Topological lumps are known to be present in gluonic fields of SU(3) gluodynamics. Near the transition temperature they were classified either as constituents of nondissociated (anti)calorons, or as constituents of (anti)dyon pairs, or as isolated (anti)dyons. In this paper we study the density and correlation functions of these objects at temperature T/T_c=0.96.
We report on our search for Kraan-van Baal calorons in finite temperature SU(2) lattice ensembles. We also discuss recent progress made in developing a caloron-anticaloron gas model decribing confinement and deconfinement in the context of trivial and non-trivial holonomy.
We study SU(2) gluodynamics at finite temperature near the deconfining phase transition. We create the lattice ensembles using the tadpole improved Luscher-Weisz action. The overlap Dirac operator is used to determine the following three aspects of vacuum structure: (i) The topological susceptibility is evaluated at various temperatures across the phase transition, (ii) the overlap fermion spectral density is determined and found to depend on the Polyakov loop above the phase transition and (iii) the corresponding localization properties of low lying eigenmodes are investigated.
We study SU(2) gluodynamics at finite temperature on both sides of the deconfining phase transition. We create the lattice ensembles using the tree-level tadpole-improved Symanzik action. The Neuberger overlap Dirac operator is used to determine the following three aspects of vacuum structure: (i) The topological susceptibility is evaluated at various temperatures across the phase transition, (ii) the overlap fermion spectral density is determined and found to depend on the Polyakov loop above the phase transition and (iii) the corresponding localization properties of low-lying eigenmodes are …
New topological objects, which we call center monopoles, naturally arise in the Maximal Center Projection of SU(3) gluodynamics. The condensate of the center monopoles is the order parameter of the theory.
The topological structure of lattice gluodynamics is studied at intermediate resolution scale in the deconfining phase with the help of a cluster analysis. UV filtered topological charge densities are determined from a fixed number of low-lying eigenmodes of the overlap Dirac operator with three types of temporal boundary conditions applied to the valence quark fields. This method usually allows to find all three distinguished (anti)dyon constituents in the gauge field of Kraan-van Baal-Lee-Lu (anti)caloron solutions. The clustering of the three topological charge densities in Monte Carlo gene…
We suggest that the topological susceptibility in gluodynamics can be found in terms of the gluon condensate using renormalizability and heavy fermion representation of the anomaly. Analogous relations can be also obtained for other zero momentum correlation functions involving the topological density operator. Using these relations, we find the theta dependence of the condensates <GG>, <G \tilde{G}> and of the partition function for small theta and an arbitrary number of colors.
Gluodynamics and two-flavor QCD at non-zero temperature are studied with the so-called overimproved cooling technique under which caloron solutions may remain stable. We consider topological configurations either at the first occuring stable plateau of topological charge or at the first (anti)selfdual plateau and find the corresponding topological susceptibility at various temperatures on both sides of the thermal transition or crossover. In pure gluodynamics the topological susceptibility drops sharply at the deconfinement temperature while in full QCD it decreases smoothly at temperatures ab…
We study the effect of cooling on the spatial distribution of the topological charge density in quenched SU(2) lattice gauge theory with overlap fermions. We show that as the gauge field configurations are cooled, the Hausdorff dimension of regions where the topological charge is localized gradually changes from d = 2..3 towards the total space dimension. Therefore, the cooling procedure destroys some of the essential properties of the topological charge distribution.
We study the topological structure of $SU(3)$ lattice gluodynamics by cluster analysis. This methodological study is meant as preparation for full QCD. The topological charge density is becoming visible in the process of overimproved gradient flow, which is monitored by means of the the Inverse Participation Ratio (IPR). The flow is stopped at the moment when calorons dissociate into dyons due to the overimproved character of the underlying action. This gives the possibility to simultaneously detect all three dyonic constituents of KvBLL calorons in the gluonic field. The behaviour of the aver…
We study various representations of infrared effective theory of SU(2) Gluodynamics as a (quantum) perfect lattice action. In particular we derive a monopole action and a string model of hadrons from SU(2) Gluodynamics. These are lattice actions which give almost cut-off independent physical quantities even on coarse lattices. The monopole action is determined by numerical simulations in the infrared region of SU(2) Gluodynamics. The string model of hadrons is derived from the monopole action by using BKT transformation. We illustrate the method and evaluate physical quantities such as the str…
We consider properties of zero and near-zero fermionic modes in lattice gluodynamics. The modes are known to be sensitive to the topology of the underlying gluonic fields in the quantum vacuum state of the gluodynamics. We find evidence that these modes are fine tuned, that is exhibit sensitivity to both physical (one can say, hadronic) scale and to the ultraviolet cutoff. Namely, the density of the states is in physical units while the localization volume of the modes tends to zero in physical units with the lattice spacing tending to zero. We discuss briefly possible theoretical implications…
Topological objects of $SU(3)$ gluodynamics are studied at the infrared scale near the transition temperature with the help of zero and near-zero modes of the overlap Dirac operator. We construct UV filtered topological charge densities corresponding to three versions of the temporal boundary condition applied to this operator, for which the zero mode is known to be located on corresponding three constituent dyons (antidyons) in the reference case of an analytical (anti)caloron solution. The clustering of the three topological charge densities marks the positions of three types of dyons and an…
We investigate recently proposed HP1 sigma-model embedding method aimed to study the topology of SU(2) gauge fields. The HP1 based topological charge is shown to be fairly compatible with various known definitions. We study the corresponding topological susceptibility and estimate its value in the continuum limit. The geometrical clarity of HP1 approach allows to investigate non-perturbative aspects of SU(2) gauge theory on qualitatively new level. In particular, we obtain numerically precise estimation of gluon condensate and its leading quadratic correction. Furthermore, we present clear evi…
In many cases, topological excitations become fundamental variables in a dual formulation of the theory. Assuming this is true in case of SU(2) gluodynamics, we look for hints on the dual variables from the lattice simulations. Two-dimensional vortices, or branes with recently established properties seem to be most natural candidates. The total area of the branes scales in the physical units while the non-Abelian action is ultraviolet divergent. The branes are populated with magnetic monopoles, or tachyonic mode.
Correlations of the topological charge Q, the electric current J^e and the magnetic current J^m in SU(2) lattice gauge theory in the Maximal Abelian projection are investigated. It occurs that the correlator <<QJ^e J^m>> is nonzero for a wide range of values of the bare charge. It is shown that: (i) the abelian monopoles in the Maximal Abelian projection are dyons which carry fluctuating electric charge; (ii) the sign of the electric charge e(x) coincides with that of the product of the monopole charge m(x) and the topological charge density Q(x).
The liquid droplet formula is applied to an analysis of the properties of geometrical (anti)clusters formed in SU(2) gluodynamics by the Polyakov loops of the same sign. Using this approach, we explain the phase transition in SU(2) gluodynamics as a transition between two liquids during which one of the liquid droplets (the largest cluster of a certain Polyakov loop sign) experiences a condensation, while the droplet of another liquid (the next to the largest cluster of the opposite sign of Polyakov loop) evaporates. The clusters of smaller sizes form two accompanying gases, which behave oppos…
Landau gauge gluon propagators are studied numerically in the SU(3) gluodynamics as well as in the full QCD with the number of flavors $n_F=2$ using efficient gauge fixing technique. We compare these propagators at temperatures very close to the transition point in two phases : confinement and deconfinement. The electric mass $m_E$ has been determined from the momentum space longitudinal gluon propagator. Gribov copy effects are found to be rather strong in the gluodynamics, while in the full QCD case they are weak ("Gribov noise"). Also we analyse finite volume dependence of the transverse an…
The properties of the thermal Abelian color-magnetic monopoles in the maximally Abelian gauge are studied in the vicinity of the confinement-deconfinement phase transition in the lattice $SU(3)$ gluodynamics and lattice QCD. We compute the density and interaction parameters of the thermal monopoles. We find that the density of the thermal monopoles $ρ(T)$ jumps up near the transition temperature $T_c$. Additionally we present new results on the percolation transition in $SU(3)$ gluodynamics which is known to coincide in gluodynamics with the confinement-deconfinement phase transition.
We study correlations between low-lying modes of the overlap Dirac operator and vacuum defects, center vortices and three-dimensional volumes, in lattice SU(2) gluodynamics. The low-lying modes are apparently sensitive to topological properties of the underlying gluon field configurations while the vacuum defects are crucial for the confinement. We find distinct positive correlation in both cases. In case of vortices the correlation is stronger.
We review the method of blocking of topological defects from continuum used as a non--perturbative tool to construct effective actions for these defects. The actions are formulated in the continuum limit while the couplings of these actions can be derived from simple observables calculated numerically on lattices with a finite lattice spacing. We demonstrate the success of the method in deriving the effective actions for Abelian monopoles in the pure SU(2) gauge models in an Abelian gauge. In particular, we discuss the gluodynamics in three and four space--time dimensions at zero and non--zero…
Based on the strong coupling expansion we obtain effective 3-dimensional models for the Polyakov loop in finite-temperature G_2 gluodynamics. The Svetitsky-Jaffe conjecture relates the resulting continuous spin models with G_2 gluodynamics near phase transition points. In the present work we analyse the effective theory in leading order with the help of a generalised mean field approximation and with detailed Monte-Carlo simulations. In addition we derive a Potts-type discrete spin model by restricting the characters of the Polyakov loops to the three extremal points of the fundamental domain …
We consider Wilson's SU(N) lattice gauge theory (without fermions) at negative values of beta= 2N/g^2 and for N=2 or 3. We show that in the limit beta -> -infinity, the path integral is dominated by configurations where links variables are set to a nontrivial element of the center on selected non intersecting lines. For N=2, these configurations can be characterized by a unique gauge invariant set of variables, while for N=3 a multiplicity growing with the volume as the number of configurations of an Ising model is observed. In general, there is a discontinuity in the average plaquette when g^…
The vacuum type of SU(2) gluodynamics is studied using Monte-Carlo simulations in maximally Abelian (MA) gauge and in Landau (LA) gauge, where the dual Meissner effect is observed to work. The dual Meissner effect is characterized by the coherence and the penetration lengths. Correlations between Wilson loops and electric fields are evaluated in order to measure the penetration length in both gauges. The coherence length is shown to be fixed in the MA gauge from measurements of the monopole density around the static quark-antiquark pair. It is also shown numerically that a dimension 2 gluon op…
We consider properties of zero and near-zero modes for overlap fermion operator in SU(2) lattice gluodynamics. The density of the states is of the order of Lambda(QCD) while the localization volume of the modes tends to zero in physical units with the lattice spacing tending to zero. The situation changes drastically when we study "vortex removed" configurations.
We revisit the question of the convergence of lattice perturbation theory for a pure SU(3) lattice gauge theory in 4 dimensions. Using a series for the average plaquette up to order 10 in the weak coupling parameter beta^{-1}, we show that the analysis of the extrapolated ratio and the extrapolated slope suggests the possibility of a non-analytical power behavior of the form (1/β-1/5.7(1))^{1.0(1)}, in agreement with another analysis based on the same asumption. This would imply that the third derivative of the free energy density diverges near beta =5.7. We show that the peak in the third der…
In a pilot study, we use the topological charge density defined by the eigenmodes of the overlap Dirac operator (with ultraviolet filtering by mode-truncation) to search for lumps of topological charge in SU(2) pure gauge theory. Augmenting this search with periodic and antiperiodic temporal boundary conditions for the overlap fermions, we demonstrate that the lumps can be classified either as calorons or as separate caloron constituents (dyons). Inside the topological charge clusters the (smeared) Polyakov loop is found to show the typical profile characteristic for calorons and dyons. This i…
Topological charge susceptibility $χ_{t}$ for pure gauge SU(3) theory at finite temperature is studied using anisotropic lattices. The over-improved stout-link smoothing method is utilized to calculate the topological charge. Near the phase transition point we find a rapid declining behavior for $χ_{t}$ with values decreasing from $(188(1)\mathrm{MeV})^{4}$ to $(67(3)\mathrm{MeV})^{4}$ as the temperature increased from zero temperature to $1.9T_{c}$ which demonstrates the existence of topological excitations far above $T_{c}$. The 4th order cumulant $c_4$ of topological charge, as well as the …
We smear quenched lattice QCD ensembles with lattice volume $32^3\times 8$ by using Wilson flow. Six ensembles at temperature near the critical temperature $T_c$ corresponding to the critical inverse coupling $β_c=6.06173(49)$ are used to investigate the localization of topological charge density. If the effective smearing radius of Wilson flow is large enough, the density, size and peak of Harrington-Shepard (HS) caloron-like topological lumps of ensembles are stable when $β\leq 6.050$, but start to change significantly when $β\geq6.055$. The inverse participation ratio (IPR) of topological c…
Global topological charge decorrelates very slowly or even freezes in fine lattice simulations. On the other hand, its local fluctuations are expected to survive and lead to the correct physical results as long as the volume is large enough. We investigate this issue on recently generated configurations including dynamical domain-wall fermions at lattice spacings a = 0.08 fm and finer. We utilize the Yang-Mills gradient flow to define the topological charge density operator and calculate its long-distance correlation, through which we propose a new method for extracting the topological suscept…
We study the topological structure of QCD by cluster analysis. The fermionic topological charge density is constructed from low lying modes of overlap Dirac operator for three types of temporal boundary conditions for fermion field. This gives the possibility to mark all three dyon constitutents of KvBLL caloron in gluonic fields. The gluonic topological charge density is appearing in the process of overimproved gradient flow process stopped at the moment when it maximally matches the fermionic topological charge density. This corresponds to the smearing of gluonic fields up to the scale set b…
We probe the SU(3) vacuum using eigenvectors of the Dirac operator with an arbitrary phase for the temporal boundary condition. We consider configurations with topological charge |Q| = 1 near the QCD phase transition and at low temperatures on a torus. For all our ensembles we show that the zero-mode of the Dirac operator changes its position as one changes the phase of the boundary condition. For ensembles near the QCD phase transition our results closely resemble the behavior of zero-modes for Kraan - van Baal solutions of the classical Yang-Mills equations where the individual lumps are int…
We investigate numerically the spectral flow introduced by Adams for the staggered Dirac operator on realistic (quenched) gauge configurations. We obtain clear numerical evidence that the definition works as expected: there is a clear separation between crossings near and far away from the origin, and the topological charge defined through the crossings near the origin agrees, for most configurations, with the one defined through the near-zero modes of large taste-singlet chirality of the staggered Dirac operator. The crossings are much closer to the origin if we improve the Dirac operator use…
In an attempt to describe the change of topological structure of pure SU(2) gauge theory near deconfinement a renormalization group inspired method is tested. Instead of cooling, blocking and subsequent inverse blocking is applied to Monte Carlo configurations to capture topological features at a well-defined scale. We check that this procedure largely conserves long range physics like string tension. UV fluctuations and lattice artefacts are removed which otherwise spoil topological charge density and Abelian monopole currents. We report the behaviour of topological susceptibility and monopol…
We compute the topological susceptibility chi_t in SU(3) lattice gauge theory using fermionic methods based on the Atiyah-Singer index theorem. Near the phase transition we find a smooth crossover behavior for chi_t with values decreasing from (191(5) MeV)^4 to (100(5) MeV)^4 as we increase the temperature from 0.88 T_c to 1.31 T_c, showing that topological excitations exist far above T_c. Our study is the first large scale analysis of the topological susceptibility at high temperature based on the index theorem and the results agree well with field theoretical methods.
We propose an implementation of a two-dimensional $\mathbb{Z}_2$ lattice gauge theory model on a shallow quantum circuit, involving a number of single and two-qubits gates comparable to what can be achieved with present-day and near-future technologies. The ground state preparation is numerically analyzed on a small lattice with a variational quantum algorithm, which requires a small number of parameters to reach high fidelities and can be efficiently scaled up on larger systems. Despite the reduced size of the lattice we consider, a transition between confined and deconfined regimes can be de…
We study the underlying topology of gauge fields in 2+1 flavor QCD with domain wall fermions on lattices of size $32^3\times 8$, at and immediately above the chiral crossover transition. Using valence overlap fermions with exact index theorem, we focus on its zero modes for different choices of periodicity phases along the temporal direction. Our studies show that the zero modes are due to fractionally charged topological objects, the instanton-dyons. We further provide qualitative study of the interactions between those and compare with the available semi-classical results, finding remarkably…
We show the existence of strong (anti)correlations between the topological hot spots and the local values of the trace of the Polyakov loop in $2+1$ flavor QCD with physical quark mass, in the vicinity of the crossover transition corresponding to the simultaneous restoration of chiral symmetry and deconfinement. Using sophisticated lattice techniques, we have carefully identified the topological hot spots using quark zero modes and measured the short-distance fluctuations of the Polyakov loop about them, showing how the latter is repelled quite strongly around the peak of the zero modes. Thoug…
The real-time topological susceptibility is studied in $(1+1)$-dimensional massive Schwinger model with a $θ$-term. We evaluate the real-time correlation function of electric field that represents the topological Chern-Pontryagin number density in $(1+1)$ dimensions. Near the parity-breaking critical point located at $θ=π$ and fermion mass $m$ to coupling $g$ ratio of $m/g \approx 0.33$, we observe a sharp maximum in the topological susceptibility. We interpret this maximum in terms of the growth of critical fluctuations near the critical point, and draw analogies between the massive Schwinger…
The experimental discovery of the fractional Hall conductivity in two-dimensional electron gases revealed new types of quantum particles, called anyons, which are beyond bosons and fermions as they possess fractionalized exchange statistics. These anyons are usually studied deep inside an insulating topological phase. It is natural to ask whether such fractionalization can be detected more broadly, say near a phase transition from a conventional to a topological phase. To answer this question, we study a strongly correlated quantum phase transition between a topological state, called a $\mathb…
