We combine the Density Matrix Renormalization Group (DMRG) with Matrix Product State tangent space concepts to construct a variational algorithm for finding ground states of one dimensional quantum lattices in the thermodynamic limit. A careful comparison of this variational uniform Matrix Product State algorithm (VUMPS) with infinite Density Matrix Renormalization Group (IDMRG) and with infinite Time Evolving Block Decimation (ITEBD) reveals substantial gains in convergence speed and precision. We also demonstrate that VUMPS works very efficiently for Hamiltonians with long range interactions and also for the simulation of two dimensional models on infinite cylinders. The new algorithm can be conveniently implemented as an extension of an already existing DMRG implementation.

We present a conjugate-gradient method for the ground-state optimization of projected entangledpair states (PEPS) in the thermodynamic limit, as a direct implementation of the variational principle within the PEPS manifold. Our optimization is based on an efficient and accurate evaluation of the gradient of the global energy functional by using effective corner environments, and is robust with respect to the initial starting points. It has the additional advantage that physical and virtual symmetries can be straightforwardly implemented. We provide the tools to compute static structure factors directly in momentum space, as well as the variance of the Hamiltonian. We benchmark our method on Ising and Heisenberg models, and show a significant improvement on the energies and order parameters as compared to algorithms based on imaginary-time evolution.

In these lecture notes we give a technical overview of tangent-space methods for matrix product states in the thermodynamic limit. We introduce the manifold of uniform matrix product states, show how to compute different types of observables, and discuss the concept of a tangent space. We explain how to variationally optimize ground-state approximations, implement real-time evolution and describe elementary excitations for a given model Hamiltonian. Also, we explain how matrix product states approximate fixed points of one-dimensional transfer matrices. We show how all these methods can be translated to the language of continuous matrix product states for one-dimensional field theories. We conclude with some extensions of the tangent-space formalism and with an outlook to new applications.

We use the formalism of tensor network states to investigate the relation between static correlation functions in the ground state of local quantum many-body Hamiltonians and the dispersion relations of the corresponding low-energy excitations. In particular, we show that the matrix product state transfer matrix (MPS-TM)-a central object in the computation of static correlation functionsprovides important information about the location and magnitude of the minima of the low-energy dispersion relation(s), and we present supporting numerical data for one-dimensional lattice and continuum models as well as two-dimensional lattice models on a cylinder. We elaborate on the peculiar structure of the MPS-TM's eigenspectrum and give several arguments for the close relation between the structure of the low-energy spectrum of the system and the form of the static correlation functions. Finally, we discuss how the MPS-TM connects to the exact quantum transfer matrix of the model at zero temperature. We present a renormalization group argument for obtaining finite bond dimension approximations of the MPS, which allows one to reinterpret variational MPS techniques (such as the density matrix renormalization group) as an application of Wilson's numerical renormalization group along the virtual (imaginary time) dimension of the system.. The overall energy scale is represented by a characteristic velocity (e.g., the Lieb-Robinson velocity related to the norm of the Hamiltonian terms, or some spin-wave velocity) in the system.In theory, the full dispersion relation can be reproduced from the ground state if the map between a local Hamiltonian and its corresponding ground state is bijective. For strictly n-local Hamiltonians (i.e., Hamiltonians for which every term acts only on a finite number n of neighboring sites), such a bijective relation is generically obtained. There the n-site reduced-density matrices (RDMs) of ground states represent extreme points in the convex set of all possible n-site RDMs. The Hamiltonian can then be represented as a hyperplane in the space of such RDMs, and the energy will necessarily be minimized for an extreme point in this set. Each of these points uniquely determines an n-local parent Hamiltonian via the tangent space to the boundary at this point, if the boundary is smooth there [4]. This argument is, however, of very limited practical use as it is computationally almost infeasible to characterize this convex set [5]. Also, the uniqueness is only obtained by restricting to a class of n-local Hamiltonians, and there might exist other + n k ( )-local (with ⩾ k 1) or quasi local Hamiltonians for which this is the exact ground state. One of the main goals of this paper is thus to identify which features of all those Hamiltonians can be captured in the ground state and its correlations.We follow a more practical approach based on local information contained within the ground state, which is naturally accessible through a tensor network representation of the same. A central local object arising in te...

Confinement is a process by which particles with fractional quantum numbers bind together to form quasiparticles with integer quantum numbers. The constituent particles are confined by an attractive interaction whose strength increases with increasing particle separation and as a consequence, individual particles are not found in isolation. This phenomenon is well known in particle physics where quarks are confined in baryons and mesons. An analogous phenomenon occurs in certain spatially anisotropic magnetic insulators. These can be thought of in terms of weakly coupled chains of spins S=1/2, and a spin flip thus carries integer spin S=1. Interestingly the collective excitations in these systems, called spinons, turn out to carry fractional spin quantum number S=1/2. Interestingly, at sufficiently low temperatures the weak coupling between chains can induce an attractive interaction between pairs of spinons that increases with their separation and thus leads to confinement. In this paper, we employ inelastic neutron scattering to investigate the spinonconfinement process in the quasi-one dimensional, spin-1/2, antiferromagnet with Heisenberg-Ising (XXZ) anisotropy SrCo2V2O8. A wide temperature range both above and below the long-range ordering temperature TN =5.2 K is explored. Spinon excitations are observed above TN in quantitative agreement with established theory. Below TN the pairs of spinons are confined and two sequences of meson-like bound states with longitudinal and transverse polarizations are observed. Several theoretical approaches are used to explain the data. These are based on a description in terms of a one-dimensional, S=1/2 XXZ antiferromagnetic spin chain, where the interchain couplings are modelled by an effective staggered magnetic mean-field. A wide range of exchange anisotropies are investigated and the parameters specific to SrCo2V2O8 are identified. A new theoretical technique based on Tangent-space Matrix Product States gives a very complete description of the data and provides good agreement not only with the energies of the bound modes but also with their intensities. We also successfully explained the effect of temperature on the excitations including the experimentally observed thermally induced resonance between longitudinal modes below TN , and the transitions between thermally excited spinon states above TN . In summary, our work establishes SrCo2V2O8 as a beautiful paradigm for spinon confinement in a quasi-one dimensional quantum magnet and provides a comprehensive picture of this process.