Thermal Tensor Network Simulations of the Heisenberg Model on the Bethe Lattice

Abstract

We have extended the canonical tree tensor network (TTN) method, which was initially introduced to simulate the zero-temperature properties of quantum lattice models on the Bethe lattice, to finite temperature simulations. By representing the thermal density matrix with a canonicalized tree tensor product operator, we optimize the TTN and accurately evaluate the thermodynamic quantities, including the internal energy, specific heat, and the spontaneous magnetization, etc, at various temperatures. By varying the anisotropic coupling constant $\Delta$, we obtain the phase diagram of the spin-1/2 Heisenberg XXZ model on the Bethe lattice, where three kinds of magnetic ordered phases, namely the ferromagnetic, XY and antiferromagnetic ordered phases, are found in low temperatures and separated from the high-$T$ paramagnetic phase by a continuous thermal phase transition at $T_c$. The XY phase is separated from the other two phases by two first-order phase transition lines at the symmetric coupling points $\Delta =\pm 1$. We have also carried out a linear spin wave calculation on the Bethe lattice, showing that the low-energy magnetic excitations are always gapped, and find the obtained magnon gaps in very good agreement with those estimated from the TTN simulations. Despite the gapped excitation spectrum, Goldstone-like transverse fluctuation modes, as a manifestation of spontaneous continuous symmetry breaking, are observed in the ordered magnetic phases with $|\Delta |\le 1$. One remarkable feature there is that the prominent transverse correlation length reaches ${\xi }_c=1/\ln (z-1)$ for $T\le T_c$, the maximal value allowed on a $z$-coordinated Bethe lattice.

arXiv comment15 pages, 13 figures

References in corpus (37)

• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Classical simulation of infinite-size quantum lattice systems in one spatial dimension
• The Bethe lattice spin glass revisited
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• Accurate determination of tensor network state of quantum lattice models in two dimensions
• The iTEBD algorithm beyond unitary evolution
• Operator space entanglement entropy in transverse Ising chain
• Entanglement scaling of operators: a conformal field theory approach, with a glimpse of simulability of long-time dynamics in 1+1d
• Cayley Trees and Bethe Lattices, a concise analysis for mathematicians and physicists
• Time Evolution of an Infinite Projected Entangled Pair State: an Efficient Algorithm
• Projected Entangled Pair States at Finite Temperature: Imaginary Time Evolution with Ancillas
• Linearized Tensor Renormalization Group Algorithm for Thermodynamics of Quantum Lattice Models
• Exponential Thermal Tensor Network Approach for Quantum Lattice Models
• Variational Approach to Projected Entangled Pair States at Finite Temperature
• Efficient simulation of infinite tree tensor network states on the Bethe lattice
• A tensor network annealing algorithm for two-dimensional thermal states
• Variational tensor network renormalization in imaginary time: benchmark results in the Hubbard model at finite temperature
• Optimized Decimation of Tensor Networks with Super-orthogonalization for Two-Dimensional Quantum Lattice Models
• Two-temperature scales in the triangular-lattice Heisenberg antiferromagnet
• Hopping on the Bethe lattice: Exact results for densities of states and dynamical mean-field theory
• Simplex valence-bond crystal in the spin-1 kagome Heisenberg antiferromagnet
• Finite correlation length scaling with infinite projected entangled pair states at finite temperature
• Exact solution of the Bose-Hubbard model on the Bethe lattice
• Series-Expansion Thermal Tensor Network Approach for Quantum Lattice Models
• Quantum Phase Transition, O(3) Universality Class and Phase Diagram of Spin-1/2 Heisenberg Antiferromagnet on Distorted Honeycomb Lattice: A Tensor Renormalization Group Study
• Heisenberg antiferromagnet on Cayley trees: low-energy spectrum and even/odd site imbalance
• Overcoming the Sign Problem at Finite Temperature: Quantum Tensor Network for the Orbital $e_g$ Model on an Infinite Square Lattice
• There are no Goldstone bosons on the Bethe lattice
• One-dimensional quantum systems at finite temperatures can be simulated efficiently on classical computers
• Bilayer Linearized Tensor Renormalization Group Approach for Thermal Tensor Networks
• Featureless quantum spin liquid, 1/3-magnetization plateau state and exotic thermodynamic properties of spin-1/2 frustrated Heisenberg antiferromagnet on an infinite Husimi lattice
• Density matrix renormalization group algorithm for Bethe lattices of spin 1/2 or 1 sites with Heisenberg antiferromagnetic exchange
• Emergent Spin Excitations in a Bethe Lattice at Percolation
• Spin ordered ground state and thermodynamic behaviors of the spin-3/2 kagome Heisenberg antiferromagnet
• Theoretical studies of the phase transition in the anisotropic 2-D square spin lattice
• The Coulomb potential V(r)=1/r and other radial problems on the Bethe lattice

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