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arXiv:1605.03274 · Physical review. B./Physical review. B

Classification of trivial spin-1 tensor network states on a square lattice

Hyun‐Yong Lee, Jung Hoon Han

Published Sep 23, 2016• 7 citations
arXivHTMLPDF

Abstract

Classification of possible quantum spin liquid (QSL) states of interacting spin-1/2's in two dimensions has been a fascinating topic of condensed matter for decades, resulting in enormous progress in our understanding of low-dimensional quantum matter. By contrast, relatively little work exists on the identification, let alone classification, of QSL phases for spin-1 systems in dimensions higher than one. Employing the powerful ideas of tensor network theory and its classification, we develop general methods for writing QSL wave functions of spin-1 respecting all the lattice symmetries, spin rotation, and time reversal with trivial gauge structure on the square lattice. We find ${2}^{5}$ distinct classes characterized by five binary quantum numbers. Several explicit constructions of such wave functions are given for bond dimensions $D$ ranging from two to four, along with thorough numerical analyses to identify their physical characters. Both gapless and gapped states are found. The topological entanglement entropy of the gapped states is close to zero, indicative of topologically trivial states. In $D=4$, several different tensors can be linearly combined to produce a family of states within the same symmetry class. A rich ``phase diagram'' can be worked out among the phases of these tensors, as well as the phase transitions among them. Among the states we identified in this putative phase diagram is the plaquette-ordered phase, gapped resonating valence bond phase, and a critical phase. A continuous transition separates the plaquette-ordered phase from the resonating valence bond phase.

DOI
10.1103/physrevb.94.115150

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Classification of possible quantum spin liquid (QSL) states of interacting spin-1/2's in two dimensions has been a fascinating topic of condensed matter for decades, resulting in enormous progress in our understanding of low-dimensional quantum matter. By contrast, relatively little work exists on the identification, let alone classification, of QSL phases for spin-1 systems in dimensions higher than one. Employing the powerful ideas of tensor network theory and its classification, we develop general methods for writing QSL wave functions of spin-1 respecting all the lattice symmetries, spin rotation, and time reversal with trivial gauge structure on the square lattice. We find 25{2}^{5}25 distinct classes characterized by five binary quantum numbers. Several explicit constructions of such wave functions are given for bond dimensions DDD ranging from two to four, along with thorough numerical analyses to identify their physical characters. Both gapless and gapped states are found. The topological entanglement entropy of the gapped states is close to zero, indicative of topologically trivial states. In D=4D=4D=4, several different tensors can be linearly combined to produce a family of states within the same symmetry class. A rich ``phase diagram'' can be worked out among the phases of these tensors, as well as the phase transitions among them. Among the states we identified in this putative phase diagram is the plaquette-ordered phase, gapped resonating valence bond phase, and a critical phase. A continuous transition separates the plaquette-ordered phase from the resonating valence bond phase.
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We elaborate a simple classification scheme of all rank-5 SU(2) spin rotational symmetric tensors according to (i) the onsite physical spin SSS, (ii) the local Hilbert space ${V}^{\ensuremath{\bigotimes}4}$ of the four virtual (composite) spins attached to each site, and (iii) the irreducible representations of the C4v{C}_{4v}C4v​ point group of the square lattice. We apply our scheme to draw a complete list of all SU(2)-symmetric translationally and rotationally invariant projected entangled pair states (PEPS) with bond dimension $D\ensuremath{\le}6$. All known SU(2)-symmetric PEPS on the square lattice are recovered and simple generalizations are provided in some cases. More generally, to each of our symmetry class can be associated a $(\mathcal{D}\ensuremath{-}1)\text{-dimensional}$ manifold of spin liquids (potentially) preserving lattice symmetries and defined in terms of D-independent\mathcal{D}\text{-independent}D-independent tensors of a given bond dimension DDD. In addition, generic (low-dimensional) families of PEPS explicitly breaking either (i) particular point-group lattice symmetries (lattice nematics) or (ii) time-reversal symmetry (chiral spin liquids) or (iii) SU(2) spin rotation symmetry down to U(1)\text{U}(1)U(1) (spin nematics or N\'eel antiferromagnets) can also be constructed. We apply this framework to search for new topological chiral spin liquids characterized by well-defined chiral edge modes, as revealed by their entanglement spectrum. In particular, we show how the symmetrization of a double-layer PEPS leads to a chiral topological state with a gapless edge described by a SU(2)2\mathrm{SU}{(2)}_{2}SU(2)2​ Wess-Zumino-Witten model.
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.
The quantum spin liquid is an emergent state of matter, which has attracted a lot of recent attention. I will review recent numerical progress based on the density matrix renormalization calculations in identifying gapped spin liquid in twodimensional frustrated spin systems. I will first focus on extended model with Heisenberg exchange couplings on kagome lattice and demonstrate a topological state with fractionalized spinon and emergent gauge field clearly shown in numerical simulations. I will present concrete results on the quantum phase diagram of the extended kagome Heisenberg model, and compare that with the phase diagrams of the square and honeycomb lattice models with the dominant plaquette valence bond phase in nonmagnetic region. I will discuss numerical effort and theoretical challenge in fully pinning down the nature of the gapped topological phase, and also the nature of the quantum phase transitions in these Heisenberg systems.
A one-parameter family of long-range resonating-valence-bond (RVB) state on the square lattice was previously proposed to describe a critical spin liquid (SL) phase of the spin-1/2 frustrated Heisenberg model. We provide evidence that this RVB state in fact also realizes a topological (long-range entangled) Z2{\mathbb{Z}}_{2}Z2​ SL, limited by two transitions to critical SL phases. The topological phase is naturally connected to the Z2{\mathbb{Z}}_{2}Z2​ gauge symmetry of the local tensor. This Rapid Communication shows that, on one hand, spin-1/2 topological SL with C4v{C}_{4v}C4v​ point-group symmetry and SU(2)SU(2)SU(2) spin rotation symmetry exists on the square lattice and, on the other hand, criticality and nonbipartiteness are compatible. We also point out that strong similarities between our phase diagram and the ones of classical interacting dimer models suggest both can be described by similar Kosterlitz-Thouless transitions. This scenario is further supported by the analysis of the one-dimensional boundary state. Forms of parent Hamiltonians hosting the Z2{\mathbb{Z}}_{2}Z2​ SL are suggested.
Investigation of the J1J_1J1​-J2J_2J2​ Heisenberg model on the triangular lattice: A study with projected entangled-pair states
The nature of the quantum spin liquid (QSL) phase in the frustrated J1J_1J1​-J2J_2J2​ Heisenberg model on the triangular lattice remains an open and actively debated problem. In this work, we employ the infinite projected entangled-pair state (PEPS) to systematically investigate the model under different symmetry constraints. Our simulations reveal a direct transition from the 120∘120^\circ120∘ Néel state to a putative QSL at J2/J1≈0.08J_2/J_1\approx 0.08J2​/J1​≈0.08, signaled by the collapse of magnetic order. We further show that, through either an appropriate unitary rotation or spontaneous spin long-range order, the stripe antiferromagnetic phase can also be accurately captured within the infinite PEPS framework. A central focus of our study is the role played by the PEPS symmetry in approximating the QSL ground-state sandwiched between the two magnetic phases. We first found that a fully-symmetric topological Z2\mathbb{Z}_2Z2​ Resonating Valence Bond state, which can be written as a simple PEPS with bond dimension D=3D=3D=3, exhibits a reasonably good variational energy. Motivated by this finding, we have further constructed generic Z2\mathbb{Z}_2Z2​-symmetric PEPS of larger bond dimension (up to D=7D=7D=7). We found that, under wavefunction optimization, spinons condense and, simultaneously, topological vison excitations get confined, hence precluding Z2\mathbb{Z}_2Z2​ topological order. This strongly indicates the gapless (or critical) nature of the QSL phase, which is most naturally consistent with a U(1) Dirac spin liquid scenario.