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 W I,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.

We introduce a strictly single-site DMRG algorithm based on the subspace expansion of the Alternating Minimal Energy (AMEn) method. The proposed new MPS basis enrichment method is sufficient to avoid local minima during the optimisation, similarly to the density matrix perturbation method, but computationally cheaper. Each application ofĤ to |Ψ in the central eigensolver is reduced in cost for a speed-up of ≈ (d + 1)/2, with d the physical site dimension. Further speed-ups result from cheaper auxiliary calculations and an often greatly improved convergence behaviour. Runtime to convergence improves by up to a factor of 2.5 on the Fermi-Hubbard model compared to the previous single-site method and by up to a factor of 3.9 compared to two-site DMRG. The method is compatible with real-space parallelisation and non-abelian symmetries.

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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.