Hopping Parameter Expansion Research Articles

High-precision analysis of the critical point in heavy-quark QCD at Nt=6

Binder-cumulant analysis of the critical point in the heavy-quark region of QCD is performed by Monte-Carlo simulations with the hopping-parameter expansion at Nt=6. We extend our previous analysis at Nt=4 to finer lattices and perform high-precision analyses on large spatial volumes up to the aspect ratio LT=Ns/Nt=18. Higher order terms in the hopping-parameter expansion are incorporated effectively up to 14th order. The numerical results show that the violation of the finite-size scaling becomes more prominent on the finer lattice at a given aspect ratio. Published by the American Physical Society 2024

Scope and convergence of the hopping parameter expansion in finite-temperature quantum chromodynamics with heavy quarks around the critical point

Abstract Hopping parameter expansion is a useful tool for investigating heavy dynamical quarks in lattice quantum chromodynamics (QCD), though its range of applicability has sometimes been questioned. We study the convergence and the valid range of the hopping parameter expansion in determining the critical point (critical quark mass) of QCD with heavy quarks at finite temperature and density. On lattices with sufficiently large spatial extent, the terms in the hopping parameter expansion are classified into Wilson loop terms and Polyakov-type loop terms. We first study the case of the worst convergence in which all the gauge link variables are unit matrices and thus the Wilson loops and the Polyakov-type loops get their maximum values. We perform explicit calculation up to more than the 100th order of the hopping parameter expansion. We show that the hopping parameter expansion is convergent up to the chiral limit of free Wilson quarks. We then perform a Monte Carlo simulation to measure correlation among Polyakov-type loop terms up to the 20th order of the hopping parameter expansion. In previous studies, strong correlation between the leading-order Polyakov loop term and the next-to-leading-order bent Polyakov loop terms was reported and used to construct an effective theory to incorporate the next-to-leading-order effect by a shift of the leading-order coupling parameter. We establish that the strong correlation among Polyakov-type loop terms also holds at higher orders of the hopping parameter expansion, and extend the effective theory to incorporate higher-order effects up to high orders. Using the effective theory, we study the truncation error of the hopping parameter expansion. We find that the previous next-to-leading-order result for the critical point for Nt = 4 is reliable. For Nt ≥ 6, we need to incorporate higher-order effects in the effective theory.

Finite-size scaling around the critical point in the heavy quark region of QCD

Finite-size scaling is investigated in detail around the critical point in the heavy-quark region of nonzero temperature QCD. Numerical simulations are performed with large spatial volumes up to the aspect ratio $N_s/N_t=12$ at a fixed lattice spacing with $N_t=4$. We show that the Binder cumulant and the distribution function of the Polyakov loop follow the finite-size scaling in the $Z(2)$ universality class for large spatial volumes with $N_s/N_t \ge 9$, while, for $N_s/N_t \le 8$, the Binder cumulant becomes inconsistent with the $Z(2)$ scaling. To realize the large-volume simulations in the heavy-quark region, we adopt the hopping parameter expansion for the quark determinant: We generate gauge configurations using the leading order action including the Polyakov loop term for $N_t=4$, and incorporate the next-to-leading order effects in the measurements by the multipoint reweighting method. We find that the use of the leading-order configurations is crucially effective in suppressing the overlapping problem in the reweighting and thus reducing the statistical errors.

End point of the first-order phase transition of QCD in the heavy quark region by reweighting from quenched QCD

We study the end point of the first-order deconfinement phase transition in two and 2+1 flavor QCD in the heavy quark region of the quark mass parameter space. We determine the location of critical point at which the first-order deconfinement phase transition changes to crossover, and calculate the pseudo-scalar meson mass at the critical point. Performing quenched QCD simulations on lattices with the temporal extents Nt=6 and 8, the effects of heavy quarks are determined using the reweighting method. We adopt the hopping parameter expansion to evaluate the quark determinants in the reweighting factor. We estimate the truncation error of the hopping parameter expansion by comparing the results of leading and next-to-leading order calculations, and study the lattice spacing dependence as well as the spatial volume dependence of the result for the critical point. The overlap problem of the reweighting method is also examined. Our results for Nt=4 and 6 suggest that the critical quark mass decreases as the lattice spacing decreases and increases as the spatial volume increases.

Exploiting the hopping parameter expansion in the hybrid Monte Carlo simulation of lattice QCD with two degenerate flavors of Wilson fermions

We show how the hopping parameter expansion at order $\kappa^2$ and $\kappa^4$ can be exploited in the simulation of lattice QCD with two flavours of degenerate Wilson fermions. A natural extension of this idea is a "UV-filtering" by using rooted polynomials. These approaches can be easily combined with, for example, mass preconditioning. First numerical tests are performed for the Wilson gauge action at $\beta=5.6$ and $\kappa=0.156$ and $0.1575$.

Hopping Parameter Expansion Technique in Noise Method for Disconnected Quark Loops**Supported by the National Natural Science Foundation of China under Grant Nos 11335001 and 11275169.

We investigate the effectiveness of the hopping parameter expansion (HPE) combined with the Z(2) noise method in the calculation of the trace of the inverse of Wilson’s Dirac operator and some other disconnected contributions. A numerical comparison of the standard deviation for the Z(2) noise method and HPE with the Z(2) noise method is carried out. It is found that there are noise reductions in all the quantities we calculated using the HPE with the Z(2) noise method. For the trace of the inverse of Wilson’s Dirac operator, the HPE can reduce the statistical error by about 60%.

Entanglement entropy for 2D gauge theories with matters

We investigate the entanglement entropy in 1+1-dimensional $SU(N)$ gauge theories with various matter fields using the lattice regularization. Here we use extended Hilbert space definition for entanglement entropy, which contains three contributions; (1) classical Shannon entropy associated with superselection sector distribution, where sectors are labelled by irreducible representations of boundary penetrating fluxes, (2) logarithm of the dimensions of their representations, which is associated with "color entanglement", and (3) EPR Bell pairs, which give "genuine" entanglement. We explicitly show that entanglement entropies (1) and (2) above indeed appear for various multiple "meson" states in gauge theories with matter fields. Furthermore, we employ transfer matrix formalism for gauge theory with fundamental matter field and analyze its ground state using hopping parameter expansion (HPE), where the hopping parameter $K$ is roughly the inverse square of the mass for the matter. We evaluate the entanglement entropy for the ground state and show that all (1), (2), (3) above appear in the HPE, though the Bell pair part (3) appears in higher order than (1) and (2) do. With these results, we discuss how the ground state entanglement entropy in the continuum limit can be understood from the lattice ground state obtained in the HPE.

New approach to canonical partition functions computation in Nf=2 lattice QCD at finite baryon density

We propose and test a new approach to computation of canonical partition functions in lattice QCD at finite density. We suggest a few steps procedure. We first compute numerically the quark number density for imaginary chemical potential $i\mu_{qI}$. Then we restore the grand canonical partition function for imaginary chemical potential using fitting procedure for the quark number density. Finally we compute the canonical partition functions using high precision numerical Fourier transformation. Additionally we compute the canonical partition functions using known method of the hopping parameter expansion and compare results obtained by two methods in the deconfining as well as in the confining phases. The agreement between two methods indicates the validity of the new method. Our numerical results are obtained in two flavor lattice QCD with clover improved Wilson fermions.

Study of lattice QCD at finite chemical potential using the canonical ensemble approach

New approach to computation of canonical partition functions in $N_f=2$ lattice QCD is presented. We compare results obtained by new method with results obtained by known method of hopping parameter expansion. We observe agreement between two methods indicating validity of the new method. We use results for the number density obtained in the confining and deconfining phases at imaginary chemical potential to determine the phase transition line at real chemical potential.

Novel approach to deriving the canonical generating functional in lattice QCD at a finite chemical potential

A novel approach to the problem of deriving the generating functional for the canonical ensemble in lattice QCD at a nonzero chemical potential is proposed. The derivation proceeds in several steps. First, the baryon density for imaginary values of the chemical potential is obtained. Then, again for imaginary values of the chemical potential, the generating functional of the grand canonical ensemble is derived. In this analysis, a fit of baryon density is employed toward simplifying the procedure of numerical integration. Finally, the generating potential for the canonical ensemble is derived using a high-precision numerical Fourier transform. The generating functional for the canonical ensemble is also derived using the known hopping-parameter expansion, and the results obtained with the two methods are compared for the deconfinement phase in the lattice QCD with two flavors.

Exploring the nature of chiral phase transition in two-flavor QCD using extra heavy quarks

Chiral phase transition of two-flavor QCD at finite quark masses is known to be a crossover except near the chiral limit, but it can turn to a first order transition when adding many extra flavors. This property is used to explore the nature of the phase transition of massless two-flavor QCD using lattice numerical simulations. The extra heavy flavors being incorporated in the form of the hopping parameter expansion through the reweighting, the number of the extra flavors and their masses appear only in a single parameter, defined by $h$. We determine the critical value of $h$, at which the first order and the crossover regions are separated, and examine its dependence on the two-flavor mass. The lattice calculations are carried out at ${N}_{t}=4$, and show that the critical value of $h$ does not depend on the two-flavor mass in the range we have studied ($0.46\ensuremath{\le}{m}_{\ensuremath{\pi}}/{m}_{\ensuremath{\rho}}\ensuremath{\le}0.66$) and appears to remain finite and positive in the chiral limit, suggesting that the phase transition of massless two-flavor QCD is of second order.

Calculation ofK→ππdecay amplitudes with improved Wilson fermion action in lattice QCD

We present our result for the $K\to\pi\pi$ decay amplitudes for both the $\Delta I=1/2$ and $3/2$ processes with the improved Wilson fermion action. Expanding on the earlier works by Bernard {\it et al.} and by Donini {\it et al.}, we show that mixings with four-fermion operators with wrong chirality are absent even for the Wilson fermion action for the parity odd process in both channels due to CPS symmetry. Therefore, after subtraction of an effect from the lower dimensional operator, a calculation of the decay amplitudes is possible without complications from operators with wrong chirality, as for the case with chirally symmetric lattice actions. As a first step to verify the possibility of calculations with the Wilson fermion action, we consider the decay amplitudes at an unphysical quark mass $m_K \sim 2 m_\pi$. Our calculations are carried out with $N_f=2+1$ gauge configurations generated with the Iwasaki gauge action and nonperturbatively $O(a)$-improved Wilson fermion action at $a=0.091\,{\rm fm}$, $m_\pi=280\,{\rm MeV}$, and $m_K=580\,{\rm MeV}$ on a $32^3\times 64$ ($La=2.9\,{\rm fm}$) lattice. For the quark loops in the penguin and disconnected contributions in the $I=0$ channel, the combined hopping parameter expansion and truncated solver method work very well for variance reduction. We obtain, for the first time with a Wilson-type fermion action, that ${\rm Re}A_0 = 60(36) \times10^{ -8}\,{\rm GeV}$ and ${\rm Im}A_0 =-67(56) \times10^{-12}\,{\rm GeV}$ for a matching scale $q^* =1/a$. The dependence on the matching scale $q^*$ for these values is weak.

Progress in complex Langevin simulations of full QCD at non-zero density

Progress in the application of the complex Langevin method to full QCD at non-zero chemical potential is reported. The method evades the sign problem which makes naive simulations at nonzero density impossible. The procedure 'gauge cooling' is used to stabilize the simulations at small enough lattice spacings. The method allows simulations also at high densities, all the way up to saturation. Simulations in a systematic hopping parameter expansion are also performed and good convergence is observed, validating the full as well as the expanded simulations.

Scalar pion form factor in two-flavor lattice QCD

We calculate the scalar form factor of the pion using two dynamical flavors of non-perturbatively $\mathcal{O}(a)$-improved Wilson fermions, including both the connected and the disconnected contribution to the relevant correlation functions. We employ the calculation of all-to-all propagators using stochastic sources and a generalized hopping parameter expansion. From the form factor data at vanishing momentum transfer, $Q^2=0$, and two non-vanishing $Q^2$ we obtain an estimate for the scalar radius $\left<r^2\right>^\pi_{_{\rm S}}$ of the pion at one value of the lattice spacing and for five different pion masses. Using Chiral Perturbation Theory at next-to-leading order, we find $\left<r^2\right>^\pi_{_{\rm S}}=0.635\pm0.016$ fm$^2$ at the physical pion mass (statistical error only). This is in good agreement with the phenomenological estimate from $\pi\pi$-scattering. The inclusion of the disconnected contribution is essential for achieving this level of agreement.

The density in the density of states method

It has been suggested that for QCD at finite baryon density the distribution of the phase angle, i.e. the angle defined as the imaginary part of the logarithm of the fermion determinant, has a simple Gaussian form. This distribution provides the density in the density of states approach to the sign problem. We calculate this phase angle distribution using i) the hadron resonance gas model; and ii) a combined strong coupling and hopping parameter expansion in lattice gauge theory. While the former model leads only to a Gaussian distribution, in the latter expansion we discover terms which cause the phase angle distribution to deviate, by relative amounts proportional to powers of the inverse lattice volume, from a simple Gaussian form. We show that despite the tiny inverse-volume deviation of the phase angle distribution from a simple Gaussian form, such non-Gaussian terms can have a substantial impact on observables computed in the density of states/reweighting approach to the sign problem.

Overview of large N QCD with chemical potential at weak and strong coupling

In this note we summarize the results from a longer article on obtaining the QCD phase diagram as a function of the temperature and chemical potential at large Nc and large Nf in the weak coupling limit λ → 0, and the strong coupling limit λ → ∞. The weak coupling phase diagram is obtained from the Polyakov line order parameter, and the quark number, calculated using 1-loop perturbation theory for QCD formulated on S1 × S3. The strong coupling phase diagram is obtained from the same observables calculated at leading order in the lattice strong coupling and hopping parameter expansions. We show that the matrix models in these two limits agree at temperatures and chemical potentials which are not too high, such that observables in the strongly-coupled theory can be obtained from the observables in the weakly-coupled theory, and vice versa, using a simple transformation of variables.

Phase transitions in heavy-quark QCD from an effective theory

With combined hopping parameter and strong coupling expansions, we calculate a dimensionally reduced Polyakov-loop effective theory valid for heavy quarks at nonzero temperature and arbitrary chemical potential. We numerically compute the critical endpoint of the deconfinement transition as a function of quark masses and number of flavours. We also investigate the applicability of the model to the low-T and high density region, specifically in terms of baryon condensation phenomena.

Strong-coupling analysis of parity phase structure in staggered-Wilson fermions

We study strong-coupling lattice QCD with staggered-Wilson fermions, with an emphasis on discrete symmetries and the possibility of their spontaneous breaking. We perform hopping parameter expansion and effective potential analyses in the strong-coupling limit. From gap equations we find the nonzero pion condensate in some range of a mass parameter, which indicates the existence of the parity-broken phase in lattice QCD with staggered-Wilson fermions. We also find massless pions and PCAC relations around the second-order phase boundary. These results suggest that we can take a chiral limit by tuning a mass parameter in lattice QCD with staggered-Wilson fermions as we would with the Wilson fermion.

Phase of quark determinant in lattice QCD with finite chemical potential

We investigate the phase of the quark determinant with finite chemical potential in lattice QCD using both analytic and numerical methods. Applying the winding expansion and the hopping parameter expansion to the logarithm of the determinant, we show that the absolute value of the phase has an upper bound that grows with the spatial volume but decreases exponentially with an increase in the temporal extent of the lattice. This analytic but approximate result is confirmed with a numerical study in four-flavor QCD in which the phase is calculated exactly. Since the phase is well controlled on lattices with larger time extents, we try the phase reweighting method in a region beyond $\ensuremath{\mu}/T=1$ where the Taylor expansion method cannot be applied. Working in four-flavor QCD, we find a first-order like behavior on a ${6}^{3}\ifmmode\times\else\texttimes\fi{}4$ lattice at $\ensuremath{\mu}/T\ensuremath{\approx}0.8$ which was previously observed by the Kentucky group with the canonical method. We also show that the winding expansion has a nice convergence property beyond $\ensuremath{\mu}/T=1$. We expect that this expansion is useful to study the high density region of the QCD phase diagram at low temperatures.

The QCD deconfinement transition for heavy quarks and all baryon chemical potentials

Using combined strong coupling and hopping parameter expansions, we derive an effective three-dimensional theory from thermal lattice QCD with heavy Wilson quarks. The theory depends on traced Polyakov loops only and correctly reflects the centre symmetry of the pure gauge sector as well as its breaking by finite mass quarks. It is valid up to certain orders in the lattice gauge coupling and hopping parameter, which can be systematically improved. To its current order it is controlled for lattices up to N_\tau\sim 6 at finite temperature. For nonzero quark chemical potentials, the effective theory has a fermionic sign problem which is mild enough to carry out simulations up to large chemical potentials. Moreover, by going to a flux representation of the partition function, the sign problem can be solved. As an application, we determine the deconfinement transition and its critical end point as a function of quark mass and all chemical potentials.