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Worldline representations were established as a powerful tool for studying bosonic lattice field theories at finite density. For fermions, however, the worldlines still may carry signs that originate from the Dirac algebra and from the Grassmann nature of the fermion fields. We show that a density of states approach can be set up to deal with this remaining sign problem, where finite density is implemented in a canonical approach by working with a fixed winding number of the fermion worldlines. We discuss the approach in detail and show first results of a numerical implementation in 2 dimensions., Contribution to the proceedings of the 39th International Symposium on Lattice Field Theory - Lattice2022, August 8 - 13, 2022, Bonn, Germany

We study the massless Schwinger model with an additional 4-fermi interaction and a topological term. For topological angle $\theta = \pi$ charge conjugation symmetry is implemented in a non-trivial way and we study the possibility of its spontaneous breaking. For the lattice discretization we use staggered fermions and the Villain action for the gauge fields, where the topological term is an integer and charge conjugation at $\theta = \pi$ is an exact symmetry. The complex action problem is overcome by a suitable worldline/worldsheet representation. We find that as a function of the 4-fermi coupling the system shows a critical point separating a weak coupling phase where charge conjugation symmetry is intact from a strong coupling phase with spontaneously broken charge conjugation symmetry., Comment: PoS(LATTICE2022)372. Contribution to the Proceedings of the 39th International Symposium on Lattice Field Theory, 8.-13. August 2022, Bonn, Germany

We study a recently proposed formulation of U(1) lattice field theory with electric and magnetic matter based on the Villain formulation. This discretization allows for a duality that gives rise to relations between weak and strong gauge coupling. There exists a self-dual value of the gauge coupling where one may study the model as a function of the remaining matter coupling. Using Monte Carlo simulations based on a worldline/worldsheet representation of the system we evaluate order parameters for spontaneous breaking of self-duality. We find that in some interval of the matter coupling self-duality becomes broken spontaneously. We determine the endpoints of this interval and study the nature of the corresponding critical points. Finally we explore the system away from the self-dual gauge coupling and show that when crossing the self-dual point a first order jump is seen in the order parameters., Contribution to: 38th International Symposium on Lattice Field Theory - Lattice2021

We present a lattice study of a 2-flavor U(1) gauge-Higgs model quantum field theory with a topological term at θ=π. Such studies are prohibitively costly in the standard lattice formulation due to the sign problem. Using a novel discretization of the model, along with an exact lattice dualization, we overcome the sign problem and reliably simulate such systems. Our work provides the first ab initio demonstration that the model is in the spin-chain universality class, and demonstrates the power of the new approach to U(1) gauge theories.

We revisit the issue of worldline formulations for the q-state Potts model and discuss a worldline representation in arbitrary dimensions which also allows for magnetic terms. For vanishing magnetic field we implement a Hodge decomposition for resolving the constraints with dual variables, which in two dimensions implies self-duality as a simple corollary. We present exploratory 2-d Monte Carlo simulations in terms of the worldlines, based on worm algorithms. We study both, vanishing and non-zero magnetic field, and explore q between q = 2 and q = 30, i.e., Potts models with continuous, as well as strong first order transitions., Discussion of statistical errors improved and various comments added. Final version to appear in Nuclear Physics B

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.

We present two new suggestions for density of states (DoS) approaches to finite density lattice QCD. Both proposals are based on the recently developed and successfully tested DoS functional fit approach (FFA) technique, which is a DoS approach for bosonic systems with a complex action problem. The two different implementations of DoS FFA we suggest for QCD make use of different representations of finite density lattice QCD in terms of suitable pseudofermion path integrals. The first proposal is based on a pseudofermion representation of the grand canonical QCD partition sum, while the second is a formulation for the canonical ensemble. We work out the details of the two proposals and discuss the results of exploratory two-dimensional test studies for free fermions at finite density, where exact reference data allow one to verify the final results and intermediate steps.

We explore the possibility of a simulation of strong coupling QCD in terms of so-called baryon bags. In this form the known representation in terms of monomers, dimers and baryon loops is reorganized such that the baryon contributions are collected in space time domains referred to as baryon bags. Within the bags three quarks propagate coherently as a baryon that is described by a free fermion, whereas the rest of the lattice is solely filled with interacting meson terms, i.e., quark and diquark monomers and dimers. We perform a simulation directly in the baryon bag language using a newly developed worm update and show first results in two dimensions., Contribution to: The 37th Annual International Symposium on Lattice Field Theory - LATTICE2019 16-22 June, 2019

Based on a complete set of $J = 0$ and $J=1$ spatial isovector correlation functions calculated with $N_F = 2$ domain wall fermions we identify an intermediate temperature regime of $T \sim 220 - 500$ MeV ($1.2T_c$--$2.8T_c$), where chiral symmetry is restored but the correlators are not yet compatible with a simple free quark behavior. More specifically, in the temperature range $T \sim 220 - 500$ MeV we identify a multiplet structure of spatial correlators that suggests emergent $SU(2)_{CS}$ and $SU(4)$ symmetries, which are not symmetries of the free Dirac action. The symmetry breaking effects in this temperature range are less than 5%. Our results indicate that at these temperatures the chromo-magnetic interaction is suppressed and the elementary degrees of freedom are chirally symmetric quarks bound into color-singlet objects by the chromo-electric component of the gluon field. At temperatures between 500 and 660 MeV the emergent $SU(2)_{CS}$ and $SU(4)$ symmetries disappear and one observes a smooth transition to the regime above $T \sim 1$ GeV where only chiral symmetries survive, which are finally compatible with quasi-free quarks., A new figure and discussion added. Accepted for publication in PRD

We discuss a particular lattice discretization of abelian gauge theories in arbitrary dimensions. The construction is based on gauging the center symmetry of a non-compact abelian gauge theory, which results in a Villain type action. We show that this construction has several benefits over the conventional $U(1)$ lattice gauge theory construction, such as electric-magnetic duality, natural coupling of the theory to magnetically charged matter in four dimensions, complete control over the monopoles and their charges in three dimensions and a natural $\theta$-term in two dimensions. Moreover we show that for bosonic matter our formulation can be mapped to a worldline/worldsheet representation where the complex action problem is solved. We illustrate our construction by explicit dualizations of the $CP(N\!-\!1)$ and the gauge Higgs model in $2d$ with a $\theta$ term, as well as the gauge Higgs model in $3d$ with constrained monopole charges. These models are of importance in low dimensional anti-ferromagnets. Further we perform a natural construction of the $\theta$-term in four dimensional gauge theories, and demonstrate the Witten effect which endows magnetic matter with a fractional electric charge. We extend this discussion to $PSU(N)=SU(N)/\mathbb Z_N$ non-abelian gauge theories and the construction of discrete $\theta$-terms on a cubic lattice., Comment: 47 pages, 1 figure. Version cleaned up. Mistake in section 5 fixed

In this contribution we revisit the lattice discretization of the topological charge for abelian lattice field theories. The construction departs from an initially non-compact discretization of the gauge fields and after absorbing $2\pi$ shifts of the gauge fields leads to a generalized Villain action that also includes the topological term. The topological charge in two, as well as in four dimensions can be expressed in terms of only the integer-valued Villain variables. We test various properties of the topological charge and in particular analyze the index theorem in two dimensions and discuss the Witten effect in 4-d. As an application of our formulation we present results from a simulation of the 2-d U(1) gauge Higgs model at vacuum angle $\theta = \pi$, where we use a suitable worldline/worldsheet representation to overcome the complex action problem at non-zero $\theta$., Comment: Combined contribution to LATTICE 2019 based on parallel talks by Mariia Anosova, Christof Gattringer and Daniel G\"oschl (19 pages)

Properties of QCD matter change significantly around the chiral crossover temperature, and the effects on $U(1)_A$ and topological susceptibilities, as well as the meson spectrum have been studied with much care. Baryons and the effect of parity doubling in this temperature range have been analyzed previously by various other groups employing different setups. Here we construct suitable operators to investigate chiral and axial $U(1)_A$ symmetries in the baryon spectrum. Measurements for different volumes and quark-masses are done with two flavors of chirally symmetric domain-wall fermions at temperatures above the critical one. The possibility of emergent $SU(4)$ and $SU(2)_{CS}$ symmetries is discussed., Comment: 7 pages, 4 figures, talk presented at the 37th International Symposium on Lattice Field Theory (Lattice 2019), 16-22 June 2019, Wuhan, China. Updated numerical data in Fig. 1 and Fig. 2

Using 2-d U(1) lattice gauge theory we study two definitions of the topological charge constructed from a generalized Villain action and analyze the implementation of the index theorem based on the overlap Dirac operator. One of the two definitions expresses the topological charge as a sum of the Villain variables and treats charge conjugation symmetry exactly, making it particularly useful for studying related physics. Our numerical analysis establishes that for both topological charge definitions the index theorem becomes exact quickly towards the continuum limit., Comment: Clarifying comments added. Final version to appear in Physics Letters B

We study 2d U(1) gauge Higgs systems with a $\theta$-term. For properly discretizing the topological charge as an integer we introduce a mixed group- and algebra-valued discretization (MGA scheme) for the gauge fields, such that the charge conjugation symmetry at $\theta = \pi$ is implemented exactly. The complex action problem from the $\theta$-term is overcome by exactly mapping the partition sum to a worldline/worldsheet representation. Using Monte Carlo simulation of the worldline/worldsheet representation we study the system at $\theta = \pi$ and show that as a function of the mass parameter the system undergoes a phase transition. Determining the critical exponents from a finite size scaling analysis we show that the transition is in the 2d Ising universality class. We furthermore study the U(1) gauge Higgs systems at $\theta = \pi$ also with charge 2 matter fields, where an additional $Z_2$ symmetry is expected to alter the phase structure. Our results indicate that for charge 2 a true phase transition is absent and only a rapid crossover separates the large and small mass regions., Comment: The 36th Annual International Symposium on Lattice Field Theory - LATTICE2018

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 Z2 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.

We study the quantum field theory of a charged $\phi^4$ 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 $\mu$ is overcome by using a worldline representation for the Monte Carlo simulation. At low temperature we study the particle number as a function of $\mu$ and observe the steps for $\mbox{1-,}$ 2- and 3-particle condensation. We determine the corresponding critical values $\mu_n^{crit}, \, n = 1,2,3$ and analyze their dependence on the spatial extent $L$ of the lattice. Linear combinations of the $\mu_n^{crit}$ give the interaction energies in the 2- and 3-particle sectors and their dependence on $L$ is related to scattering data by L\"uscher'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., Comment: Comments and two references added. Final version to appear in Physical Review Letters

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., Comment: Final version to appear in Phys. Rev. D

We study $\phi^4$ lattice field theory at finite chemical potential $\mu$ in two and four dimensions, using a worldline representation that overcomes the complex action problem. We compute the particle number at very low temperature as a function of $\mu$ and determine the first three condensation thresholds, where the system condenses 1, 2 and 3 particles. The corresponding critical values of the chemical potential can be related to the 1-, 2- and 3-particle energies of the system, and we check this relation with a direct spectroscopy determination of the $n$-particle energies from $2n$-point functions. We analyze the thresholds as a function of the spatial size of the system and use the known finite volume results for the $n$-particle energies to relate the thresholds to scattering data. For four dimensions we determine the scattering length from the 2-particle threshold, while in two dimensions the full scattering phase shift can be determined. In both cases the scattering data computed from the 2-particle threshold already allow one to determine the 3-particle energy. In both, two and four dimensions we find very good agreement of this ''prediction'' with direct determinations of the 3-particle energy from either the thresholds or the 6-point functions. The results show that low temperature condensation is indeed governed by scattering data., Comment: Contribution to the proceedings of LATTICE2018

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., Comment: Footnote added. Final version to appear in Phys. Rev. D

We use the complex $\phi^4$ 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 $\phi^4$ 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., Comment: Contribution to Lattice 2017

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., Contribution to LATTICE 2017, 16 pages

We analyze a complex scalar field with ϕ4 interaction and a chemical potential μ 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 μ for zero and finite temperature. It is shown that at zero temperature thermodynamical observables are independent of μ up to a critical value μc (Silver Blaze phenomenon). In a spectroscopy calculation we cross-check that μc 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 μ-dependence.

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.

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)., Comment: Final version to appear in Nucl. Phys. B

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., Comment: Comments and a new plot for the relative errors added. Version to appear in Physics Letters B

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., Comment: 16 pages, 17 figures; finite size analysis and a second version of the lattice phase diagram added, minor modifications of the text, conclusions unchanged

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 $\mu$ - 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 $\mu$ 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., Comment: Plot, reference and comments added. Final version to appear in Nucl. Phys. B

We present lattice results for simulations of the O(3) non-linear sigma model at finite chemical potential. The complex action problem is overcome by a dual variable representation of the model. We discuss two aspects of the theory at finite density: 1) The relation of the finite density data to scattering phases and wave-functions. 2) The phase-structure of the theory as a function of chemical potential and the possibility of a Kosterlitz-Thouless phase transition.

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., mini-review, 20 pages, 4 figures

We study the behavior of center sectors in pure SU(4) lattice gauge theory at finite temperature. The center sectors are defined as spatial clusters of neighboring sites with values of their local Polyakov loops near the same center elements. We study the connectedness and percolation properties of the center clusters across the deconfinement transition. We show that for SU(4) gauge theory deconfinement cannot be described as a percolation transition of center clusters, a finding which is different from pure SU(2) or pure SU(3) Yang Mills theory, where the percolation description even allows for a continuum limit., Comment: Three figures and discussion added. Final version to appear in Physics Letters B

We analyze canonical fermion determinants, i.e., fermion determinants projected to a fixed quark number q. The canonical determinants are computed using a dimensional reduction formula and are studied for pure SU(3) gauge configurations in a wide range of temperatures. It is demonstrated that the center sectors of the Polyakov loop very strongly manifest themselves in the behavior of the canonical determinants in the deconfined phase, and we discuss physical implications of this finding. Furthermore the distribution of the quark sectors is studied as a function of the temperature., Comments on the physical interpretation added. Final version to appear in Physics Letters B

We study the expectation value of the phase of the fermion determinant for Wilson lattice fermions with chemical potential. We use quenched SU(3) ensembles and implement a recently proposed exact dimensional reduction of the fermion determinant. Ensembles at several temperatures below and above the phase transition are studied and we analyze the role of the quark mass, the temperature, the volume and the topological sectors. We compare our numerical results to predictions from chiral perturbation theory., Comment: Statistical errors for fit parameters and a footnote added. Final version to appear in Physics Letters B