SciPost Phys. Lect. Notes 5 (2018) Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy) Johannes Hauschild1? and Frank Pollmann1 1 Department of Physics, TFK, Technische Universität München, James-Franck-Straße 1, D-85748 Garching, Germany ? [email protected] Abstract Tensor product state (TPS) based methods are powerful tools to efficiently simulate quantum many-body systems in and out of equilibrium. In particular, the one-dimensional matrix-product (MPS) formalism is by now an established tool in con- densed matter theory and quantum chemistry. In these lecture notes, we combine a compact review of basic TPS concepts with the introduction of a versatile tensor library for Python (TeNPy) [1]. As concrete examples, we consider the MPS based time-evolving block decimation and the density matrix renormalization group algorithm. Moreover, we provide a practical guide on how to implement abelian symmetries (e.g., a particle number conservation) to accelerate tensor operations. Copyright J. Hauschild and F. Pollmann Received 08-05-2018 This work is licensed under the Creative Commons Accepted 27-09-2018 Check for Attribution 4.0 International License. Published 08-10-2018 updates Published by the SciPost Foundation. doi:10.21468/SciPostPhysLectNotes.5 Contents 1 Introduction2 2 Entanglement in quantum many-body systems3 2.1 Area law4 3 Finite systems in one dimension5 3.1 Matrix Product States (MPS)6 3.2 Canonical form8 3.3 Time Evolving Block Decimation (TEBD) 10 3.4 Matrix Product Operators (MPO) 14 3.5 Density Matrix Renormalization Group (DMRG) 15 4 Infinite systems in one dimension 18 4.1 Infinite Time Evolving Block Decimation (iTEBD) 19 4.2 Infinite Density Matrix Renormalization Group (iDMRG) 21 5 Charge conservation 22 5.1 Definition of charges 22 5.2 Basic operations on tensors 24 1 SciPost Phys. Lect. Notes 5 (2018) 6 Conclusion 27 References 28 1 Introduction The interplay of quantum fluctuations and correlations in quantum many-body systems can lead to exciting phenomena. Celebrated examples are the fractional quantum Hall effect [2,3 ], the Haldane phase in quantum spin chains [4,5 ], quantum spin liquids [6], and high- temperature superconductivity [7]. Understanding the emergent properties of such challeng- ing quantum many-body systems is a problem of central importance in theoretical physics. The main difficulty in investigating quantum many-body problems lies in the fact that the Hilbert space spanned by the possible microstates grows exponentially with the system size. To unravel the physics of microscopic model systems and to study the robustness of novel quantum phases of matter, large scale numerical simulations are essential. In certain systems where the infamous sign problem can be cured, efficient quantum Monte Carlo (QMC) meth- ods can be applied. In a large class of quantum many-body systems (most notably, ones that involve fermionic degrees of freedom or geometric frustration), however, these QMC sampling techniques cannot be used effectively. In this case, tensor-product state based methods have been shown to be a powerful tool to efficiently simulate quantum many-body systems. The most prominent algorithm in this context is the density matrix renormalization group (DMRG) method [8] which was originally conceived as an algorithm to study ground state properties of one-dimensional (1D) systems. The success of the DMRG method was later found to be based on the fact that quantum ground states of interest are often only slightly entangled (area law), and thus can be represented efficiently using matrix-product states (MPS) [9–11]. More recently it has been demonstrated that the DMRG method is also a useful tool to study the physics of two-dimensional (2D) systems using geometries such as a cylinder of finite circum- ference so that the quasi-2D problem can be mapped to a 1D one [12]. The DMRG algorithm has been successively improved and made more efficient. For example, the inclusion of Abelian and non-Abelian symmetries, [13–17], the introduction of single-site optimization with density matrix perturbation [18, 19], hybrid real-momentum space representation [20, 21], and the development of real-space parallelization [22] have increased the convergence speed and de- creased the requirements of computational resources. An infinite version of the algorithm [23] has facilitated the investigation of translationally invariant systems. The success of DMRG was extended to also simulate real-time evolution allowing to study transport and non-equilibrium phenomena, [24–29]. However, the bipartite entanglement of pure states generically grows linearly with time, leading to a rapid exponential growth of the computational cost. This limits time evolution to rather short times. An exciting recent development is the generalization of DMRG to obtain highly excited states of many-body localized systems [30–32] (see also [33] for a different approach). Tensor-product states (TPS) or equivalently projected entangled pair states (PEPS), are a generalization of MPS to higher dimensions [34,35]. This class of states is believed to efficiently describe a wide range of ground states of two-dimensional local Hamil- tonians. TPS serve as variational wave functions that can approximate ground states of model Hamiltonians. For this several algorithms have been proposed, including the Corner Trans- fer Matrix Renormalization Group Method [36], Tensor Renormalization Group (TRG) [37], Tensor Network Renormalization (TNR) [38], and loop optimizations [39]. A number of very useful review articles on different tensor network related topics appeared 2 SciPost Phys. Lect. Notes 5 (2018) over the past couple of years. Here we mention a few: Ref. [11] provides a pedagogical intro- duction to MPS and DMRG algorithms with detailed discussions regarding their implementa- tion. In Ref. [40], a practical introduction to tensor networks including MPS and TPS is given. Applications of DMRG in quantum chemistry are discussed in Ref. [41]. In these lecture notes we combine a pedagogical review of basic MPS and TPS based al- gorithms for both finite and infinite systems with the introduction of a versatile tensor library for Python (TeNPy) [1]. In the following section, we motivate the ansatz of TPS with the area law of entanglement entropy. In section3 we introduce the MPS ansatz for finite systems and explain the time evolving block decimation (TEBD) [24] and the DMRG method [8] as promi- nent examples for algorithms working with MPS. In section4 we explain the generalization of these algorithms to the thermodynamic limit. For each algorithm, we give a short exam- ple code showing how to call it from the TeNPy library. Finally, we provide a practical guide on how to implement abelian symmetries (e.g., a particle number conservation) to accelerate tensor operations in section5. 2 Entanglement in quantum many-body systems Entanglement is one of the fundamental phenomena in quantum mechanics and implies that different degrees of freedom of a quantum system cannot be described independently. Over the past decades it was realized that the entanglement in quantum many-body systems can give access to a lot of useful information about quantum states. First, entanglement related quan- tities provide powerful tools to extract universal properties of quantum states. For example, scaling properties of the entanglement entropy help to characterize critical systems [42–45], and entanglement is the basis for the classification of topological orders [46,47]. Second, the understanding of entanglement helped to develop new numerical methods to efficiently sim- ulate quantum many-body systems [11, 48]. In the following, we give a short introduction to entanglement in 1D systems and then focus on the MPS representation. Let us consider the bipartition of the Hilbert space H = HL HR of a 1D system as illus- ⊗ trated in Fig.1(a), where HL (HR) describes all the states defined on the left (right) of a given bond. In the so called Schmidt decomposition, a (pure) state Ψ H is decomposed as X j i 2 H Ψ = Λα α L α R , α L(R) L(R), (1) j i α j i ⊗ j i j i 2 H H where the states α L(R) form an orthonormal basis of (the relevant subspace of) L ( R) fj i g and Λα 0. The Schmidt decomposition is unique up to degeneracies and for a normalized ≥ P 2 state Ψ we find that α Λα = 1. Anj importanti aspect of the Schmidt decomposition is that it gives direct insight into the bipartite entanglement (i.e., the entanglement between degrees of freedom in HL and HR) of a state, as we explain in the following. The amount of entanglement is measured by the R R entanglement entropy, which is defined as the von-Neumann entropy S = Tr ρ log(ρ ) of the reduced density matrix ρR. The reduced density matrix of an entangled− (pure) quantum state is the density matrix of a mixed state defined on the subsystem, j i R ρ TrL ( ) . (2) ≡ j i h j A simple calculation shows that it has the Schmidt states α R as eigenstates and the Schmidt j i R P 2 coefficients are the square roots of the corresponding eigenvalues, i.e., ρ = α Λα α R α R (equivalently for ρL). Hence, the entanglement entropy can be expressed in termsj i ofh thej Schmidt values Λα, R R X 2 2 S Tr ρ log(ρ ) = Λα log Λα. (3) ≡ − − α 3 SciPost Phys. Lect. Notes 5 (2018) Figure 1: (a): Bipartition of a 1D system into two half chains. (b): Significant quan- tum fluctuations in gapped ground states occur only on short length scales. (c): 1D area law states make up a very small fraction of the many-body Hilbert space but contain all gapped ground states. (d): Comparison of the largest Schmidt values of the ground state of the transverse field Ising model (g = 1.5) and a random state for a system consisting of N = 16 spins.