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SU(2) spin-rotation symmetric tensor network state for spin-1/2 Heisenberg model on kagome lattice and its modular matrices

Jia‐Wei Mei, Ji-Yao Chen, Huan He, Xiao-Gang Wen

Published Jun 30, 2016• 2 citations
OpenAlex

Abstract

We apply symmetric tensor network state (TNS) to study the nearest neighbor spin-1/2 antiferromagnetic Heisenberg model on kagome lattice. We develop a method to keep track of symmetries in TNS update procedure and coarse-graining tensor renormalization group (TRG). The symmetric update generates symmetric TNS that has $SU(2)$ spin rotation, time reversal and translation symmetries. The variational energy (per site) is -0.4355 for TNS with bond dimension $D=20$. The global SU(2) spin-rotation symmetry implies $\mathbb{Z}_2$ gauge symmetry in TNS. We use symmetric TRG to compute the modular matrices and confirm that $\mathbb{Z}_2$ gauge symmetry deconfines and the SU(2) spin-rotation symmetric TNS we find has the toric code topological order.

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Physical Review Letters•Aug 7, 2012•635 citations
We apply symmetric tensor network state (TNS) to study the nearest neighbor spin-1/2 antiferromagnetic Heisenberg model on kagome lattice. We develop a method to keep track of symmetries in TNS update procedure and coarse-graining tensor renormalization group (TRG). The symmetric update generates symmetric TNS that has SU(2)SU(2)SU(2) spin rotation, time reversal and translation symmetries. The variational energy (per site) is -0.4355 for TNS with bond dimension D=20D=20D=20. The global SU(2) spin-rotation symmetry implies Z2\mathbb{Z}_2Z2​ gauge symmetry in TNS. We use symmetric TRG to compute the modular matrices and confirm that Z2\mathbb{Z}_2Z2​ gauge symmetry deconfines and the SU(2) spin-rotation symmetric TNS we find has the toric code topological order.
We apply the symmetric tensor network state (TNS) to study the nearest-neighbor spin-1/2 antiferromagnetic Heisenberg model on the kagome lattice. Our method keeps track of the global and gauge symmetries in the TNS update procedure and in tensor renormalization group (TRG) calculations. We also introduce a very sensitive probe for the gap of the ground state---the modular matrices, which can also determine the topological order if the ground state is gapped. We find that the ground state of the Heisenberg model on the kagome lattice is a gapped spin liquid with the Z2{\mathbb{Z}}_{2}Z2​ topological order (or toric code type), which has a long correlation length $\ensuremath{\xi}\ensuremath{\sim}10$ unit cells. We justify that the TRG method can handle very large systems with thousands of spins. Such a long $\ensuremath{\xi}$ explains the gapless behaviors observed in simulations on smaller systems with less than 300 spins or shorter than the length of 10 unit cells. We also discuss experimental implications of the topological excitations encoded in our symmetric tensors.
We apply the symmetric tensor network state (TNS) to study the nearest-neighbor spin-1/2 antiferromagnetic Heisenberg model on the kagome lattice. Our method keeps track of the global and gauge symmetries in the TNS update procedure and in tensor renormalization group (TRG) calculations. We also introduce a very sensitive probe for the gap of the ground state—the modular matrices, which can also determine the topological order if the ground state is gapped. We find that the ground state of the Heisenberg model on the kagome lattice is a gapped spin liquid with the Z[subscript 2] topological order (or toric code type), which has a long correlation length ξ∼10 unit cells. We justify that the TRG method can handle very large systems with thousands of spins. Such a long ξ explains the gapless behaviors observed in simulations on smaller systems with less than 300 spins or shorter than the length of 10 unit cells. We also discuss experimental implications of the topological excitations encoded in our symmetric tensors.
We investigate the ground-state properties of a spin-1 kagome antiferromagnetic Heisenberg model using tensor-network (TN) methods. We obtain the energy per site ${e}_{0}=\ensuremath{-}1.410\phantom{\rule{0.16em}{0ex}}90(2)$, with D∗=8{D}^{*}=8D∗=8 multiplets retained (i.e., a bond dimension of D=24)D=24)D=24), and ${e}_{0}=\ensuremath{-}1.4116(4)$ from large-DDD extrapolation, by accurate TN calculations directly in the thermodynamic limit. The symmetry between the two kinds of triangles is spontaneously broken, with a relative energy difference of $\ensuremath{\delta}\ensuremath{\approx}19%$, i.e, there is a trimerization (simplex) valence-bond order in the ground state. The spin-spin, dimer-dimer, and chirality-chirality correlation functions are found to decay exponentially with a rather short correlation length, showing that the ground state is gapped. We thus identify the ground state to be a simplex valence-bond crystal. We also discuss the spin-1 bilinear-biquadratic Heisenberg model on a kagome lattice, and determine its ground-state phase diagram. Moreover, we implement non-Abelian symmetries, here spin SU(2), in the TN algorithm, which improves the efficiency greatly and provides insight into the tensor structures.
We perform a density-matrix renormalization group (DMRG) study of the S=1/2 Heisenberg antiferromagnet on the kagome lattice to identify the conjectured spin liquid ground state. Exploiting SU(2) spin symmetry, which allows us to keep up to 16,000 DMRG states, we consider cylinders with circumferences up to 17 lattice spacings and find a spin liquid ground state with an estimated per site energy of -0.4386(5), a spin gap of 0.13(1), very short-range decay in spin, dimer and chiral correlation functions and finite topological entanglement gamma consistent with gamma=ld 2, ruling out gapless, chiral or non-topological spin liquids. At the same time, DMRG results provide strong evidence for a gapped topological Z2 spin liquid.
We study the antiferromagnetic kagome Heisenberg model with an additional scalar-chiral interaction by using the infinite projected entangled-pair state (iPEPS) ansatz. We discuss in detail the implementation of the optimization algorithm in the framework of the single-layer tensor network based on the corner transfer matrix technique. Our benchmark based on the full-update algorithm shows that the single-layer algorithm is stable, which leads to the same level of accuracy as the double-layer ansatz but with much less computation time. We further apply this algorithm to study the nature of the kagome Heisenberg model with a scalar-chiral interaction by computing the bond dimension scaling of magnetization, bond energy difference, chiral order parameter, and correlation length. In particular, we find that for strong chiral coupling the correlation length, which is extracted from the transfer matrix, saturates to a finite value for large bond dimension, representing a gapped spin-liquid state. Further comparison with density matrix renormalization group results supports that our iPEPS faithfully represents the time-reversal symmetry breaking chiral state. Our iPEPS simulation results shed new light on constructing PEPS for describing gapped chiral topological states.
We perform a density-matrix renormalization group (DMRG) study of the S=1/2 Heisenberg antiferromagnet on the kagome lattice to identify the conjectured spin liquid ground state. Exploiting SU(2) spin symmetry, which allows us to keep up to 16,000 DMRG states, we consider cylinders with circumferences up to 17 lattice spacings and find a spin liquid ground state with an estimated per site energy of -0.4386(5), a spin gap of 0.13(1), very short-range decay in spin, dimer and chiral correlation functions, and finite topological entanglement γ consistent with γ=log(2)2, ruling out gapless, chiral, or nontopological spin liquids in favor of a topological spin liquid of quantum dimension 2, with strong evidence for a gapped topological Z(2) spin liquid.