Monte Carlo Simulation of the $SU(2)/\mathbb{Z}_{2}$ Yang–Mills Theory

Recently, consideration on the Yang–Mills partition function with the ’t Hooft flux tHooft:1979rtg has revived Kitano:2017jng ; Tanizaki:2022ngt ; Nguyen:2023fun , largely motivated by the perspective of the generalized symmetries Gaiotto:2014kfa , in particular in connection with the study in Ref. Gaiotto:2017yup . See also recent related studies Yamazaki:2017ulc ; Hayashi:2023wwi ; Hayashi:2024qkm ; Hayashi:2024yjc ; Hayashi:2024psa , in which the ’t Hooft flux plays the crucial role in analyses of the low-energy dynamics of gauge theory.

In the present paper, partially motivated by the above developments and partially motivated by the possibility of a geometrical definition of the fractional topological charge Abe:2023ncy in the $SU(N)$ Yang–Mills theory with the ’t Hooft flux tHooft:1981nnx ; vanBaal:1982ag , we carry out a hybrid Monte Carlo (HMC) lattice simulation of the $SU(2)/\mathbb{Z}_{2}$ Yang–Mills theory, in which the ’t Hooft flux, the 2-form gauge field coupled to the $\mathbb{Z}_{2}$ 1-form symmetry in the modern language, is dynamical. Traditionally, lattice simulations of the $SU(N)/\mathbb{Z}_{N}$ Yang–Mills theory are carried out by adopting the Wilson plaquette action in the adjoint representation, which is blind on the center $\mathbb{Z}_{N}$ Halliday:1981te ; Creutz:1982ga ; in this way, nontrivial $SU(N)/\mathbb{Z}_{N}$ bundle structures are effectively summed over. See also Refs. Edwards:1998dj ; deForcrand:2002vs . In this paper, instead, we treat the $\mathbb{Z}_{N}$ 2-form flat gauge field explicitly as one of the dynamical variables; therefore, each gauge field configuration in the Monte Carlo simulation explicitly possesses the value of the $\mathbb{Z}_{N}$ 2-form gauge field. In this sense, our idea is similar in spirit to the study in Ref. Kovacs:2000sy , in which the $SU(2)$ Yang–Mills partition function with a given fixed ’t Hooft flux is computed, although we make the ’t Hooft flux dynamical; see also Ref. Halliday:1981tm for an earlier Monte Carlo study with an explicit dynamical $\mathbb{Z}_{N}$ 2-form gauge field. Here, we adopt (a variant of) the HMC simulation algorithm Duane:1987de having future applications of our method to lattice quantum chromodynamics (QCD) [$SU(3)$ Yang–Mills theory with fundamental quarks] in mind.

Meanwhile, for many years, it has been known that the conventional HMC algorithm on the periodic lattice suffers from the topological freezing problem, a phenomenon whereby the Monte Carlo update is stuck within a particular topological sector toward the continuum limit DelDebbio:2002xa ; Schaefer:2010hu ; see Ref. Bonanno:2024zyn and references therein for recent analyses. The HMC simulation with open boundary conditions Luscher:2011kk , by activating in/out flows of topological charges from lattice boundaries, appears to remove topological barriers in between otherwise topological sectors and solve this problem. Here, we see an analogue with the HMC simulation in the $SU(N)/\mathbb{Z}_{N}$ gauge theory, in which the topological charge is shuffled by random dynamics of the $\mathbb{Z}_{N}$ 2-form gauge field.

In continuum theory, the topological charge $Q$ in the $SU(N)$ gauge theory on a 4-torus, $T^{4}$, is defined by¹¹1Throughout this paper, we take a convention that the field strength $F_{\mu\nu}(x)$ is a hermitian matrix.

$$Q:=\int_{T^{4}}d^{4}x\,\frac{1}{32\pi^{2}}\varepsilon_{\mu\nu\rho\sigma}\tr\left[F_{\mu\nu}(x)F_{\rho\sigma}(x)\right].$$

(1.1)

In the $SU(N)$ Yang–Mills theory, $Q\in\mathbb{Z}$. However, in the $SU(N)/\mathbb{Z}_{N}$ theory, $Q$ can be fractional as tHooft:1981nnx ; vanBaal:1982ag

$$Q=-\frac{1}{N}\frac{\varepsilon_{\mu\nu\rho\sigma}B_{\mu\nu}B_{\rho\sigma}}{8}+\mathbb{Z},$$

(1.2)

where $B_{\mu\nu}\in\mathbb{Z}$ is the ’t Hooft flux. Since $B_{\mu\nu}$ takes discrete values, we make a random choice of $B_{\mu\nu}$ in the HMC simulation of $SU(N)/\mathbb{Z}_{N}$ theory (see below). Then, this random choice inevitably shuffles the topological charge $Q$ and we may expect that the topological sectors are efficiently sampled. We will observe that this expectation is actually true.

This paper is organized as follows. In Section 2, we briefly summarize basic facts about the $SU(N)/\mathbb{Z}_{N}$ theory, mainly to set up our lattice formulation on the periodic lattice of size $L$, $\Gamma:=(\mathbb{Z}/L\mathbb{Z})^{4}$. Under a certain gauge fixing of the $\mathbb{Z}_{N}$ 1-form gauge symmetry, the 2-form $\mathbb{Z}_{N}$ gauge field takes a particular form (2.10) of the “$B$-field” over which we carry out the “functional integral.” Section 3 is the main part of this paper. Since the $B$-field in Eq. (2.10) takes values in $\mathbb{Z}$ and has no obvious conjugate momentum, we have to appropriately modify the HMC algorithm which is based on the molecular dynamics (MD) of continuous variables. In Section 3.1, we present our “halfway-updating” HMC algorithm which fulfills the detailed balance. The proof of the detailed balance is deferred to Appendix A. In Section 3.2, we study the autocorrelation function of the topological charge in the HMC history; we compare it in the $SU(2)$ theory (i.e., without the $B$-field) with the one in the $SU(2)/\mathbb{Z}_{2}$ theory (with the $B$-field). For the lattice topological charge, we employ the tree-level improved definition in Ref. Alexandrou:2015yba , in which the lattice field smeared by the gradient flow Luscher:2010iy is substituted. We observe a drastic reduction of the autocorrelation in the HMC simulation in the latter theory as anticipated above. We also observe a similar reduction of the autocorrelation in the “energy-operator” $E(t)$ defined by the gradient flow Luscher:2010iy . As shown in Section 2, the difference between $SU(N)/\mathbb{Z}_{N}$ and $SU(N)$ can be understood as the difference in boundary conditions (and the sum over them) and thus local observables are expected to be insensitive to the difference between $SU(N)$ and $SU(N)/\mathbb{Z}_{N}$ in the large volume limit in these gapped theories. To have some idea on this point, we carry out an exploratory study on the finite size effect in the continuum extrapolation of the topological susceptibility. In Section 4, we present a possible method to incorporate fermions in the fundamental representation of $SU(N)$ (quarks) in this framework. This is achieved by gauging the baryon number $U(1)$, $U(1)_{B}$, and embedding $\mathbb{Z}_{N}$ into $SU(N)\times U(1)_{B}$. The $U(1)_{B}$ gauge boson unwanted for QCD is made super-heavy by the Stückelberg mechanism on lattice. Section 5 is devoted to a conclusion. In Appendix B, we present HMC histories and histograms of the topological charge for various lattice parameters.

The $SU(N)/\mathbb{Z}_{N}$ Yang–Mills theory on $T^{4}$ of size $L$ in continuum can be defined as follows. See Refs. Kapustin:2014gua ; Tanizaki:2022ngt . We define boundary conditions of the $SU(N)$ gauge potential 1-form $a$ on $T^{4}$ by

$$a(x+L\hat{\mu})=g_{\mu}(x)^{\dagger}a(x)g_{\mu}(x)-ig_{\mu}(x)^{\dagger}\mathrm{d}g_{\mu}(x),$$

(2.1)

where $x_{\mu}=0$ and $\hat{\mu}$ is the unit vector in the $\mu$ direction; the transition functions $g_{\mu}(x)$ are $SU(N)$-valued. Here, since the gauge potential behaves as the adjoint representation under the gauge transformation, one may relax the cocycle condition for the transition functions at $x_{\mu}=x_{\nu}=0$ by $\mathbb{Z}_{N}$ factors as

$$g_{\mu}(x)g_{\nu}(x+L\hat{\mu})g_{\mu}(x+L\hat{\nu})^{\dagger}g_{\nu}(x)^{\dagger}=e^{2\pi iB_{\mu\nu}(x)/N}\bm{1},$$

(2.2)

where $B_{\mu\nu}(x)\in\mathbb{Z}$ and $B_{\mu\nu}(x)=-B_{\nu\mu}(x)$. For the consistency of transition functions among “quadruple” overlaps, it is required that $B_{\mu\nu}(x)$ is flat modulo $N$, $\mathrm{d}B=0\bmod N$. This defines the $SU(N)/\mathbb{Z}_{N}$ principal bundle and the $SU(N)/\mathbb{Z}_{N}$ Yang–Mills theory over $T^{4}$. It can be shown that we may take constant $B_{\mu\nu}(x)$, $B_{\mu\nu}$. These integers are called the ’t Hooft fluxes.

It is well-understood how to implement this gauge theory on $T^{4}$ as a lattice gauge theory Mack:1978kr ; Ukawa:1979yv ; Seiler:1982pw ; see also Appendix A.4 of Ref. Tanizaki:2022ngt for a nice exposition. One first introduces link variables as $\tilde{U}(x,\mu)\sim\exp(i\int_{x}^{x+\hat{\mu}}a)$. Boundary conditions on $T^{4}$ are then defined by, as a lattice counterpart of Eq. (2.1),²²2Here is a side remark on the Wilson line wrapping around a cycle of $T^{4}$: Under the ordinary 0-form gauge transformation, $$\tilde{U}(x,\nu)\to\Omega(x)^{\dagger}\tilde{U}(x,\nu)\Omega(x+\hat{\nu}),$$ (2.3) where $\Omega(x)$ is not necessarily periodic, the transition functions are transformed as $$g_{\mu}(x)\to\Omega(x)g_{\mu}(x)\Omega(x+L\hat{\mu})^{\dagger}$$ (2.4) and one sees that the cocycle condition (2.2) is invariant under this. The Wilson line of $\tilde{U}(x,\mu)$ thus should not contain the transition functions at the boundary, because under Eq. (2.3), $\displaystyle\dotsb\tilde{U}(x+(L-1)\hat{\mu},\mu)\tilde{U}(x+L\hat{\mu},\mu)\dotsb$ $\displaystyle\to\dotsb\Omega(x+(L-1)\hat{\mu})^{\dagger}\tilde{U}(x+(L-1)\hat{\mu},\mu)\tilde{U}(x+L\hat{\mu},\mu)\Omega(x+(L+1)\hat{\mu})\dotsb.$ (2.5)

$$\tilde{U}(x+L\hat{\mu},\nu)=g_{\mu}(x)^{\dagger}\tilde{U}(x,\nu)g_{\mu}(x+\hat{\nu}),$$

(2.6)

where $x_{\mu}=0$. We first regard link variables with $x_{\rho}=L$ for a certain $\rho$ are all expressed by link variables with $x_{\rho}=0$ by the boundary conditions (2.6). Then, we make the change of link variables from $\tilde{U}\to U$ by

$$\tilde{U}(x,\mu)=\begin{cases}U(x,\mu)g_{\mu}(x)&\text{for $x_{\mu}=L-1$},\\ U(x,\mu)&\text{otherwise}.\\ \end{cases}$$

(2.7)

The new variables $U$ are regarded as obeying periodic boundary conditions, $U(x+L\hat{\mu},\nu)=U(x,\nu)$, where $x_{\mu}=0$. Under this change of variables, one finds that the Boltzmann weight defined by the Wilson plaquette action acquires $\mathbb{Z}_{N}$ factors as (letting $\beta$ the lattice bare gauge coupling),

	$\displaystyle\exp\left\{\beta\sum_{x\in\Gamma}\sum_{\mu<\nu}\frac{1}{N}\real\tr\left[\tilde{P}(x,\mu,\nu)-\bm{1}\right]\right\}$
	$\displaystyle=\exp\left\{\beta\sum_{x\in\Gamma}\sum_{\mu<\nu}\frac{1}{N}\real\tr\left[e^{-2\pi iB_{\mu\nu}(x)/N}P(x,\mu,\nu)-\bm{1}\right]\right\},$		(2.8)

where plaquette variables are

	$\displaystyle\tilde{P}(x,\mu,\nu)$	$\displaystyle:=\tilde{U}(x,\mu)\tilde{U}(x+\hat{\mu},\nu)\tilde{U}(x+\hat{\nu},\mu)^{\dagger}\tilde{U}(x,\nu)^{\dagger},$
	$\displaystyle P(x,\mu,\nu)$	$\displaystyle:=U(x,\mu)U(x+\hat{\mu},\nu)U(x+\hat{\nu},\mu)^{\dagger}U(x,\nu)^{\dagger},$		(2.9)

and the integer field $B_{\mu\nu}(x)$ is given by the ’t Hooft fluxes as

$$B_{\mu\nu}(x)=\begin{cases}B_{\mu\nu}&\text{for $x_{\mu}=L-1$ and~{}$x_{\nu}=L-1$},\\ 0&\text{otherwise}.\\ \end{cases}$$

(2.10)

Note that this field is flat modulo $N$, $\partial_{[\rho}B_{\mu\nu]}(x)=0\bmod N$, where $\partial_{\rho}$ denotes the lattice derivative. The particular form of $B_{\mu\nu}(x)$ in Eq. (2.10) is not unique and can be changed by the $\mathbb{Z}_{N}$ 1-form gauge transformation, $U(x,\mu)\to e^{2\pi iz_{\mu}(x)/N}U(x,\mu)$, where $z_{\mu}(x)\in\mathbb{Z}$. The invariant characterization of $B_{\mu\nu}(x)$ is the total flux, $B_{\mu\nu}=\sum_{s,t=0}^{L-1}B_{\mu\nu}(x+s\hat{\mu}+t\hat{\nu})\bmod N$. In what follows, we call the $\mathbb{Z}_{N}$ 2-form gauge field and/or the ’t Hooft flux simply, “$B$-field” in a moderate abuse of language.

In the next section, we carry out an HMC simulation of the system defined by the second line of Eq. (2.8), employing configurations of the $B$-field of the particular form shown in Eq. (2.10). This implies that we work in a particular gauge of the $\mathbb{Z}_{N}$ 1-form gauge symmetry; the observables thus should be invariant under the $\mathbb{Z}_{N}$ 1-form gauge transformation, $U(x,\mu)\to e^{2\pi iz_{\mu}(x)/N}U(x,\mu)$ and $B_{\mu\nu}(x)\to B_{\mu\nu}(x)+\partial_{\mu}z_{\nu}(x)-\partial_{\nu}z_{\mu}(x)\bmod N$.

One clearly wants to incorporate matter fields in the present framework. The incorporation of the adjoint matter fields such as the gaugino in the $\mathcal{N}=1$ supersymmetric Yang–Mills theory would be straightforward at least in principle because it is blind to the center part of the gauge group $SU(N)$, $\mathbb{Z}_{N}$; thus $SU(N)/\mathbb{Z}_{N}$ can be gauged. A better control on errors in the topological susceptibility could be quite useful for finding a supersymmetric point of the bare lattice gluino mass parameter Curci:1986sm , since a massless fermion implies the vanishing of the topological susceptibility.

It is, however, not obvious whether it is possible to incorporate “quarks,” fermions in the fundamental representation of $SU(N)$ and this point can be an obstacle to extending the present framework to lattice QCD. In what follows, we consider a possible method to avoid this obstruction.

First, when the $B$-field is fixed, i.e., when it is not dynamical, one may incorporate quarks as follows Tanizaki:2022ngt : We first define boundary conditions of the fundamental fermions by

	$\displaystyle\tilde{\psi}(x+L\hat{\mu})$	$\displaystyle=e^{-i\alpha_{\mu}(x)/N}g_{\mu}(x)^{\dagger}\tilde{\psi}(x),$
	$\displaystyle\tilde{\bar{\psi}}(x+L\hat{\mu})$	$\displaystyle=\tilde{\bar{\psi}}(x)g_{\mu}(x)e^{i\alpha_{\mu}(x)/N},$		(4.1)

where $x_{\mu}=0$. Here, transition functions $g_{\mu}(x)$ are identified with those in Eq. (2.1) and $e^{-i\alpha_{\mu}(x)/N}$ is the transition functions of the baryon number $U(1)$ ($U(1)_{B}$) principal bundle; the factor $1/N$ arises since we set the quark $U(1)_{B}$ charge $1/N$. Then, postulating the cocycle condition at $x_{\mu}=x_{\nu}=0$,

$$e^{i\alpha_{\mu}(x)/N}e^{i\alpha_{\nu}(x+L\hat{\mu})/N}e^{-i\alpha_{\mu}(x+L\hat{\nu})/N}e^{-i\alpha_{\nu}(x)/N}=e^{-2\pi iB_{\mu\nu}/N},$$

(4.2)

the $\mathbb{Z}_{N}$ factors in this relation cancel the $\mathbb{Z}_{N}$ factors in Eq. (2.2); the fermion fields can thus be consistently defined on $T^{4}$ although it belongs to the fundamental representation of $SU(N)$ Tanizaki:2022ngt . Corresponding to Eq. (4.1), the $U(1)_{B}$ gauge potential 1-form $A_{B}$ in the charge $1/N$ representation possess the boundary conditions

$$A_{B}(x+L\hat{\mu})=A_{B}(x)+\frac{1}{N}\mathrm{d}\alpha_{\mu}(x),$$

(4.3)

where $x_{\mu}=0$. This $U(1)_{B}$ bundle is characterized by the first Chern numbers,

$$\int_{\text{$\mu\nu$ plane}}\mathrm{d}A_{B}=2\pi\left(\frac{B_{\mu\nu}}{N}+\mathbb{Z}\right),$$

(4.4)

where we have used Eqs. (4.3) and (4.2). Note that we can introduce mass terms for quarks in this setup as far as they preserve the $U(1)_{B}$ symmetry, not necessarily the flavor $SU(N_{f})$ Tanizaki:2022ngt . Thus, this system is quite different from the $\mathbb{Z}_{N}$ QCD Kouno:2012zz ; Iritani:2015ara with the $SU(N)$ flavor symmetry.

In lattice gauge theory, we introduce the $U(1)_{B}$ link variables as $\tilde{U}_{B}(x,\mu)\sim\exp(i\int_{x}^{x+\hat{\mu}}A_{B})$. Corresponding to Eq. (4.3), $\tilde{U}_{B}(x,\mu)$ obey boundary conditions

$$\tilde{U}_{B}(x+L\hat{\mu},\nu)=e^{-i\alpha_{\mu}(x)/N}\tilde{U}_{B}(x,\nu)e^{i\alpha_{\mu}(x+\hat{\nu})/N}.$$

(4.5)

Under the change of link variables analogous to Eq. (2.7),

$$\tilde{U}_{B}(x,\mu)=\begin{cases}U_{B}(x,\mu)e^{i\alpha_{\mu}(x)/N}&\text{for $x_{\mu}=L-1$},\\ U_{B}(x,\mu)&\text{otherwise},\\ \end{cases}$$

(4.6)

we find that

$$\tilde{P}_{B}(x,\mu,\mu)=e^{2\pi iB_{\mu\nu}(x)/N}P_{B}(x,\mu,\nu),$$

(4.7)

where

	$\displaystyle\tilde{P}_{B}(x,\mu,\nu)$	$\displaystyle:=\tilde{U}_{B}(x,\mu)\tilde{U}_{B}(x+\hat{\mu},\nu)\tilde{U}_{B}(x+\hat{\nu},\mu)^{*}\tilde{U}_{B}(x,\nu)^{*},$
	$\displaystyle P_{B}(x,\mu,\nu)$	$\displaystyle:=U_{B}(x,\mu)U_{B}(x+\hat{\mu},\nu)U_{B}(x+\hat{\nu},\mu)^{*}U_{B}(x,\nu)^{*},$		(4.8)

and $B_{\mu\nu}(x)$ is given by Eq. (2.10). The variables $U_{B}$ are regarded as obeying periodic boundary conditions, $U_{B}(x+L\hat{\mu},\nu)=U_{B}(x,\nu)$, where $x_{\mu}=0$. Using the boundary conditions, Eqs. (4.1), (4.5) and (2.3), and then the change of variables, Eqs. (2.7) and (4.6), we find that lattice hopping terms of the fundamental fermion take the following forms,

	$\displaystyle\tilde{\bar{\psi}}(x)\tilde{U}_{B}(x,\mu)\tilde{U}(x,\mu)\tilde{\psi}(x+\hat{\mu})$	$\displaystyle=\bar{\psi}(x)U_{B}(x,\mu)U(x,\mu)\psi(x+\hat{\mu}),$
	$\displaystyle\tilde{\bar{\psi}}(x)\tilde{U}(x-\hat{\mu},\mu)^{\dagger}\tilde{U}_{B}(x-\hat{\mu},\mu)^{*}\tilde{\psi}(x-\hat{\mu})$	$\displaystyle=\bar{\psi}(x)U(x-\hat{\mu},\mu)^{\dagger}U_{B}(x-\hat{\mu},\mu)^{*}\psi(x-\hat{\mu}),$		(4.9)

everywhere on the lattice $\Gamma$ including boundaries. In this expression, fermion variables $\psi(x)$ and $\bar{\psi}(x)$ are understood to obey the periodic boundary conditions, $\psi(x+L\hat{\mu})=\psi(x)$ and $\bar{\psi}(x+L\hat{\mu})=\bar{\psi}(x)$ for $x_{\mu}=0$.

So far, the $U(1)_{B}$ gauge field $\tilde{U}_{B}(x,\mu)$ is regarded as a non-dynamical background. However, since the $B$-field is dynamical in the $SU(N)/\mathbb{Z}_{N}$ theory and $\tilde{U}_{B}(x,\mu)$ depends on the $B$-field through the boundary conditions (4.5), $\tilde{U}_{B}(x,\mu)$ inevitably becomes dynamical in the $SU(N)/\mathbb{Z}_{N}$ theory.

The Wilson plaquette action for the $U(1)_{B}$ gauge field would be

	$\displaystyle\exp\left\{\beta_{B}\sum_{x\in\Gamma}\sum_{\mu<\nu}\real\left[\tilde{P}_{B}(x,\mu,\nu)-1\right]\right\}$
	$\displaystyle=\exp\left\{\beta_{B}\sum_{x\in\Gamma}\sum_{\mu<\nu}\real\left[e^{2\pi iB_{\mu\nu}(x)/N}P_{B}(x,\mu,\nu)-1\right]\right\},$		(4.10)

where $\beta_{B}$ is the bare coupling. However, of course, the dynamical $U(1)_{B}$ gauge field is unwanted for the sake to simulate QCD. For a possible solution on this point, we make the $U(1)_{B}$ gauge boson super-heavy by the Stückelberg mechanism on the lattice. That is, we introduce a $U(1)$-valued dynamical scalar field $\Omega(x)\in U(1)$ and add the lattice action

	$\displaystyle S_{\text{St\"{u}ckelberg}}$
	$\displaystyle:=\frac{1}{2}\sum_{x\in\Gamma}\sum_{\mu}\left[1-\tilde{\Omega}(x+\hat{\mu})^{*}\tilde{U}_{B}(x,\mu)^{N*}\tilde{\Omega}(x)\right]\left[\tilde{\Omega}(x)^{*}\tilde{U}_{B}(x,\mu)^{N}\tilde{\Omega}(x+\hat{\mu})-1\right]$
	$\displaystyle=\frac{1}{2}\sum_{x\in\Gamma}\sum_{\mu}\left[1-\Omega(x+\hat{\mu})^{*}U_{B}(x,\mu)^{N*}\Omega(x)\right]\left[\Omega(x)^{*}U_{B}(x,\mu)^{N}\Omega(x+\hat{\mu})-1\right].$		(4.11)

In this expression, $\tilde{\Omega}(x)$ obeys the twisted boundary conditions, $\tilde{\Omega}(x+L\hat{\mu})=e^{-i\alpha_{\mu}(x)}\tilde{\Omega}(x)$ ($x_{\mu}=0$), while $\Omega(x)$ obeys the periodic boundary conditions, $\Omega(x+L\hat{\mu})=\Omega(x)$. $S_{\text{St\"{u}ckelberg}}$ is invariant under the $U(1)_{B}$ 0-form and $\mathbb{Z}_{N}$ 1-form gauge transformations. The gauge fixing of $U(1)_{B}$ of the form $\Omega(x)=1$ then shows that the $U(1)_{B}$ gauge boson acquires the mass of $O(a^{-2})$ and we expect that this freedom decouples in the continuum limit.

In this paper, we carried out an HMC simulation of the $SU(2)/\mathbb{Z}_{2}$ Yang–Mills theory, in which the $\mathbb{Z}_{N}$ 2-form flat gauge field (the ’t Hooft flux) is treated as a dynamical variable. We observed that our HMC algorithm in the $SU(2)/\mathbb{Z}_{2}$ theory drastically reduces the autocorrelation lengths of the topological charge and of the “energy-operator” defined by the gradient flow. We thus infer that, provided that sufficiently large lattice volumes are available, the HMC simulation of the $SU(N)/\mathbb{Z}_{N}$ theory could be employed as an alternative for the simulation of the $SU(N)$ Yang–Mills theory with a very efficient sampling of topological sectors. Toward applications in lattice QCD, we also presented a possible method to incorporate quarks. Further tests on these ideas are to be awaited.

We would like to thank Yui Hayashi, Yuki Nagai, Yuya Tanizaki, Akio Tomiya, and Hiromasa Watanabe for helpful discussions. We appreciate the opportunity of the discussion during the YITP–RIKEN iTHEMS conference “Generalized symmetries in QFT 2024” (YITP-W-24-15) in execution of this work. Numerical computations in this paper were carried out on Genkai, a supercomputer system of the Research Institute for Information Technology (RIIT), Kyushu University. The work of M.A. was supported by Kyushu University Innovator Fellowship Program in Quantum Science Area. O.M. acknowledges the RIKEN Special Postdoctoral Researcher Program. The work of H.S. was partially supported by Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research, JP23K03418.