The curse of dimensionality associated with the Hilbert space of spin systems provides a significant obstruction to the study of condensed matter systems. Tensor networks have proven an important ...tool in attempting to overcome this difficulty in both the numerical and analytic regimes. These notes form the basis for a seven lecture course, introducing the basics of a range of common tensor networks and algorithms. In particular, we cover: introductory tensor network notation, applications to quantum information, basic properties of matrix product states, a classification of quantum phases using tensor networks, algorithms for finding matrix product states, basic properties of projected entangled pair states, and multiscale entanglement renormalisation ansatz states. The lectures are intended to be generally accessible, although the relevance of many of the examples may be lost on students without a background in many-body physics/quantum information. For each lecture, several problems are given, with worked solutions in an ancillary file.
Matrix-product states have become the de facto standard for the representation of one-dimensional quantum many body states. During the last few years, numerous new methods have been introduced to ...evaluate the time evolution of a matrix-product state. Here, we will review and summarize the recent work on this topic as applied to finite quantum systems. We will explain and compare the different methods available to construct a time-evolved matrix-product state, namely the time-evolving block decimation, the MPO WI,II method, the global Krylov method, the local Krylov method and the one- and two-site time-dependent variational principle. We will also apply these methods to four different representative examples of current problem settings in condensed matter physics.
•Introductory review of time-evolution methods for matrix-product states.•Detailed description of five state-of-the-art algorithms.•Extended comparison of numerical examples.•Extended error analysis of each algorithm.•Pseudo-code examples towards an implementation where necessary.
We investigate the effect of uniaxial heterostrain on the interacting phase diagram of magic-angle twisted bilayer graphene. Using both self-consistent Hartree-Fock and density-matrix renormalization ...group calculations, we find that small strain values (ε ∼ 0.1 % – 0.2 %) drive a zero-temperature phase transition between the symmetry-broken "Kramers intervalley-coherent" insulator and a nematic semimetal. The critical strain lies within the range of experimentally observed strain values, and we therefore predict that strain is at least partly responsible for the sample-dependent experimental observations.
It is shown that from the point of view of the generalized pairing Hamiltonian, the atomic nucleus is a system with small entanglement and can thus be described efficiently using a 1D tensor network ...(matrix‐product state) despite the presence of long‐range interactions. The ground state can be obtained using the density‐matrix renormalization group (DMRG) algorithm, which is accurate up to machine precision even for large nuclei, is numerically as cheap as the widely used Bardeen‐Cooper‐Schrieffer (BCS) approach, and does not suffer from any mean‐field artifacts. This framework is applied to compute the even‐odd mass differences of all known lead isotopes from 178Pb$^{178}\text{Pb}$ to 220Pb$^{220}\text{Pb}$ in a very large configuration space of 13 shells between the neutron magic numbers 82 and 184 (i.e., two major shells) and find good agreement with the experiment. Pairing with non‐zero angular momentum is also considered and the lowest excited states in the full configuration space of one major shell is determined, which is demonstrated for the N=126$N=126$, Z≥82$Z\ge 82$ isotones. To demonstrate the capabilities of the method beyond low‐lying excitations, the first 100 excited states of 208Pb$^{208}\text{Pb}$ with singlet pairing and the two‐neutron removal spectral function of 210Pb$^{210}\text{Pb}$ are calculated, which relate to a two‐neutron pickup experiment.
It is shown that from the point of view of the generalized pairing Hamiltonian, the atomic nucleus is a system with small entanglement and can thus be described efficiently using a 1D tensor network (matrix‐product state) despite the presence of long‐range interactions. The ground state can be obtained using the density‐matrix renormalization group (DMRG) algorithm, which is accurate up to machine precision even for large nuclei, is numerically as cheap as the widely used BCS approach, and does not suffer from any mean‐field artifacts.
We determine in a closed form all scalar one-point functions of the defect CFT dual to the D3–D5 probe brane system with k units of flux which amounts to calculating the overlap between a Bethe ...eigenstate of the integrable SO(6) spin chain and a certain matrix product state of bond dimension k. In particular, we show that the matrix product state is annihilated by all the parity odd charges of the spin chain which has recently been suggested as the criterion for such a state to correspond to an integrable initial state. Finally, we discuss the properties of the analogous matrix product state for the SO(5) symmetric D3–D7 probe brane set-up.
Motivated by conjectures in holography relating the entanglement of purification and reflected entropy to the entanglement wedge cross section, we introduce two related non-negative measures of ...tripartite entanglement g and h. We prove structure theorems which show that states with nonzero g or h have nontrivial tripartite entanglement. We then establish that in one dimension these tripartite entanglement measures are universal quantities that depend only on the emergent low-energy theory. For a gapped system, we argue that either g≠0 and h=0 or g=h=0, depending on whether the ground state has long-range order. For a critical system, we develop a numerical algorithm for computing g and h from a lattice model. We compute g and h for various CFTs and show that h depends only on the central charge whereas g depends on the whole operator content.
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