Differentiable Programming Tensor Networks

Differentiable programming is a fresh programming paradigm which composes parameterized algorithmic components and trains them using automatic differentiation (AD). The concept emerges from deep learning but is not only limited to training neural networks. We present 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 AD. We present essential techniques to differentiate through the tensor networks contractions, including stable AD for tensor decomposition 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 quantum antiferromagnetic Heisenberg model and obtain start-of-the-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.Comment: Typos corrected, discussion and refs added; revised version accepted for publication in PRX. Source code available at https://github.com/wangleiphy/tensorgra

Dynamic Gardner crossover in a simple structural glass

The criticality of the jamming transition responsible for amorphous solidification has been theoretically linked to the marginal stability of a thermodynamic Gardner phase. While the critical exponents of jamming appear independent of the preparation history, the pertinence of Gardner physics far from equilibrium is an open question. To fill this gap, we numerically study the nonequilibrium dynamics of hard disks compressed towards the jamming transition using a broad variety of protocols. We show that dynamic signatures of Gardner physics can be disentangled from the aging relaxation dynamics. We thus define a generic dynamic Gardner crossover regardless of the history. Our results show that the jamming transition is always accessed by exploring increasingly complex landscape, resulting in the anomalous microscopic relaxation dynamics that remains to be understood theoretically

Spin Excitation Spectra of Anisotropic Spin-1/21/2 Triangular Lattice Heisenberg Antiferromagnets

Investigation of dynamical excitations is difficult but crucial to the understanding of many exotic quantum phenomena discovered in quantum materials. This is particularly true for highly frustrated quantum antiferromagnets whose dynamical properties deviate strongly from theoretical predictions made based on the spin-wave or other approximations. Here we present a large-scale numerical calculation on the dynamical correlation functions of spin-$1/2$ triangular Heisenberg model using a state-of-the-art tensor network renormalization group method. The calculated results allow us to gain for the first time a comprehensive picture on the nature of spin excitation spectra in this highly frustrated quantum system. It provides a quantitative account for all the key features of the dynamical spectra disclosed by inelastic neutron scattering measurements for $\rm Ba_3CoSb_2O_9$, revealing the importance of the interplay between low- and high-energy excitations and its renormalization effect to the low-energy magnon bands and high-energy continuums. We identify the longitudinal Higgs modes in the intermediate-energy scale and predict the energy and momentum dependence of spectral functions along the three principal axes that can be verified by polarized neutron scattering experiments. Furthermore, we find that the spin excitation spectra weakly depend on the anisotropic ratio of the antiferromagnetic interaction.Comment: 6 pages, 3 figures, and a Supplemental Materia

A Novel Completely Local Repairable Code Algorithm Based on Erasure Code

Hadoop Distributed File System (HDFS) is widely used in massive data storage. Because of the disadvantage of the multi-copy strategy, the hardware expansion of HDFS cannot keep up with the continuous volume of big data. Now, the traditional data replication strategy has been gradually replaced by Erasure Code due to its smaller redundancy rate and storage overhead. However, compared with replicas, Erasure Code needs to read a certain amount of data blocks during the process of data recovery, resulting in a large amount of overhead for I/O and network. Based on the Reed-Solomon (RS) algorithm, we propose a novel Completely Local Repairable Code (CLRC) algorithm. By grouping RS coded blocks and generating local check blocks, CLRC algorithm can optimize the locality of the RS algorithm, which can reduce the cost of data recovery. Evaluations show that the CLRC algorithm can reduce the bandwidth and I/O consumption during the process of data recovery when a single block is damaged. What\u27s more, the cost of decoding time is only 59% of the RS algorithm

Differentiable programming tensor networks for Kitaev magnets

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 $K$-$\Gamma$ model. Furthermore, in the case of the realistic $K$-$J$-$\Gamma$-$\Gamma'$ model for the Kitaev material $\alpha$-RuCl$_3$, we discovered a non-colinear 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

Projected d-wave superconducting state: a fermionic projected entangled pair state study

We investigate the physics of projected d-wave pairing states using their fermionic projected entangled pair state (fPEPS) representation. First, we approximate a d-wave Bardeen-Cooper-Schrieffer state using the Gaussian fPEPS. Next, we translate the resulting state into fPEPS tensors and implement the Gutzwiller projection which removes double occupancy by modifying the local tensor elements. The tensor network representation of the projected d-wave pairing state allows us to evaluate physical quantities in the thermodynamic limit without employing the Gutzwiller approximation. Despite having very few variational parameters, such physically motivated tensor network states are shown to exhibit competitive energies for the doped t-J model. We expect that such construction offers useful initial states and guidance for variational tensor network calculations.Comment: 9 pages, 7 figure