Narrow Results

The minimization of the action of a QFT with a constraint dictated by the block averaging procedure is an important part of Bałaban’s approach to renormalization. It is particularly interesting for QFTs with non-trivial target spaces, such as gauge theories or non-linear sigma models on a lattice. We analyze this step for the $O(4)$ non-linear sigma model in two dimensions and demonstrate, in this case, how various ingredients of Bałaban’s approach play together. First, using variational calculus on Lie groups, the equation for the critical point is derived. Then, this non-linear equation is solved by the Banach contraction mapping theorem. This step requires detailed control of lattice Green functions and their integral kernels via random walk expansions.

We discuss the properties of the Casimir energy on the lattice spacetime. In general, the Casimir energy is defined with a regularization to eliminate the divergence of zero-point energy for a quantum field. Here, we apply the lattice regularization to define the Casimir energy, where lattice effects are implemented as long as the lattice spacing is nonzero. First, we demonstrate the calculation procedure of the Casimir energies using the naive and Wilson fermions. If the lattice effect appears only at a small number of lattices, we can take the continuum limit to obtain the Casimir energy in the continuum spacetime. As examples of physical systems, we investigate the Casimir energy for electron fields in Dirac/Weyl semimetals, where the lattice effect is correctly contained, and for photon fields in axion electrodynamics, where the continuum result is correctly reproduced.

In the study of QCD dynamics, C^⋆ boundary conditions are physically relevant in certain cases. In this paper, we study the implementation of these boundary conditions in the lattice formulation of full QCD with staggered fermions. In particular, we show that the usual even-odd partition trick to avoid the redoubling of the fermion matrix is still valid in this case. We give an explicit implementation of these boundary conditions for the Hybrid Monte Carlo algorithm.

We calculate the equation of state of strongly coupled Hamiltonian lattice QCD at finite density by constructing a solution to the equation of motion corresponding to an effective Hamiltonian describing the ground state of the theory with Wilson fermions. To lowest order in N_c we find that the pressure of the quark Fermi sea can be negative indicating its mechanical instability. This new mean field result is in qualitative agreement with continuum models and should be verifiable by future numerical simulations.

In this brief review, we summarize the current theoretical knowledge in supersymmetry on the lattice, with special emphasis on recent results in the framework of N=1 supersymmetric Yang–Mills theory, Wess–Zumino model and Yang–Mills theory with extended supersymmetries.

We study the critical behavior of lattice Quantum Chromodynamics (QCD) in the strong coupling approximation with Kogut–Susskind and Wilson fermions at finite temperature (T) and zero chemical potential. Using the Hamiltonian formulation we construct a mean field solution to the equation of motion at finite T and use it to study the elementary thermal excitations and to extract some critical exponents characterizing the observed second-order phase transition. We find similar critical behaviors for Kogut–Susskind and Wilson fermions at finite T.

A general introduction into the subject aimed at a general theoretical physics audience. We introduce the sign problem posed by finite density lattice QCD, and we discuss the main methods proposed to circumvent it, with emphasis on the imaginary chemical potential approach. The interrelation between Taylor expansion and analytic continuation from imaginary chemical potential is discussed in detail. The main applications to the calculation of the critical line, and to the thermodynamics of the hot and normal phase are reviewed.

A method to apply the tensor renormalization group to a lattice boson model is proposed. It is based on the truncated singular value decomposition of a compact operator. We demonstrate it using the (1+1)-dimensional ϕ⁴ model. The evaluated critical points on the lattice are consistent with the Monte Carlo result.

Causal Dynamical Triangulations (CDT) provide a non-perturbative formulation of Quantum Gravity assuming the existence of a global time foliation. In our earlier study we analyzed the effect of including d copies of a massless scalar field in the two-dimensional CDT model with imaginary time. For d > 1 we observed the formation of a "blob", somewhat similar to that observed in four-dimensional CDT without matter. In the two-dimensional case the "blob" has a Hausdorff dimension D_H = 3. In this paper, we study the spectral dimension D_S of the two-dimensional CDT-universe, both for d = 0 (pure gravity) and d = 4. We show that in both cases the spectral dimension is consistent with D_S = 2.

Lattice models with complex actions are important for the understanding of matter at finite densities, but not accessible by the standard Monte Carlo techniques due to the sign problem. Here, we propose a new approach aiming to avoid the complex action/sign problem, by extending the method of convergent series (CS) with a non-Gaussian initial approximation. The main properties of the new series are demonstrated on the example of the two-dimensional oscillating integral.

In this work, we reexamine the holographic dark energy concept proposed already for cosmological applications. By considering, more precisely, the bounds on the entropy arising from lattice field theory on one side and Bekenstein–Hawking entropy of black holes on another side, it is shown that the so-called holographic dark energy cannot be mimicked as easily as claimed in the literature. In addition, the limits on the electron $(g-2)$ experiments are taken into account again. It is shown that the corrections to the electron magnetic momentum are of the order of ${𝒪}(10^{-23})$.

This paper contains both a review of recent approaches to supersymmetric lattice field theories and some new results on the deconstruction approach. The essential reason for the complex phase problem of the fermion determinant is shown to be derivative interactions that are not present in the continuum. These irrelevant operators violate the self-conjugacy of the fermion action that is present in the continuum. It is explained why this complex phase problem does not disappear in the continuum limit. The fermion determinant suppression of various branches of the classical moduli space is explored, and found to be supportive of previous claims regarding the continuum limit.

Following the approach developed by some of the authors in recent papers and using a matrix representation for the superfields, we formulate an exact supersymmetric theory with two supercharges on a one-dimensional lattice. In the superfield formalism supersymmetry transformations are uniquely defined and do not suffer of the ambiguities recently pointed out by some authors. The action can be written in a unique way and it is invariant under all supercharges. A modified Leibniz rule applies when supercharges act on a superfield product and the corresponding Ward identities take a modified form but hold exactly at least at the tree level, while their validity in presence of radiative corrections is still an open problem and is not considered here.

We study a class of lattice field theories in two dimensions that includes gauge theories. We show that in these theories it is possible to implement a broader notion of local symmetry, based on semisimple Hopf algebras. A character expansion is developed for the quasitopological field theories, and partition functions are calculated with this tool. Expected values of generalized Wilson loops are defined and studied with the character expansion.

We discuss our results on QCD with a number of fundamental fermions ranging from zero to sixteen. These theories exhibit a wide array of fascinating phenomena which have been under close scrutiny, especially in recent years, first and foremost is the approach to conformality. To keep this review focused we have chosen scale generation, or lack thereof, as a guiding theme, however the discussion will be set in the general framework of the analysis of the phases and phase transitions of strong interactions at zero and nonzero temperature.

Different series expansions in the chemical potential μ are studied and compared for an effective theory of QCD which has a flux representation where the complex action is overcome. In particular we consider fugacity series, Taylor expansion and a modified Taylor expansion and compare the outcome of these series to the reference results from a Monte Carlo simulation in the flux representation where arbitrary μ is accessible. It is shown that for most parameter values the fugacity expansion gives the best approximation to the data from the flux simulation, followed by our newly proposed modified Taylor expansion. For the conventional Taylor expansion we find that the results coincide with the flux data only for very small μ.

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 forty 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, focusing on two more recent methods: dualization to worldline type of representations and the density-of-states approach.

Supersymmetric lattice Ward–Takahashi identities are investigated perturbatively up to two-loop corrections for super doubler approach of $N=2$ lattice Wess–Zumino models in 1- and 2-dimensions. In this approach, notorious chiral fermion doublers are treated as physical particles and momentum conservation is modified in such a way that lattice Leibniz rule is satisfied. The two major difficulties to keep exact lattice supersymmetry are overcome. This formulation defines, however, nonlocal field theory. Nevertheless we confirm that exact supersymmetry on the lattice is realized for all supercharges at the quantum level. Delicate issues of associativity are also discussed.

The dynamical origin of electroweak symmetry breaking is an open question with many possible theoretical explanations. Strongly coupled systems predicting the Higgs boson as a bound state of a new gauge-fermion interaction form one class of candidate models. Due to increased statistics, LHC run II will further constrain the phenomenologically viable models in the near future. In the meanwhile it is important to understand the general properties and specific features of the different competing models.

In this work we discuss many-flavor gauge-fermion systems that contain both massless (light) and massive fermions. The former provide Goldstone bosons and trigger electroweak symmetry breaking, while the latter indirectly influence the infrared dynamics. Numerical results reveal that such systems can exhibit a light $0^{++}$ isosinglet scalar, well separated from the rest of the spectrum. Further, when we set the scale via the vev of electroweak symmetry breaking, we predict a 2 TeV vector resonance which could be a generic feature of SU(3) gauge theories.

In this paper, we explore the perspectives for canonical simulations in the worldline formulation of a lattice field theory. Using the charged ${\varphi }^4$ field in two dimensions as an example, we present the details of the canonical formulation based on worldlines and outline the 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.