Duality between coherent quantum phase slip and Josephson junction in a nanosheet determined by the dual Hamiltonian method

In recent decades, two types of phenomena, which are considered to be dual states in superconductivity, have attracted attention. One of them is a phenomenon called coherent quantum phase slip, which is mainly known as a dual phenomenon of a Josephson junction ($J\!J$) in a one-dimensional superconducting system. The other is a phenomenon called superinsulator, which is mainly known as a dual phenomenon to superconductors (SC). A theoretical study of coherent quantum phase slip was submitted by Mooij et al.Mooij et al. (2005)-Mooij (2015) As flux quanta through nano superconducting wires, and an experimental demonstration of coherent quantum phase slip was performed by Astafiev et al.Astafiev (2012) which embedded in a large superconducting loop with InOx Achieved in wire. The concept of superinsulator was first conceived in 1978 by ’t HooftHooft (1978) as a theory explaining quark confinement. The superinsulator in the case of condensed matter was rediscovered by Diamantini et al.Diamantini et al. (1996)in 1996 as a periodic mixed Chern-Simons Abelian gauge theory describing charge-vortex coupling equivalent to planar Josephson junction arrays in a self-dual approximation. Then, in 1998, Krämer and Doniach Doniach et al. (1998) also rediscovered superinsulators using a phenomenological model in which vortices had a finite mass and traveled in a dissipative environment. Furthermore, in 2012, Yoneda et al. Yoneda et al. (2012) rediscovered superinsulators from a superinsulator / superconductor / superinsulator junction that is as the dual junction to Josephson junction. In previous studies, TiNVinokur et al. (2008), InOOvadia et al. (2015) and NbTiNMironov et al. (2018) films have been experimentally observed to be superinsulating materials. Recently, superinsulators have attracted attention as a powerful desktop environment for QCD phenomenaDiamantini et al. (2018, 2019) in order to realize a single-color version of quantum chromodynamics and elucidate quark confinement and asymptotic freedom. In a previous paperYoneda et al (2019), we introduced two Hamiltonians dual to each other for a one-dimensional nanowire-based Josephson junction and a quantum phase slip junction ($Q\!P\!S\!J$), and applied dual conditions between the current and voltage of the electric circuit. A general theory to construct a dual system, called the dual Hamiltonian method, was proposed. Furthermore, a general discussion of the superconductor-superinsulator transition in a $1+1$ dimensional ($1+1d$) system at zero temperature from these two Hamiltonians was conducted. Our results show that in the $1+1d$ system at zero temperature, coherent quantum phase slip and superinsulator are completely equivalent phenomena. The main purpose of this work was to extend the theory of $1+1d$ systems on nanowires shown in the previous paper to the theory of $2+1d$ systems on nanosheets. The rest of this paper is organized as follows: In Section 2, we use the dual Hamiltonian method for between the JJ model and the $Q\!P\!S\!J$ model on a nanosheet at zero temperature to derive a between the phase and the amplitude dual relations, and a dual relations between the various constants. In Section 3, based on the nonlinear Legendre transformation between the Lagrangians and the Hamiltonians with canonical conjugate variables of infinite order in a compact lattice space, and between the phase and the amplitude relationship derived in the previous section, we show that there is exact duality between the $J\!J$ model and the $Q\!P\!S\!J$ model. Additionally, the phase diagram between the $J\!J$ state (superconducting state) and the $Q\!P\!S$ state (superinsulating state) was discussed. In Section 4, we prove the validity of the results of the previous section by deriving a dual transformation from the anisotropic $X\!Y(A\!X\!Y)$ model to the gauged dual anisotropic $X\!Y(D\!A\!X\!Y)$ model by the Villain approximation in the $2+1d$ system. In Section 5, contrary to Section 4, we derive a dual transformation from the $D\!A\!X\!Y$ model to the $A\!X\!Y$ model with gauge by the Villain approximation in a $2+1d$ system. In Section 6, the dual Ginzburg-Landau ($D\!G\!L$) theory was derived from the mean field approximation of the gauged $Q\!P\!S\!J$ model, and the possibility of confinement of electric flux in the superinsulator was shown. In Section 7, we calculated the critical value of the $Q\!P\!S$ amplitude by the effective energy approach. In the summary and discussion in Section 8, we summarize the conclusions of this paper, discuss whether the minimum unit of charge confinement in a superinsulator is $2e$ or $e$, and, finally, we describe the possibility of dual BCS theory. In Appendix A, the numerical evaluation of the anisotropic massless lattice Green’s function is shown at the origin of $x\!=\!0$, which is necessary for loop correction. In Appendix B, the effective energy approach for the $Q\!P\!S\!J$ model is explained.

Consider a checkered nanosheet consisting of superconductors and superinsulators as shown in Figure 1. In such a $2d$ checkered nanosheet, when the superconductor region acts strongly, the $J\!J$ state becomes strong, and the Hamiltonian in this case is as follows:

$\displaystyle\scalebox{0.85}{$\displaystyle{H_{\!J\!J}}={E_{c}}\!\sum\limits_{\mathbf{x}=1}^{M}{\left[N_{\theta}\left(x\right)\right]}^{2}+{E\!_{J}}\!\sum\limits_{\mathbf{x}=1}^{M}\sum\limits_{j=1}^{2}{\left[1-\cos{{\nabla\!}_{j}}\theta\left(x\right)\right]}$},$

(1)

where, ${N_{\theta}}\!\left(x\right)\!\equiv\!{N_{\theta}}\!\left(\mathbf{x},\tau\right)$ means the number of the particles in the Cooper pair that is the canonical conjugate to the phase $\theta$ of the Cooper pair, and the space differenceYoneda et al (2019); Diamantini et al (2017) of the phase $\theta\left(x\right)\!\equiv\!\theta\left(\mathbf{x},\tau\right)$ is defined by ${{\nabla}_{j}}\theta\!\left(\mathbf{x},\tau\right)\!\equiv\!\theta\!\left(\mathbf{x},\tau\right)\!-\!\theta\left(\mathbf{x}\!-\!a\mathbf{e},\tau\right)$ ; $\tau\!\equiv\hbar\beta$ ( $\beta\!\equiv\!1/{{k_{B}}T}\;$ is the reverse temperature), $\mathbf{x}\!=\!\left(x_{1},x_{2}\right)$ and $M\!\equiv\!{l^{2}}/{a^{2}}$ ($l$ and $a$are the space length and lattice spacing, respectively)are the imaginary time, the lattice coordinate points, and the total number of lattices in the two-dimensional lattice space, respectively; and ${E_{c}}\!\equiv\!{{\left(2e\right)}^{2}}/{2C}$ and ${E\!_{J}}\!\equiv\!{{\Phi_{0}}{I_{c}}}/{2\pi}$ are the charging energy per Cooper pair and the Josephson energy, respectively, where $C$,${I_{c}}$ and ${\Phi_{0}}\!=\!h/2e$ are the capacitance, the critical current and the magnetic flux-quantum, respectively. From the Hamiltonian of Eq. (2), Josephson’s equations are as follows:

	$\displaystyle\scalebox{0.9}{$\displaystyle V\left(t\right)=i\frac{\hbar}{2e}\frac{\partial\theta\left(x\right)}{\partial\tau}=\frac{2{N_{\theta}}\left(x\right)}{2e}{E_{c}}$},$
			(2)

where, $V\left(x\right)$ and ${I_{j}}\left(x\right)$ ($j\!=\!1,2$) are the voltage and the $j$ components of the current in the $J\!J$, respectively. On the other hand, when the superinsulator region acts strongly, the $Q\!P\!S\!J$ state becomes strong, and the Hamiltonian in this case is as follows:

$\displaystyle\scalebox{0.85}{$\displaystyle{H_{Q\!P\!S}}={E_{L}}\sum\limits_{\mathbf{x}=1}^{M}{{{\left[{\tilde{N}_{\tilde{\theta}}}\left(x\right)\right]}^{2}}}+{E_{S}}\sum\limits_{j=1}^{2}{\sum\limits_{\mathbf{x}=1}^{M}{\left[1-\cos{{\nabla}_{j}}\tilde{\theta}\left(x\right)\right]}}$},$

(3)

where, ${{\tilde{N}}_{\tilde{\theta}}}\left(x\right)\equiv{\tilde{N}_{\tilde{\theta}}}\left(\mathbf{x},\tau\right)$ means the magnetic flux quantum numbers $\tilde{\theta}\left(x\right)\equiv\tilde{\theta}\left(\mathbf{x},\tau\right)$ of the magnetic flux quantum field. $\,{E_{L}}\equiv{{{\Phi}_{0}}^{2}}/{2L}\;$ and $\,{E_{S}}\equiv{2e{V_{c}}}/{2\pi}\;$ are the inductive energy per magnetic flux quantum and the $Q\!P\!S$ amplitude, respectively, where $\,{V_{c}}$ is the critical voltage. From the Hamiltonian of Eq.(3), the dual Josephson equations are as follows:

	$\displaystyle\scalebox{0.9}{$\displaystyle\tilde{V}\left(x\right)=i\frac{\hbar}{{\Phi}_{0}}\frac{\partial\tilde{\theta}\left(x\right)}{\partial\tau}=\frac{2{\tilde{N}_{\tilde{\theta}}}\left(x\right)}{\Phi_{0}}{E_{L}}$},$
			(4)

where, $\tilde{V}\left(x\right)$ and ${\tilde{I}_{j}}\left(x\right)$ ($j\!=\!1,2$) are the dual voltage and the j components of dual current in the $Q\!P\!S\!J$, respectively. As the first step of the dual Hamiltonian method, two dual conditionsYoneda et al. (2012); Yoneda et al (2019) between Eq.(2) and the dual equations of Eq.(2) are assumed as follows:

$\displaystyle\scalebox{0.9}{$\displaystyle V\left(x\right)\equiv\tilde{I}\left(x\right),\text{ }I\left(x\right)\equiv\tilde{V}\left(x\right),$}.$

(5)

where, $I\!\equiv\!\sqrt{{I_{1}}\!^{2}+{I_{2}}\!^{2}}$ and $\tilde{I}\!\equiv\!\sqrt{{\tilde{I}_{1}}^{2}+{\tilde{I}_{2}}^{2}}$ are the intensity of the current in the $J\!J$ and the intensity of the dual current in the $Q\!P\!S\!J$, respectively. When the condition of Eq.(5) is imposed between Eqs.(2) and (2), the following two relational expressions between the phase and the number of particles between dual systems are derived as shown below. One of them is the relationship between the phase $\theta$ of the Cooper pair and the magnetic flux quantum numbers ${\tilde{N}_{\tilde{\theta}}}$, and the other is the relationship between the phase $\tilde{\theta}$ of the magnetic flux quantum and the number ${N_{\theta}}$ of the Cooper pair, as follows:

	$\displaystyle\scalebox{0.85}{$\displaystyle{{N}_{\theta}}\left(x\right)=\frac{1}{2\pi}\sqrt{\sum\limits_{j=1}^{2}{{{\sin}^{2}}{\nabla_{j}}\tilde{\theta}\left(x\right)}}$},$
	$\displaystyle\scalebox{0.85}{$\displaystyle{{\tilde{N}}_{\tilde{\theta}}}\left(x\right)=\frac{1}{2\pi}\sqrt{\sum\limits_{j=1}^{2}{{{\sin}^{2}}{\nabla_{j}}\theta\left(x\right)}}$},$		(6)

In Eq.(2), the leftmost equation shows that the Cooper pair number is proportional to the strength of the flux quantum current, and the next equation Is shown that the flux quantum number is proportional to the strength of the Cooper pair current. Since the Cooper pair current and the flux quantum current form a closed loop, the number of Cooper pairs and the flux quantum number can be considered as the winding number of the current loop formed from each dual current. If the relationships described in Eq.(2) are satisfied, the relationship between the $Q\!P\!S$ amplitude and the charging energy per single charge, and the relationship between the Josephson energy and the inductive energy per magnetic flux-quantum, are as follows:

$\displaystyle\scalebox{0.95}{$\displaystyle{E_{S}}=\frac{E_{c}}{2{\pi}^{2}},\;\;\;{E_{J}}=\frac{E_{L}}{2{\pi}^{2}}$}.$

(7)

Furthermore, the inductance and capacitance are related to the critical current and the critical voltage, respectively, as follows:

$\displaystyle\scalebox{0.95}{$\displaystyle L=\frac{\Phi_{0}}{2\pi{I_{c}}},\;\;\;C=\frac{2e}{2\pi{V_{c}}}$},$

(8)

The Lagrangians at zero temperature in Eqs.(1) and Eq.(2) are as follows:

(9)

$\displaystyle\scalebox{0.8}{$\displaystyle{L\!_{Q\!P\!S}}\!=\!-\!\sum\limits_{\mathbf{x}=1}^{M}\!{\left\{\frac{E\!_{S}^{0}}{2}{{\left[{{\nabla}_{\tau}}\tilde{\theta}\left(x\right)\right]}^{2}}\!\!+\!{E\!_{S}}\!\sum\limits_{j=1}^{2}{\left[1-\cos{{\nabla}_{j}}\tilde{\theta}\left(x\right)\right]}\right\}}$},$

(10)

where $E_{J}^{0}$ and $E_{S}^{0}$ can be considered as the imaginary time components of the $J\!J$ and the $Q\!P\!S\!J$, respectively, and are defined as follows:

(11)

where ${a_{0}}\equiv{\tau}_{\max}/M_{\tau}$ is an imaginary time spacing in the time dimension; and ${\tau}_{\max}$ and $M_{\tau}$ are an imaginary time length and an imaginary time division number, respectively. Eqs.(7) and (11) can be summarized as a relationship between the Josephson energy and the $Q\!P\!S\!J$ energy as follows:

(12)

where ${{E^{\prime}}\!_{J}}\equiv{{E_{J}}{a_{0}}}/{\hbar}$, ${E^{\prime}}\!_{J}^{0}\equiv{E_{J}^{0}{a_{0}}}/{\hbar}$, ${{E^{\prime}}\!_{S}}\equiv{{E_{S}}{a_{0}}}/{\hbar}$ and ${E^{\prime}}\!_{S}^{0}\equiv{E_{S}^{0}{a_{0}}}/{\hbar}$ represent the nondimension energies, respectively.The first terms of Eq.(9) and (10) are expressed in a quadratic form for the imaginary time difference of each phase, however, the rightmost terms in Eqs.(9) and (10) are expressed in a cosine form for the spatial difference of each phase. Here, the cosine form also introduces approximately to the leftmost terms of Eqs.(9) and (10) in consideration of the periodicity in the lattice space as follows:

$\displaystyle\scalebox{0.75}{$\displaystyle{L\!_{A\!X\!Y}}\!=\!-\!\sum\limits_{\mathbf{x}=1}^{M}{\left\{\!E_{J}^{0}\Bigl{[}1-\cos{{\nabla}_{\tau}}\theta\left(x\right)\Bigr{]}\!+\!{E_{J}}\sum\limits_{j=1}^{2}{\Bigl{[}1-\cos{{\nabla}_{j}}\theta\left(x\right)\Bigr{]}}\right\}}$},$

(13)

$\displaystyle\scalebox{0.75}{$\displaystyle{L_{D\!A\!X\!Y}}\!=\!-\!\sum\limits_{\mathbf{x}=1}^{M}{\left\{\!E_{S}^{0}\left[1-\cos{{\nabla}_{\tau}}\tilde{\theta}\left(x\right)\right]\!+\!{E_{S}}\sum\limits_{j=1}^{2}{\left[1-\cos{{\nabla}_{x}}\tilde{\theta}\left(x\right)\right]}\right\}}$}.$

(14)

where, ${L_{A\!X\!Y}}$ and ${L_{D\!A\!X\!Y}}$ are equivalent to ${L_{J\!J}}$ and ${L_{Q\!P\!S}}$, respectively, and the $A\!X\!Y$ and $D\!A\!X\!Y$ model, respectively. From the Lagrangian in Eq.(13) the partition function of the $J\!J$ state (superconducting state) at zero temperature is as follows:

	$\displaystyle\scalebox{0.8}{$\displaystyle Z\!_{A\!X\!Y}\equiv\exp\left\{-\Bigl{(}{E^{\prime}}\!_{J}^{0}+{E^{\prime}}\!_{J}\Bigr{)}\!M{M_{\tau}}\right\}{{Z^{\prime}}\!_{A\!X\!Y}}$},\quad\quad\;\;\;$
	$\displaystyle\scalebox{0.8}{$\displaystyle{Z^{\prime}}\!_{A\!X\!Y}\!\!\equiv\!\!\int\!\!{D\theta}\exp\!\sum\limits_{x,\tau}\!{\left[{E^{\prime}}_{J}^{0}\cos{\nabla\!_{\tau}}\theta\left(x\right)+{E^{\prime}}\!_{J}\sum\limits_{j=1}^{2}{\cos{\nabla\!_{j}}\theta\left(x\right)}\right]}$}.$		(15)

where $\sum\limits_{x}\!\!\!\equiv\!\!\sum\limits_{\tau\!=\!1}^{M_{\tau}}\!{\sum\limits_{\mathbf{x}\!=\!1}^{M}}$ and $\int\!\!{D\theta\!}\!\!\equiv\!\!\prod\limits_{\tau\!=\!1}^{M_{\tau}}{\!\prod\limits_{\mathbf{x}\!=\!1}^{M}{\!\!\int\limits_{-\pi}^{\pi}\!\!{\frac{d\theta\left(\mathbf{x},\tau\right)}{2\pi}}}}$ are sums and path integrals, respectively, in $2+1d$ lattice space $x\!=\!(x,\tau)$. Similarly, from the Lagrangian in Eq.(14), the partition function of the $Q\!P\!S\!J$ state (superinsulating state) at zero temperature is as follows:

	$\displaystyle\scalebox{0.8}{$\displaystyle{Z_{D\!A\!X\!Y}}\equiv\exp\left\{-\!\left({E^{\prime}}_{S}^{0}+{E^{\prime}}_{S}\right)\!{M_{\tau}}{M_{x}}\right\}{{Z^{\prime}}\!_{D\!A\!X\!Y}}$},\quad\quad$
	$\displaystyle\scalebox{0.8}{$\displaystyle{{Z^{\prime}}\!_{D\!A\!X\!Y}}\!\!\equiv\!\!\int\!\!{D\tilde{\theta}}\exp\!\sum\limits_{x}\!{\left[{E^{\prime}}_{S}^{0}\cos{\nabla\!_{\tau}}\tilde{\theta}\left(x\right)+{E^{\prime}}\!_{S}\!\sum\limits_{j=1}^{2}{\cos{\nabla\!_{j}}\tilde{\theta}\left(x\right)}\right]}$}.$		(16)

The partition functions $Z_{A\!X\!Y}$ and $Z_{D\!A\!X\!Y}$ in Eqs.(2) and (2) are the starting points for the analysis in the following sections.

This section describes the nonlinear Legendre transformation between the Hamiltonian and the Lagrangian with canonical conjugate variables of the infinite order in a compact $2+1d$ lattice space on a nanosheet. Herein, we show that there is exact duality between the $J\!J$ model and the $Q\!P\!S\!J$ model. For the partition functions ${Z\!_{J\!J}}$ in the Lagrangian of Eq.(9), the auxiliary field $N\left(x\right)$ by Hubbard-Stratonovich transformation is introduced as follows:

	$\displaystyle\scalebox{0.9}{$\displaystyle\Bigl{.}-\!{{E^{\prime}}\!_{J}}\!\sum\limits_{j=1}^{2}{\left[1\!-\!\cos{{\nabla}\!_{j}}\theta\left(x\right)\right]}\Bigr{\}}$},\quad$		(17)

It can be seen that the auxiliary field $N\left(x\right)$ is the same as the number ${N_{\theta}}\left(x\right)$ of the Cooper pairs introduced in Eq.(1).The canonical conjugate momentum ${{p}_{\theta}}\left(x\right)$ with respect to $\theta\left(x\right)$ in the Lagrangian of the lattice space of Eq.(9) is defined as follows:.

$\displaystyle\scalebox{0.98}{$\displaystyle i{p_{\theta}}\!\left(x\right)\!\equiv\!i\hbar{N}\!\left(x\right)\!=\!-{a_{0}}E_{J}^{0}{\nabla_{\tau}}\theta\left(x\right)$}.$

(18)

On the other hand, for the partition functions $Z_{A\!X\!Y}$ in Eq.(2), the auxiliary field $N\!\left(x\right)$ by Hubbard–Stratonovich transformation is introduced as follows:

	$\displaystyle\scalebox{0.83}{$\displaystyle\Bigl{.}-{{E^{\prime}}\!_{J}}\sum\limits_{j=1}^{2}{\left[1\!-\!\cos{\nabla\!_{j}}\theta\!\left(x\right)\right]}\Bigr{\}}$},\quad$		(19)

Again, it can be seen that the auxiliary field $N\left(x\right)$ is the same as the number ${N_{\theta}}\left(x\right)$ of the Cooper pairs introduced in Eq.(1). The canonical conjugate momentum ${{p}_{\theta}}\!\left(x\right)$ with respect to $\theta\!\left(x\right)$ in the Lagrangian of the compact lattice space of Eq.(13) is defined as follows:

$\displaystyle\scalebox{0.95}{$\displaystyle i{p_{\theta}}\left(x\right)\equiv i\hbar{N_{\theta}}\left(x\right)=-2{a_{0}}E_{J}^{0}\sin\Bigl{[}{\nabla_{\tau}\theta\left(x\right)}/2\Bigr{]}$},$

(20)

In Eq.(20), if the linear approximation of $\sin\!\left({\nabla\!_{\tau}\theta\!}/2\right)\!\!\simeq\!\!{{\nabla_{\tau}}\!\theta\!}/2$ holds, it matches the canonical conjugate momentum in Eq.(18).Therefore, the contents of the curly bracket of the exponential function between Eqs.(2) and (3) can be considered as a “nonlinear Legendre transformation” introduced by the canonical conjugate momentum of Eq.(20). Similarly, for the partition function $Z\!_{D\!A\!X\!Y}$ in Eq.(2), the auxiliary field $\tilde{N}\left(x\right)$ by Hubbard–Stratonovich transformation is introduced as follows:

	$\displaystyle\scalebox{0.83}{$\displaystyle\Bigl{.}-{{E^{\prime}}\!_{S}}\sum\limits_{j=1}^{2}\!{\left[1\!-\!\cos{\nabla\!_{j}}\tilde{\theta}\!\left(x\right)\right]}\Bigr{\}}$},\quad$		(21)

It can be seen that the auxiliary field $\tilde{N}\left(x\right)$ is the same as the magnetic flux quantum numbers $\tilde{N}_{\tilde{\theta}}\left(x\right)$ in Eq.(3). The canonical conjugate momentum ${\tilde{p}_{\tilde{\theta}}}\left(x\right)$ with respect to $\tilde{\theta}\left(x\right)$ in the Lagrangian of the compact lattice space of Eq.(14) is defined as follows:

$\displaystyle\scalebox{0.95}{$\displaystyle i{\tilde{p}_{\tilde{\theta}}}\left(x\right)\equiv i\hbar{\tilde{N}_{\tilde{\theta}}}\left(x\right)=-2{a_{0}}E_{S}^{0}\sin\!\left[{\nabla_{\tau}\tilde{\theta}\left(x\right)}/2\right]$},$

(22)

Therefore, the contents of the curly brackets of the exponential function between Eqs.(2) and (3) can also be considered as the“nonlinear Legendre transformation” introduced by the canonical conjugate momentum of Eq.(22). Since Eqs.(20) and (22) have a half-angle relationship, a half-angle version of Eq.(2) is introduced as follows:

	$\displaystyle\scalebox{0.82}{$\displaystyle{N_{\theta}}\left(x\right)=\frac{1}{2\pi}\sqrt{\sum\limits_{j=1}^{2}{{\sin}^{2}\left[{{\nabla_{j}}\tilde{\theta}\left(x\right)}/2\right]}}$},$
	$\displaystyle\scalebox{0.82}{$\displaystyle\tilde{N}_{\tilde{\theta}}\left(x\right)=\frac{1}{2\pi}\sqrt{\sum\limits_{j=1}^{2}{{\sin}^{2}\Bigl{[}{{\nabla_{j}}\theta\left(x\right)}/2\Bigr{]}}}$},$		(23)

Eq.(3) is equivalent to Eq.(2) within the range of linear approximation. From Eqs.(20), (3), and (3), the relationship between the phase $\theta\left(x\right)$ of the Cooper pair and the phase $\tilde{\theta}\left(x\right)$ of the magnetic flux quantum field is as follows:

	$\displaystyle\scalebox{0.82}{$\displaystyle N_{\theta}^{2}\!\left(x\right)\!=\!-\frac{4{\left({a_{0}}E_{J}^{0}\right)}^{2}}{\hbar^{2}}{\sin^{2}}\Bigl{[}\nabla_{\tau}\theta\left(x\right)/2\Bigr{]}\!=\!\frac{1}{\pi^{2}}\!\sum\limits_{j=1}^{2}{{\sin^{2}}\left[{{\nabla}_{j}\tilde{\theta}\left(x\right)}/2\right]}$},$
	$\displaystyle\scalebox{0.82}{$\displaystyle\tilde{N}_{\tilde{\theta}}^{2}\!\left(x\right)\!=\!-\frac{4{\left({a_{0}}E_{S}^{0}\right)}^{2}}{\hbar^{2}}{\sin^{2}}\left[{\nabla_{\tau}\tilde{\theta}\left(x\right)}/2\right]\!=\!\frac{1}{\pi^{2}}\!\sum\limits_{j=1}^{2}{{\sin^{2}}\Bigl{[}{\nabla_{j}}\theta\left(x\right)/2\Bigr{]}}$},$		(24)

Using Eqs.(7), (11) and (3), the following relationships are obtained:

$\displaystyle\scalebox{0.75}{$\displaystyle E_{c}\!\sum\limits_{\mathbf{x}}{{\left[N_{\theta}\left(x\right)\right]}^{2}}\!\!=\!\!-E_{J}^{0}\sum\limits_{\mathbf{x}}\!{\Bigl{[}1\!-\!\cos{\nabla_{\tau}}\theta\left(x\right)\Bigr{]}}\!=\!E_{S}\!\sum\limits_{\mathbf{x}}{\sum\limits_{j=1}^{2}{\left[1\!-\!\cos{\nabla_{j}}\tilde{\theta}\left(x\right)\right]}}$},$

(25)

$\displaystyle\scalebox{0.75}{$\displaystyle E_{L}\!\sum\limits_{\mathbf{x}}{{\left[\tilde{N}_{\tilde{\theta}}\left(x\right)\right]}^{2}}\!\!\!\!=\!\!-E_{S}^{0}\sum\limits_{\mathbf{x}}\!{\left[1\!-\!\cos{\nabla_{\tau}}\tilde{\theta}\left(x\right)\right]}\!=\!E_{J}\sum\limits_{\mathbf{x}}{\sum\limits_{j=1}^{2}{\Bigl{[}1\!-\!\cos{{\nabla}_{j}}\theta\left(x\right)\Bigr{]}}}$}.$

(26)

Eq.(25) means that the charging energy by the charge $2e$ in the $J\!J$ is equal to the $Q\!P\!S\!J$ energy, that is, the condensation energy of the magnetic flux. Eq.(26) means that the charging energy by the flux quantum $\Phi_{0}$ in the $Q\!P\!S\!J$ is equal to the $J\!J$ energy, that is, the condensation energy of the Cooper pair. By establishing the relationship between Eqs.(25) and (26), between the Hamiltonian Eqs.(1) and (3), and, between the Lagrangian Eqs.(13) and (14), it can be shown that each of them is self-dual. Thus, from the canonical conjugate momentum of Eqs.(20) and (22), and the relationship between the phase and amplitude of Eq.(2) derived in the previous section, an exact duality between the $J\!J$ and $Q\!P\!S\!J$ models has been proven. From the Josephson’s equations Eq.(2) and the dual Josephson’s equations Eq.(2), the electrical conductance $G$ can be derived as followsYoneda et al. (2012); Yoneda et al (2019):

$\displaystyle\scalebox{0.85}{$\displaystyle G=-{G_{Q}}\frac{d{N_{\theta}}\left(x\right)}{d{\tilde{N}_{\tilde{\theta}}}\left(x\right)}={G_{Q}}\frac{E_{J}}{E_{S}}\frac{{\tilde{N}_{\tilde{\theta}}}\left(x\right)}{N_{\theta}\left(x\right)}$},$

(27)

where ${G_{Q}}\equiv{{\left(2e\right)}^{2}}/{h}$ is the quantum conductance. From Eq.(27), the following elliptical orbit can be drawn:

	$\displaystyle\scalebox{1.0}{$\displaystyle g^{-1}{N_{\theta}}^{2}+g\tilde{N}_{\tilde{\theta}}^{2}=\frac{1}{\eta}$},\;\;\;\;\;\;\;\;\;\;\;$
	$\displaystyle\scalebox{1.0}{$\displaystyle g\equiv{\left(E\!_{J}/E\!_{S}\right)}^{1/2},\;\eta\equiv\sqrt{{E\!_{J}}{E\!_{S}}}/{\Gamma}$},$		(28)

where $\Gamma$ is an arbitrary integral constant with an energy dimension. From Eq.(27), one can draw an elliptical orbit as shown in Figure 2.

Figure 2 (a) shows a $J\!J$ junction state (superconducting state) for $E_{J}\gg E_{S}$, Figure 2 (b) shows a quantum Hall state (Bose semiconductor state) for $E_{J}=E_{S}$, and Figure 2 (c) shows a $Q\!P\!S$ junction state (superinsulating state) for $E_{J}\ll E_{S}$.The results shown in Figure 2 are very similar to those in referencesDiamantini et al. (2018)’Diamantini et al (2017)’Diamantini et al. (2019). Compare the result of Eq.(27) with the result of the quantum Hall effect shown below:.

$\displaystyle\scalebox{0.85}{$\displaystyle G={G_{Q}}\nu$},$

(29)

where $\nu$ is the Landau level occupancy and is defined as:

$\displaystyle\scalebox{0.85}{$\displaystyle\nu\equiv\frac{N_{e}}{N_{\phi}}$},$

(30)

where $N_{e}$ and $N_{\theta}$ are the electron number and the number of magnetic fluxes (vortex number), respectively. In our model, the correspondence is ${N_{e}}\!\to\!N_{\theta}$ and $N_{\phi}\!\to\!\tilde{N}_{\tilde{\theta}}$. When the Hall conductivity of the quantum Hall effect in the bulk is derived using the Kubo formula, the Landau level occupancy $\nu$ can also be expressed as the topological quantum number, also called the Chern number. From Eqs.(3), (27), (29) and (30), it can be seen that the quantum Hall state is obtained if the following relational expression is satisfied:

$\displaystyle\scalebox{0.85}{$\displaystyle E_{J}\Bigl{[}1-\cos\theta\left(x\right)\Bigr{]}=E_{S}\left[1-\cos\tilde{\theta}\left(x\right)\right]$},$

(31)

Eq.(31) means that the energy of the Josephson junction is equal to the energy of the quantum phase slip. In the vicinity of the region where the Landau level occupancy $\nu$ is represented by the Chern number, it is expected that not only the quantum Hall phase but also a topological insulator phase and a topological metal phase exist. If $\eta>1$ and $g\!\equiv\!\left(E_{J}/E_{S}\right)^{1/2}\!\leqq\!1$ , the region means Bose insulatorDiamantini et al. (2019)-Zhang and Schilling (2018), and if $\eta>1$ and $g\equiv\left(E_{J}/E_{S}\right)^{1/2}\geqq 1$, the region means Bose metalDiamantini et al. (2019); Das and Doniach (1999)-Phillips (2019). Figure 3 shows a schematic phase diagram for $g$ versus $\eta$.

In this section, we perform the dual transformation between the $A\!X\!Y$ model and the $D\!A\!X\!Y$ model from the Villain approximation.Kleinert (1989)-Janke and Kleinert (1986).First, we apply the Villain approximation to ${Z^{\prime}}\!_{A\!X\!Y}$ introduced in Eq.(2) as follows:

	$\displaystyle\scalebox{0.88}{$\displaystyle\Bigl{.}\!+\!\frac{-\left({E^{\prime}}_{J}\right)_{v}}{2}\sum\limits_{j=1}^{2}{\Bigl{(}\nabla_{j}\theta-2\pi{n_{j}}\Bigr{)}}^{2}\Bigr{\}}$},$		(32)

where $Z\!_{Q\!V}$ is the Villain approximation of the partition function ${Z^{\prime}}\!_{A\!X\!Y}$ , and $R\!_{Q\!V}\!\!\equiv\!\!{[R_{v}{\left(E_{J}\right)^{2}}R_{v}({E^{\prime}}_{J}^{0})]^{MM_{\tau}}}$ is the Villain’s normalization parameter,, where ${R_{v}}\!\left(E\right)\!\!\equiv\!\!\sqrt{2\pi{(E)_{v}}}{I_{0}}\!\left(E\right)$ , and $(E)_{v}\!\!\equiv\!\!{\left\{-2\ln[I_{0}(E)/I_{1}(E)]\right\}}^{-1}$ . ${I_{0}}\!\left(E\right)$ and ${I_{1}}\!\left(E\right)$ represents the modified Bessel functions of order zero and order one, respectively. The summation symbols $\sum\limits_{\left\{n\right\}}\!\!\equiv\!\!\prod\limits_{x,\tau}\!\prod\limits_{j=0}^{2}\sum\limits_{n_{j}\left(x,\tau\right)=-\infty}^{\infty}$ are used for the integer fields $n_{j}\left(x,\tau\right)$ . The partition function in Eq.(4) is equivalent to the Euclidean version of the quantum vortex dynamics in a film of superfluid helium introduced by KleinertKleinert (1989, 1985). For Eq.(4) the following identities associated with the Jacobi theta function are used:

$\displaystyle\scalebox{0.84}{$\displaystyle\sum\limits_{n=-\infty}^{\infty}\!\!\!\exp\!\left\{\frac{-E}{2}\left(\theta-2\pi n\right)^{2}\right\}\!=\!\!\!\sum\limits_{l=-\infty}^{\infty}\!\!\!{\frac{1}{\sqrt{2\pi E}}\!\exp\!\left(\frac{-b^{2}}{2E}+ib\theta\right)}$},$

(33)

As a result, Eq.(4) can be rewritten as follows:

$\displaystyle\scalebox{0.85}{$\displaystyle Z\!_{QV}\!=\!C\!_{QV}\!\!\sum\limits_{\left\{b\right\}}{\delta_{{\nabla_{j}}b_{j},0}}\!\exp\!\sum\limits_{x,\tau}\!{\left[\frac{-b_{0}^{2}\left(x\right)}{2{{\left({E^{\prime}}\!_{J}^{0}\right)}_{v}}}+\sum\limits_{j=1}^{2}\frac{-b_{j}^{2}\left(x\right)}{2{{\left({E^{\prime}}\!_{J}\right)}_{v}}}\right]}$},$

(34)

where ${C_{QV}}$ is a normalization parameter defined by ${{[{I_{0}}{{\left({E^{\prime}}_{J}\right)}^{2}}{I_{0}}({E^{\prime}}_{J}^{0})]}^{MM_{\tau}}}$ , and ${b_{i}}\left(x\right)$ represents auxiliary magnetic fields with integer values. The integer dual vector potentials ${\tilde{a}_{i}}\!\left(x\right)$ ($i=0,1,2$) is introduced as follows Villain (1975):

$\displaystyle\scalebox{0.98}{$\displaystyle b_{i}\!\left(x\right)={{\varepsilon}_{ijl}}{\nabla_{j}}{\tilde{a}_{l}}\!\left(x\right)={\left(\nabla\times\mathbf{\tilde{a}}\right)_{i}}\!\left(x\right)$},$

(35)

where ${\varepsilon_{ijl}}$ is the Levi–Civita symbol of three dimensions. By using the dual transformations of Eq.(35), the following Eq.(36) can be obtained:

$\displaystyle\scalebox{0.73}{$\displaystyle Z\!_{Q\!V}\!\equiv\!{C\!_{Q\!V}}\!\!\sum\limits_{\left\{{\tilde{a}}\right\}}{\delta_{{\nabla_{j}}{\varepsilon_{ijl}}{\nabla_{j}}{\tilde{a}_{l}},0}}\!\exp\!\sum\limits_{x}\left[\frac{-{{\left(\nabla\times\mathbf{\tilde{a}}\right)}_{0}}^{2}}{2{{\left({E^{\prime}}_{J}^{0}\right)}_{v}}}\!+\!\sum\limits_{j=1}^{2}\frac{-{\left(\nabla\times\mathbf{\tilde{a}}\right)_{j}}^{2}}{2{{\left({{E^{\prime}}_{J}}\right)}_{v}}}\right]$},$

(36)

Substituting Poisson’s formula in Eq.(37) to (36):

$\displaystyle\scalebox{0.8}{$\displaystyle\sum\limits_{\left({\tilde{a}_{j}}\right)\!=\!-\infty}^{\infty}\!\!\!\!\!\!f\left({\tilde{a}_{j}}\right)\!=\!\!\!\!\int\limits_{-\infty}^{\infty}\!\!\!D{{\tilde{\alpha}^{\prime}}_{j}}f\!(\tilde{\alpha^{\prime}}_{j})\!\!\!\!\!\!\!\sum\limits_{\left({\tilde{l}_{j}}\right)=-\infty}^{\infty}\!\!\!\!\!\!{\delta_{{\nabla_{j}}{\tilde{l}_{j}},0}\!\exp\!\sum\limits_{x}\!\!{\left(i2\pi\sum\limits_{j=0}^{2}{{\tilde{l}_{j}}{\tilde{\alpha^{\prime}}_{j}}}\right)}}$},$

(37)

then Eq..(4) is as follows:

	$\displaystyle\scalebox{0.8}{$\displaystyle\left.+\sum\limits_{j=1}^{2}\frac{-\left(\nabla\times\mathbf{\tilde{\alpha^{\prime}}}\right)_{j}^{2}\left(x\right)}{2{{\left({{E^{\prime}}_{J}}\right)}_{v}}}\!+i2\pi\!\sum\limits_{j=0}^{2}{\tilde{l}_{j}\left(x\right)\tilde{\alpha^{\prime}}_{j}\left(x\right)}\right\}$},$		(38)

Poisson’s formula in Eq.(37) converts the integer-valued vector potentials $\tilde{a}_{i}$ to the continuous-valued vector potentials ${\tilde{\alpha}^{\prime}}_{i}$. Following the quantum vortex dynamics Kleinert (1989, 1985), the Euclidean Lagrangian density of ${\tilde{\alpha}^{\prime}}_{i}$ is as follows:

$\displaystyle\scalebox{0.85}{$\displaystyle L_{QV}\!\!\left(x\right)\!=\!\frac{\tilde{\beta}_{0}^{2}\left(x\right)}{2{{\left({E^{\prime}}_{J}^{0}\right)}_{v}}}+\sum\limits_{i=1}^{2}\frac{\tilde{\beta}_{i}^{2}\left(x\right)}{2{{\left({E^{\prime}}_{J}\right)}_{v}}}-i2\pi\sum\limits_{j=0}^{2}\!{{\tilde{l}_{j}}\!\left(x\right){{\tilde{\alpha^{\prime}}}_{j}}\!\left(x\right)}$},$

(39)

where, ${\tilde{\beta}_{i}}$($i=1,2$) and ${\tilde{\beta}_{0}}$ can be considered as a dual electric field and a dual magnetic field in a $2+1d$ dual electromagnetic field, respectively, and are defined as follows:

	$\displaystyle\scalebox{0.9}{$\displaystyle{\tilde{\beta}_{0}}\left(x\right)\equiv{\nabla_{1}}{\tilde{\alpha^{\prime}}_{2}}\left(x\right)-{{\nabla}_{2}}{\tilde{\alpha^{\prime}}_{1}}\left(x\right)$},$
	$\displaystyle\scalebox{0.9}{$\displaystyle{\tilde{\beta}_{1}}\left(x\right)\equiv{\nabla_{2}}{\tilde{\alpha^{\prime}}_{0}}\left(x\right)-{{\nabla}_{0}}{\tilde{\alpha^{\prime}}_{2}}\left(x\right)$},$
	$\displaystyle\scalebox{0.9}{$\displaystyle{\tilde{\beta}_{2}}\left(x\right)\equiv{\nabla_{0}}{\tilde{\alpha^{\prime}}_{1}}\left(x\right)-{{\nabla}_{1}}{\tilde{\alpha^{\prime}}_{0}}\left(x\right)$},$		(40)

If the 1,2 components ${\tilde{e}_{1}}$ and ${\tilde{e}_{2}}$ of the dual electric field are set as ${\tilde{e}_{1}}\equiv{\tilde{\beta}_{2}}$, and ${\tilde{e}_{2}}\equiv-{\tilde{\beta}_{1}}$, respectively, the dual Maxwell’s equations from Lagrangian of Eq.(39) are as follows:

	$\displaystyle\scalebox{0.98}{$\displaystyle\frac{1}{\left({E^{\prime}}\!_{J}\right)_{v}}\left[{\nabla_{1}}{\tilde{e}_{1}}\left(x\right)+{\nabla_{2}}{\tilde{e}_{2}}\left(x\right)\right]=i2\pi{\tilde{l}_{0}}\left(x\right)$},$
	$\displaystyle\scalebox{0.98}{$\displaystyle\frac{1}{\left({E^{\prime}}\!_{J}^{0}\right)_{v}}{\nabla_{2}}{\tilde{\beta}_{0}}\left(x\right)-\frac{1}{\left({E^{\prime}}\!_{J}\right)_{v}}{\nabla_{0}}{\tilde{e}_{1}}\left(x\right)=i2\pi{\tilde{l}_{1}}\left(x\right)$},$
	$\displaystyle\scalebox{0.98}{$\displaystyle-\frac{1}{\left({E^{\prime}}\!_{J}\right)_{v}}{\nabla_{0}}{\tilde{e}_{2}}\left(x\right)-\frac{1}{\left({E^{\prime}}\!_{J}^{0}\right)_{v}}{\nabla_{1}}{\tilde{\beta}_{0}}\left(x\right)=i2\pi{\tilde{l}_{2}}\left(x\right)$},$		(41)

For Eq.(4), in order to integrate out for continuous-valued vector potentials $\tilde{\alpha^{\prime}}_{i}$, the partition function when the axial gauge-fixing condition $\tilde{\alpha^{\prime}}_{0}=0$ is imposed is as follows:

	$\displaystyle\scalebox{0.82}{$\displaystyle\Bigl{.}\times\sum\limits_{x,x^{\prime}}{\tilde{\alpha^{\prime}}_{i}}^{\bot}\left(x\right)D_{ij}^{\bot}\left(x,{x^{\prime}}\right){\tilde{\alpha^{\prime}}_{j}}^{\bot}\left(x^{\prime}\right)\!+\!i2\pi\!\sum\limits_{j=0}^{2}{{\tilde{l}_{j}}^{\bot}\!\left(x\right){\tilde{\alpha^{\prime}}_{i}}^{\bot}\left(x\right)}\Bigr{]}$},$
	$\displaystyle\scalebox{0.82}{$\displaystyle D_{ij}^{\bot}\left(x,x^{\prime}\right)\equiv-{{\delta}_{ij}}^{\bot}{{g}^{ab}}{{\bar{\nabla}}_{a}}{{\nabla}_{b}}+{{\bar{\nabla}}_{i}}^{\bot}{{\nabla}_{j}}^{\bot}$},$		(42)

where the orthogonal symbol $\bot$ as a superscript indicates that only the components ${{\tilde{\alpha}^{\prime}}_{1}}$ and ${{\tilde{\alpha}^{\prime}}_{2}}$ exist, and the metric $g^{ab}$ is defined as follows:

$\displaystyle\scalebox{0.8}{$\displaystyle g^{ab}\equiv\begin{pmatrix}\gamma&0&0\\[-12.0pt] 0&1&0\\[-12.0pt] 0&0&1\\ \end{pmatrix},\;\;\;\gamma\equiv\frac{\left(E_{J}^{0}\right)_{v}}{\left(E_{J}\right)_{v}}$},$

(43)

where $\gamma$ is an anisotropic parameter in the $J\!J$ model. Integrating over the continuous-valued gauge fields ${{\tilde{\alpha}^{\prime}}_{1}}$ and ${{\tilde{\alpha}^{\prime}}_{2}}$ for Eq.(4) yields following the partition function:

	$\displaystyle\scalebox{0.78}{$\displaystyle\Bigl{.}-2{{\pi}^{2}}{{({E^{\prime}}_{J})}_{v}}{\tilde{l}_{0}}\left(x\right){{\cal{V}}_{0}}\left(x-x^{\prime}\right){\tilde{l}_{0}}\left(x^{\prime}\right)\Bigr{]}$},$		(44)

where ${C^{\prime}_{QV}}\equiv{C_{QV}}{{\left[\det\left(-{{\eta}^{ab}}{\bar{\nabla}_{a}}{\nabla_{b}}\right)\right]}^{\frac{-1}{2}}}{{\left[\det\left(-{\bar{\nabla}_{0}}{\nabla_{0}}\right)\right]}^{\frac{-1}{2}}}$ , and the anisotropic massless lattice potential (or lattice Green’s function)Kleinert (1989) ${{\cal{V}}_{0}}\left(x\right)$ is defined as:

$\displaystyle\scalebox{0.85}{$\displaystyle{{\cal{V}}_{0}}\left(x\right)\equiv\frac{-1}{g^{ab}\bar{\nabla}_{a}\nabla_{b}}\left(x\right)$},\quad$

(45)

Now, from this lattice potential, the ”split lattice potential” ${{\cal{V^{\prime}}}_{0}}\left(x\right)$ obtained by dividing the ”core lattice potential” ${\cal{V}}_{0}\left(0\right)$ and ”split difference operator”${{\nabla^{\prime}}_{i}}$ are introduced as follows:

			(46)

Thus, we have the following identity:

	$\displaystyle\scalebox{0.64}{$\displaystyle\!=\!C^{\prime}\!\!\!\int{D\tilde{\alpha^{\prime}}_{i}}\exp\!\sum\limits_{x}{\left[\frac{-\left({\nabla^{\prime}}\times\mathbf{\tilde{\alpha^{\prime}}}\right)_{0}^{2}}{2\left({E^{\prime}}_{J}^{0}\right)_{v}}+\sum\limits_{j=1}^{2}\frac{-{\left({\nabla^{\prime}}\times\mathbf{\tilde{\alpha^{\prime}}}\right)_{j}^{2}}}{2\left({E^{\prime}}_{J}\right)_{v}}\right]}$},\quad$		(47)

where $C^{\prime}\!\equiv\!{{\left[\det\left(-\tilde{\bar{\nabla^{\prime}}}_{l}\tilde{\nabla^{\prime}}_{l}\right)\right]}^{\frac{1}{2}}}{{\left[\det\left(-\tilde{\bar{\nabla^{\prime}}}_{0}{\tilde{\nabla}^{\prime}}_{0}\right)\right]}^{\frac{1}{2}}}$ . Substituting Eqs.(4) and (4) in Eq.(4) gives:

	$\displaystyle\scalebox{0.7}{$\displaystyle\!\times\!\exp\!\sum\limits_{x}{\left[-2{{\pi}^{2}}{{\cal{V}}_{0}}\left(0\right){{\left({E^{\prime}}_{J}^{0}\right)}_{v}}{{\tilde{l}}_{j}}^{2}-2{{\pi}^{2}}{{\cal{V}}_{0}}\left(0\right){{\Bigl{(}{E^{\prime}}_{J}\Bigr{)}}_{v}}{\tilde{l}_{0}}^{2}+i2\pi\sum\limits_{j=0}^{2}{{{\tilde{l}}_{j}}{{{\tilde{\alpha^{\prime}}}}_{j}}}\right]}$},$		(48)

where ${C^{\prime\prime}_{QV}}\!\equiv\!{C^{\prime}_{QV}}{{\left[\det\left(-{\tilde{\bar{\nabla^{\prime}}}_{l}}{\tilde{\nabla^{\prime}}_{l}}\right)\right]}^{\frac{1}{2}}}{{\left[\det\left(-{\tilde{\bar{\nabla^{\prime}}}_{0}}{\tilde{\nabla^{\prime}}\!_{0}}\right)\right]}^{\frac{1}{2}}}$ , and the metric ${f^{ab}}$ is defined as::

$\displaystyle\scalebox{0.9}{$\displaystyle f^{ab}\equiv\begin{pmatrix}1&0&0\\[-12.0pt] 0&\gamma&0\\[-12.0pt] 0&0&\gamma\end{pmatrix}$}.$

(49)

Furthermore, from Eqs.(2), (4), and (34), the following identity can be obtained:

	$\displaystyle\scalebox{0.62}{$\displaystyle\approx\!\!\!{\int\!\!D\tilde{\theta}}\exp\!\left(-1\right)\!\sum\limits_{x}\!{\left\{\frac{\left[1-\cos\left({\nabla_{0}}\tilde{\theta}-2\pi{\tilde{\alpha^{\prime}}_{0}}\right)\right]\!\!}{4{\pi}^{2}{{\cal{V}}_{0}}\left(0\right){\Bigl{(}{{E^{\prime}}_{J}}\Bigr{)}}_{v}}+\frac{\sum\limits_{j=1}^{2}{\left[1-\cos\left({\nabla}_{j}\tilde{\theta}-2\pi{\tilde{\alpha^{\prime}}_{j}}\right)\right]\!\!}}{4{{\pi}^{2}}{{\cal{V}}_{0}}\left(0\right){{\Bigl{(}{E^{\prime}}_{J}^{0}\Bigr{)}}_{v}}}\right\}}$},$		(50)

As an analogy to the relational expression between the $J\!J$ energy and the $Q\!P\!S\!J$ energy in Eq.(12),the following relational expression is obtained from Eq.(4).

(51)

Eqs.(12) and (51) differ only in the coefficients of the core lattice potential ${V_{0}}\left(0\right)$, therefore, Eq.(51) is a reasonable result. Substituting Eqs.(4) and (51) in Eq.(4) gives:

	$\displaystyle\scalebox{0.66}{$\displaystyle\times\!\sum\limits_{x}{\left\{\!\!{({E^{\prime}}_{S}^{0})}_{v}\!\!\left[1\!-\!\cos\left({\nabla_{0}}\tilde{\theta}-2\pi{\tilde{\alpha^{\prime}}_{0}}\right)\right]\!+\!{{({E^{\prime}}\!_{S})}_{v}}\!\!\sum\limits_{j=1}^{2}{\left[1\!-\!\cos\!\left({\nabla}_{j}\tilde{\theta}-2\pi{{\tilde{\alpha^{\prime}}}_{j}}\right)\right]}\right\}}$},$		(52)

Eq.(4) represents the gauge-coupled $Q\!P\!S\!J$ partition function by the dual gauge field $\tilde{\alpha^{\prime}}_{i}$. For the coefficient ${{\left({E^{\prime}}\!_{J}^{0}\right)}\!_{v}}$ of the dual gauge field energy, use the relationships in Eqs.(51) and (7), and next, when these non-dimensional energy constants have been converted to the original energy dimension, Eq. (and next, when these non-dimensional energy constants have been converted to the original energy dimension, Eq. (52) will be as follows:) will be as follows:

	$\displaystyle\scalebox{0.62}{$\displaystyle\times\!\!\exp\!\!\left(\!\frac{-a_{0}}{\hbar}\!\right)\!\!\sum\limits_{\tau,\bm{x}}{\!\left\{\!(E_{S}^{0})_{v}\!\!\left[\!1\!-\!\cos\!\left({{\nabla}_{0}}\tilde{\theta}\!-\!2\pi{{\tilde{\alpha^{\prime}}}_{0}}\right)\!\right]\!+\!(E_{S})_{v}\!\!\sum\limits_{j=1}^{2}\!{\left[\!1\!-\!\cos\!\left({{\nabla}_{j}}\tilde{\theta}\!-\!2\pi{{\tilde{\alpha^{\prime}}}_{j}}\right)\!\right]}\!\right\}}$},$		(53)

Furthermore, by scaling $2e$ to the non-dimensional dual gauge field $\tilde{\alpha^{\prime}}\!_{i}$, we introduce a new dual gauge field $\tilde{\alpha}_{i}$ as follows:

$\displaystyle\scalebox{0.95}{$\displaystyle 2e{\tilde{\alpha^{\prime}}_{i}}\left(x\right)\equiv{\tilde{\alpha}_{i}}\left(x\right)$},$

(54)

By the transformation of Eq.(54), Eq.(4) is transformed as follows:

	$\displaystyle\scalebox{0.7}{$\displaystyle\times\!\sum\limits_{\tau,\bm{x}}\!\left\{E_{S}^{0}\!\left[1\!-\!\cos\!\left({\nabla}_{0}\tilde{\theta}-q_{m}{\tilde{\alpha}_{0}}\right)\right]\!+\!{E_{S}}\!\sum\limits_{j=1}^{2}{\left[1\!-\!\cos\!\left({\nabla}_{j}\tilde{\theta}-q_{m}{\tilde{\alpha}_{j}}\right)\right]}\!\right\}$},$		(55)

where $\tilde{\mu^{\prime\prime}}\!\equiv\!C/V_{0}\!\left(0\right)$ , and $q_{m}\!\equiv\!2\pi\!/2e\!=\!\Phi_{0}\!/\hbar$ represents a unit magnetic charge. Eq.(4) shows that the pure $A\!X\!Y$ model ( $J\!J$ model without gauge coupling) has been dual transformed to the gauged $D\!A\!X\!Y$ model (gauged $Q\!P\!S\!J$ model) by the Villain approximation in the $2+1d$ system. In other words, the gauge $Q\!P\!S\!J$ model is a “frozen lattice dual superconductor”Kleinert (1989); Herbut (2007)-Neuhaus et al. (2003), for the $J\!J$ model without gauge coupling.