model at non-zero chemical potential and temperature, which were computed using the higher-order tensor-renormalization-group method (HOTRG). This necessitated enhancements to the HOTRG blocking procedure to reduce the truncation error in the case of anisotropic tensors. Moreover, the construction of the truncated vector spaces was adapted to strongly reduce the effect of systematic errors in the computation of observables using the finite-difference method. Our (improved) HOTRG results for the evolution of the number density with the chemical potential are in agreement with results obtained with the worm algorithm, and both the Silver Blaze phenomenon at zero temperature and the temperature dependence of the number density can be adequately reproduced.

DOI: 10.48550/arxiv.2112.01414

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Tensor-network study of the 3d

O(2)

model at non-zero chemical potential and temperature

We present results of tensor-network simulations of the three-dimensional

O(2)

We present results of tensor-network simulations of the three-dimensional

O(2)

We calculate thermodynamic potentials and their derivatives for the three-dimensional

O(2)

model using tensor-network methods to investigate the well-known second-order phase transition. We also consider the model at nonzero chemical potential to study the Silver Blaze phenomenon, which is related to the particle number density at zero temperature. Furthermore, the temperature dependence of the number density is explored using asymmetric lattices. Our results for both zero and nonzero magnetic field, temperature, and chemical potential are consistent with those obtained using other methods.

D=70

, the absolute errors of the critical temperature

{T}_{c}

and the critical exponent $\ensuremath{\nu}$, $\ensuremath{\beta}$ are at the order of ${10}^{\ensuremath{-}7},{10}^{\ensuremath{-}5}$, ${10}^{\ensuremath{-}4}$, respectively. Furthermore, the results can be systematically improved by increasing the bond dimension of the HOTRG method. Finally, we study the length scale induced by the finite cutoff in bond dimension and elucidate its physical meaning in this context.

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

\mu

, 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

\mu

, reweighting, complex Langevin, thimbles or path optimization. Nevertheless, none of the proposed remedies can successfully reach regimes in which

\mu/T>1

, and therefore even today the full QCD phase diagram in the

\mu-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

Z_n

of the partition function

Z=\sum_nZ_n\beta^n

can be calculated separately. Finally, numerical results in two dimensions are studied.