Introduction
Quantum Many-Body Problem
This is more commonly accepted in the quantum physics community, where high-level numerical theory is used to gain insight into the underlying mechanism. In the remainder of this chapter, we will briefly present the opportunities we have identified in this dynamic.
Precision Simulation for the Condensed Phase
Alternatively, we can consider a paradigm shift to wavefunction-based approaches widely adopted in the quantum chemistry community. We are interested in investigating the performance of paired clustering in the solid-state interconnected region.
Tensor Contraction with Symmetry Groups
After balancing all the above factors, coupled cluster theory is arguably the most promising candidate for solid state application. Indeed, pioneering work in the field in recent years has suggested promising results from applying low-level coupled cluster theory to weakly correlated crystalline materials.
Beyond Bohr-Oppenheimer Approximation
In the solid state, crystal vibrations are usually characterized by phonons representing the collective excitations of the underlying lattice. At the other end of the spectrum, method (2) often treats coupling as a confusing term for the electronic problem as a computational trade-off.
Tensor Network Methods for Fermion Simulation
The 𝑇1 metric is the Frobenius norm (normalized by the number of correlated electrons) of the 𝑡1 amplitudes. Our results clearly validate the enormous potential of the coupled cluster framework in the correlated region of the solid state.
Electronic Structure of Crystalline Materials from Coupled Cluster
Abstract
We present the ground and excited state electronic structure of two prototypical transition metal oxides, MnO and NiO using coupled clustering in its single and double approximation (CCSD and EOM-CCSD). Moreover, our wave function representation allows for a detailed analysis of the charge transfer/Mott insulating character and atomic nature of the electronic bands of the two materials.
Introduction
The coupled cluster (CC) theory is a theoretical framework that originated in quantum chemistry and nuclear physics [29, 30] and has recently emerged as a new way of understanding the electronic structure in solids at the many-dimensional level. treat particles [31, 32]. In section 2.4 we present the CCSD results on NiO and MnO, where a detailed analysis is given of the numerical convergence, ground state properties and the nature of the insulation states.
Theory
A detailed comparison of the diagrammatic content of GW and excited-state EOM-CCSD can be found in [37]. Here, 𝐸𝑛 is the energy of the 𝑛th excited state and 𝜇 indexes an element of the excitation operator𝑅.
Results
The CC ground state moments reported in Table 2.2 are significantly reduced from those of the UHF reference. Figure 2.7 shows graphs of the true density of the quasiparticle orbitals on the VBM and CBM.
Conclusion
Spectral functions of the uniform electron gas via coupled cluster theory and comparison with the G W and related approaches”. The graphical notation of fermion tensor contraction is also slightly modified to take into account the order of the two operators. Our fermionic tensor backend is coupled with the Quimb package[18] to take advantage of the tensor network infrastructure.
Efficient Contraction Scheme for Tensors with Cyclic Group
Abstract
In this work, we describe how we can accelerate tensor contractions involving block-sparse tensors whose structure is induced by a cyclic group symmetry or a product of cyclic group symmetries. Intuitively aided by tensor diagrams, we present our irreducible representation alignment technique, which allows efficient processing of such block sparsity structures via only dense tensor operations. The algorithm is implemented in Python and we perform benchmark calculations on a variety of representative contractions from quantum chemistry and tensor network methods.
Introduction
We call such tensors tensors with cyclic group symmetry, or cyclic group tensors for short. State-of-the-art sequential and parallel libraries for handling cyclic group symmetry, both in specific physical applications and domain-agnostic settings, typically iterate over non-zero blocks stored in a block-sparse tensor format [1–12 ]. We study the effectiveness of this new method for cyclic group tensor contractions occurring in periodic coupled cluster theory and tensor network methods.
Theory
The standard reduced form is primarily called the reduced form indexed by 3 symmetry modes, which are a subset of the symmetry modes of the original tensors. The standard minified form provides an implicit representation of the non-preserved symmetry mode due to symmetry conservation and can easily be used to implement the blockwise shrinking approach that dominates many libraries. The symmetry mode𝑄 is chosen so that it can serve as part of the reduced forms of each of U, V, and W.
Implementation
As introduced in section 3.3, the main operations in our irrep alignment algorithm consist of the transformation of the reduced form and the contraction of reduced forms. Symtensor chooses the least costly version of the irrep alignment algorithm from a space of variants defined by different choices of the implicitly represented modes of the three tensors in symmetry-aligned reduced form. This allows both the transformations and the reduced shape contraction to be performed as dense tensor sum operations with the desired backend.
Benchmarks
First, we examine the performance of the irrep alignment algorithm for three contractions as a function of increasing𝐺. We now illustrate the parallelism of the irrep alignment algorithm by studying multi-node scalability with distributed memory. The solid lines in Figure 3.11 show the strong scaling behavior (fixed problem size) of the Symtensor implementation on up to eight nodes.
Conclusion
The fermionic part of the Hamiltonian is a Hubbard model with jumps 𝑡 and on-site reflection𝑈. Importantly, many-body electronic effects have been shown to be important for direct-gap ZPR [18] and dynamical effects have been shown to be important for capturing some qualitative features of EPI [79] . The nature of the spin-fluid ground state of the Heisenberg model S= 1/2 on the kagome lattice.
Coupled Cluster Ansatz for Electrons and Phonons
Abstract
Benchmarks in the Hubbard-Holstein model allow us to evaluate the strengths and weaknesses of various coupled set approximations, which generally perform well for weak to moderate coupling. Finally, we report progress towards an implementation for ab initio calculations in solids and present some preliminary results on finite-size diamond models with a linear electron-phonon coupling. We also report the implementation of electron-phonon coupling matrix elements from Gaussian-type crystal orbitals (cGTO) within the PySCF program package.
Introduction
In this work we describe a coupled cluster theory and the corresponding scaling equation of motion for interacting electrons and phonons. This theory is similar to several combined cluster theories for cavity polaritons that were developed independently around the same time[36, 37]. In Section 4.3, we describe the ground-state coupled electron-phonon cluster theory and the EOM formalism for excited states for coupled electron-phonon systems.
Theory
We will refer to these types of theories as electron-phonon coupled cluster (ep-CC). All theories include single and double numbers for the pure electronic part of the 𝑇𝑒𝑙 operator (not shown), and we've omitted them here for simplicity. Note that our ep-CCSD-1-S1 method is the same as the QED-CCSD-1 method presented in Ref.
Benchmark on Hubbard-Holstein Model
This transformation diagonalizes the effective phonic Hamiltonian obtained by normal ordering the electronic part of the EPI term. We then focus on the accuracy of excited state properties of EOM variant of the theory. As 𝜆 increases, we find that the results are qualitatively correct, although the closing of the gap at 𝜆 = 0.6 and the Peierls insulating condition at 𝜆 > 0.6 are not captured by this approximation.
Application to Periodic Solids
Second, we find that the bandgap renormalization becomes unexpectedly large as the size of the basis set increases. These results are not limited to calculating the direct gap, so the results for 2x2x2 and 3x3x3 supercells should be viewed as finite-size approximations to the indirect bandgap ZPR. The slow and oscillatory convergence of the ZPR of supercell-sized diamond is a well-known problem.
Conclusion
Phase diagram of the one-dimensional molecular crystal model with Coulomb interactions: half-filled band sector”. One of the most fundamental principles of the fermionic tensor network is that only tensors that are adjacently ordered (connected by the brown arrow) can be directly contracted. It has been shown that the order of fermion tensors can be reversed by introducing a phase into each block of the tensor inputs[11].
Fermionic Tensor Network Simulation with Arbitrary Geometry . 73
Tensor Network Theory
Fermionic operators on the other hand, follow anti-commuting rules due to their anti-symmetry nature. In particular, we implement anti-motion tensor algebras at the last level so that they behave like fermionic operators. Intuitively, our "tensor" object now represents a collection of fermionic operators: i,viis the pure tensor object that obeys commutative algebraic rules and ˆ𝑂s. i are the fermionic operators acting in physical space and virtual space respectively.
Linear Algebras for Fermionic Tensors
This phase is in practice absorbed into the permuted tensor data to form a new fermion tensor object. Notably, in tensor network theories, the decomposition routines are mainly called for canonicalization or compression between two tensors, so it is natural to adopt the convention that the two output tensors each take the same total symmetry as the two input tensors . In our implementation, once the output symmetry splitting is determined, we can calculate all possible indices for the common index of the output tensors.
Rules for Fermionic Tensor Network
As we will prove here, compression and canonicalization in our construction can be done in-place without the need for a swap. The whole compression operation can be divided into three steps: (1) The swap operation is performed between 𝐵and𝐶so that the new state becomes𝐴𝐶𝐵˜. 2) Compression is performed on adjacent 𝐴𝐶, leading to the new state ¯𝐴𝐶¯𝐵˜. We can easily generalize to cases with any number of operators between 𝐴and𝐶, since all the operators in between can be seen as one big, contracted 𝐵.
Results
The second type is referred to as cluster approach as shown in Figure 5.9, which is inspired by the cluster update method in TEBD[23]. Therefore, we will calculate the energies using the group approximation method shown in Figure 5.9. We therefore used a radius of 3 to calculate the approximate energies and the results are shown in Figure 5.15.
Conclusion
The first entry in the SU column is calculated from compressed contraction and the second from cluster approximation. In the future, we expect more quantum chemical methods to be transferred to the solid state, which can potentially provide Our results indicate that in the highly correlated region, tensor network methods can be at least as competitive as coupled cluster methods.
Summary and Future Outlook
