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arXiv:2304.01551 · Physical review. B./Physical review. B

Differentiable programming tensor networks for Kitaev magnets

Xing-Yu Zhang, Shuang Liang, Haijun Liao, Wei Li, Lei Wang

Published Aug 3, 2023• 11 citations
arXivHTMLPDF

Abstract

We present a general computational framework to investigate ground-state properties of quantum spin models on infinite two-dimensional lattices using automatic differentiation-based gradient optimization of infinite projected entangled-pair states. The approach exploits the variational uniform matrix product states to contract infinite tensor networks with unit cell structure and incorporates automatic differentiation to optimize the local tensors. We applied this framework to the Kitaev-type model, which involves complex interactions and competing ground states. To evaluate the accuracy of this method, we compared the results with exact solutions for the Kitaev model and found that it has a better agreement for various observables compared to previous tensor network calculations based on imaginary-time projection. Additionally, by finding out the ground state with lower variational energy compared to previous studies, we provided convincing evidence for the existence of nematic paramagnetic phases and 18-site configuration in the phase diagram of the $K\text{\ensuremath{-}}\mathrm{\ensuremath{\Gamma}}$ model. Furthermore, in the case of the realistic $K\text{\ensuremath{-}}J\text{\ensuremath{-}}\mathrm{\ensuremath{\Gamma}}\text{\ensuremath{-}}{\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ model for the Kitaev material $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$, we discovered a noncollinear zigzag ground state. Lastly, we also find that the strength of the critical out-of-plane magnetic field that suppresses such a zigzag state has a lower transition field value than the previous finite-cylinder calculations. The framework is versatile and will be useful for a quick scan of phase diagrams for a broad class of quantum spin models.

DOI
10.1103/physrevb.108.085103

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We present a general computational framework to investigate ground-state properties of quantum spin models on infinite two-dimensional lattices using automatic differentiation-based gradient optimization of infinite projected entangled-pair states. The approach exploits the variational uniform matrix product states to contract infinite tensor networks with unit cell structure and incorporates automatic differentiation to optimize the local tensors. We applied this framework to the Kitaev-type model, which involves complex interactions and competing ground states. To evaluate the accuracy of this method, we compared the results with exact solutions for the Kitaev model and found that it has a better agreement for various observables compared to previous tensor network calculations based on imaginary-time projection. Additionally, by finding out the ground state with lower variational energy compared to previous studies, we provided convincing evidence for the existence of nematic paramagnetic phases and 18-site configuration in the phase diagram of the K\ensuremath-\ensuremathΓK\text{\ensuremath{-}}\mathrm{\ensuremath{\Gamma}}K\ensuremath-\ensuremathΓ model. Furthermore, in the case of the realistic K\ensuremath-J\ensuremath-\ensuremathΓ\ensuremath-\ensuremathΓ\ensuremath′K\text{\ensuremath{-}}J\text{\ensuremath{-}}\mathrm{\ensuremath{\Gamma}}\text{\ensuremath{-}}{\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}K\ensuremath-J\ensuremath-\ensuremathΓ\ensuremath-\ensuremathΓ\ensuremath′ model for the Kitaev material \ensuremathα\ensuremath-RuCl3\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}\ensuremathα\ensuremath-RuCl3​, we discovered a noncollinear zigzag ground state. Lastly, we also find that the strength of the critical out-of-plane magnetic field that suppresses such a zigzag state has a lower transition field value than the previous finite-cylinder calculations. The framework is versatile and will be useful for a quick scan of phase diagrams for a broad class of quantum spin models.
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Infinite Projected entangled-pair state (iPEPS) is a two-dimensional tensor network (TN) ansatz targeting the ground states of lattice models directly in the thermodynamic limit. It is the most successful among the true two-dimensional TN methods, challenging the more common density matrix renormalization group computations conducted on a finite-width cylinders. iPEPS, being a variational method does not suffer from the infamous sign problem, and hence could be readily applied to both fermionic problems off the half-filling and frustrated spin system. Using the imaginary-time optimization, the so called Full Update, led to the celebrated results of iPEPS such as the analysis of stripes in doped t-J and Hubbard models or the insight into the magnetization plateaus in Shastry-Sutherland model. However, the physics of the continuous phase transitions from the symmetry-broken Neel phase into the valence-bond solids or spin-liquids in frustrated magnets is still eluding iPEPS. The main difficulty lies in the optimization, since Full Update is just not precise enough. In the pursuit of the best variational iPEPS states, the gradient methods were put forward in 2016 by Corboz and Vanderstraeten. However, due to their complex nature, they have not been widely adopted especially for iPEPS ansatze with larger unit cells. We will introduce an alternative approach first applied in the context of iPEPS by Liao et al., based on the well established method of Algorithmic differentiation (AD) which allows us to evaluate gradients for iPEPS in a robust and conceptually simple way. Finally, we will present the results of our investigation into the physics of frustrated magnets putting AD to work.
We develop a method of variational optimization of the infinite projected entangled pair states on the honeycomb lattice. The method is based on the automatic differentiation of the honeycomb-lattice corner transfer matrix renormalization group. We apply the approach to the antiferromagnetic Heisenberg spin-1/2 and ferromagnetic Kitaev models on the honeycomb lattice. The developed formalism gives quantitatively accurate results for the main physical observables and has a necessary potential for further extensions.
We present a general graph-based projected entangled-pair state (gPEPS) algorithm to approximate ground states of nearest-neighbor local Hamiltonians on any lattice or graph of infinite size. By introducing the structural matrix, which codifies the details of tensor networks on any graphs in any dimension ddd, we are able to produce a code that can be essentially launched to simulate any lattice. We further introduce an optimized algorithm to compute simple tensor updates as well as expectation values and correlators with a mean-field-like effective environments. Though not being variational, this strategy allows to cope with PEPS of very large bond dimension (e.g., D=100D=100D=100) and produces remarkably accurate results in the thermodynamic limit in many situations, and specially when the correlation length is small and the connectivity of the lattice is large. We prove the validity of our approach by benchmarking the algorithm against known results for several models, i.e., the antiferromagnetic Heisenberg model on a chain, star and cubic lattices, the hardcore Bose-Hubbard model on square lattice, the ferromagnetic Heisenberg model in a field on the pyrochlore lattice, as well as the three-state quantum Potts model in field on the kagome lattice and the spin-1 bilinear-biquadratic Heisenberg model on the triangular lattice. We further demonstrate the performance of gPEPS by studying the quantum phase transition of the 2d2d2d quantum Ising model in transverse magnetic field on the square lattice, and the phase diagram of the Kitaev-Heisenberg model on the hyperhoneycomb lattice. Our results are in excellent agreement with previous studies.
The efficient simulation of a quantum many-body system is often incredibly difficult due to the exponential scaling of the Hilbert space with increasing number of interacting particles. In practice, it is only possible to exactly simulate small clusters. In many cases, tensor-network-based methods are capable of overcoming this challenge, providing reliable simulations of much larger or even infinite-sized systems through a low-rank factorization of the state coefficients. Tensor operations are typically highly parallelizable, and therefore capable of utilizing the high performance of modern general-purpose graphics processing units (GPU). A particular tensor-network ansatz called infinite projected-entangled pair states (iPEPS) may be used to simulate infinite two-dimensional lattice systems, and are the primary subject of this thesis. As part of this work, a GPU-based open-source software library called Ace-TN was developed to accelerate iPEPS calculations. Benchmarks are presented showing efficient single- and muli-gpu execution of the core iPEPS simulation techniques. iPEPS simulations are performed for quantum spin systems with bond directional interactions, referred to as quantum compass models (QCM) or Kitaev models depending on pre-established naming conventions. The QCM defined on square and kagome lattices are considered with the addition of an external magnetic field. Each system possesses one-dimensional gauge-like symmetries in zero field, relating subextensively-many ground states that are capable of being represented by iPEPS. These states are shown to be metastable in non-zero field through adiabatic evolution from the zero-field ground state, each separated by a small energy gap. Several field-dependent phases and phase transitions are found in each system. Prominent nematic phases are found in both the square- and kagome-lattice systems. The square-lattice system contains a field value that admits an exact solution of the ground state which is characteristic of the surrounding phase. The inclusion of isotropic Heisenberg coupling to the kagome-lattice system is also explored, which produces a frustrated parameter regime between two dual quantum spin-liquid phases.
Differentiable programming is a fresh programming paradigm which composes parameterized algorithmic components and optimizes them using gradient search.The concept emerges from deep learning but is not limited to training neural networks.We present the theory and practice of programming tensor network algorithms in a fully differentiable way.By formulating the tensor network algorithm as a computation graph, one can compute higher-order derivatives of the program accurately and efficiently using automatic differentiation.We present essential techniques to differentiate through the tensor networks contraction algorithms, including numerical stable differentiation for tensor decompositions and efficient backpropagation through fixed-point iterations.As a demonstration, we compute the specific heat of the Ising model directly by taking the second-order derivative of the free energy obtained in the tensor renormalization group calculation.Next, we perform gradient-based variational optimization of infinite projected entangled pair states for the quantum antiferromagnetic Heisenberg model and obtain state-ofthe-art variational energy and magnetization with moderate efforts.Differentiable programming removes laborious human efforts in deriving and implementing analytical gradients for tensor network programs, which opens the door to more innovations in tensor network algorithms and applications.