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arXiv:1309.5669 · Physical Review B

Obtaining topological degenerate ground states by the density matrix renormalization group

Yin-Chen He, D. N. Sheng, Yan Chen

Published Feb 11, 2014• 46 citations
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Abstract

We develop the density matrix renormalization group approach to systematically identify the topological order of the quantum spin liquid (QSL) through adiabatically obtaining different topological degenerate sectors of the QSL on an infinite cylinder. As an application, we study the anisotropic kagome Heisenberg model known for hosting a ${Z}_{2}$ QSL, however no numerical simulations have been able to access all four sectors before. We obtain the complete set of four topological degenerate ground states distinguished by the presence or absence of the spinon and vison quasiparticle line, which fully characterizes the topological nature of the quantum phase. We have also studied the kagome Heisenberg model, which has recently attracted a lot of attention. We find two topological sectors accurately and also estimate various properties of the other topological sectors, where the larger correlation length is found indicating the possible proximity to another phase.

DOI
10.1103/physrevb.89.075110

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We develop the density matrix renormalization group approach to systematically identify the topological order of the quantum spin liquid (QSL) through adiabatically obtaining different topological degenerate sectors of the QSL on an infinite cylinder. As an application, we study the anisotropic kagome Heisenberg model known for hosting a Z2{Z}_{2}Z2​ QSL, however no numerical simulations have been able to access all four sectors before. We obtain the complete set of four topological degenerate ground states distinguished by the presence or absence of the spinon and vison quasiparticle line, which fully characterizes the topological nature of the quantum phase. We have also studied the kagome Heisenberg model, which has recently attracted a lot of attention. We find two topological sectors accurately and also estimate various properties of the other topological sectors, where the larger correlation length is found indicating the possible proximity to another phase.
We apply density matrix renormalization group methods to study the topological nature of quan- tum spin liquid (QSL). A systematical and controlled way is proposed to get different topological sectors of QSL on an infinite cylinder. Applying our method to easy axis kagome model where a Z2 QSL is expected, all the four topological degenerate states are straightforwardly found. The more tough and interesting model, kagome Heisenberg model (KHM) is also studied. In KHM, we can only get two topological sectors exactly. However, the obtained results allow us to estimate various properties of other topological sectors, where we find that the new topological sector has much larger correlation length comparing to the lowest energy sector.
Topological Spin Liquid Phase in the Spin-1/21/21/2 Triangular Heisenberg Model
We study the spin-1/2 Heisenberg model on the triangular lattice with the antiferromagnetic first (J1) and second nearest-neighbor (J2) interactions using density matrix renormalization group. By studying the spin correlation function, we find a 120◦ magnetic order phase for J2 . 0.08J1 and a stripe antiferromagnetic phase for J2 & 0.16J2. Between these two phases, we identify a gapped spin liquid phase characterized by the exponential decaying spin and dimer correlations, as well as finite spin singlet and triplet excitation gaps. We also find two topological degenerating ground states in two topological sectors. Furthermore, we detect the signature of the fractionalization by following the evolution of different ground states with inserting spin flux into the cylinder system. While the sector with spinon is found to respect the time reveral symmetry, the even sector without a spinon may break such a symmetry. We discuss the implications of numerical results for the nature of the topological spin liquid phase.
It is generally believed that the spin- 1 2 triangular-lattice J 1 − J 2 Heisenberg model hosts a quantum spin liquid in the intermediate regime between the 120 ∘ and stripe ordered phases. Density matrix renormalization group studies on cylinders have consistently found two nearly degenerate ground states, commonly interpreted as distinct topological sectors. Using state-of-the-art matrix product state simulations on YC6 cylinders, we compare the static and dynamical properties of these two sectors at J 2 / J 1 = 0.125 . Noticeable differences appear already in static correlations; moreover, high-resolution dynamical structure factors reveal qualitatively distinct low-energy excitations. These results suggest that the two ground states cannot be understood as merely topologically distinct sectors of a gapped Z 2 spin liquid.
Using density-matrix renormalization-group calculations for infinite cylinders, we elucidate the properties of the spin-liquid phase of the spin-$\frac{1}{2}\phantom{\rule{4pt}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the triangular lattice. We find four distinct ground states characteristic of a nonchiral, Z2{Z}_{2}Z2​ topologically ordered state with vison and spinon excitations. We shed light on the interplay of topological ordering and global symmetries in the model by detecting fractionalization of time-reversal and space-group dihedral symmetries in the anyonic sectors, which leads to the coexistence of symmetry protected and intrinsic topological order. The anyonic sectors, and information on the particle statistics, can be characterized by degeneracy patterns and symmetries of the entanglement spectrum. We demonstrate the ground states on finite-width cylinders are short-range correlated and gapped; however, some features in the entanglement spectrum suggest that the system develops gapless spinonlike edge excitations in the large-width limit.
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 perform highly accurate density matrix renormalization group (DMRG) simulations to investigate the ground-state properties of the spin-1/21/21/2 antiferromagnetic square lattice Heisenberg J1{J}_{1}J1​-J2{J}_{2}J2​ model. Based on studies of numerous long cylinders with circumferences of up to 14 lattice spacings, we obtain strong evidence for a topological quantum spin liquid state in the region $0.41\ensuremath{\le}{J}_{2}/{J}_{1}\ensuremath{\le}0.62$, separating conventional N\'eel and striped antiferromagnetic states for smaller and larger J2/J1{J}_{2}/{J}_{1}J2​/J1​, respectively. The quantum spin liquid is characterized numerically by the absence of magnetic or valence bond solid order, and nonzero singlet and triplet energy gaps. Furthermore, we positively identify its topological nature by measuring a nonzero topological entanglement entropy $\ensuremath{\gamma}=0.70\ifmmode\pm\else\textpm\fi{}0.02$, extremely close to $\ensuremath{\gamma}=\mathrm{ln}(2)\ensuremath{\approx}0.69$ (expected for a Z2{Z}_{2}Z2​ quantum spin liquid) and a nontrivial finite size dimerization effect depending upon the parity of the circumference of the cylinder. We also point out that a valence bond solid, and indeed any discrete symmetry breaking state, would be expected to show a constant correction to the entanglement entropy of opposite sign to the topological entanglement entropy.