arXiv subjects

Christof Gattringer

Publications and source records attributed to Christof Gattringer.

At least 37 records · Page 2Linked to original sources

arXiv2018

Dual simulation of the 2d U(1) gauge Higgs model at topological angle $θ= π\,$: Critical endpoint behavior

We simulate the 2d U(1) gauge Higgs model on the lattice with a topological angle $θ$. The corresponding complex action problem is overcome by using a dual representation based on the Villain action appropriately endowed with a $θ$-term. The Villain action is interpreted as a non-compact gauge theory whose center symmetry is gauged and has the advantage that the topological term is correctly quantized so that $2π$ periodicity in $θ$ is intact. Because of this the $θ= π$ theory has an exact $Z_2$ charge-conjugation symmetry $C$, which is spontaneously broken when the mass-squared of the scalars is large and positive. Lowering the mass squared the symmetry becomes restored in a second order phase transition. Simulating the system at $θ= π$ in its dual form we determine the corresponding critical endpoint as a function of the mass parameter. Using a finite size scaling analysis we determine the critical exponents and show that the transition is in the 2d Ising universality class, as expected.

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arXiv2018

Finite density condensation and scattering data - a study in $ϕ^4$ lattice field theory

We study the quantum field theory of a charged $ϕ^4$ field in lattice regularization at finite density and low temperature in 2 and 4 dimensions with the goal of analyzing the connection of condensation phenomena to scattering data in a non-perturbative way. The sign problem of the theory at non-zero chemical potential $μ$ is overcome by using a worldline representation for the Monte Carlo simulation. At low temperature we study the particle number as a function of $μ$ and observe the steps for $\mbox{1-,}$ 2- and 3-particle condensation. We determine the corresponding critical values $μ_n^{crit}, \, n = 1,2,3$ and analyze their dependence on the spatial extent $L$ of the lattice. Linear combinations of the $μ_n^{crit}$ give the interaction energies in the 2- and 3-particle sectors and their dependence on $L$ is related to scattering data by Lüscher's formula and its generalizations to three particles. For 2-$d$ we determine the scattering phase shift and for 4-$d$ the scattering length. We cross-check our results with a determination of the mass and the 2- and 3-particle energies from conventional 2-, 4-, and 6-point correlators at zero chemical potential. The letter demonstrates that the physics of condensation at finite density and low temperature is closely related to scattering data of a quantum field theory.

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arXiv2018

Baryon bags in strong coupling QCD

We discuss lattice QCD with one flavor of staggered fermions and show that in the path integral the baryon contributions can be fully separated from quark and diquark contributions. The baryonic degrees of freedom are independent of the gauge field and the corresponding free fermion action describes the baryons through the joint propagation of three quarks. The non-baryonic dynamics is described by quark and diquark terms that couple to the gauge field. When evaluating the quark and diquark contributions in the strong coupling limit the partition function completely factorizes into baryon bags and a complementary domain. Baryon bags are regions in space-time where the dynamics is described by a single free fermion made out of three quarks propagating coherently as a baryon. Outside the baryon bags the relevant degrees of freedom are monomers and dimers for quarks and diquarks. The partition sum is a sum over all baryon bag configurations and for each bag a free fermion determinant appears as a weight factor.

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arXiv2018

Dual representation of lattice QCD with worldlines and worldsheets of abelian color fluxes

We present a new dual representation for lattice QCD in terms of wordlines and worldsheets. The exact reformulation is carried out using the recently developed abelian color flux method where the action is decomposed into commuting minimal terms that connect different colors on neighboring sites. Expanding the Boltzmann factors for these commuting terms allows one to reorganize the gauge field contributions according to links such that the gauge fields can be integrated out in closed form. The emerging constraints give the dual variables the structure of worldlines for the fermions and worldsheets for the gauge degrees of freedom. The partition sum has the form of a strong coupling expansion and with the abelian color flux approach discussed here all coefficients of the expansion are known in closed form. We present the dual form for three cases: pure SU(3) lattice gauge theory, strong coupling QCD and full QCD, and discuss in detail the constraints for the color fluxes and their physical interpretation.

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arXiv2018

Kramers-Wannier duality and worldline representation for the SU(2) principal chiral model

In this letter we explore different representations of the SU(2) principal chiral model on the lattice. We couple chemical potentials to two of the conserved charges to induce finite density. This leads to a complex action such that the conventional field representation cannot be used for a Monte Carlo simulation. Using the recently developed Abelian color flux approach we derive a new worldline representation where the partition sum has only real and positive weights, such that a Monte Carlo simulation is possible. In a second step we transform the model to new dual variables in the Kramers-Wannier (KW) sense, such that the constraints are automatically fulfilled, and we obtain a second representation free of the complex action problem. We implement exploratory Monte Carlo simulations for both, the worldline, as well as the KW-dual form, for cross-checking the two dualizations and a first assessment of their potential for dual simulations.

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arXiv2017

New techniques and results for worldline simulations of lattice field theories

We use the complex $ϕ^4$ field at finite density as a model system for developing further techniques based on worldline formulations of lattice field theories. More specifically we: 1) Discuss new variants of the worm algorithm for updating the $ϕ^4$ theory and related systems with site weights. 2) Explore the possibility of canonical simulations in the worldline formulation. 3) Study the connection of 2-particle condensation at low temperature to scattering parameters of the theory.

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arXiv2017

Worldlines and worldsheets for non-abelian lattice field theories: Abelian color fluxes and Abelian color cycles

We discuss recent developments for exact reformulations of lattice field theories in terms of worldlines and worldsheets. In particular we focus on a strategy which is applicable also to non-abelian theories: traces and matrix/vector products are written as explicit sums over color indices and a dual variable is introduced for each individual term. These dual variables correspond to fluxes in both, space-time and color for matter fields (Abelian color fluxes), or to fluxes in color space around space-time plaquettes for gauge fields (Abelian color cycles). Subsequently all original degrees of freedom, i.e., matter fields and gauge links, can be integrated out. Integrating over complex phases of matter fields gives rise to constraints that enforce conservation of matter flux on all sites. Integrating out phases of gauge fields enforces vanishing combined flux of matter- and gauge degrees of freedom. The constraints give rise to a system of worldlines and worldsheets. Integrating over the factors that are not phases (e.g., radial degrees of freedom or contributions from the Haar measure) generates additional weight factors that together with the constraints implement the full symmetry of the conventional formulation, now in the language of worldlines and worldsheets. We discuss the Abelian color flux and Abelian color cycle strategies for three examples: the SU(2) principal chiral model with chemical potential coupled to two of the Noether charges, SU(2) lattice gauge theory coupled to staggered fermions, as well as full lattice QCD with staggered fermions. For the principal chiral model we present some simulation results that illustrate properties of the worldline dynamics at finite chemical potentials.

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arXiv2017

Density of States FFA analysis of SU(3) lattice gauge theory at a finite density of color sources

We present a Density of States calculation with the Functional Fit Approach (DoS FFA) in SU(3) lattice gauge theory with a finite density of static color sources. The DoS FFA uses a parameterized density of states and determines the parameters of the density by fitting data from restricted Monte Carlo simulations with an analytically known function. We discuss the implementation of DoS FFA and the results for a qualitative picture of the phase diagram in a model which is a further step towards implementing DoS FFA in full QCD. We determine the curvature $κ$ in the $μ$-$T$ phase diagram and find a value close to the results published for full QCD.

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arXiv2017

Canonical simulations with worldlines: an exploratory study in $ϕ^{4}_{2}$ lattice field theory

In this letter we explore the perspectives for canonical simulations in the worldline formulation of a lattice field theory. Using the charged $ϕ^4$ field in two dimensions as an example we present the details of the canonical formulation based on worldlines and outline algorithmic strategies for canonical worldline simulations. We discuss the steps for converting the data from the canonical approach to the grand canonical picture which we use for cross-checking our results. The canonical approach presented here can easily be generalized to other lattice field theories with a worldline representation.

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arXiv2017

Simulation strategies for the massless lattice Schwinger model in the dual formulation

The dual form of the massless Schwinger model on the lattice overcomes the complex action problems from two sources: a topological term, as well as non-zero chemical potential, making these physically interesting cases accessible to Monte Carlo simulations. The partition function is represented as a sum over fermion loops, dimers and plaquette-surfaces such that all contributions are real and positive. However, these new variables constitute a highly constrained system and suitable update strategies have to be developed. In this exploratory study we present an approach based on locally growing plaquette-surfaces surrounded by fermion loop segments combined with a worm based strategy for updating chains of dimers, as well as winding fermion loops. The update strategy is checked with conventional simulations as well as reference data from exact summation on small volumes and we discuss some physical implications of the results.

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arXiv2017

Remarks on the construction of worm algorithms for lattice field theories in worldline representation

We introduce a generalized worldline model where the partition function is a sum over configurations of a conserved flux on a d-dimensional lattice. The weights for the configurations of the corresponding worldlines have factors living on the links of the lattice, as well as terms which live on the sites x and depend on all fluxes attached to x. The model represents a general class of worldline systems, among them the dual representation of the relativistic Bose gas at finite density. We construct a suitable worm algorithm and show how to correctly distribute the site weights in the various Metropolis probabilities that determine the worm. We analyze the algorithm in detail and give a proof of detailed balance. Our algorithm admits the introduction of an amplitude parameter A that can be chosen freely. Using a numerical simulation of the relativistic Bose gas we demonstrate that A allows one to influence the starting and terminating probabilities and thus the average length and the efficiency of the worm.

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arXiv2017

Lattice study of the Silver Blaze phenomenon for a charged scalar phi-4 field

We analyze a complex scalar field with phi-4 interaction and a chemical potential mu on the lattice. An exact flux representation of the partition sum is used which avoids the complex action problem and based on a generalized worm algorithm we can run Monte Carlo simulations at arbitrary densities. We study thermodynamical quantities as a function of the chemical potential mu for zero- and finite temperature. It is shown that at zero temperature thermodynamical observables are independent of mu up to a critical value mu_c (Silver Blaze phenomenon). In a spectroscopy calculation we cross-check that mu_c agrees with the mass m of the scalar field. The Silver Blaze region ends in a second order phase transition and we show that for low temperatures the second order phase boundary persists and separates a pseudo Silver Blaze region from a condensed phase with strong mu-dependence.

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arXiv2017

Abelian color cycles: a new approach to strong coupling expansion and dual representations for non-abelian lattice gauge theory

We propose a new approach to strong coupling series and dual representations for non-abelian lattice gauge theories using the SU(2) case as an example. The Wilson gauge action is written as a sum over "abelian color cycles" (ACC) which correspond to loops in color space around plaquettes. The ACCs are complex numbers which can be commuted freely such that the strong coupling series and the dual representation can be obtained as in the abelian case. Using a suitable representation of the SU(2) gauge variables we integrate out all original gauge links and identify the constraints for the dual variables in the SU(2) case. We show that the construction can be generalized to the case of SU(2) gauge fields with staggered fermions. The result is a strong coupling series where all gauge integrals are known in closed form and we discuss its applicability for possible dual simulations. The abelian color cycle concept can be generalized to other non-abelian gauge groups such as SU(3).

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arXiv2016

Phase diagram of the two-dimensional O(3) model from dual lattice simulations

We have simulated the asymptotically free two-dimensional O(3) model at nonzero chemical potential using the model's dual representation. We first demonstrate how the latter solves the sign (complex action) problem. The system displays a crossover at nonzero temperature, while at zero temperature it undergoes a quantum phase transition when mu reaches the particle mass (generated dynamically similar to QCD). The density follows a square root behavior universal for repulsive bosons in one spatial dimension. We have also measured the spin stiffness, known to be sensitive to the spatial correlation length, using different scaling trajectories to zero temperature and infinite size. It points to a dynamical critical exponent z=2. Comparisons to thermodynamic Bethe ansaetze are shown as well.

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arXiv2016

Dualization of non-abelian lattice gauge theory with Abelian Color Cycles (ACC)

We discuss a new approach to strong coupling expansion and dual representations for non-abelian lattice gauge theories. The Wilson gauge action is decomposed into a sum over "abelian color cycles" (ACC), which are loops around plaquettes visiting different colors at the corners. ACCs are complex numbers and thus commute such that a dual representation of a non-abelian theory can be obtained as in the abelian case. We apply the ACC approach to SU(2) and SU(3) lattice gauge theory and exactly rewrite the two partition sums in a strong coupling series where all gauge integrals are known in closed form.

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arXiv2016

Developing and testing the density of states FFA method in the SU(3) spin model

The Density of States Functional Fit Approach (DoS FFA) is a recently proposed modern density of states technique suitable for calculations in lattice field theories with a complex action problem. In this article we present an exploratory implementation of DoS FFA for the SU(3) spin system at finite chemical potential $μ$ - an effective theory for the Polyakov loop. This model has a complex action problem similar to the one of QCD but also allows for a dual simulation in terms of worldlines where the complex action problem is solved. Thus we can compare the DoS FFA results to the reference data from the dual simulation and assess the performance of the new approach. We find that the method reproduces the observables from the dual simulation for a large range of $μ$ values, including also phase transitions, illustrating that DoS FFA is an interesting approach for exploring phase diagrams of lattice field theories with a complex action problem.

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arXiv2016

The two-dimensional O(3) model at nonzero density: from dual lattice simulations to repulsive bosons

We discuss the thermodynamics of the O(3) nonlinear sigma model in 1+1 dimensions at nonzero chemical potential (equivalent to a magnetic field). In its conventional field theory representation the model suffers from a sign problem. By dualizing the lattice model we present, for the first time, nonzero density data of an asymptotically free theory with dynamical mass-gap. We find a quantum phase transition at zero temperature where as a function of the chemical potential the density assumes a nonzero value. We present evidence for a corresponding dynamical critical exponent z close to 2. The low energy O(3) model is conjectured to be described by a massive boson triplet with repulsive interactions. We confirm the universal square root behavior expected for such a system at low density (and temperature) and compare our data to the results of Bethe ansatz solutions of the relativistic and non-relativistic one-dimensional Bose gas. We also comment on a potential Berezinskii-Kosterlitz-Thouless transition at nonzero density.

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arXiv2016

Approaches to the sign problem in lattice field theory

Quantum field theories (QFTs) at finite densities of matter generically involve complex actions. Standard Monte-Carlo simulations based upon importance sampling, which have been producing quantitative first principle results in particle physics for almost fourty years, cannot be applied in this case. Various strategies to overcome this so-called Sign Problem or Complex Action Problem were proposed during the last thirty years. We here review the sign problem in lattice field theories, focussing on two more recent methods: Dualization to world-line type of representations and the density-of-states approach.

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