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Infinite Density Matrix Renormalisation Group for Symmetric Systems at High Filling Fraction, and Anyons

Robert N. C. Pfeifer

Published May 23, 2015• 0 citations
OpenAlex

Abstract

The infinite Density Matrix Renormalisation Group (iDMRG) algorithm is a highly successful numerical algorithm for the study of low-dimensional quantum systems. However, it may fail at high filling fractions when exploiting global symmetries such as particle number conservation to increase simulation speed. This paper introduces a modified iDMRG algorithm capable of exploiting these symmetries while still producing accurate results at high filling fractions, and at a substantially reduced computational cost.

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The infinite Density Matrix Renormalisation Group (iDMRG) algorithm is a highly successful numerical algorithm for the study of low-dimensional quantum systems. However, it may fail at high filling fractions when exploiting global symmetries such as particle number conservation to increase simulation speed. This paper introduces a modified iDMRG algorithm capable of exploiting these symmetries while still producing accurate results at high filling fractions, and at a substantially reduced computational cost.
The infinite density matrix renormalization group (iDMRG) algorithm is a highly successful numerical algorithm for the study of low-dimensional quantum systems, and is also frequently used to initialize the more popular finite DMRG algorithm. Implementations of both finite and infinite DMRG frequently incorporate support for the protection and exploitation of symmetries of the Hamiltonian. In common with other variational tensor network algorithms, convergence of iDMRG to the ground state is not guaranteed, with the risk that the algorithm may become stuck in a local minimum. In this paper, I demonstrate the existence of a particularly harmful class of physically irrelevant local minima affecting both iDMRG and to a lesser extent also infinite time-evolving block decimation (iTEBD), for which the ground state is compatible with the protected symmetries of the Hamiltonian but cannot be reached using the conventional iDMRG or iTEBD algorithms. I describe a modified iDMRG algorithm which evades these local minima, and which also admits a natural interpretation on topologically ordered systems with a boundary.
Substantial improvements in the computational effort in a density matrix renormalisation program can be made by utilising symmetries of the Hamiltonian. Extra quantum numbers are always desirable to include in the calculation, since it allows the Hilbert space of the superblock to be refined. Since the speed of the calculation is approximately O ( n 3 ) in superblock states, the speed increase in targeting a specific total spin state can be considerable. In this paper a new density matrix renormalisation algorithm is presented which conserves the total spin. The general procedure obtained works for any operator, even operators that do not commute with the Hamiltonian.
In this study, we report a parallel algorithm for the infinite-size density matrix renormalization group (iDMRG) that is applicable to one-dimensional (1D) quantum systems with ℓ\ellℓ-site periods, where ℓ\ellℓ is an even number. It combines Hida's iDMRG applied to random 1D spin systems with a variant of McCulloch's wavefunction prediction. This allows us to apply ℓ/2\ell/2ℓ/2 times of computational power to accelerate the investigation of multi-leg frustrated quantum systems in the thermodynamic limit, which is a challenging simulation. We performed benchmark calculations for a spin-1/2 Heisenberg model on a Kagome cylinder YC8 using the parallel iDMRG, and found that the proposed iDMRG was efficiently parallelized for shared memory and distributed memory systems, and provided such bulk physical quantities as total energy, bond strength on nearest neighbor spins, and spin--spin correlation functions and their correlation lengths without finite-size effects. Moreover, the variant of the wavefunction prediction sped up Lanczos methods in the parallel iDMRG by approximately three times.
I revisit the infinite-size variant of the Density Matrix Renormalization Group (iDMRG) algorithm for obtaining a fixed-point translationally invariant matrix product wavefunction in the context of one-dimensional quantum systems. A crucial ingredient of this algorithm is an efficient transformation for obtaining the matrix elements of the wavefunction as the lattice size is increased, and I introduce here a versatile transformation that is demonstrated to be much more effective than previous versions. The resulting algorithm has a surprisingly close relationship to Vidal's Time Evolving Block Decimation for infinite systems, but allows much faster convergence. Access to a translationally invariant matrix product state allows the calculation of correlation functions based on the transfer matrix, which directly gives the spectrum of all correlation lengths. I also show some advantages of the Matrix Product Operator (MPO) technique for constructing expectation values of higher moments, such as the exact variance $$.
The numerical study of anyonic systems is known to be highly challenging due to their nonbosonic, nonfermionic particle-exchange statistics, and with the exception of certain models for which analytical solutions exist, very little is known about their collective behavior as a result. Meanwhile, the density matrix renormalization group (DMRG) algorithm is an exceptionally powerful numerical technique for calculating the ground state of a low-dimensional lattice Hamiltonian, and has been applied to the study of bosonic, fermionic, and group-symmetric systems. The recent development of a tensor network formulation for anyonic systems opened up the possibility of studying these systems using algorithms such as DMRG, though this has proved challenging in terms of both programming complexity and computational cost. This paper presents the implementation of DMRG for finite anyonic systems, including a detailed scheme for the implementation of anyonic tensors with optimal scaling of computational cost. The anyonic DMRG algorithm is demonstrated by calculating the ground-state energy of the ``golden chain,'' which has become the benchmark system for the numerical study of anyons and is shown to produce results comparable to those of the anyonic time-evolving block decimation algorithm and superior to the variationally optimized anyonic multiscale renormalization ansatz, at far lesser computational cost.
We develop the density-matrix renormalization group (DMRG) technique for numerically studying incompressible fractional quantum Hall (FQH) states on the sphere. We calculate accurate estimates for ground-state energies and excitation gaps at FQH filling fractions nu=1/3 and nu=5/2 for systems that are considerably larger than the largest ever studied by exact diagonalization. We establish, by carefully comparing with existing numerical results on smaller systems, that DMRG is a highly effective numerical tool for studying incompressible FQH states.