Session R40: Precision many-body physics VI: Novel methods and algorithms

DMRG Approach to Optimizing Two-Dimensional Tensor Networks

Tensor network algorithms have been remarkably successful solving a variety of problems in quantum many-body physics. However, algorithms to optimize two-dimensional tensor networks known as PEPS lack many of the aspects that make the seminal density matrix renormalization group (DMRG) algorithm so powerful for optimizing one-dimensional tensor networks known as matrix product states. We implement a framework for optimizing two-dimensional PEPS tensor networks which includes all of steps that make DMRG so successful for optimizing one-dimension tensor networks. We present results for several 2D spin models and discuss possible extensions and applications.   

Diagrammatic Monte Carlo for the Hubbard Model: Recent developments

In this talk, I will present recent developments of the diagrammatic Monte Carlo method applied to the Hubbard model. Diagrammatic Monte Carlo methods are based on the construction of a perturbation series for physical observables. They face two main challenges: The first challenge is the accurate stochastic computation of the series coefficients that suffers from the fermionic sign problem. The second difficulty is the resummation of the series whose success depends on the structure of the physical observable expressed in the complex plane of the expansion parameter, e.g. the onsite repulsion U in the Hubbard model. I will discuss how these two issues can be addressed and, in particular, how the freedom to choose the non-interacting starting point of the theory can be used to construct more efficient perturbation series. I will illustrate these developments with calculations performed in different regimes of the Hubbard model.   

Differentiable Programming Tensor Networks

Differentiable programming is a fresh programming paradigm which composes parameterized algorithmic components and trains them using automatic differentiation (AD). 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.   

Simulating open quantum many-body systems using matrix product state purifications

Although we usually try to isolate quantum systems in the lab as effectively as possible, some degree of external noise and interaction with the environment is inevitable. This generally leads to decoherence, dissipation, and can alter the critical behavior in many-body systems. One can also try to engineer the environment coupling to achieve new physical phenomena. We consider Markovian systems, evolving according to Lindblad master equations. We introduce a new algorithm based on matrix product state purifications to simulate open many-body systems. This resolves a fundamental problem in simulations using matrix product density operators (MPDO) which generally lose positivity in truncations, leading to nonphysical states. We test and demonstrate our algorithm for spin chains and fermionic systems, comparing to exact diagonalization, analytical solutions, and MPDO simulations.   

Compressing Infinite Matrix Product Operators with an Application To Twisted Bilayer Graphene

We present a new method for compressing matrix product operators (MPOs) which represent sums of local terms, such as Hamiltonians. Just as with area law states, such local operators may be fully specified with a small amount of information per site. Standard matrix product state (MPS) tools are ill-suited to this case, due to extensive Schmidt values that coexist with intensive ones, and Jordan blocks in the transfer matrix. We ameliorate these issues by introducing an "almost Schmidt decomposition" that respects locality. Our method is "ε-close" to the accuracy of MPS-based methods for finite MPOs, and extends seamlessly to the thermodynamic limit, where MPS techniques are inapplicable. As an application, we compress the Hamiltonian for a model of twisted bilayer graphene --- whose naive bond dimension is tens of thousands --- down to a size where its ground state may be computed.

This talk is based on arXiv: 1909.06341.
  

Embedding via the Exact Factorization Approach

We present a quantum electronic embedding method derived from the exact factorization approach to calculate static properties of a many-electron system. The method is exact in principle but the practical power lies in utilizing input from a low-level calculation on the entire system in a high-level method computed on a small fragment, as in other embedding methods. Here, the exact factorization approach defines an embedding Hamiltonian on the fragment. Various Hubbard models demonstrate that remarkably accurate ground-state energies are obtained over the full range of weak to strongly correlated systems.   

Putting modern many-body methods to the test: the two-dimensional Hubbard model at weak coupling

We provide a detailed synopsis and comparison of a comprehensive set of state-of-the art many body techniques for the weak-coupling regime of the two dimensional half-filled Hubbard model on a square lattice. We put each of these methods to the test, for both one- and two-particle observables, in relation to the salient physical crossovers of this model: upon cooling from the high-temperature incoherent regime, coherent quasiparticles are formed below T_QP. At lower T, magnetic correlations stemming from the antiferromagnetically ordered phase at T=0 are gradually enhanced, resulting in the opening of an electronic (pseudo-)gap at T_*. By covering the range of modern many-body techniques available today, from numerically exact benchmark methods (determinantal and diagrammatic QMC [CDet]), over (dynamical) mean field theory (RPA, DMFT) and its cluster (DCA) and vertex based extensions (DΓA, TRILEX, DF, DB) to well-known approximations like parquet approximation (PA), the two-particle-self-consistent approach (TPSC) and the functional renormalization group (fRG), the realm of their applicability is put into perspective and a reference and agenda for future improvements is provided.   

A time-dependent tunneling approach to RIXS and other core-hole spectroscopies

We introduce a numerical approach to calculate the Resonant Inelastic X-Ray Scattering (RIXS) in strongly correlated systems. Since accounting for intermediate processes involved in RIXS requires the knowledge of all eigenstates of the Hamiltonian, these computations have remained very challenging. We recast the calculation as a tunneling problem that can be readily solved by means of the time-dependent DMRG method and does not require a full knowledge of the excitations. This powerful formulation overcomes all the hurdles imposed by other techniques that rely on explicitly obtaining dynamical spectral functions. Results for the Hubbard model are achieved with minimal effort on large systems using a fraction of the states --and simulation time-- required by the dynamical DMRG formulation. These ideas can be readily applied without modification away from equilibrium.   

A time and momentum resolved tunneling spectroscopy approach to non-equilibrium phenomena in correlated systems

We introduce a numerical method to calculate the spectral density of correlated systems using a tunneling approach efficiently implemented through the time-dependent Density Matrix Renormalization-Group (tDMRG). By using an extended probe, which is basically a copy of the system of interest, we are able to extract the time-resolved spectrum with the momentum information of the excitations. We illustrate our ideas by calculating the time-resolved spectrum of a Mott-insulating extended Hubbard chain after a sudden quench. Our results demonstrate that the system realizes a non-thermal state that contains an admixture of spin and charge density excitations, with corresponding signatures recognizable as in-gap subbands. In particular, we identify a band of excitons and one of stable anti-bound states at high energies that gains enhanced visibility after the pump. We do not appreciate noticeable relaxation within the time-scales considered, which is attributed to the lack of decay channels due to spin-charge separation.   

Effect of Electron-Phonon Interactions in the Holstein Model on a Staggered-Flux Square Lattice

The effect of electron-phonon coupling on Dirac fermions raises interesting questions concerning charge-density wave (CDW) formation in a semi-metal, which have recently been explored on a honeycomb lattice[1,2]. Here, we use the unbiased determinant Quantum Monte Carlo (DQMC) method to study the Holstein model on a half-filled staggered-flux square lattice, and compare with the results on the honeycomb lattice. Our work analyzes a range of phonon frequencies, 0.1 ≤ ω ≤ 2.0. We find that interactions give rise to charge-density wave order but only above a finite coupling strength λ_crit. The transition temperature is evaluated and presented in a T_c - λ phase diagram. An accompanying mean-field theory (MFT) calculation also reveals the existence of quantum phase transition, although at a smaller critical coupling strength than found in DQMC.

[1] "Charge Order in the Holstein Model on a Honeycomb Lattice," Y.-X. Zhang, W.-T. Chiu, N.C. Costa, G.G. Batrouni, and R.T. Scalettar, Phys. Rev. Lett. 122, 077602 (2019).
[2] "Charge-Density-Wave Transitions of Dirac Fermions Coupled to Phonons," Chuang Chen, Xiao Yan Xu, Zi Yang Meng, and Martin Hohenadler, Phys. Rev. Lett. 122, 077601 (2019).   

The effect of disorder on the phase diagrams of hard-core lattice bosons with cavity-mediated long-range and nearest-neighbor interactions

We use quantum Monte Carlo simulations with the worm algorithm to study the phase diagram of a two-dimensional Bose-Hubbard model with cavity-mediated long-range interactions and uncorrelated disorder in the hard-core limit. Our study shows the system is in a supersolid phase at weak disorder and a disordered solid phase at stronger disorder. Due to long-range interactions, a large region of metastable states exists in both clean and disordered systems. By comparing the phase diagrams for both clean and disordered systems, we find that disorder suppresses metastable states and superfluidity. We compare these results with the phase diagram of the extended Bose-Hubbard model with nearest-neighbor interactions. Here, the supersolid phase does not exist even at weak disorder. We identify two kinds of glassy phases: a Bose glass phase and a disordered solid phase. The glassy phases intervene between the density-wave and superfluid phases as the Griffiths phase of the Bose-Hubbard model. The disordered solid phase intervenes between the density-wave and Bose glass phases since both have a finite structure factor.