Combined unitary and symmetric group approach applied to low-dimensional spin systems

Abstract

A novel combined unitary and symmetric group approach is used to study the spin-$\frac{1}{2}$ Heisenberg model and related Fermionic systems in a spin-adapted representation, using a linearly-parameterised Ansatz for the many-body wave function. We show that a more compact ground state wave function representation is obtained when combining the symmetric group, ${\mathcal{S}}_n$, in the form of permutations of the underlying lattice site ordering, with the cumulative spin-coupling based on the unitary group, $\mathrm{U}(n)$. In one-dimensional systems the observed compression of the wave function is reminiscent of block-spin renormalization group approaches, and allows us to study larger lattices (here taken up to 80 sites) with the spin-adapted full configuration interaction quantum Monte Carlo method, which benefits from the sparsity of the Hamiltonian matrix and the corresponding sampled eigenstates that emerge from the reordering. We find that in an optimal lattice ordering the configuration state function with highest weight already captures with high accuracy the spin-spin correlation function of the exact ground state wave function. This feature is found for more general lattice models, such as the Hubbard model, and ab initio quantum chemical models, in this work exemplified by a one-dimensional hydrogen chain. We also provide numerical evidence that the optimal lattice ordering for the unitary group approach is not generally equivalent to the optimal ordering obtained for methods based on matrix-product states, such as the density-matrix renormalization group approach.

References in corpus (72)

• The density-matrix renormalization group in the age of matrix product states
• The density-matrix renormalization group
• Entanglement in quantum critical phenomena
• Solving the Quantum Many-Body Problem with Artificial Neural Networks
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Entropy and Area
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Surprises on the Way from 1D to 2D Quantum Magnets: the Novel Ladder Materials
• Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations
• Entanglement renormalization
• Thermodynamic limit of the density matrix renormalization for the spin-1 Heisenberg chain
• An Area Law for One Dimensional Quantum Systems
• Quantum Monte Carlo with Directed Loops
• Stochastic series expansion method with operator-loop update
• Computational Studies of Quantum Spin Systems
• Susceptibility of the Spin 1/2 Heisenberg Antiferromagnetic Chain
• Quantum Entanglement in Neural Network States
• Exact, Complete, and Universal Continuous-Time Worldline Monte Carlo Approach to the Statistics of Discrete Quantum Systems
• Cluster Algorithm for Vertex Models
• Quantum Monte Carlo method using phase-free random walks with Slater determinants
• Orbital Optimization in the Density Matrix Renormalization Group, with applications to polyenes and β-carotene
• Restricted-Boltzmann-Machine Learning for Solving Strongly Correlated Quantum Systems
• A spin-adapted Density Matrix Renormalization Group algorithm for quantum chemistry
• The Loop Algorithm
• Measuring orbital interaction using quantum information theory
• Symmetries and many-body excited states with neural-network quantum states
• Towards the solution of the many-electron problem in real materials: equation of state of the hydrogen chain with state-of-the-art many-body methods
• Equivalence of restricted Boltzmann machines and tensor network states
• Study of the Two-Dimensional Frustrated J1-J2 Model with Neural Network Quantum States
• The Non-Abelian Density Matrix Renormalization Group Algorithm
• Variational Quantum Monte Carlo Method with a Neural-Network Ansatz for Open Quantum Systems
• Quantum information analysis of electronic states at different molecular structures
• Neural-Network Quantum States, String-Bond States, and Chiral Topological States
• CheMPS2: a free open-source spin-adapted implementation of the density matrix renormalization group for ab initio quantum chemistry
• Exploring frustrated spin-systems using Projected Entangled Pair States (PEPS)
• Variational Monte Carlo Method Combined with Quantum-Number Projection and Multi-Variable Optimization
• Unbiased Reduced Density Matrices and Electronic Properties from Full Configuration Interaction Quantum Monte Carlo
• The $S=1/2$ Kagome Heisenberg Antiferromagnet Revisited
• Tensor network states and algorithms in the presence of a global SU(2) symmetry
• An excited-state approach within full configuration interaction quantum Monte Carlo
• Solving frustrated quantum many-particle models with convolutional neural networks
• A tensor product state approach to spin-1/2 square $J_1$-$J_2$ antiferromagnetic Heisenberg model: evidence for deconfined quantum criticality
• Correlation Amplitudes for the spin-1/2 XXZ chain in a magnetic field
• NECI: N-Electron Configuration Interaction with emphasis on state-of-the-art stochastic methods
• Quantum Lattice Model Solver $\mathcal{H}\Phi$
• Ground-state properties of the hydrogen chain: insulator-to-metal transition, dimerization, and magnetic phases
• Efficient Formulation of Full Configuration Interaction Quantum Monte Carlo in a Spin Eigenbasis via the Graphical Unitary Group Approach
• Efficient Matrix Product State Method for periodic boundary conditions
• The antiferromagnetic spin-1/2 Heisenberg model on the square lattice in a magnetic field
• A two-dimensional algorithm of the density matrix renormalization group
• Introduction to the Bethe ansatz II
• Critical exponents of the quantum phase transition in a planar antiferromagnet
• Compact numerical solutions to the two-dimensional repulsive Hubbard model obtained via non-unitary similarity transformations
• The Density Matrix Renormalization Group Method applied to Interaction Round a Face Hamiltonians
• Density matrices in full configuration interaction quantum Monte Carlo: Excited states, transition dipole moments and parallel distribution
• Matrix product decomposition and classical simulation of quantum dynamics in the presence of a symmetry
• Determining Factors for the Accuracy of DMRG in Chemistry
• Deep Learning-Enhanced Variational Monte Carlo Method for Quantum Many-Body Physics
• DMRG simulations of SU(N) Heisenberg chains using standard Young tableaux: fundamental representation and comparison with finite-size Bethe ansatz
• Correlation functions of the spin-1/2 anti-ferromagnetic Heisenberg chain: exact calculation via the generating function
• Population Control Bias and Importance Sampling in Full Configuration Interaction Quantum Monte Carlo
• Combined use of translational and spin-rotational invariance for spin systems
• Nested Cluster Algorithm for Frustrated Quantum Antiferromagnets
• Time propagation and spectroscopy of Fermionic systems using a stochastic technique
• An Efficient Density Matrix Renormalization Group Algorithm for Chains with Periodic Boundary Condition
• Benchmark study of Nagaoka ferromagnetism by spin-adapted full configuration interaction quantum Monte Carlo
• Symmetry adapted finite-cluster solver for quantum Heisenberg model in two-dimensions: a real-space renormalization approach
• Diagrammatics for SU(2) invariant matrix product states
• A variational lower bound on the ground state of a many-body system and the squaring parametrization of density matrices
• Interaction-round-a-face density-matrix renormalization-group method
• Entanglement Perturbation Theory for Antiferromagnetic Heisenberg Spin Chains
• Merits of using density matrices instead of wave functions in the stationary Schrödinger equation for systems with symmetries

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