Quantum Lattice Models for Low-Dimensional Systems

We calculate the topological entanglement entropy (TEE) for a three-dimensional hyperhoneycomb lattice generalization of Kitaev's honeycomb lattice spin model. We find that for this model TEE is not directly determined by the total quantum dimension of the system. This is in contrast to general two-dimensional systems and many three-dimensional models, where TEE is related to the total quantum dimension. Our calculation also provides TEE for a three-dimensional toric-code-type Hamiltonian that emerges as the effective low-energy theory for the Kitaev model in a particular limit.We calculate the topological entanglement entropy (TEE) for a three-dimensional hyperhoneycomb lattice generalization of Kitaev's honeycomb lattice spin model. We find that for this model TEE is not directly determined by the total quantum dimension of the system. This is in contrast to general two-dimensional systems and many three-dimensional models, where TEE is related to the total quantum dimension. Our calculation also provides TEE for a three-dimensional toric-code-type Hamiltonian that emerges as the effective low-energy theory for the Kitaev model in a particular limit.

arXiv:1804.02249Physical review. B./Physical review. BSep 20, 20189 citations

Microscopic Theories of Quantum Lattice Systems

R. F. Bishop

Recent Progress in Many-Body TheoriesJan 1, 19953 citations

Numerical study of low-dimensional quantum systems

Tôru Sakai

Jan 1, 19990 citations

Spectroscopy of two-dimensional interacting lattice electrons using symmetry-aware neural backflow transformations

Imelda Romero, Jannes Nys, Giuseppe Carleo

Neural networks have shown to be a powerful tool to represent the ground state of quantum many-body systems, including fermionic systems. However, efficiently integrating lattice symmetries into neural representations remains a significant challenge. In this work, we introduce a framework for embedding lattice symmetries in fermionic wavefunctions and demonstrate its ability to target both ground states and low-lying excitations. Using group-equivariant neural backflow transformations, we study the t-V model on a square lattice away from half-filling. Our symmetry-aware backflow significantly improves ground-state energies and yields accurate low-energy excitations for lattices up to 10 × 10. We also compute accurate two-point density-correlation functions and the structure factor to identify phase transitions and critical points. These findings introduce a symmetry-aware framework important for studying quantum materials and phase transitions. In this manuscript, the authors introduce a method for numerically modelling fermionic systems using a neural network wavefunction ansatz. They demonstrate that this method can efficiently and accurately find ground and low-lying excited states in two-dimensional models, outperforming existing approaches.Neural networks have shown to be a powerful tool to represent the ground state of quantum many-body systems, including fermionic systems. However, efficiently integrating lattice symmetries into neural representations remains a significant challenge. In this work, we introduce a framework for embedding lattice symmetries in fermionic wavefunctions and demonstrate its ability to target both ground states and low-lying excitations. Using group-equivariant neural backflow transformations, we study the t-V model on a square lattice away from half-filling. Our symmetry-aware backflow significantly improves ground-state energies and yields accurate low-energy excitations for lattices up to 10 × 10. We also compute accurate two-point density-correlation functions and the structure factor to identify phase transitions and critical points. These findings introduce a symmetry-aware framework important for studying quantum materials and phase transitions. In this manuscript, the authors introduce a method for numerically modelling fermionic systems using a neural network wavefunction ansatz. They demonstrate that this method can efficiently and accurately find ground and low-lying excited states in two-dimensional models, outperforming existing approaches.

arXiv:2406.09077Communications PhysicsJan 30, 20257 citations

Low- and high-temperature series expansions for quantum coupled oscillators systems or quantum fields on a lattice

F. T. Hioe

In this paper, a new formulation which can be used to systematically extend and study the low- and high-temperature series for the thermodynamic functions of systems of coupled quantum oscillators corresponding to the quantum $\ensuremath{\lambda}{\ensuremath{\varphi}}^{2m}$ fields, $m=2,3$ ,..., on a lattice is proposed. In this formulation, the low- and high-temperature series for the partition function of the systems are given in the forms $Z={Z}_{h}(T)[1+{A}_{1}(\ensuremath{\lambda}{T}^{m\ensuremath{-}1})+{A}_{2}{(\ensuremath{\lambda}{T}^{m\ensuremath{-}1})}^{2}+\ensuremath{\cdots}]$ and $Z={Z}_{a}(T)[1+{B}_{1}{(\ensuremath{\lambda}{T}^{m\ensuremath{-}1})}^{\frac{\ensuremath{-}1}{m}}+{B}_{2}{(\ensuremath{\lambda}{T}^{m\ensuremath{-}1})}^{\frac{\ensuremath{-}2}{m}}+\ensuremath{\cdots}] ,$ respectively, where ${A}_{n}$ and ${B}_{n}$ are certain combinatorial numbers which can be expressed in terms of the numbers of ways of placing certain diagrams on the $d$ -dimensional $d=1,2,3$ ,... discrete lattice. Note that the forms of these series are new and they are different from those of the more extensively studied classical models. A systematic study of these series will, as in the corresponding cases of the Ising and Heisenberg models, reveal the critical behaviors of this class of quantum-field models which is not only interesting in its own right, but is also important in our overall understanding of the critical behaviors of many other related models from the point of view of the renormalization-group approach.In this paper, a new formulation which can be used to systematically extend and study the low- and high-temperature series for the thermodynamic functions of systems of coupled quantum oscillators corresponding to the quantum $\ensuremath{\lambda}{\ensuremath{\varphi}}^{2m}$ fields, $m=2,3$,..., on a lattice is proposed. In this formulation, the low- and high-temperature series for the partition function of the systems are given in the forms $Z={Z}_{h}(T)[1+{A}_{1}(\ensuremath{\lambda}{T}^{m\ensuremath{-}1})+{A}_{2}{(\ensuremath{\lambda}{T}^{m\ensuremath{-}1})}^{2}+\ensuremath{\cdots}]$ and $Z={Z}_{a}(T)[1+{B}_{1}{(\ensuremath{\lambda}{T}^{m\ensuremath{-}1})}^{\frac{\ensuremath{-}1}{m}}+{B}_{2}{(\ensuremath{\lambda}{T}^{m\ensuremath{-}1})}^{\frac{\ensuremath{-}2}{m}}+\ensuremath{\cdots}] ,$ respectively, where ${A}_{n}$ and ${B}_{n}$ are certain combinatorial numbers which can be expressed in terms of the numbers of ways of placing certain diagrams on the $d$-dimensional $d=1,2,3$,... discrete lattice. Note that the forms of these series are new and they are different from those of the more extensively studied classical models. A systematic study of these series will, as in the corresponding cases of the Ising and Heisenberg models, reveal the critical behaviors of this class of quantum-field models which is not only interesting in its own right, but is also important in our overall understanding of the critical behaviors of many other related models from the point of view of the renormalization-group approach.

Physical review. B, Solid stateNov 1, 19777 citations

Study of low dimensional lattice systems

Kush Saha

Jan 1, 20130 citations

m=2,3

,..., on a lattice is proposed. In this formulation, the low- and high-temperature series for the partition function of the systems are given in the forms $Z={Z}_{h}(T)[1+{A}_{1}(\ensuremath{\lambda}{T}^{m\ensuremath{-}1})+{A}_{2}{(\ensuremath{\lambda}{T}^{m\ensuremath{-}1})}^{2}+\ensuremath{\cdots}]$ and $Z={Z}_{a}(T)[1+{B}_{1}{(\ensuremath{\lambda}{T}^{m\ensuremath{-}1})}^{\frac{\ensuremath{-}1}{m}}+{B}_{2}{(\ensuremath{\lambda}{T}^{m\ensuremath{-}1})}^{\frac{\ensuremath{-}2}{m}}+\ensuremath{\cdots}] ,$ respectively, where

{A}_{n}

and

{B}_{n}

are certain combinatorial numbers which can be expressed in terms of the numbers of ways of placing certain diagrams on the

d

-dimensional

d=1,2,3

,... discrete lattice. Note that the forms of these series are new and they are different from those of the more extensively studied classical models. A systematic study of these series will, as in the corresponding cases of the Ising and Heisenberg models, reveal the critical behaviors of this class of quantum-field models which is not only interesting in its own right, but is also important in our overall understanding of the critical behaviors of many other related models from the point of view of the renormalization-group approach.