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arXiv:1308.3463 · Physical Review B

Field-induced superfluids and Bose liquids in projected entangled pair states

Didier Poilblanc, Norbert Schuch, J. I. Cirac

Published Oct 17, 2013• 13 citations
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Abstract

In two-dimensional incompressible quantum spin liquids, a large enough magnetic field generically induces ``doping'' of polarized $S=1$ triplons or $S=1/2$ spinons. We review a number of cases such as spin-3/2 Affleck-Kennedy-Lieb-Tasaki (AKLT) or spin-1/2 resonating valence bond (RVB) liquids where the projected entangled pair states (PEPS) framework provides very simple and comprehensive pictures. On the bipartite honeycomb lattice, simple PEPS can describe Bose condensed triplon (AKLT) or spinon (RVB) superfluids with transverse staggered (N\'eel) magnetic order. On the kagome lattice, doping the RVB state with deconfined spinons or triplons (i.e., spinon bound pairs) yields uncondensed Bose liquids preserving $U(1)$ spin-rotation symmetry. We find that spinon (triplon) doping destroys (preserves) the topological ${\mathbb{Z}}_{2}$ symmetry of the underlying RVB state. We also find that spinon doping induces longer range interactions in the entanglement Hamiltonian, suggesting the emergence of (additive) log corrections to the entanglement entropy.

DOI
10.1103/physrevb.88.144414

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In two-dimensional incompressible quantum spin liquids, a large enough magnetic field generically induces ``doping'' of polarized S=1S=1S=1 triplons or S=1/2S=1/2S=1/2 spinons. We review a number of cases such as spin-3/2 Affleck-Kennedy-Lieb-Tasaki (AKLT) or spin-1/2 resonating valence bond (RVB) liquids where the projected entangled pair states (PEPS) framework provides very simple and comprehensive pictures. On the bipartite honeycomb lattice, simple PEPS can describe Bose condensed triplon (AKLT) or spinon (RVB) superfluids with transverse staggered (N'eel) magnetic order. On the kagome lattice, doping the RVB state with deconfined spinons or triplons (i.e., spinon bound pairs) yields uncondensed Bose liquids preserving U(1)U(1)U(1) spin-rotation symmetry. We find that spinon (triplon) doping destroys (preserves) the topological Z2{\mathbb{Z}}_{2}Z2​ symmetry of the underlying RVB state. We also find that spinon doping induces longer range interactions in the entanglement Hamiltonian, suggesting the emergence of (additive) log corrections to the entanglement entropy.
Gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquids have been proposed as candidates for the ground state of the S=1/2S=1/2S=1/2 quantum antiferromagnet on the kagome lattice. We extend the use of projected entangled pair states to construct (on the cylinder) resonating valence bond (RVB) states including both nearest-neighbor and next-nearest-neighbor singlet bonds. Our ansatz---dubbed ``simplex spin liquid''---allows for an asymmetry between the two types of triangles (of order 22%--3%2 in the energy density after optimization) leading to the breaking of inversion symmetry. We show that the topological Z2{\mathbb{Z}}_{2}Z2​ structure is still preserved and, by considering the presence or the absence of spinon and vison lines along an infinite cylinder, we explicitly construct four orthogonal RVB minimally entangled states. The spinon and vison coherence lengths are extracted from a finite size scaling with regard to the cylinder perimeter of the energy splittings of the four sectors and are found to be of the order of the lattice spacing. The entanglement spectrum of a partitioned (infinite) cylinder is found to be gapless, suggesting the occurrence, on a cylinder with real open boundaries, of gapless edge modes formally similar to Luttinger liquid (nonchiral) spin and charge modes. When inversion symmetry is spontaneously broken, the RVB spin liquid exhibits an extra Ising degeneracy, which might have been observed in recent exact diagonalization studies.
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 study the response of critical resonating valence bond (RVB) spin liquids to doping with longer-range singlets, and more generally of U ( 1 ) -symmetric tensor networks to nonsymmetric perturbations. Using a field theory description, we find that in the RVB, doping constitutes a relevant perturbation that immediately opens up a gap, contrary to previous observations. Our analysis predicts a very large correlation length even at significant doping, which we verify using high-accuracy numerical simulations. This emphasizes the need for careful analysis, but it also justifies the use of such states as a variational ansatz for critical systems. Finally, we give an example of a projected entangled pair state where nonsymmetric perturbations do not open up a gap and the U ( 1 ) symmetry reemerges. Published by the American Physical Society 2024
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.
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.
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.