Anomalous U(1)A couplings and the Columbia plot

This work ties in with the investigation of the intermediate valent states and valence fluctuations in certain europium based intermetallic systems. Valence fluctuations are a property of the electronic system of a compound that is possibly accompanied by structural effects, which, in some cases, are quite noticable. By assuming how the changes in the electronic system and in the crystal lattice are connected, valence _uctuations of europium are believed to be a possible probe for the theory of quantum critical elasticity, which is investigated on by the SFB TRR 288 (Frankfurt, Mainz, Karlsruhe, Bochum, Dresden). Here, the proceedings in growing single crystals of di_erent compounds related to this _eld of research are reported. This includes the ThCr2Si2 (122) type compounds EuPd2Si2 as well as the doping series EuPd2(Si1-xGex)2, the Europium based ternary Phosphides EuFe2P2, EuCo2P2, EuNi2P2 and EuRu2P2, and attempts to grow compounds of a derived 1144 structure by ordered substitution of half the Europium, EuKRu4P4. The largest part of this work focusses on the EuPd2Si2 system, which exhibits intermediate valent europium and a temperature dependent transition between two di_erent intermediate valent states of europium. Crystals of this system were grown using the Czochralski method with a levitating melt and an europium excess flux after a two step prereaction process. Also, explorations of a PdSi-rich flux and external flux methods are reported. Ten Czochralski grown experiments, in six generations iteratevely seeded by the previous generation, were prepared. Thermodynamical and structural analyses of the crystals located the transition between the di_erent intermediate valent states of europium between 140K and 165 K, transitioning from a high temperature Eu2.3+ state to a low temperature Eu2.7+ state, and classified it as a second order transition. To this transition a lattice anomaly of the a-parameter collapsing about 2% is connected, while the c-parameter remains largely unaffected. Large differences between individual samples can be explained by combining thermodynamical and structural analyses with compositional analysis, revealing the valence transition temperature as strongly dependent on the sample composition and Pd-Si site interchanges. Searching to change the character of the valence transition to first order, silicon was substituted by germanium to introduce negative pressure. Germanium substituted samples of EuPd2(Si1-xGex)2 were grown using the Czochralski method with the optimized parameters from the growth experiments for the undoped compound. Samples were prepared with a nominal substitution of x = 0.05, x = 0.10, x = 0.15, x = 0.20 (twice) and x = 0.30. For the EuPd2(Si1-xGex)2 system, a phase diagram for the europium valence states is derived from chemical and thermodynamical characterizations. n ternary europium phosphides EuT2P2, the position of the compounds in the generalized phase diagram and the question of long range magnetic order or valence transition appear connected to an isostructural transition of the tetragonal crystal structure, drastically decreasing the length of the c-parameter while establishing covalent bonds between phosphorus atoms of different interlayers of the structure, the so called ‚collapse‘. While EuFe2P2, EuT2P2 and EuCo2P2 display both long range magnetic order and a non-collapsed crystal structure, EuNi2P2 shows both a valence transition between two intermediate valent states at a characteristic temperature of 36K - accompanied by a small lattice anomaly of the a-parameter shrinking about 0.2% - and a collapsed crystal structure. Samples of EuFe2P2, EuCo2P2 and EuNi2P2 were grown in tin flux and using solid-solid sintering approaches. Single crystals of EuFe2P2, EuCo2P2 and EuRu2P2 were investigated at ESRF in Grenoble with single crystal X-ray di_ractometry on a pressure range up to 15GPa and at temperatures down to 15K to investigate the nature of the structural transitions in the compounds. While in EuCo2P2 the structural transition occurs as a transition of first order at all temperatures (e.g. at 2GPa for 15 K), in EuFe2P2 and EuRu2P2 the structural collapse evolves over a broad pressure range up to 8GPa and as a transition of second order troughout the temperature ranges, albeit seeming to sharpen at lower temperatures. From the crystallographic data, elastic constants of the compounds could be derived, revealing EuFe2P2 and EuRu2P2 as unexpectedly elastic materials. In order to probe the structural collapse at more accessible pressures, crystals with a sturcture derived from the 122 structure, but with ordered 50% substitution of europium and hence altering the symmetry from I4/mmm to P4/mmm in a 1144 structure, were exploratively pursued. Different experiments to obtain EuAT4P4 (with A = K, Rb, Cs and T = Fe, Ru) from binary or ternary prereactants or directly from the elements remained largely unsuccessful.

We discuss the influence of a helicity imbalance on the phase diagram of dense QCD at finite temperature. We argue that the helical chemical potential is a thermodynamically relevant quantity in theories with the mass gap generation. Using the linear sigma model coupled to quarks, we show that the presence of the helical density substantially affects the phase diagram of dense quark matter. A moderate helical density makes the chiral phase transition softer while shifting the critical endpoint towards lower temperatures and higher baryon chemical potentials. As the helical density increases, the segment of the first-order transition disappears, and the chiral transition becomes a soft crossover. At even higher helical chemical potentials, the first-order transition reappears again at the zero-density finite-temperature transition and extends into the interior of the phase diagram. This evolution of the chiral transition reflects the existence of a thermodynamic duality between helical and vector (baryonic) chemical potentials. We also show that the presence of the helicity imbalance of quark matter increases the curvature of the chiral pseudocritical line in QCD.

We present a detailed calculation of the transition temperature in QCD with two light and one heavier (strange) quark mass on lattices with temporal extent ${N}_{\ensuremath{\tau}}=4$ and 6. Calculations with improved staggered fermions have been performed for various light to strange quark mass ratios in the range, $0.05\ensuremath{\le}{\stackrel{^}{m}}_{l}/{\stackrel{^}{m}}_{s}\ensuremath{\le}0.5$, and with a strange quark mass fixed close to its physical value. From a combined extrapolation to the chiral (${\stackrel{^}{m}}_{l}\ensuremath{\rightarrow}0$) and continuum ($aT\ensuremath{\equiv}1/{N}_{\ensuremath{\tau}}\ensuremath{\rightarrow}0$) limits we find for the transition temperature at the physical point ${T}_{c}{r}_{0}=0.457(7)$ where the scale is set by the Sommer-scale parameter ${r}_{0}$ defined as the distance in the static quark potential at which the slope takes on the value, $(\mathrm{d}{V}_{\overline{q}q}(r)/\mathrm{d}r{)}_{r={r}_{0}}=1.65/{r}_{0}^{2}$. Using the currently best known value for ${r}_{0}$ this translates to a transition temperature ${T}_{c}=192(7)(4)\text{ }\text{ }\mathrm{MeV}$. The transition temperature in the chiral limit is about 3% smaller. We discuss current ambiguities in the determination of ${T}_{c}$ in physical units and also comment on the universal scaling behavior of thermodynamic quantities in the chiral limit.

As an extension of $QCD$, consider a theory with ``$2+1$'' flavors, where the current quark masses are held in a fixed ratio as the overall scale of the quark masses is varied. At nonzero temperature and baryon density it is expected that in the chiral limit the chiral phase transition is of first order. Increasing the quark mass from zero, the chiral transition becomes more weakly first order, and can end in a chiral critical point. We show that the only massless field at the chiral critical point is a sigma meson, with the universality class that of the Ising model. Present day lattice simulations indicate that $QCD$ is (relatively) near to the chiral critical point.

The infrared regularized and the large Nc standard U(3) chiral perturbation theory (ChPT) predicted scaling of the pion, kaon decay constants fπ, fK, and Mη2=mη2+mη′2, has been used in the renormalized 2+1 flavor quark-meson (RQM) model to find a consistent path to the chiral limit as the π and K meson masses approach zero. Using the inputs from the above two prescriptions of the ChPT, the Columbia plots are drawn and compared in the mπ−mK and the light strange quark mass mud−ms planes, free from any ambiguity or heuristic adjustment in the model parameter fixing away from the physical point. The on-shell renormalization of the parameters and consistent treatment of the divergent quark one-loop vacuum fluctuations in the RQM model make the axial UA(1) anomaly significantly stronger, and the light (strange) explicit chiral symmetry breaking strength becomes weaker by a small (relatively large) amount. The first-order chiral transition region in the mπ−mK and μ−mK planes of the Columbia plots generated using the inputs from both versions of the ChPT increases due to the above novel features of the RQM model. Comparing our results (in the presence of the axial UA(1) anomaly) regarding the softening effect of the quark one-loop vacuum fluctuations on the chiral transition, it looks overestimated in a recent functional renormalization group study. The UA(1) symmetry and the SUA(3) chiral symmetry restoration in the chiral limit takes place at the same transition temperature.

Paradigme shift in gauge topology, from instantons to their constituents – instanton-dyons – has recently lead to very significant advances. Like instantons, they have fermionic zero modes, and their collectivization at sufficiently high density explains the chiral symmetry breaking. Unlike instantons, these objects have electric and magnetic charges. Their back reaction on the mean value of the Polyakov line (holonomy) allows to explain the deconfinement transition. The talk summarizes recent works on the dyon ensemble, done in the mean field approximation (MFA), and also by direct numerical statistical simulation. Introduction of non-trivial quark periodicity conditions leads to drastic changes in both deconfinement and chiral transitions. In particulaly, in the so called Z (Nc ) – QCD model the former gets much stronger, while the latter does not seem to occur at all.

Coupling the deconfining and chiral transitions

Tuning the strange quark mass in lattice simulations

Ferromagnetic system with the first order transition and applicability of the Maxwell relation to its magnetocaloric effect

The region of the Columbia plot with two light quark flavors is not yet conclusively understood. Non-perturbative effects, e.g. the magnitude of the anomalous U(1) axial symmetry breaking, decides on the nature of the phase transition in this region. We report on our study of this region of the Columbia plot using lattice techniques. We use gauge field ensembles generated within the Highly Improved Staggered Quark discretization scheme, with the strange quark mass fixed at its physical value and the light quark mass varied such that $m_l=m_s/27$ and $m_s/40$, where $m_l=m_s/27$ corresponds to the physical light quark mass. We study the eigenvalue spectrum of QCD using the overlap Dirac operator on these gauge field ensembles at finite temperature around the chiral transition temperature $T_c$, as the light quark masses approach the chiral limit, and infer about the fate of the anomalous $U_A(1)$ symmetry breaking.

We study the chiral phase transition with 2 + 1 quark flavors in an improved soft-wall AdS/QCD model, which can produce the light meson spectrum and many other low-energy quantities consistent with experiments in the two-flavor case. The chiral transition behaviors of the quark condensates at different quark masses are analysed in detail, and the (m_{u,d}, m_s) phase diagram for the quark sector has been obtained from the improved soft-wall model. We find that the features of the calculated phase diagram are completely consistent with the standard scenario, which is supported by lattice simulations and theoretical arguments. The evidence of a tricritical point on the m_{u,d} = 0 boundary of the (m_{u,d}, m_s) phase diagram is first clearly presented in the bottom-up AdS/QCD.

We study the Blume-Emery-Griffiths model in a random crystal field in two and three dimensions, through a real-space renormalization-group approach and a mean-field approximation, respectively. According to the two-dimensional renormalization-group calculation, non-symmetry-breaking first-order phase transitions are eliminated and symmetry-breaking discontinuous transitions are replaced by continuous ones, when disorder is introduced. On the other hand, the mean-field calculation predicts that first-order transitions are not eliminated by disorder, although some changes are introduced in the phase diagrams. We make some comments on the consequences of a degeneracy parameter, which may be relevant in martensitic transitions.

We discuss a unique system that allows exact analytical investigation of first- and second-order transitions with finite-size effects: mechanical desorption of an ideal lattice polymer chain grafted with one end to a solid substrate with a pulling force applied to the other end. We exploit the analogy with a continuum model and use accurate mapping between the parameters in continuum and lattice descriptions, which leads to a fully analytical partition function as a function of chain length, temperature (or adsorption strength), and pulling force. The adsorption-desorption phase diagram, which gives the critical force as a function of temperature, is nonmonotonic and gives rise to re-entrance. We analyze the chain length dependence of several chain properties (bound fraction, chain extension, and heat capacity) for different cross sections of the phase diagram. Close to the transition a single parameter (the product of the chain length N and the deviation from the transition point) describes all thermodynamic properties. We discuss finite-size effects at the second-order transition (adsorption without force) and at the first-order transition (mechanical desorption). The first-order transition has some unusual features: The heat capacity in the transition region increases anomalously with temperature as a power law, metastable states are completely absent, and instead of a bimodal distribution there is a flat region that becomes more pronounced with increasing chain length. The reason for this anomaly is the absence of an excess surface energy for the boundary between adsorbed and stretched coexisting phases (this boundary is one segment only): The two states strongly fluctuate in the transition point. The relation between mechanical desorption and mechanical unzipping of DNA is discussed.

Chapter 1 - Phase diagrams

Motivated by recent claims questioning the existence of strong CP violation, we present a pedagogical review of CP violation in Quantum Chromodynamics (QCD). Using fundamental properties of the QCD partition function, we analyze the dependence of the chiral quark and CP violating gluon condensates on the $\theta$ parameter and the quark masses in the chiral limit. We show explicitly how CP violation arises, clarify the role of the axial U(1) anomaly and the ordering of the infinite-volume limit, and discuss the conditions under which CP symmetry may or may not be realized, including in the large-$N_c$ framework. Our results reaffirm the presence of strong CP violation for physically relevant parameters and thus the theoretical basis of the strong CP problem and axion physics.