We review recent developments in tensor network approaches, focusing on renormalization group methods. Since they are free from the negative sign and complex action problems, there is growing interest in their application to lattice field theories, particularly with a view toward future studies of quantum chromodynamics (QCD) at finite temperature and density. They are also of broad interest in quantum field theory, with recent advances in approaches that allow one to directly investigate universal aspects of critical behavior by making use of theoretical insights from conformal field theory. We highlight several recently explored topics that are expected to play important roles in forthcoming tensor-network studies of QCD.
We review recent developments in tensor network approaches, focusing on renormalization group methods. Since they are free from the negative sign and complex action problems, there is growing interest in their application to lattice field theories, particularly with a view toward future studies of quantum chromodynamics (QCD) at finite temperature and density. They are also of broad interest in quantum field theory, with recent advances in approaches that allow one to directly investigate universal aspects of critical behavior by making use of theoretical insights from conformal field theory. We highlight several recently explored topics that are expected to play important roles in forthcoming tensor-network studies of QCD.
Formulating gauge theories on a lattice offers a genuinely non-perturbative way of studying quantum field theories, and has led to impressive achievements. In particular, it significantly deepened our understanding of quantum chromodynamics. Yet, some very relevant problems remain inherently challenging, such as real time evolution, or the presence of a chemical potential, cases in which Monte Carlo simulations are hindered by a sign problem. In the last few years, a number of possible alternatives have been put forward, based on quantum information ideas, which could potentially open the access to areas of research that have so far eluded more standard methods. They include tensor network calculations, quantum simulations with different physical platforms and quantum computations, and constitute nowadays a vibrant research area. Experts from different fields, including experimental and theoretical high energy physics, condensed matter, and quantum information, are turning their attention to these interdisciplinary possibilities, and driving the progress of the field. The aim of this article is to review the status and perspectives of these new avenues for the exploration of lattice gauge theories.
We discuss the successes and limitations of statistical sampling for a sequence of models studied in the context of lattice QCD and emphasize the need for new methods to deal with finite-density and real-time evolution. We show that these lattice models can be reformulated using tensorial methods where the field integrations in the path-integral formalism are replaced by discrete sums. These formulations involve various types of duality and provide exact coarse-graining formulas which can be combined with truncations to obtain practical implementations of the Wilson renormalization group program. Tensor reformulations are naturally discrete and provide manageable transfer matrices. Combining truncations with the time continuum limit, we derive Hamiltonians suitable to perform quantum simulation experiments, for instance using cold atoms, or to be programmed on existing quantum computers. We review recent progress concerning the tensor field theory treatment of non-compact scalar models, supersymmetric models, economical four-dimensional algorithms, noise-robust enforcement of Gauss's law, symmetry preserving truncations and topological considerations.
The phase diagram of QCD is a central research topic in modern particle physics. Its calculation, however, is notoriously difficult especially when not only nonzero temperatures
T but also non-vanishing baryon densities are considered – as is relevant for heavy‑ion collisions or inside large neutron stars. At the theoretical level, nonzero baryon densities are achieved by introducing a nonzero chemical potential
μ, which creates a particle-antiparticle asymmetry. It turns out that for nonzero chemical potential, the fermion determinant is no longer real but complex, and therefore standard approaches based on importance sampling break down. In the literature, this is commonly referred to as the sign problem. In order to circumvent this sign problem, a variety of methods has been proposed. Examples include a Taylor expansion in
μ, reweighting, complex Langevin, thimbles or path optimization. Nevertheless, none of the proposed remedies can successfully reach regimes in which
μ/T>1, and therefore even today the full QCD phase diagram in the
μ−T plane cannot be calculated. There is a promising further approach, called tensor-network approach, that differs crucially from all of the aforementioned approaches. In this approach, the partition function is written as a contraction of a large number of local tensors. Of special relevance is the higher order tensor renormalization group (HOTRG) approach, which can be applied for arbitrary number of dimensions, and in particular its generalization, the Grassmann HOTRG (GHOTRG) approach, which is applicable to theories including fermions. Recently, GHOTRG was tailored to strong-coupling QCD using one flavor of staggered quarks in two dimensions. It was observed that, for a non‑zero chemical potential, no numerical instabilities arise. In this thesis, we go beyond both the strong-coupling limit and the single-flavor case. We express the partition function of lattice QCD with staggered fermions in the strong-coupling expansion in terms of a tensor network. This is done for an arbitrary number of dimensions, flavors, and orders of the strong-coupling expansion. Further, the GHOTRG approach is extended such that every coefficient
Zn of the partition function
Z=∑nZnβn can be calculated separately. Finally, numerical results in two dimensions are studied.