Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1209.2387 · Physical Review B

Efficient simulation of infinite tree tensor network states on the Bethe lattice

Wei Li, Jan von Delft, Tao Xiang

Published Nov 26, 2012• 84 citations
arXivHTMLPDF

Abstract

We show that the simple update approach proposed by Jiang et al. [H. C. Jiang, Z. Y. Weng, and T. Xiang, Phys. Rev. Lett. 101, 090603 (2008)] is an efficient and accurate method for determining the infinite tree tensor network states on the Bethe lattice. Ground-state properties of the quantum transverse Ising model and the Heisenberg XXZ model on the Bethe lattice are studied. The transverse Ising model is found to undergo a second-order quantum phase transition with a diverging magnetic susceptibility but a finite correlation length which is upper bounded by $1/\mathrm{ln}(q\ensuremath{-}1)$ even at the transition point ($q$ is the coordinate number of the Bethe lattice). An intuitive explanation on this peculiar ``critical'' phenomenon is given. The XXZ model on the Bethe lattice undergoes a first-order quantum phase transition at the isotropic point. Furthermore, the simple update scheme is found to be related with the Bethe approximation. Finally, by applying the simple update to various tree tensor clusters, we can obtain rather nice and scalable approximations for two-dimensional lattices.

DOI
10.1103/physrevb.86.195137

Related papers

Thermal tensor network simulations of the Heisenberg model on the Bethe lattice

Dai-Wei Qu, Wei Li, Tao Xiang

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 compute accurately the thermodynamic quantities, including the internal energy, specific heat, spontaneous magnetization, etc., at various temperatures. By varying the anisotropic coupling constant $\mathrm{\ensuremath{\Delta}}$, we obtain the phase diagram of the spin-1/2 Heisenberg XXZ model on the Bethe lattice, where three magnetic ordered phases, namely the ferromagnetic, XY, and antiferromagnetic 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 $\mathrm{\ensuremath{\Delta}}=\ifmmode\pm\else\textpm\fi{}1$. 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 $|\mathrm{\ensuremath{\Delta}}|\ensuremath{\le}1$. One remarkable feature there is that the transverse correlation length reaches ${\ensuremath{\xi}}_{c}=1/ln(z\ensuremath{-}1)$ for $T\ensuremath{\le}{T}_{c}$, the maximal value allowed on a $z$-coordinated Bethe lattice.

arXiv:1905.12478•Physical review. B./Physical review. B•Sep 10, 2019•8 citations

Tree Tensor Network States

Tao Xiang

This chapter discusses the properties of tree tensor network states and the methods for evaluating the ground state and thermodynamic properties of quantum lattice models on a Bethe lattice or, more generally, a Husimi lattice. It starts with a brief discussion of the canonical form of a tree tensor network state. Then, a canonicalization scheme is proposed. To calculate the ground state through the imaginary time evolution, the full and simple update methods are introduced to renormalize the local tensors. Finally, as the correlation length of a quantum system is finite even at a critical point, an accurate and efficient method is described to compute the thermodynamic quantities of quantum lattice models on the Bethe lattice.

Cambridge University Press eBooks•Aug 31, 2023•0 citations

Schmidt gap and quantum phase transitions in the XXZ model under external field on the Bethe lattice

Guanghua Liu, Wei Li, Wen‐Long You +2 more

\n\nAlong the line developed in a recent paper by one of the authors (Li W., von Delft J. and Xiang T., Phys. Rev. B, 86 (2012) 195137), a simple update scheme, as an optimal method for optimizing the infinite tree tensor network (iTTN) states, is utilized to study the XXZ model under external field on the Bethe lattice. The ground-state energy, bipartite entanglement, order parameters, and nearest-neighbor correlation functions are calculated in the framework of the iTTN states. With increasing external field, two quantum phase transitions (QPTs) (a first-order and a second-order one) take place successively. Between the antiferromagnetic phase and the saturated ferromagnetic phase, there exists another intermediate (canted Néel) phase, in which both x-axis antiferromagnetic order and unsaturated z-axis ferromagnetic order coexist. An interesting spin-flop transition between antiferromagnetic phase and canted Néel phase is observed. The model-independent bipartite entanglement is found to be capable of describing all the QPTs. It is interesting that the Schmidt gap acts as a local order parameter, and can also be used to describe the QPTs.\n

Europhysics Letters (EPL)•Mar 1, 2013•4 citations

Quantum Ising transition on the Cayley tree: boundary criticality, universality, and an effective Bethe line

Pietro Richelli, Natalia Chepiga

We investigate quantum criticality in the transverse-field Ising model on finite Cayley trees and demonstrate that their extensive boundary fundamentally changes the nature of the quantum phase transition compared with the infinite, boundary-free Bethe lattice. Using tree tensor-network simulations for systems with up to $N=12286$ sites, we identify a boundary-driven continuous transition at $h_x^c\approx1.76$, distinct from the mean-field Bethe-lattice transition at $h_x^c\approx2.23$. We characterize this transition through finite-size scaling of the magnetization and magnetic susceptibility and extract a consistent set of critical exponents. By introducing next-nearest-neighbor interactions, we demonstrate the robustness of the extracted critical exponents, providing numerical argument for the Ising boundary universality class on the Cayley tree. Furthermore, we show that the physics of the boundary-free Bethe lattice can be efficiently recovered on a finite Cayley tree by fine-tuning a local boundary field. The resulting \textit{effective Bethe line}, whose existence is tied to the mean-field nature of the bulk transition, effectively screens the extensive boundary and restores both the critical point and the mean-field critical scaling of the infinite Bethe lattice.

arXiv:2610.16774•Oct 12, 2026•0 citations

Quantum transverse-field Ising model on an infinite tree from matrix product states

Daniel Nagaj, Edward Farhi, Jeffrey Goldstone +2 more

We give a generalization to an infinite tree geometry of Vidal's infinite time-evolving block decimation (iTEBD) algorithm [G. Vidal, Phys. Rev. Lett. 98, 070201 (2007)] for simulating an infinite line of quantum spins. We numerically investigate the quantum Ising model in a transverse field on the Bethe lattice using the matrix product state ansatz. We observe a second order phase transition, with certain key differences from the transverse field Ising model on an infinite spin chain. We also investigate a transverse field Ising model with a specific longitudinal field. When the transverse field is turned off, this model has a highly degenerate ground state as opposed to the pure Ising model whose ground state is only doubly degenerate.

arXiv:0712.1806•Physical Review B•Jun 25, 2008•69 citations

Cluster update for tensor network states

Ling Wang, Frank Verstraete

We propose a novel recursive way of updating the tensors in projected entangled pair states by evolving the tensor in imaginary time evolution on clusters of different sizes. This generalizes the so- called simple update method of Jiang et al. [Phys. Rev. Lett. 101, 090603 (2008)] and the updating schemes in the single layer picture of Pižorn et al. [Phys. Rev. A 83, 052321 (2011)]. A finite-size scaling of the observables as a function of the cluster size provides a remarkable improvement in the accuracy as compared to the simple update scheme. We benchmark our results on the hand of the spin 1/2 staggered dimerized antiferromagnetic model on the square lattice, and accurate results for the magnetization and the critical exponents are determined.

arXiv:1110.4362•Bulletin of the American Physical Society•Oct 19, 2011•21 citations
We show that the simple update approach proposed by Jiang et al. [H. C. Jiang, Z. Y. Weng, and T. Xiang, Phys. Rev. Lett. 101, 090603 (2008)] is an efficient and accurate method for determining the infinite tree tensor network states on the Bethe lattice. Ground-state properties of the quantum transverse Ising model and the Heisenberg XXZ model on the Bethe lattice are studied. The transverse Ising model is found to undergo a second-order quantum phase transition with a diverging magnetic susceptibility but a finite correlation length which is upper bounded by 1/ln(q\ensuremath−1)1/\mathrm{ln}(q\ensuremath{-}1)1/ln(q\ensuremath−1) even at the transition point (qqq is the coordinate number of the Bethe lattice). An intuitive explanation on this peculiar ``critical'' phenomenon is given. The XXZ model on the Bethe lattice undergoes a first-order quantum phase transition at the isotropic point. Furthermore, the simple update scheme is found to be related with the Bethe approximation. Finally, by applying the simple update to various tree tensor clusters, we can obtain rather nice and scalable approximations for two-dimensional lattices.
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 compute accurately the thermodynamic quantities, including the internal energy, specific heat, spontaneous magnetization, etc., at various temperatures. By varying the anisotropic coupling constant $\mathrm{\ensuremath{\Delta}}$, we obtain the phase diagram of the spin-1/2 Heisenberg XXZ model on the Bethe lattice, where three magnetic ordered phases, namely the ferromagnetic, XY, and antiferromagnetic phases, are found in low temperatures and separated from the high-TTT paramagnetic phase by a continuous thermal phase transition at Tc{T}_{c}Tc​. The XY phase is separated from the other two phases by two first-order phase transition lines at the symmetric coupling points $\mathrm{\ensuremath{\Delta}}=\ifmmode\pm\else\textpm\fi{}1$. 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 $|\mathrm{\ensuremath{\Delta}}|\ensuremath{\le}1$. One remarkable feature there is that the transverse correlation length reaches ${\ensuremath{\xi}}_{c}=1/ln(z\ensuremath{-}1)$ for $T\ensuremath{\le}{T}_{c}$, the maximal value allowed on a zzz-coordinated Bethe lattice.
This chapter discusses the properties of tree tensor network states and the methods for evaluating the ground state and thermodynamic properties of quantum lattice models on a Bethe lattice or, more generally, a Husimi lattice. It starts with a brief discussion of the canonical form of a tree tensor network state. Then, a canonicalization scheme is proposed. To calculate the ground state through the imaginary time evolution, the full and simple update methods are introduced to renormalize the local tensors. Finally, as the correlation length of a quantum system is finite even at a critical point, an accurate and efficient method is described to compute the thermodynamic quantities of quantum lattice models on the Bethe lattice.
\n\nAlong the line developed in a recent paper by one of the authors (Li W., von Delft J. and Xiang T., Phys. Rev. B, 86 (2012) 195137), a simple update scheme, as an optimal method for optimizing the infinite tree tensor network (iTTN) states, is utilized to study the XXZ model under external field on the Bethe lattice. The ground-state energy, bipartite entanglement, order parameters, and nearest-neighbor correlation functions are calculated in the framework of the iTTN states. With increasing external field, two quantum phase transitions (QPTs) (a first-order and a second-order one) take place successively. Between the antiferromagnetic phase and the saturated ferromagnetic phase, there exists another intermediate (canted Néel) phase, in which both x-axis antiferromagnetic order and unsaturated z-axis ferromagnetic order coexist. An interesting spin-flop transition between antiferromagnetic phase and canted Néel phase is observed. The model-independent bipartite entanglement is found to be capable of describing all the QPTs. It is interesting that the Schmidt gap acts as a local order parameter, and can also be used to describe the QPTs.\n
We investigate quantum criticality in the transverse-field Ising model on finite Cayley trees and demonstrate that their extensive boundary fundamentally changes the nature of the quantum phase transition compared with the infinite, boundary-free Bethe lattice. Using tree tensor-network simulations for systems with up to N=12286N=12286N=12286 sites, we identify a boundary-driven continuous transition at hxc≈1.76h_x^c\approx1.76hxc​≈1.76, distinct from the mean-field Bethe-lattice transition at hxc≈2.23h_x^c\approx2.23hxc​≈2.23. We characterize this transition through finite-size scaling of the magnetization and magnetic susceptibility and extract a consistent set of critical exponents. By introducing next-nearest-neighbor interactions, we demonstrate the robustness of the extracted critical exponents, providing numerical argument for the Ising boundary universality class on the Cayley tree. Furthermore, we show that the physics of the boundary-free Bethe lattice can be efficiently recovered on a finite Cayley tree by fine-tuning a local boundary field. The resulting \textit{effective Bethe line}, whose existence is tied to the mean-field nature of the bulk transition, effectively screens the extensive boundary and restores both the critical point and the mean-field critical scaling of the infinite Bethe lattice.
We give a generalization to an infinite tree geometry of Vidal's infinite time-evolving block decimation (iTEBD) algorithm [G. Vidal, Phys. Rev. Lett. 98, 070201 (2007)] for simulating an infinite line of quantum spins. We numerically investigate the quantum Ising model in a transverse field on the Bethe lattice using the matrix product state ansatz. We observe a second order phase transition, with certain key differences from the transverse field Ising model on an infinite spin chain. We also investigate a transverse field Ising model with a specific longitudinal field. When the transverse field is turned off, this model has a highly degenerate ground state as opposed to the pure Ising model whose ground state is only doubly degenerate.
We propose a novel recursive way of updating the tensors in projected entangled pair states by evolving the tensor in imaginary time evolution on clusters of different sizes. This generalizes the so- called simple update method of Jiang et al. [Phys. Rev. Lett. 101, 090603 (2008)] and the updating schemes in the single layer picture of Pižorn et al. [Phys. Rev. A 83, 052321 (2011)]. A finite-size scaling of the observables as a function of the cluster size provides a remarkable improvement in the accuracy as compared to the simple update scheme. We benchmark our results on the hand of the spin 1/2 staggered dimerized antiferromagnetic model on the square lattice, and accurate results for the magnetization and the critical exponents are determined.