arXiv subjects

Jacques Bloch

Publications and source records attributed to Jacques Bloch.

At least 19 recordsLinked to original sources

arXiv2026

Order-separated tensor-network method for QCD in the strong-coupling expansion

We introduce the order-separated Grassmann higher-order tensor renormalization group (OS-GHOTRG) method for QCD with staggered quarks in the strong-coupling expansion. The method allows us to determine the expansion coefficients of the partition function, from which we can obtain the strong-coupling expansions of thermodynamical observables. We use the method in two dimensions to compute the free energy, the particle-number density, and the chiral condensate as a function of the chemical potential up to third order in the inverse coupling $\beta$. Although near the phase transition the expansion is only a good approximation to the full theory at small $\beta$, we show that the range of applicability can be greatly extended by fits to judiciously chosen transition functions.

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arXiv2025

Tensor-network formulation of QCD in the strong-coupling expansion

We present a tensor-network formulation for the strong-coupling expansion of QCD with staggered quarks at nonzero chemical potential, for arbitrary number of dimensions, colors, and flavors. We integrate out the gauge and quark degrees of freedom and rewrite the partition function as the complete trace of a tensor network. This network consists of local tensors that contain a numerical and a Grassmann part. We truncate the initial tensor at a fixed order in the inverse coupling $\beta$ and compute analytical results for the partition function, the free energy, and the chiral condensate on a $2\times2$ lattice up to order $\beta^4$. In a follow-up paper we will introduce an enhanced tensor-network method, order-separated GHOTRG, to explicitly compute the expansion coefficients of the partition function for larger lattices. To demonstrate its potential, first results obtained with this new method are already presented here.

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arXiv2025

Grassmann tensor approach for two-dimensional QCD in the strong-coupling expansion

We present a tensor-network approach for the strong-coupling expansion of two-dimensional QCD with staggered quarks at non-zero chemical potential. After expanding the Boltzmann factor in the gauge and fermion actions, all gauge fields can be integrated out exactly and the partition function can be evaluated using the Grassmann higher-order tensor renormalization group approach. The method is modified to compute the $\mu$ dependence of the quark number density and the chiral condensate up to order $\beta^3$ with complete absence of higher-order terms infiltrating the result. Although the expansion itself is only a good approximation to the full theory at small $\beta<0.1$, the range can be extended, by using judiciously chosen fits. Moreover, these fits also yield a valuable expansion in $\beta$ for the critical chemical potential.

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arXiv2022

Grassmann tensor-network method for strong-coupling QCD

We present a tensor-network method for strong-coupling QCD with staggered quarks at nonzero chemical potential. After integrating out the gauge fields at infinite coupling, the partition function can be written as a full contraction of a tensor network consisting of coupled local numeric and Grassmann tensors. To evaluate the partition function and to compute observables, we develop a Grassmann higher-order tensor renormalization group method, specifically tailored for this model. We apply the method to the two-dimensional case and validate it by comparing results for the partition function, the chiral condensate and the baryon density with exact analytical expressions on small lattices up to volumes of $4\times4$. For larger two-dimensional volumes, we present tensor results for the chiral condensate as a function of the mass and volume, and observe that the chiral symmetry is not broken dynamically in two dimensions. Furthermore, our results for the number density as a function of the chemical potential hint at a first-order phase transition. Finally, we present some preliminary tensor results for three-dimensional strong-coupling QCD.

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arXiv2022

Improved local truncation schemes for the higher-order tensor renormalization group method

The higher-order tensor renormalization group is a tensor-network method providing estimates for the partition function and thermodynamical observables of classical and quantum systems in thermal equilibrium. At every step of the iterative blocking procedure, the coarse-grid tensor is truncated to keep the tensor dimension under control. For a consistent tensor blocking procedure, it is crucial that the forward and backward tensor modes are projected on the same lower dimensional subspaces. In this paper we present two methods, the SuperQ and the iterative SuperQ method, to construct tensor truncations that reduce or even minimize the local approximation errors, while satisfying this constraint.

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arXiv2022

Grassmann higher-order tensor renormalization group approach for two-dimensional strong-coupling QCD

We present a tensor-network approach for two-dimensional strong-coupling QCD with staggered quarks at nonzero chemical potential. After integrating out the gauge fields at infinite coupling, the partition function can be written as a full contraction of a tensor network consisting of coupled local numeric and Grassmann tensors. To evaluate the partition function and to compute observables, we develop a Grassmann higher-order tensor renormalization group method, specifically tailored for this model. During the coarsening procedure, the blocking of adjacent Grassmann tensors is performed analytically, and the total number of Grassmann variables in the tensor network is reduced by a factor of two at each coarsening step. The coarse-site numeric tensors are truncated using higher-order singular value decompositions. The method is validated by comparing the partition function, the chiral condensate and the baryon density computed with the tensor method with exact analytical results on small lattices up to volumes of $4\times4$. For larger volumes, we present first tensor results for the chiral condensate as a function of the mass and volume, and observe that the chiral symmetry is not broken dynamically in two dimensions. We also present tensor results for the number density as a function of the chemical potential, which hint at a first-order phase transition.

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arXiv2021

Tensor-network simulation of the strong-coupling $U(N)$ model

We apply tensor network methods to study the strong-coupling $U(N)$ model in its dimer formulation. In three and four dimensions, we investigate the chiral condensate as a function of the quark mass and the degree of the symmetry group, and find good agreement with Monte Carlo simulations. Particularly interesting is the study of chiral symmetry breaking as a function of the mass and the volume, which clearly shows that this symmetry is spontaneously broken in the limit of infinite volume and zero mass.

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arXiv2021

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 $O(2)$ 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.

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arXiv2021

Effective $\mathbb{Z}_{3}$ model for finite-density QCD with tensor networks

The tensor renormalization group is a promising numerical method used to study lattice statistical field theories. However, this approach is computationally expensive in 2+1 and 3+1 dimensions. Here we use tensor renormalization group methods to study an effective three-dimensional $\mathbb{Z}_{3}$ model for the heavy-quark, high-temperature, strong-coupling limit of single-flavor 3+1 dimensional quantum chromodynamics. Our results are cross-checked using the worm Monte Carlo algorithm. We present the phase diagram of the model through the measurement of the Polyakov loop, the nearest-neighbor Polyakov loop correlator, and their susceptibilities. The tensor renormalization group results are in good agreement with the literature

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arXiv2021

Tensor renormalization group study of the 3d $O(2)$ model

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 non-zero 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 non-zero magnetic field, temperature, and chemical potential are consistent with those obtained using other methods.

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arXiv2018

Anatomy of a strong residual sign problem on the thimbles

Using a simple Gaussian-like Ansatz for the phase distribution of a theory with a complex action, we show how the thimble integration for the average phase factor can be plagued by a strong residual sign problem when the phase of the complex integration measure conspires with the constant phase of the integrand along the thimble. This strong sign problem prohibits the accurate computation of the average phase factor when it becomes exponentially small, and causes a strong sensitivity to the parameters describing the phase distribution.

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arXiv2018

Progress on Complex Langevin simulations of a finite density matrix model for QCD

We study the Stephanov model, which is an RMT model for QCD at finite density, using the Complex Langevin algorithm. Naive implementation of the algorithm shows convergence towards the phase quenched or quenched theory rather than to intended theory with dynamical quarks. A detailed analysis of this issue and a potential resolution of the failure of this algorithm are discussed. We study the effect of gauge cooling on the Dirac eigenvalue distribution and time evolution of the norm for various cooling norms, which were specifically designed to remove the pathologies of the complex Langevin evolution. The cooling is further supplemented with a shifted representation for the random matrices. Unfortunately, none of these modifications generate a substantial improvement on the complex Langevin evolution and the final results still do not agree with the analytical predictions.

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arXiv2017

Selected inversion as key to a stable Langevin evolution across the QCD phase boundary

We present new results of full QCD at nonzero chemical potential. In PRD 92, 094516 (2015) the complex Langevin method was shown to break down when the inverse coupling decreases and enters the transition region from the deconfined to the confined phase. We found that the stochastic technique used to estimate the drift term can be very unstable for indefinite matrices. This may be avoided by using the full inverse of the Dirac operator, which is, however, too costly for four-dimensional lattices. The major breakthrough in this work was achieved by realizing that the inverse elements necessary for the drift term can be computed efficiently using the selected inversion technique provided by the parallel sparse direct solver package PARDISO. In our new study we show that no breakdown of the complex Langevin method is encountered and that simulations can be performed across the phase boundary.

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arXiv2017

Reweighting complex Langevin trajectories

Although the complex Langevin method can solve the sign problem in simulations of theories with complex actions, the method will yield the wrong results if known validity conditions are not satisfied. We present a novel method to compute observables for a target ensemble by reweighting complex trajectories generated with the complex Langevin method for an auxiliary ensemble having itself a complex action. While it is imperative that the validity conditions be satisfied for the auxiliary ensemble, there are no such requirements for the target ensemble. This allows us to enlarge the applicability range of the complex Langevin method. We illustrate this at the hand of a one-dimensional partition function and two-dimensional strong-coupling QCD.

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arXiv2017

Reweighted complex Langevin and its application to two-dimensional QCD

We present the reweighted complex Langevin method, which enlarges the applicability range of the complex Langevin method by reweighting the complex trajectories. In this reweighting procedure both the auxiliary and target ensembles have a complex action. We validate the method by applying it to two-dimensional strong-coupling QCD at nonzero chemical potential, and observe that it gives access to parameter regions that could otherwise not be reached with the complex Langevin method.

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arXiv2016

Complex Langevin simulations of a finite density matrix model for QCD

We study a random matrix model for QCD at finite density via complex Langevin dynamics. This model has a phase transition to a phase with nonzero baryon density. We study the convergence of the algorithm as a function of the quark mass and the chemical potential and focus on two main observables: the baryon density and the chiral condensate. For simulations close to the chiral limit, the algorithm has wrong convergence properties when the quark mass is in the spectral domain of the Dirac operator. A possible solution of this problem is discussed.

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arXiv2016

Complex Langevin Dynamics in 1+1d QCD at Non-Zero Densities

We present our results obtained from gauge cooled complex Langevin simulations in 1+1d QCD at non-zero densities in the strong coupling regime with unrooted staggered fermions. For small quark masses there are regions of the chemical potential where this method fails to reproduce correct results. In these parameter ranges we studied the effect of different gauge cooling schemes on the distributions of the fermion determinant as well as of observables.

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arXiv2016

Positivity of center subsets for QCD

We further pursue an approach to the sign problem of quantum chromodynamics at nonzero chemical potential, in which configurations of the lattice path integral are gathered into subsets. In the subset construction we multiply each temporal link by center elements independently and in a first step neglect the gauge action. The positivity of the subset weights -- shown for 0+1 dimensions in an earlier study -- extends to larger lattices: for two sites in the temporal direction and arbitrary spatial extent we give a proof of the positivity by decomposing the subset weight in positive summands. From numerical evidence we conjecture that the positivity persists on larger lattices and that the gauge action can be reintroduced through mild reweighting. First results on the quark number obtained with this method in two dimensions are shown as well.

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