Chiral Order Parameter Research Articles

Multiplicity fluctuations at the quark-hadron phase transition from a fluid dynamical model

The region of large net-baryon densities in the QCD phase diagram is expected to exhibit a first-order phase transition. Experimentally, its study will be one of the primary objectives for the upcoming FAIR accelerator. We model the transition between quarks and hadrons in a heavy-ion collision using a fluid which is coupled to the explicit dynamics of the chiral order parameter and a dilaton field. This allows us to investigate signals stemming from the nonequilibrium evolution during the expansion of the hot plasma. Special emphasis is put on an event-by-event analysis of baryon number fluctuations which have long since been claimed to be sensitive to a critical point.

Constraints on 3-flavor QCD order parameters and light quark mass difference from [formula omitted] decays

The η→3π decays are a valuable source of information on low energy QCD. Yet they were not used for an extraction of the chiral symmetry breaking order parameters until now. We use a bayesian approach in the framework of resummed chiral perturbation theory to obtain constraints on the quark condensate and pseudoscalar decay constant in the chiral limit, as well as the mass difference of light quarks. We compare our results with recent χPT and lattice QCD fits and find some tension, as the η→3π data seem to prefer a larger ratio of the chiral order parameters. The results also seem to exclude a large value of the chiral decay constant, which was found by some recent works.

On SU(3) Effective Models and Chiral Phase Transition

Sensitivity of Polyakov Nambu-Jona-Lasinio (PNJL) model and Polyakov linear sigma-model (PLSM) has been utilized in studying QCD phase-diagram. From quasi-particle model (QPM) a gluonic sector is integrated into LSM. The hadron resonance gas (HRG) model is used in calculating the thermal and dense dependence of quark-antiquark condensate. We review these four models with respect to their descriptions for the chiral phase transition. We analyze the chiral order parameter, normalized net-strange condensate, and chiral phase-diagram and compare the results with recent lattice calculations. We find that PLSM chiral boundary is located in upper band of the lattice QCD calculations and agree well with the freeze-out results deduced from various high-energy experiments and thermal models. Also, we find that the chiral temperature calculated from HRG is larger than that from PLSM. This is also larger than the freeze-out temperatures calculated in lattice QCD and deduced from experiments and thermal models. The corresponding temperature and chemical potential are very similar to that of PLSM. Although the results from PNJL and QLSM keep the same behavior, their chiral temperature is higher than that of PLSM and HRG. This might be interpreted due the very heavy quark masses implemented in both models.

Discovery of metastable states in a finite-size classical one-dimensional planar spin chain with competing nearest- and next-nearest-neighbor exchange couplings

A theoretical method recently developed is used to find all possible equilibrium magnetic states of a finite-size classical one-dimensional planar spin chain with competing nearest-neighbor (nn) and next-nearest-neighbor (nnn) exchange interactions. The energy of a classical planar model with $N$ spins is a function of $N$ absolute orientational angles or equivalently, due to the absence of in-plane anisotropy, of $(N\ensuremath{-}1)$ relative orientational angles. The lowest energy stable state (ground state) corresponds to a global minimum of the energy in the $(N\ensuremath{-}1)$-dimensional space, while metastable states correspond to local minima. For a given value of the ratio, $\ensuremath{\gamma}$, between nnn and nn exchange couplings, all the equilibrium configurations of the model were calculated with great accuracy for $N\ensuremath{\le}16$, and a stability analysis was subsequently performed. For any value of $N$, the ground state was found to be ``symmetric'' with respect to the middle of the chain in the relative angles representation. For the chosen value of $\ensuremath{\gamma}$, the ground state consists of a helix whose chirality is constant in sign along the chain (i.e., all the spins turn clockwise, or all anticlockwise), but whose pitch varies owing to finite-size effects; e.g., for positive chirality we found that the chiral order parameter $\ensuremath{\chi}(N)g0$ increases monotonically with increasing $N$, approaching the value $(\ensuremath{\chi}=1)$ pertinent to the ground state in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$. For finite but not too small values of $N$, we found metastable states characterized by one reversal of chirality, either localized just in the middle of the chain [``antisymmetric'' state, with chiral order parameter $\ensuremath{\chi}(N)=0$], or shifted away from the middle of the chain, to the right or to the left [pairs of ``ugly'' states, with equal and opposite values of $\ensuremath{\chi}(N)\ensuremath{\ne}0$; the attribute ``ugly'' refers to the absence of a definite symmetry in the relative angles representation]. Concerning the stability of these states with one reversal of chirality, two main results were found. First, the ``antisymmetric'' state is metastable for even $N$ and unstable for odd $N$. Second, an additional pair of ``ugly'' states is found whenever the number of spins in the chain is increased by 1; the states in each additional pair are unstable for even $N$ and metastable for odd $N$. Analysis of stable and metastable configurations in the framework of a discrete nonlinear mapping approach provides further support for the above results.

On the Form Factors of Local Operators in the Bazhanov–Stroganov and Chiral Potts Models

We consider general cyclic representations of the six-vertex Yang–Baxter algebra and analyze the associated quantum integrable systems, the Bazhanov–Stroganov model and the corresponding chiral Potts model on finite size lattices. We first determine the propagator operator in terms of the chiral Potts transfer matrices and we compute the scalar product of separate states (including the transfer matrix eigenstates) as a single determinant formulae in the framework of Sklyanin’s quantum separation of variables. Then, we solve the quantum inverse problem and reconstruct the local operators in terms of the separate variables. We also determine a basis of operators whose form factors are characterized by a single determinant formulae. This implies that the form factors of any local operator are expressed as finite sums of determinants. Among these form factors written in determinant form are in particular those which will reproduce the chiral Potts order parameters in the thermodynamic limit. The results presented here are the generalization to the present models associated to the most general cyclic representations of the six-vertex Yang–Baxter algebra of those we derived for the lattice sine–Gordon model.

Critical behavior of three-dimensional frustrated helimagnets

The critical properties of classical frustrated helimagnets with different numbers of chiral order parameters are studied in three dimensions. The model of an antiferromagnet on a simple cubic lattice is considered with additional competing exchange interactions between spins of the first and the third range orders. In this model, helicoidal ordered phases exist with one, two, and three independent chiral order parameters. In all cases, it was found that a transition from an ordered to a disordered phase is a single first-order transition in the absence of partially ordered phases.

Is ρ-meson melting compatible with chiral restoration?

Utilizing in-medium vector spectral functions which describe dilepton data in ultra-relativistic heavy-ion collisions, we conduct a comprehensive evaluation of QCD and Weinberg sum rules at finite temperature. The starting point is our recent study in vacuum, where the sum rules have been quantitatively satisfied using phenomenological vector and axial-vector spectral functions which describe hadronic τ-decay data. In the medium, the temperature dependence of condensates and chiral order parameters is taken from thermal lattice QCD where available, and otherwise is estimated from a hadron resonance gas. Since little is known about the in-medium axial-vector spectral function, we model it with a Breit–Wigner ansatz allowing for smooth temperature variations of its width and mass parameters. Our study thus amounts to testing the compatibility of the ρ-broadening found in dilepton experiments with (the approach toward) chiral restoration, and thereby searching for viable in-medium axial-vector spectral functions.

Quark–gluon mixed condensate for the [formula omitted] light-flavor sector at finite temperature

We investigate the quark–gluon mixed condensate 〈q¯σ⋅Gq〉≡m02〈q¯q〉 for the SU(2) light-flavor sector at finite temperature (T). Relevant model parameters, such as the average (anti)instanton size, inter-(anti)instanton distance, and constituent-quark mass at zero virtuality, are modified as functions of T, employing the trivial-holonomy caloron solution. By doing that, we observe correct chiral restoration patterns depending on the current-quark mass m, i.e. the second-order and crossover chiral phase transitions for the zero and finite current-quark masses, respectively. We also perform the two-loop renormalization-group (RG) evolution for the both condensates by increasing the renormalization scale μ=(0.6→2.0) GeV. It turns out that the mixed condensate is insensitive to the RG evolution, whereas the quark condensate become larger considerably by the evolution. Numerically, we obtain −〈q¯σ⋅Gq〉1/5=(0.45–0.46) GeV at T=0 within the present theoretical framework, and the mixed condensate plays the role of the chiral order parameter for finite T. The ratio of the two condensates m02 is almost flat below the chiral transition T (T0), and increases rapidly beyond it. From a simple linear parametrization, we obtain m02(T)/m02(0)≈(0.07,0.47)T/T0+(1,0.6) for (T≲T0,T≳T0) at μ=0.6 GeV. The present results are compared with other theoretical ones including the lattice QCD simulations, and show qualitatively good agreement with them.

Proposed Chiral Texture of the Magnetic Moments of Unit-Cell Loop Currents in the Pseudogap Phase of Cuprate Superconductors

We propose a novel chiral order parameter to explain the unusual polar Kerr effect in underdoped cuprates. It is based on the loop-current model by Varma, which is characterized by the in-plane anapole moment N and exhibits the magnetoelectric effect. We propose a helical structure where the vector N(n) in the layer n is twisted by the angle π/2 relative to N(n-1), thus breaking inversion symmetry. We show that coupling between magnetoelectric terms in the neighboring layers for this structure produces optical gyrotropy, which results in circular dichroism and the polar Kerr effect.

QCD magnetic susceptibility at finite temperature beyond the chiral limit

We investigate the QCD magnetic susceptibility chi_q for flavor SU(2) at finite temperature (T) beyond the chiral limit, using the liquid instanton model, defined in Euclidean space and modified by the T-dependent caloron solution. The background electromagnetic fields are induced to the QCD vacuum, employing the Schwinger method. We first compute the scalar (chiral) and tensor condensates as functions of T as well the current-quark mass m, signaling the correct universal chiral restoration patterns. It turns out that chi_q, given by the ratio of the two condensates, is a smoothly decreasing function of T, showing about 20% reduction of its strength at the chiral transition T = T_0, in comparison to that at T = 0, and decreases almost linearly beyond T_0 for m > 0. We observe that the present numerical results are in qualitatively good agreement with other theoretical results, including the lattice simulations. Finally, we examine the effects of the external magnetic field on the tensor-polarization VEV, resulting in that it plays the role of the chiral order parameter.

P6chiral resonating valence bonds in the kagome antiferromagnet

The Kagome Heisenberg antiferromagnet is mapped onto an effective Hamiltonian on the star superlattice by Contractor Renormalization. Comparison of ground state energies on large lattices to Density Matrix Renormalization Group justifies truncation of effective interactions at range 3. Within our accuracy, magnetic and translational symmetries are not broken (i.e. a spin liquid ground state). However, we discover doublet spectral degeneracies which signal the onset of p6 - chirality symmetry breaking. This is understood by simple mean field analysis. Experimentally, the p6 chiral order parameter should split the optical phonons degeneracy near the zone center. Addition of weak next to nearest neighbor coupling is discussed.

Colloquium: Homochirality: Symmetry breaking in systems driven far from equilibrium

Subsequent to the discovery of chirality of organic molecules by Pasteur, living organisms have been found to utilize biomolecules of only one handedness. The origin of this homochirality in life still remains unknown. It is believed that homochirality is attained in two stages: the initial creation of a chirality bias and its subsequent amplification to pure chirality. In the last two decades, two novel experiments have established the second stage in different fields: Soai and co-workers achieved the amplification of enantiomeric excess in the production of chiral organic molecules, and Viedma obtained homochirality in the solution growth of sodium chlorate crystals. These experiments are explained by a theory with a nonlinear evolution equation for the chiral order parameter; nonlinear processes in reactions or in crystal growth induce enantiomeric excess amplification, and the recycling of achiral elements ensures homochirality. Recycling drives the system to a state far from equilibrium with a free energy higher than that of the equilibrium state.

The density of states and thermopower in disordered carbon nanotubes

The calculated values of thermopower and density of states are presented as a function of temperature, tube chirality, impurity concentration and short-range order parameter for disordered carbon nanotubes. A possible interpretation of the features of low-temperature behavior of the properties under study is presented.

Update on Chiral Symmetry Restoration in the Context of Dilepton Data

We evaluate currently available information on low-mass dilepton and direct-photon emission spectra measured in ultrarelativistic heavy-ion collisions. In the first part an attempt is made to develop a consistent picture of the in-medium effects on the electromagnetic spectral function and pinpoint its emission history by utilizing its radial and elliptic flow signatures. In the second part we elaborate on the implications of the empirical information on the nature of chiral symmetry restoration. We indicate how the melting of the ρ resonance in hot and dense matter is related to, and compatible with, the reduction of chiral order parameters as "measured" in thermal lattice QCD.

Chiral spin liquid in a two-dimensional helical XY magnet with two chiral order parameters

The critical properties of an XY helimagnet on a square lattice with two chiral order parameters are studied by Monte Carlo simulations. This model is a modification of the J1-J2-J3 model with J2 = 0. The case of different third range order interactions J3 are considered, J3a ≠ J3b. A first order transition is found away from the Lifshitz points 4J3a= J1 and 4 J3b= J1. It is pointed out that a chiral spin liquid phase possibly exists near the Lifshitz points.

Possible chiral spin-liquid phase in noncentrosymmetricRBaCo4O7

Based on a symmetry approach, we propose a possible explanation of the weak ferromagnetic component recently observed in YBaCo${}_{3}$FeO${}_{7}$ [M. Valldor et al., Phys. Rev. B 84, 224426 (2011)] and other isostructural compounds in the high-temperature spin-liquid phase. Due to the polar nature of their crystal structure, a coupling between time-odd scalar spin chirality which we suggest as the primary order parameter and macroscopic magnetization is possible as follows from the general form of the appropriate free-energy invariant. The deduced pseudoproper coupling between both physical quantities provides a unique possibility to study the critical behavior of the chiral order parameter.

Edge mass current and the role of Majorana fermions inA-phase superfluid3He

The total angular momentum associated with the edge mass current flowing at the boundary in the superfluid ${}^{3}$He $A$ phase confined in a disk is proved to be $L=N\ensuremath{\hbar}/2$, consisting of ${L}^{\mathit{MJ}}=N\ensuremath{\hbar}$ from the Majorana quasiparticles (QPs) and ${L}^{\mathrm{cont}}=\ensuremath{-}N\ensuremath{\hbar}/2$ from the continuum state. We show this based on an analytic solution of the chiral order parameter for the quasiclassical Eilenberger equation. Important analytic expressions are obtained for mass current, angular momentum, and density of states (DOS). Notably the DOS of the Majorana QPs is exactly ${N}_{0}/2$ (${N}_{0}$: normal state DOS) responsible for the factor 2 difference between ${L}^{\mathit{MJ}}$ and ${L}^{\mathrm{cont}}$. The current decreases as ${E}^{\ensuremath{-}3}$ against the energy $E$, and $L(T)\ensuremath{\propto}\ensuremath{-}{T}^{2}$. This analytic solution is fully backed up by numerically solving the Eilenberger equation. We touch on the so-called intrinsic angular momentum problem.

ΛMS¯QCDfrom renormalization group optimized perturbation

A recent extension of a variationally optimized perturbation, combined with renormalization group properties in a straightforward way, can provide approximations to nonperturbative quantities such as the chiral symmetry breaking order parameters typically. We apply this to evaluate, up to third order in this modified perturbation, the ratio Fpi/Lambda, where Fpi is the pion decay constant and Lambda the basic QCD scale in the modified MS scheme. Using experimental Fpi input value we obtain Lambda(nf=2) ~ 255_{-15}^{+40} MeV, where quoted errors are estimates of theoretical uncertainties of the method. This compares reasonably well with some recent lattice simulation results. We briefly discuss prospects (and obstacles) for extrapolation to alpha_S(mu) at perturbative mu values.

An effective thermodynamic potential from the instanton vacuum with the Polyakov loop

In this talk, we report our recent studies on an effective thermodynamic potential (Omega_eff) at finite temperature (T>0) and zero quark-chemical potential (mu_R=0), using the singular-gauge instanton solution and Matsubara formula for N_c=3 and N_f=2 in the chiral limit, i.e. m_q=0. The momentum-dependent constituent-quark mass is computed as a function of T, together with the Harrington-Shepard caloron solution in the large-N_c limit. In addition, we take into account the imaginary quark-chemical potential mu_I = A_4, indentified as the traced Polayakov-loop (Phi) as an order parameter for the Z(N_c) symmetry, characterizing the confinement (intact) and deconfinement (spontaneously broken) phases. As a consequence, we observe the crossover of the chiral (chi) order parameter sigma^2 and Phi. It also turns out that the critical temperature for the deconfinement phase transition, T^Z_c is lowered by about (5~10) % in comparison to the case with the constant constituent-quark mass. This behavior can be understood by considerable effects from the partial chiral restoration and nontrivial QCD vacuum on the Phi. Numerical results show that the crossover transitions occur at (T^chi_c,T^Z_c) ~ (216,227) MeV.

Transitions in three-dimensional XY magnets with two chiral order parameters

Two models of classic XY antiferromagnets in three dimensions are studied by Monte Carlo simulation: the model on a simple cubic lattice with two extra intralayer exchanges and the model on a stackedtriangular lattice with an extra interlayer exchange. In suggested models, the order parameters are magnetization and two chiral parameters. A transition corresponds to breaking ℤ2 ⊗ ℤ2 ⊗ SO(2) symmetry. A distinct first order transition is found in both models.