S=2S=2, it is a simplex-solid state with a spin gap and no symmetry breaking; both integer-spin simplex-solid states are characterized by specific degeneracies in the entanglement spectrum. For S=32S=\frac{3}{2}S=23​, and indeed for all spin values $S\ensuremath{\ge}\frac{5}{2}$, the ground states have ${120}^{\ensuremath{\circ}}$ antiferromagnetic order. In a finite magnetic field, we find that, irrespective of the value of SSS, there is always a plateau in the magnetization at m=13m=\frac{1}{3}m=31​.
The nature of the 1/9-magnetization plateau of the spin-12\frac{1}{2}21​ kagome Heisenberg antiferromagnet remains controversial due to the exotic physical properties and high complexity induced by the geometrical frustration. Instead of a Z3{\mathbb{Z}}_{3}Z3​ quantum spin liquid revealed on a cylinder, we show on an infinite-size lattice that the 1/9 plateau can be described by a valence bond crystal (VBC) that breaks spatial translational invariance. Consistent results are achieved by two accurate tensor network methods, namely the full-update infinite projected-entangled pair states and the projected-entangled simplex states. The VBC exhibits an hourglass pattern with $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ spatial symmetry, demonstrated by magnetizations, bond energies, and three-body correlators. The spatial inversion symmetry in the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ VBC is instantly broken with the presence of the difference between the coupling strengths in the up and down triangles, suggesting the existence of gapless excitations. The gapless nature of the 1/9 plateau is further indicated by the scaling behaviors of the entanglement entropy and the correlation length, which indicate a c=1c=1c=1 conformal field theory.
We study numerically the spin-1/2 XXZ model in a field on an infinite kagome lattice. We use different algorithms based on infinite projected entangled pair states (iPEPSs) for this, namely, (i) an approach with simplex tensors and a 9-site unit cell, and (ii) an approach based on coarse-graining three spins in the kagome lattice and mapping it to a square-lattice model with local and nearest-neighbor interactions, with the usual PEPS tensors, 6- and 12-site unit cells. Similarly to our previous calculation at the SU(2)-symmetric point (Heisenberg Hamiltonian), for any anisotropy from the Ising limit to the XY limit, we also observe the emergence of magnetization plateaus as a function of the magnetic field, at mz=13{m}_{z}=\frac{1}{3}mz​=31​ using 6-, 9-, and 12-site PEPS unit cells, and at mz=19,59{m}_{z}=\frac{1}{9},\phantom{\rule{0.28em}{0ex}}\frac{5}{9}mz​=91​,95​, and 79\frac{7}{9}97​ using a 9-site PEPS unit cell, the latter setup being able to accommodate $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ solid order. We also find that, at mz=13{m}_{z}=\frac{1}{3}mz​=31​, (lattice) nematic and $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ VBC-order states are degenerate within the accuracy of the nine-site simplex method, for all anisotropy. The 6- and 12-site coarse-grained PEPS methods produce almost-degenerate nematic and $1\ifmmode\times\else\texttimes\fi{}2$ VBC-solid orders. We also find that, within our accuracy, the six-site coarse-grained PEPS method gives slightly lower energies, which can be explained by the larger amount of entanglement this approach can handle, even in cases where the PEPS unit cell is not commensurate with the expected ground-state unit cell. Furthermore, we do not observe chiral spin liquid behaviors at and close to the XY point, as has been recently proposed. Our results are the first tensor network investigations of the XXZ model in a field and reveal the subtle competition between nearby magnetic orders in numerical simulations of frustrated quantum antiferromagnets, as well as the delicate interplay between energy optimization and symmetry in tensor network numerical simulations.
Due to strong geometric frustration and quantum fluctuation, the S=1/2S=1/2S=1/2 quantum Heisenberg antiferromagnet on the kagome\text{kagome}kagome lattice has long been considered as an ideal platform to realize a spin liquid (SL), a phase exhibiting fractionalized excitations without any symmetry breaking. A recent numerical study (Yan et al., e-print arXiv:1011.6114) of the Heisenberg S=1/2,kagomeS=1/2,\text{kagome}S=1/2,kagome lattice model (HKLM) shows, in contrast to earlier results, that the ground state is a singlet-gapped SL with signatures of Z2{\mathbb{Z}}_{2}Z2​ topological order. Motivated by this numerical discovery, we use the projective symmetry group to classify all 20 possible Schwinger fermion mean-field states of Z2{\mathbb{Z}}_{2}Z2​ SLs on the kagome\text{kagome}kagome lattice. Among them we found only one gapped Z2{\mathbb{Z}}_{2}Z2​ SL (which we call the ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state) in the neighborhood of the U(1) Dirac SL state. Since its parent state, i.e., the U(1) Dirac SL, was found [Ran et al., Phys. Rev. Lett. 98, 117205 (2007)] to be the lowest among many other candidate U(1) SLs, including the uniform resonating-valence-bond states, we propose this ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state to be the numerically discovered SL ground state of the HKLM.