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arXiv:1303.0305 · Physical Review B

Exact real-space renormalization method and applications

Adrian E. Feiguin, Rolando D. Somma, Cristian D. Batista

Published Aug 30, 2013• 2 citations
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Abstract

We present a numerical method based on real-space renormalization that outputs the exact ground space of ``frustration-free'' Hamiltonians. The complexity of our method is polynomial in the degeneracy of the ground spaces of the Hamiltonians involved in the renormalization steps. We apply the method to obtain the full ground spaces of two spin systems. The first system is a spin-1/2 Heisenberg model with four-spin cyclic-exchange interactions defined on a square lattice. In this case, we study finite lattices of up to 160 spins and find a triplet ground state that differs from the singlet ground states obtained by Batista and Trugman, [Phys. Rev. Lett. 93, 217202 (2004)]. We characterize such a triplet state as consisting of a triplon that propagates in a background of fluctuating singlet dimers. The second system is a family of spin-1/2 Heisenberg chains with uniaxial exchange anisotropy and next-nearest-neighbor interactions. In this case, the method finds a ground-space degeneracy that scales quadratically with the system size and outputs the full ground space efficiently. Our method can substantially outperform methods based on exact diagonalization and is more efficient than other renormalization methods when the ground-space degeneracy is large.

DOI
10.1103/physrevb.88.075145

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We present a numerical method based on real-space renormalization that outputs the exact ground space of ``frustration-free'' Hamiltonians. The complexity of our method is polynomial in the degeneracy of the ground spaces of the Hamiltonians involved in the renormalization steps. We apply the method to obtain the full ground spaces of two spin systems. The first system is a spin-1/2 Heisenberg model with four-spin cyclic-exchange interactions defined on a square lattice. In this case, we study finite lattices of up to 160 spins and find a triplet ground state that differs from the singlet ground states obtained by Batista and Trugman, [Phys. Rev. Lett. 93, 217202 (2004)]. We characterize such a triplet state as consisting of a triplon that propagates in a background of fluctuating singlet dimers. The second system is a family of spin-1/2 Heisenberg chains with uniaxial exchange anisotropy and next-nearest-neighbor interactions. In this case, the method finds a ground-space degeneracy that scales quadratically with the system size and outputs the full ground space efficiently. Our method can substantially outperform methods based on exact diagonalization and is more efficient than other renormalization methods when the ground-space degeneracy is large.
We identify a large class of quantum many-body systems that can be solved exactly: natural frustration-free spin-1/2 nearest-neighbor Hamiltonians on arbitrary lattices. We show that the entire ground-state manifold of such models can be found exactly by a tensor network of isometries acting on a space locally isomorphic to the symmetric subspace. Thus, for this wide class of models, real-space renormalization can be made exact. Our findings also imply that every such frustration-free spin model satisfies an area law for the entanglement entropy of the ground state, establishing a novel large class of models for which an area law is known. Finally, we show that our approach gives rise to an ansatz class useful for the simulation of almost frustration-free models in a simple fashion, outperforming mean-field theory.
Real-space renormalization methods which simply shift and rescale the interaction parameters in the nearest-neighbor Heisenberg model are investigated for two-dimensional lattices of equivalent sites. Results are presented for the hexagonal, square-planar, triangular, and Kagome lattices via three different renormalization techniques. The first, which has been studied for some time, uses perturbation theory to evaluate the renormalized interactions. The second uses the variational method to improve on the perturbative results. The third method is based on a cluster expansion and is found to give much improved agreement with numerical results form Monte Carlo calculations, but it does not provide a variational bound to the exact solution. The dependence of all three methods on the size and shape of the renormalized block of sites is also investigated.
The application of real space renormalization group methods to quantum lattice models has become a topic of great interest following the development of the density matrix renormalization group by White.This method has been used to find the ground and low-lying excited state energies and wave functions of quantum spin models in which the form of the ground state is not clear, for instance because the interactions are frustrated.It has also been applied to fermion problems where the tendency for localization due to the strong Coulomb repulsion is opposed by the lowering of the kinetic energy which occurs as a result of electron transfer.The approach is particularly suitable for one-, or quasi-one-dimensional problems.The method involves truncating the Hilbert space in a systematic and optimised manner.Results for the ground state energy are thus variational bounds.The results for low-lying energies and correlation functions for one-dimensional systems have unprecedented accuracy and the method has become the method of choice for solving one-dimensional quantum spin problems.We review the method and results obtained for the spin-1 chain with biquadratic exchange as well as the spin-1/2 model with competing nearest and next nearest neighbour exchange will be described.More recently, the density matrix renormalization group has been applied to reformulate the coupling constant renormalization group approach which is appropriate for the study of critical properties.This approach has been applied to the anisotropic spin-1/2 Heisenberg chain.Finally, we discuss recent work which has borne promising applications in two dimensions -the Ising model and the two-dimensional Hubbard model.
In spin lattices, frustration happens when there exists a competition between different terms of the Hamiltonian, which cannot be simultaneously minimized in all local physical bases. Frustration can destabilize the bipartite Neel order, which is the conventional groundstate of antiferromagnetic Heisenberg-type models in two dimensions. Hence, the focus of two-dimensional quantum magnetism studies has been directed toward elucidating the zero-temperature properties of Heisenberg-type models on the Archimedean lattices that exhibit geometrical frustration. This type of frustration is of great interest since it can lead to exotic forms of the quantum matter, such as topological spin-liquid (SL) and many-sublattice long-range-ordered phases. The spin-1/2 triangular Heisenberg model (THM) is a prototypical model with geometrical frustration, which has the highest coordination number of all Archimedean lattices. However, there exist two major, distinct difficulties for numerical methods to efficiently determine the groundstate properties of the THM: the immense dimension of the Hilbert space and the geometrical frustration phenomenon. In this thesis, to deal with such difficulties, we employ the state-of-the- art non-Abelian matrix product states (MPS) and density matrix renormalization group (DMRG) methods. We exploit the symmetries of the Hamiltonian to reduce the working dimension of the Hilbert space and, to lessen the effects of the geometrical frustration, we depend on the local entanglement (area-law) nature of the MPS as it scales with the size of a lattice subspace with reduced spatial dimensions. In such a way, we thoroughly examine the phase diagram of the THM with nearest and next-nearest neighbor (NN and NNN) exchange couplings on finite- and infinite-length cylinders of widths up to 12 sites. In addition, we develop new numerical tools to analyze topological and symmetry-broken phases in the context of the SU(2)-symmetric translation-invariant MPS algorithm. We establish that on infinite cylinders, the anyonic sectors of a topological order can be classified through the fractionalization of global symmetries and degeneracy patterns of the entanglement spectrum (ES). On the other hand, we detect symmetry-breaking phase transitions by a combination of the correlation lengths and second and fourth cumulants of the magnetic order parameters, even though symmetry implies that the order parameter itself is strictly zero. Furthermore, the appearance of Nambu-Goldstone modes in the excitation spectrum can be detected using “tower of states” (TOS) levels in the momentum-resolved ES. By applying the new tools to the THM, we discover a variety of exotic groundstates such as a Z2-gauge toric-code-type topological order (with vison and spinon excitations), a NNN Majumdar-Gosh, and a long-range 120◦-ordered state.
A real-space renormalization-group transformation is constructed between blocks of linear dimensions bL\mathrm{bL}bL and LLL to study the ground-state properties of quantum lattice models. In this scheme it is possible (i) to estimate the errors due to the irrelevant variables, including boundary effects, and (ii) to test scaling explicitly, so that an accurate calculation can be performed over the entire range of the parameter space. Application to the one-dimensional Ising model in a tranverse field yields excellent results.
We derive the exact ground space of a family of spin-1/2 Heisenberg chains with uniaxial exchange anisotropy (XXZ) and interactions between nearest and next-nearest-neighbor spins. The Hamiltonian family, H(eff)(Q), is parametrized by a single variable Q. By using a generalized Jordan-Wigner transformation that maps spins into anyons, we show that the exact ground states of H(eff)(Q) correspond to a condensation of anyons with a statistical phase φ=-4Q. We also provide matrix-product state representations of some ground states that allow for the efficient computation of spin-spin correlation functions.