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arXiv:1406.6841 · Physical Review B

Striped critical spin liquid in a spin-orbital entangled RVB state in a projected entangled-pair state representation

Piotr Czarnik, Jacek Dziarmaga

Published Jan 5, 2015• 5 citations
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Abstract

We introduce a spin-orbital entangled (SOE) resonating valence bond (RVB) state on a square lattice of spins $\frac{1}{2}$ and orbitals represented by pseudospins $\frac{1}{2}$. Like the standard RVB state, it is a superposition of nearest-neighbor hard-core coverings of the lattice by spin singlets, but adjacent singlets are favored to have perpendicular orientations and, more importantly, an orientation of each singlet is entangled with orbitals state on its two lattice sites. The SOE-RVB state can be represented by a projected entangled pair state (PEPS) with a bond dimension $D=4$. This representation helps to reveal that the state is a superposition of striped coverings conserving a topological quantum number. The stripes are a critical quantum spin liquid. We propose a spin-orbital Hamiltonian supporting a SOE-RVB ground state.

DOI
10.1103/physrevb.91.045101

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We introduce a spin-orbital entangled (SOE) resonating valence bond (RVB) state on a square lattice of spins 12\frac{1}{2}21​ and orbitals represented by pseudospins 12\frac{1}{2}21​. Like the standard RVB state, it is a superposition of nearest-neighbor hard-core coverings of the lattice by spin singlets, but adjacent singlets are favored to have perpendicular orientations and, more importantly, an orientation of each singlet is entangled with orbitals state on its two lattice sites. The SOE-RVB state can be represented by a projected entangled pair state (PEPS) with a bond dimension D=4D=4D=4. This representation helps to reveal that the state is a superposition of striped coverings conserving a topological quantum number. The stripes are a critical quantum spin liquid. We propose a spin-orbital Hamiltonian supporting a SOE-RVB ground state.
We generalize the construction of the spin-1/2 SU(2) resonating valence bond (RVB) state to the case of the self-conjugate 6\mathbf{6}6 representation of SU(4). As for the case of SU(2) [J.-Y. Chen and D. Poilblanc, Phys. Rev. B 97, 161107(R) (2018)], we use the projected entangled pair state (PEPS) formalism to derive a simple (two-dimensional) family of generalized SU(4) RVB states on the square lattice. We show that, when longer-range SU(4)-singlet bonds are included, a local gauge symmetry is broken down from U(1) to Z2{\mathbb{Z}}_{2}Z2​, leading to the emergence of a short-range spin liquid. Evidence for the topological nature of this spin liquid is provided by the investigation of the Renyi entanglement entropy of infinitely long cylinders and of the modular matrices. Relevance to microscopic models and experiments of ultracold atoms is discussed.
We construct a family of simple fermionic projected entangled pair states (fPEPS) on the square lattice with bond dimension D=3D=3D=3 which are exactly hole-doped resonating valence bond (RVB) wave functions with short-range singlet bonds. Under doping the insulating RVB spin liquid evolves immediately into a superconductor with mixed d+isd+isd+is pairing symmetry whose pair amplitude grows as the square root of the doping. The relative weight between sss-wave and ddd-wave components can be controlled by a single variational parameter ccc. We optimize our ansatz with respect to ccc for the frustrated $t\text{\ensuremath{-}}{J}_{1}\text{\ensuremath{-}}{J}_{2}$ model (including both nearest and next-nearest neighbor antiferromagnetic interactions J1{J}_{1}J1​ and J2{J}_{2}J2​, respectively) for ${J}_{2}\ensuremath{\simeq}{J}_{1}/2$ and obtain an energy very close to the infinite-PEPS state (using full update optimization and same bond dimension). The orbital symmetry of the optimized RVB superconductor has predominant ddd-wave character, although we argue a residual (complex sss-wave) time reversal symmetry breaking component should always be present. Connections of the results to the physics of superconducting cuprates and pnictides are outlined.
We construct a short-range resonating-valence-bond (RVB) state on the ruby lattice, using projected entangled-pair states with bond dimension D=3D=3D=3. By introducing nonlocal moves to the dimer patterns on the torus, we distinguish four distinct sectors in the space of dimer coverings, which is a signature of the topological nature of the RVB wave function. Furthermore, by calculating the reduced density matrix of a bipartition of the RVB state on an infinite cylinder and exploring its entanglement entropy, we confirm the topological nature of the RVB wave function by obtaining a nonzero topological contribution, $\ensuremath{\gamma}=\ensuremath{-}\mathrm{ln}\phantom{\rule{4pt}{0ex}}2$, consistent with that of a Z2{\mathbb{Z}}_{2}Z2​ topological quantum spin liquid. We also calculate the ground-state energy of the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg model on the ruby lattice and compare it with the RVB energy. Finally, we construct a quantum dimer model for the ruby lattice and discuss it as a possible parent Hamiltonian for the RVB wave function.
Topological chiral phases are ubiquitous in the physics of the fractional quantum Hall effect. Nonchiral topological spin liquids are also well known. Here, using the framework of projected entangled pair states (PEPS), we construct a family of chiral spin liquids on the square lattice, which are generalized spin-1/2 resonating valence bond (RVB) states obtained from deformed local tensors with d+idd+i\phantom{\rule{0.16em}{0ex}}dd+id symmetry. On a cylinder, we construct four topological sectors with even or odd number of spinons on the boundary and even or odd number of (Z2{\mathbb{Z}}_{2}Z2​) fluxes penetrating the cylinder, which, we argue, remain orthogonal in the limit of infinite perimeter. The analysis of the transfer matrix provides evidence of short-range (long-range) triplet (singlet) correlations as for the critical (nonchiral) RVB state. The entanglement spectrum exhibits chiral edge modes, which we confront to predictions of the conformal field theory, and the corresponding entanglement Hamiltonian is shown to be long-ranged.
We use Monte Carlo simulations to study properties of Anderson's resonating-valence-bond (RVB) spin-liquid state on the square lattice (i.e., the equal superposition of all pairing of spins into nearest-neighbor singlet pairs) and compare with the classical dimer model (CDM). The latter system also corresponds to the ground state of the Rokhsar-Kivelson quantum dimer model at its critical point. We find that, although spin-spin correlations decay exponentially in the RVB, four-spin valence-bond-solid correlations are critical, qualitatively like the well-known dimer-dimer correlations of the CDM, but decaying more slowly (as $1/{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}\ensuremath{\approx}1.20$, compared with $\ensuremath{\alpha}=2$ for the CDM). We also compute the distribution of monomer (defect) pair separations, which decay by a larger exponent in the RVB than in the CDM. We further study both models in their different winding-number sectors and evaluate the relative weights of different sectors. Like the CDM, all the observed RVB behaviors can be understood in the framework of a mapping to a ``height'' model characterized by a gradient-squared stiffness constant KKK. Four independent measurements consistently show a value ${K}_{\mathrm{RVB}}\ensuremath{\approx}1.6{K}_{\mathrm{CDM}}$, with the same kinds of numerical evaluations of KCDM{K}_{\mathrm{CDM}}KCDM​ giving results in agreement with the rigorously known value ${K}_{\mathrm{CDM}}=\ensuremath{\pi}/16$. The background of a nonzero winding-number gradient W/LW/LW/L introduces spatial anisotropies and an increase in the effective KKK, both of which can be understood as a consequence of anharmonic terms in the height-model free energy, which are of relevance to the recently proposed scenario of ``Cantor deconfinement'' in extended quantum dimer models. In addition to the standard case of short bonds only, we also studied ensembles in which fourth-neighbor (bipartite) bonds are allowed at a density controlled by a tunable fugacity, resulting (as expected) in a smooth reduction of KKK.
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.