A Worm Algorithm for the Lattice CP(N-1) Model

TL;DR

This paper introduces a new worm algorithm for efficiently simulating the 2D lattice CP(N-1) model, a simplified analog of QCD, especially effective at finite density and in a dual flux-variables representation.

Contribution

The paper presents a novel worm algorithm tailored for the lattice CP(N-1) model that operates efficiently at finite density in a dual flux-variables framework.

Findings

Algorithm successfully simulates the model at finite density.

Efficient handling of chemical potential without complications.

Potential for application to related lattice gauge theories.

Abstract

The CP(N-1) model in 2D is an interesting toy model for 4D QCD as it possesses confinement, asymptotic freedom and a non-trivial vacuum structure. Due to the lower dimensionality and the absence of fermions, the computational cost for simulating 2D CP(N-1) on the lattice is much lower than the one for simulating 4D QCD. However to our knowledge, no efficient algorithm for simulating the lattice CP(N-1) model has been tested so far, which also works at finite density. To this end we propose and test a new type of worm algorithm which is appropriate to simulate the lattice CP(N-1) model in a dual, flux-variables based representation, in which the introduction of a chemical potential does not give rise to any complications.

Figures

Equations

$$S_{A}\,=\,-\frac{1}{g}\int\mathrm{d}^{d}x\,\mathopen{}\mathclose{{}\left(\mathrm{D}_{\mu}z}\right)^{\dagger}\cdot\mathopen{}\mathclose{{}\left(\mathrm{D}_{\mu}z}\right)\ ,$$

$$S_{Q}\,=\,-\frac{1}{g}\int\mathrm{d}^{d}x\,\biggl{(}\bigg{.}(\partial_{\mu}z^{\dagger})\cdot(\partial_{\mu}z)\,+\,\frac{1}{4}\bigl{(}\big{.}z^{\dagger}\cdot(\partial_{\mu}z)\,-\,(\partial_{\mu}z^{\dagger})\cdot{z}\big{.}\bigr{)}^{2}\bigg{.}\biggr{)}\ ,$$

$$S_{A}\,=\,-\beta\,\sum\limits_{x,\mu}\mathopen{}\mathclose{{}\left(z^{\dagger}\mathopen{}\mathclose{{}\left(x}\right)U_{\mu}\mathopen{}\mathclose{{}\left(x}\right)z\mathopen{}\mathclose{{}\left(x+\widehat{\mu}}\right)\,+\,z^{\dagger}\mathopen{}\mathclose{{}\left(x}\right)U^{\dagger}_{\mu}\mathopen{}\mathclose{{}\left(x-\widehat{\mu}}\right)z\mathopen{}\mathclose{{}\left(x-\widehat{\mu}}\right)\,-\,2}\right)\ ,$$

$$Z_{A}\,=\,\int\mathcal{D}\bigl{[}z^{\dagger},z,U\bigr{]}\,\exp\biggl{(}\bigg{.}\beta\,\sum\limits_{x,\mu}\mathopen{}\mathclose{{}\left(z^{\dagger}\mathopen{}\mathclose{{}\left(x}\right)U_{\mu}\mathopen{}\mathclose{{}\left(x}\right)z\mathopen{}\mathclose{{}\left(x+\widehat{\mu}}\right)\,+\,z^{\dagger}\mathopen{}\mathclose{{}\left(x}\right)U^{\dagger}_{\mu}\mathopen{}\mathclose{{}\left(x-\widehat{\mu}}\right)z\mathopen{}\mathclose{{}\left(x-\widehat{\mu}}\right)}\right)\bigg{.}\biggr{)}\ ,$$

$$S_{Q}\,=\,-\beta\,\sum\limits_{x,\mu}\big{|}z^{\dagger}\mathopen{}\mathclose{{}\left(x}\right)\cdot z\mathopen{}\mathclose{{}\left(x+\widehat{\mu}}\right)\big{|}^{2}$$

$$,\quad Z_{Q}\,=\,\int\mathcal{D}\bigl{[}z^{\dagger},z\bigr{]}\exp\biggl{(}\bigg{.}\beta\sum\limits_{x,\mu}\big{|}z^{\dagger}\mathopen{}\mathclose{{}\left(x}\right)\cdot z\mathopen{}\mathclose{{}\left(x+\widehat{\mu}}\right)\big{|}^{2}\bigg{.}\biggr{)}\ .$$

$$Z_{Q}\,=\,\int\mathcal{D}\bigl{[}z^{\dagger},z\bigr{]}\,\exp\biggl{(}\bigg{.}\beta\,\sum\limits_{x}\sum\limits_{\mu=1}^{d}\sum\limits_{a,b=1}^{N}\bigl{(}\big{.}\ThisStyle{\hbox to0.0pt{\hskip 4.57985pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\SavedStyle z$}}}_{a}\mathopen{}\mathclose{{}\left(x}\right)z_{b}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}\bigl{(}\big{.}\ThisStyle{\hbox to0.0pt{\hskip 4.57985pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\SavedStyle z$}}}_{b}\mathopen{}\mathclose{{}\left(x+\widehat{\mu}}\right)z_{a}\mathopen{}\mathclose{{}\left(x+\widehat{\mu}}\right)\big{.}\bigr{)}\bigg{.}\biggr{)}\ .$$

$$Z_{Q}\,=\,\int\mathcal{D}\bigl{[}z^{\dagger},z\bigr{]}\,\prod\limits_{x}\prod\limits_{\mu=1}^{d}\prod\limits_{a,b=1}^{N}\,\sum\limits_{n_{x,\mu}^{a\,b}=0}^{\infty}\,\biggl{\{}\bigg{.}\frac{\beta^{n_{x,\mu}^{a\,b}}}{n_{x,\mu}^{a\,b}!}\mathopen{}\mathclose{{}\left(\bigl{(}\big{.}\ThisStyle{\hbox to0.0pt{\hskip 4.57985pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\SavedStyle z$}}}_{a}\mathopen{}\mathclose{{}\left(x}\right)z_{b}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}\bigl{(}\big{.}\ThisStyle{\hbox to0.0pt{\hskip 4.57985pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\SavedStyle z$}}}_{b}\mathopen{}\mathclose{{}\left(x+\widehat{\mu}}\right)z_{a}\mathopen{}\mathclose{{}\left(x+\widehat{\mu}}\right)\big{.}\bigr{)}}\right)^{n_{x,\mu}^{a\,b}}\bigg{.}\biggr{\}}\ .$$

$$Z_{Q}\,=\,\sum\limits_{\mathopen{}\mathclose{{}\left\{}\right.k,l\mathopen{}\mathclose{{}\left.}\right\}}\prod\limits_{x}\biggl{\{}\bigg{.}\biggl{(}\bigg{.}\prod\limits_{\mu=1}^{d}\prod\limits_{a,b=1}^{N}\frac{\beta^{\frac{1}{2}(|k_{x,\mu}^{a\,b}|+k_{x,\mu}^{a\,b})+l_{x,\mu}^{a\,b}}}{\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(|k_{x,\mu}^{a\,b}|+k_{x,\mu}^{a\,b}}\right)+l_{x,\mu}^{a\,b}}\right)!}\bigg{.}\biggr{)}\\ \frac{\prod\limits_{a}^{N}\biggl{(}\bigg{.}\delta\bigl{(}\big{.}\sum\limits_{\mu=1}^{d}\sum\limits_{b=1}^{N}\bigl{(}\big{.}k_{x,\mu}^{a\,b}-k_{x-\widehat{\mu},\mu}^{a\,b}\big{.}\bigr{)}\big{.}\bigr{)}\bigl{(}\big{.}\sum\limits_{\mu=1}^{d}\sum\limits_{b=1}^{N}\bigl{(}\big{.}\frac{1}{2}(|k_{x,\mu}^{a\,b}|+|k_{x-\widehat{\mu},\mu}^{a\,b}|)+l_{x,\mu}^{a\,b}+l_{x-\widehat{\mu},\mu}^{a\,b}\big{.}\bigr{)}\big{.}\bigr{)}!\bigg{.}\biggr{)}}{\bigl{(}\big{.}N-1+\sum\limits_{\mu=1}^{d}\sum\limits_{c,b=1}^{N}\bigl{(}\big{.}\frac{1}{2}(|k_{x,\mu}^{c\,b}|+|k_{x-\widehat{\mu},\mu}^{c\,b}|)+l_{x,\mu}^{c\,b}+l_{x-\widehat{\mu},\mu}^{c\,b}\big{.}\bigr{)}\big{.}\bigr{)}!}\bigg{.}\biggr{\}}\ .$$

$$w_{n_{x,\mu}}\mathopen{}\mathclose{{}\left(\beta}\right)=\frac{\beta^{n_{x,\mu}}\operatorname{e}^{-2\,\beta}}{n_{x,\mu}!}\quad,\quad\text{where}\quad n_{x,\mu}=\sum\limits_{a,b=1}^{N}\bigl{(}\big{.}\tfrac{1}{2}\big{|}k_{x,\mu}^{a\,b}\big{|}+l_{x,\mu}^{a\,b}\big{.}\bigr{)}\ ,$$

$$Z_{A}\,=\,\sum\limits_{\mathopen{}\mathclose{{}\left\{}\right.k,l\mathopen{}\mathclose{{}\left.}\right\}}\prod\limits_{x}\biggl{\{}\bigg{.}\biggl{(}\bigg{.}\prod\limits_{\mu=1}^{d}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\frac{\operatorname{e}^{-2\,\beta}}{\bigl{(}\big{.}\sum\limits_{a,b=1}^{N}\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left|k_{x,\mu}^{a\,b}}\right|+l_{x,\mu}^{a\,b}}\right)\big{.}\bigr{)}!}}\biggl{(}\bigg{.}\prod\limits_{a,b=1}^{N}\frac{\beta^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2}\,(\frac{1}{2}(|k_{x,\mu}^{a\,b}|+k_{x,\mu}^{a\,b})+l_{x,\mu}^{a\,b})}}{\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(|k_{x,\mu}^{a\,b}|+k_{x,\mu}^{a\,b}}\right)+l_{x,\mu}^{a\,b}}\right)!}\bigg{.}\biggr{)}\bigg{.}\biggr{)}\\ \frac{\prod\limits_{a}^{N}\delta\bigl{(}\big{.}\sum\limits_{\mu=1}^{d}\sum\limits_{b=1}^{N}\bigl{(}\big{.}k_{x,\mu}^{a\,b}-k_{x-\widehat{\mu},\mu}^{a\,b}\big{.}\bigr{)}\big{.}\bigr{)}\bigl{(}\big{.}\sum\limits_{\mu=1}^{d}\sum\limits_{b=1}^{N}\bigl{(}\big{.}\frac{1}{2}(|k_{x,\mu}^{a\,b}|+|k_{x-\widehat{\mu},\mu}^{a\,b}|)+l_{x,\mu}^{a\,b}+l_{x-\widehat{\mu},\mu}^{a\,b}\big{.}\bigr{)}\big{.}\bigr{)}!}{\bigl{(}\big{.}N-1+\sum\limits_{\mu=1}^{d}\sum\limits_{c,b=1}^{N}\bigl{(}\big{.}\frac{1}{2}(|k_{x,\mu}^{c\,b}|+|k_{x-\widehat{\mu},\mu}^{c\,b}|)+l_{x,\mu}^{c\,b}+l_{x-\widehat{\mu},\mu}^{c\,b}\big{.}\bigr{)}\big{.}\bigr{)}!}\bigg{.}\biggr{\}}\ ,$$

$$\tilde{Z}_{A}\,=\,\sum\limits_{\mathopen{}\mathclose{{}\left\{}\right.k,\,l\mathopen{}\mathclose{{}\left.}\right\}}\biggl{\{}\bigg{.}\prod\limits_{x}\biggl{(}\bigg{.}\prod\limits_{\nu=1}^{d}\,\operatorname{e}^{-2\,\beta}\delta\bigl{(}\big{.}\sum\limits_{a=1}^{N}k_{x,\nu}^{a}\big{.}\bigr{)}\prod\limits_{a=1}^{N}\,\frac{\beta^{|k_{x,\nu}^{a}|+2\,l_{x,\nu}^{a}}}{(|k_{x,\nu}^{a}|+l_{x,\nu}^{a})!\,l_{x,\nu}^{a}!}\bigg{.}\biggr{)}\\ \cdot\frac{\prod\limits_{a=1}^{N}\,\delta\bigl{(}\big{.}\sum\limits_{\nu=1}^{d}\bigl{(}\big{.}k_{x,\nu}^{a}-k_{x-\widehat{\nu},\nu}^{a}\big{.}\bigr{)}\big{.}\bigr{)}\bigl{(}\big{.}\sum\limits_{\nu=1}^{d}\bigl{(}\big{.}\frac{1}{2}\bigl{(}\big{.}|k_{x,\nu}^{a}|+|k_{x-\widehat{\nu},\nu}^{a}|\big{.}\bigr{)}+l_{x,\nu}^{a}+l_{x-\widehat{\nu},\nu}^{a}\big{.}\bigr{)}\big{.}\bigr{)}!}{\bigl{(}\big{.}N-1+\sum\limits_{a=1}^{N}\sum\limits_{\nu=1}^{d}\bigl{(}\big{.}\frac{1}{2}\bigl{(}\big{.}|k_{x,\nu}^{a}|+|k_{x-\widehat{\nu},\nu}^{a}|\big{.}\bigr{)}+l_{x,\nu}^{a}+l_{x-\widehat{\nu},\nu}^{a}\big{.}\bigr{)}\big{.}\bigr{)}!}\bigg{.}\biggr{\}}\ ,$$

$$Z_{A,2}^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a_{0}\,b_{0}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathopen{}\mathclose{{}\left(x,y}\right)}\,=\,Z_{A}\cdot\mathopen{}\mathclose{{}\left\langle\phi^{a_{0}\,b_{0}}\mathopen{}\mathclose{{}\left(x}\right)\phi^{b_{0}\,a_{0}}\mathopen{}\mathclose{{}\left(y}\right)}\right\rangle_{Z_{A}}\,=\,\sum\limits_{\mathopen{}\mathclose{{}\left\{}\right.k,l\mathopen{}\mathclose{{}\left.}\right\}}\prod\limits_{z}\biggl{\{}\bigg{.}\biggl{(}\bigg{.}\prod\limits_{\mu=1}^{d}\,\frac{\operatorname{e}^{-2\,\beta}}{\bigl{(}\big{.}\sum\limits_{a,b=1}^{N}\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left|k_{x,\mu}^{a\,b}}\right|+l_{x,\mu}^{a\,b}}\right)\big{.}\bigr{)}!}\prod\limits_{a,b=1}^{N}\frac{\beta^{|k_{z,\mu}^{a\,b}|+k_{z,\mu}^{a\,b}+2\,l_{z,\mu}^{a\,b}}}{\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(|k_{z,\mu}^{a\,b}|+k_{z,\mu}^{a\,b}}\right)+l_{z,\mu}^{a\,b}}\right)!}\bigg{.}\biggr{)}\\ \cdot\prod\limits_{a}^{N}\delta\bigl{(}\big{.}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bigl{(}\big{.}\delta^{a,b_{0}}-\delta^{a,a_{0}}\big{.}\bigr{)}\bigl{(}\big{.}\delta_{x,z}-\delta_{y,z}\big{.}\bigr{)}}+\sum\limits_{\mu=1}^{d}\sum\limits_{b=1}^{N}\bigl{(}\big{.}k_{z,\mu}^{a\,b}-k_{z-\widehat{\mu},\mu}^{a\,b}\big{.}\bigr{)}\big{.}\bigr{)}\\ \cdot\frac{\prod\limits_{a}^{N}\bigl{(}\big{.}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\frac{1}{2}\bigl{(}\big{.}\delta^{a,a_{0}}+\delta^{a,b_{0}}\big{.}\bigr{)}\bigl{(}\big{.}\delta_{x,z}+\delta_{y,z}\big{.}\bigr{)}}+\sum\limits_{\mu=1}^{d}\sum\limits_{b=1}^{N}\bigl{(}\big{.}\frac{1}{2}(|k_{z,\mu}^{a\,b}|+|k_{z-\widehat{\mu},\mu}^{a\,b}|)+l_{z,\mu}^{a\,b}+l_{z-\widehat{\mu},\mu}^{a\,b}\big{.}\bigr{)}\big{.}\bigr{)}!}{\bigl{(}\big{.}N-1\,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}+\,\delta_{x,z}+\delta_{y,z}}+\sum\limits_{\mu=1}^{d}\sum\limits_{c,b=1}^{N}\bigl{(}\big{.}\frac{1}{2}(|k_{z,\mu}^{c\,b}|+|k_{z-\widehat{\mu},\mu}^{c\,b}|)+l_{z,\mu}^{c\,b}+l_{z-\widehat{\mu},\mu}^{c\,b}\big{.}\bigr{)}\big{.}\bigr{)}!}\bigg{.}\biggr{\}}\ ,$$

$$\mathopen{}\mathclose{{}\left\langle E}\right\rangle\,=\,-\frac{1}{V}\frac{\partial\log\mathopen{}\mathclose{{}\left(Z}\right)}{\partial\beta}\$$

$$\text{and}\quad\chi_{m}\,=\,\frac{1}{V}\sum\limits_{x,y}\biggl{(}\bigg{.}\sum\limits_{a,b}\mathopen{}\mathclose{{}\left\langle\phi^{a\,b}\mathopen{}\mathclose{{}\left(x}\right)\phi^{b\,a}\mathopen{}\mathclose{{}\left(y}\right)}\right\rangle-\frac{1}{N}\bigg{.}\biggr{)}\ ,$$

$$\xi_{G}\,=\,\frac{1}{2\,\sin\mathopen{}\mathclose{{}\left(\frac{\pi}{L}}\right)}\biggl{(}\bigg{.}\tfrac{\sum\limits_{x,y}\bigl{(}\big{.}\sum\limits_{a,b}\bigl{\langle}\phi^{a\,b}\mathopen{}\mathclose{{}\left(x}\right)\phi^{b\,a}\mathopen{}\mathclose{{}\left(y}\right)\bigr{\rangle}-\frac{1}{N}\big{.}\bigr{)}}{\sum\limits_{\ThisStyle{\hbox to0.0pt{\hskip 1.19617pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\SavedStyle x$}}},\ThisStyle{\hbox to0.0pt{\hskip 1.08273pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\SavedStyle y$}}},t_{x},t_{y}}\operatorname{e}^{\frac{2\,\pi\,\mathrm{i}\,(t_{y}-t_{x})}{L}}\bigl{(}\big{.}\sum\limits_{a,b}\bigl{\langle}\phi^{a\,b}\mathopen{}\mathclose{{}\left(\ThisStyle{\hbox to0.0pt{\hskip 2.67464pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\SavedStyle x$}}},t_{x}}\right)\phi^{b\,a}\mathopen{}\mathclose{{}\left(\ThisStyle{\hbox to0.0pt{\hskip 2.51584pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\SavedStyle y$}}},t_{y}}\right)\bigr{\rangle}-\frac{1}{N}\big{.}\bigr{)}}-1\bigg{.}\biggr{)}^{1/2}\ ,$$

Full text

1 Introduction

The two-dimensional $\operatorname{\mathbb{C}P}^{N-1}$ model [4, 5] has a non-trivial vacuum structure with stable instanton solutions and possesses asymptotic freedom as well as confinement and is therefore an interesting toy-model for four-dimensional \operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right) Yang-Mills theories.

Unfortunately, it turns out that simulations using standard lattice formulation of the model suffer from exponential critical slowing down [8] of topological modes, which could also not be overcome by the development of cluster [9] and loop algorithms [13]222In the loop algorithm [13], the slowing down is caused by a different mechanism (update of long lists) and could possibly be overcome..

A first worm algorithm, based on a new dual formulation of the $\operatorname{\mathbb{C}P}^{N-1}$ model [11], has then been proposed and tested in [12], where it was unfortunately found that the algorithm suffers from an ergodicity problem in relevant parts of parameter space.

In this work, we implement ergodic worm algorithms for both, a more recent (and simpler) dual formulation of the $\operatorname{\mathbb{C}P}^{N-1}$ partition function [15], as well as for the dual formulation from [11], and discuss their performance. This proceeding is based on our full article [1].

1.1 The $\operatorname{\mathbb{C}P}^{N-1}$ model

The $\operatorname{\mathbb{C}P}^{N-1}$ model in $d$ dimensional continuous Euclidean space can be seen as a \operatorname{U}\mathopen{}\mathclose{{}\left(1}\right) gauged non-linear \operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right) sigma model (local \operatorname{U}\mathopen{}\mathclose{{}\left(1}\right) and global \operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right) symmetry)[2, 5, 7]:

[TABLE]

where $z\in\mathbb{C}^{N}$ is an $N$-component complex scalar field subject to the constraint $z^{\dagger}\,z\,=\,1$, $\mathrm{D}_{\mu}\,=\,\partial_{\mu}\,+\,\mathrm{i}\,A_{\mu}$ is the covariant derivative with respect to an auxiliary \operatorname{U}\mathopen{}\mathclose{{}\left(1}\right) gauge field $A_{\mu}$ and $g$ is the corresponding coupling strength. By substituting $A_{\mu}\,=\,\frac{\mathrm{i}}{2}(z^{\dagger}\cdot(\partial_{\mu}z)\,-\,(\partial_{\mu}z^{\dagger})\cdot z)$, which is the solution to the classical equation of motion for $A_{\mu}$, into (1.1), one finds the alternative, quartic action [3, 5, 7],

[TABLE]

Classically, (1.1) and (1.2) are equivalent, but in the quantum theory, quantum fluctuations generate a kinetic term for $A_{\mu}$ in (1.1) and turn it into a dynamic field [5].

1.2 Lattice formulation

Applying the standard discretization strategy to the action $S_{A}$ from (1.1) one finds

[TABLE]

where $\beta$ is the lattice version of the inverse coupling $\frac{1}{g}$, different lattice sites are labeled by $x$, $\widehat{\mu}$ is the vector that points from one lattice site to its nearest neighbor in $\mu$-direction and U_{\mu}\mathopen{}\mathclose{{}\left(x}\right)\in\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right) is the parallel transporter with respect to the gauge field $A_{\mu}$ from site $x$ to site $x+\widehat{\mu}$ along the corresponding link. The partition function for (1.3) then reads

[TABLE]

where \mathcal{D}\bigl{[}z^{\dagger},z,U\bigr{]}=\mathcal{D}\bigl{[}z^{\dagger},z\bigr{]}\mathcal{D}\bigl{[}U\bigr{]}, with \mathcal{D}\bigl{[}z^{\dagger},z\bigr{]}=\prod\limits_{x}\delta\bigl{(}\big{.}\mathopen{}\mathclose{{}\left|z\mathopen{}\mathclose{{}\left(x}\right)}\right|^{2}-1\big{.}\bigr{)}\mathrm{d}^{N}\ThisStyle{\hbox to0.0pt{\hskip 4.57985pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@color\SavedStyle z}}}\mathopen{}\mathclose{{}\left(x}\right)\,\mathrm{d}^{N}z\mathopen{}\mathclose{{}\left(x}\right) and \mathcal{D}\bigl{[}U\bigr{]}=\prod\limits_{x,\mu}\mathrm{d}U_{\mu}\mathopen{}\mathclose{{}\left(x}\right), where \mathrm{d}U_{\mu}\mathopen{}\mathclose{{}\left(x}\right) is the \operatorname{U}\mathopen{}\mathclose{{}\left(1}\right) Haar measure for the link variable U_{\mu}\mathopen{}\mathclose{{}\left(x}\right)=\operatorname{e}^{\mathrm{i}\,\theta_{x,\mu}}.

Discretizing (1.2), yields the quartic lattice action and corresponding partition function:

S_{Q}\,=\,-\beta\,\sum\limits_{x,\mu}\big{|}z^{\dagger}\mathopen{}\mathclose{{}\left(x}\right)\cdot z\mathopen{}\mathclose{{}\left(x+\widehat{\mu}}\right)\big{|}^{2}

(1.5)

,\quad Z_{Q}\,=\,\int\mathcal{D}\bigl{[}z^{\dagger},z\bigr{]}\exp\biggl{(}\bigg{.}\beta\sum\limits_{x,\mu}\big{|}z^{\dagger}\mathopen{}\mathclose{{}\left(x}\right)\cdot z\mathopen{}\mathclose{{}\left(x+\widehat{\mu}}\right)\big{|}^{2}\bigg{.}\biggr{)}\ .

(1.6)

1.3 Dual formulation

To obtain the dual formulation of the partition function (1.6), we follow the derivations in [11, 12] and start by explicitly writing out all sums in the exponential of (1.6) :

[TABLE]

Now we write the exponential of the summed terms in (1.7) as the product of exponentials of the individual terms and then use the power series representation for each of these exponentials to find:

[TABLE]

After integrating out the z_{a}\mathopen{}\mathclose{{}\left(x}\right) and defining new integer variables $k_{x,\mu}^{a\,b}=n_{x,\mu}^{a\,b}-n_{x,\mu}^{b\,a}$ and $l_{x,\mu}^{a\,b}=\frac{1}{2}(n_{x,\mu}^{a\,b}+n_{x,\mu}^{b\,a}-|k_{x,\mu}^{a\,b}|)$, so that $k_{x,\mu}^{a\,b}\in\mathbb{Z}$, $l_{x,\mu}^{a\,b}\in\mathbb{N}_{0}$ and $n_{x,\mu}^{a\,b}=\frac{1}{2}(|k_{x,\mu}^{a\,b}|+k_{x,\mu}^{a\,b})+l_{x,\mu}^{a\,b}$, we find for (1.8):

[TABLE]

In this version of dual $\operatorname{\mathbb{C}P}^{N-1}$ partition function, we have in total $N^{2}$ degrees of freedom per link which are given by the independent components of the $k$- and $l$-variables: the N\mathopen{}\mathclose{{}\left(N-1}\right)/2 independent components of the anti-symmetric $k_{x,\mu}^{a\,b}$ are subject to on-site constraints, while the N\mathopen{}\mathclose{{}\left(N+1}\right)/2 independent components of the symmetric $l_{x,\mu}^{a\,b}$ are unconstrained.

The dual version of the auxiliary \operatorname{U}\mathopen{}\mathclose{{}\left(1}\right) partition function $Z_{A}$ from (1.4) can be obtained from (1.9) by introducing additional link-weight factors (see [1] for more details),

[TABLE]

after which one finds

[TABLE]

where the changes caused by the link weight (1.10) are marked in red and the number of constrained and unconstrained degrees of freedom remains unchanged. We would like to stress that both, (1.9) and (1.11) can be coupled to \mathopen{}\mathclose{{}\left(N-1}\right) independent chemical potentials without introducing a sign-problem (see [1] for more details).

An alternative dual formulation of (1.4), which relies on only $2\,N$ degrees of freedom per link, and which also remains sign-problem free after the introduction of chemical potentials, has been given in [15] and can be written as

[TABLE]

where we have set the chemical potentials to zero, as we will not discuss finite density here.

2 The worm algorithm

The general idea behind a worm algorithm[14] is to update configuration variables for a partition function $Z$, which are subject to on-site constraints as those in (1.12) and (1.9), by generating configurations that contribute to some partition functions Z_{2}\mathopen{}\mathclose{{}\left(x,y}\right) instead of $Z$, with Z_{2}\mathopen{}\mathclose{{}\left(x,y}\right) being a two-point partition function for the same system that is described by $Z$ itself, but in the presence of an external source at $x$ and an external sink at $y$, which contribute an additional $-1$ or $+1$ to the constraint for the respective site. The constrained configuration variables are then updated one after another by moving the external sink from site to neighboring site and whenever source and sink meet again, a new configuration that contributes to $Z$ can be obtained by removing again the source/sink pair.

This strategy works well for the partition function (1.12) based on $2\,N$ degrees of freedom per link (see [1]), but for the partition functions (1.9) and (1.11), which are based on $N^{2}$ degrees of freedom per link, it has been found in [12] that an "ordinary" worm algorithm is not sufficiently ergodic to sample the most relevant configurations efficiently. The reason for this ergodicy problem with the ordinary worm is the additional freedom that one has in the $N^{2}$ d.o.f. per link formulation to compensate for a displacement of a particular external sink (see Fig. 2). In the remainder of this section we sketch how this additional freedom can be taken into account by generalizing the ordinary worm algorithm to an internal space sub-worm algorithm.

2.1 Internal space sub-worm algorithm

We will discuss the internal space sub-worm (ISSW) algorithm on the example of (1.11), in which case the relevant two-point partition functions, mentioned above, that allow for updates of individual $k$-variables, are given by:

[TABLE]

with \phi^{ab}\mathopen{}\mathclose{{}\left(x}\right)=z_{a}\mathopen{}\mathclose{{}\left(x}\right)\,\ThisStyle{\hbox to0.0pt{\hskip 4.57985pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@color\SavedStyle z}}}_{b}\mathopen{}\mathclose{{}\left(x}\right). The ISSW algorithm works very similarly to an ordinary worm algorithm, but whenever it is proposed to move the head of the worm from a site $x$ to a neighboring site $x+\widehat{\nu}$, a sub-worm cycle as described in Fig. 3 is started instead of simply proposing to update a single $k^{ab}$-variable. For more details see [1].

3 Results

We have run extensive tests to verify that our worm and internal space sub-worm algorithms correctly sample configurations for (1.12) (\mathcal{O}\big{(}2\,N\big{)} d.o.f. per link) and (1.11) (\mathcal{O}\big{(}N^{2}\big{)} d.o.f. per link), respectively, for arbitrary $N$, arbitary system sizes $V$ and in arbitrary dimensions $d$.

To test the efficiency of our algorithms, we compare in Fig. 2 for $\operatorname{\mathbb{C}P}^{9}$ in \mathopen{}\mathclose{{}\left(1+1}\right) dimensions their dynamical critical exponents $z$ for the integrated auto-correlation times of the average energy and magnetic susceptibility:

\mathopen{}\mathclose{{}\left\langle E}\right\rangle\,=\,-\frac{1}{V}\frac{\partial\log\mathopen{}\mathclose{{}\left(Z}\right)}{\partial\beta}\

(3.1)

\text{and}\quad\chi_{m}\,=\,\frac{1}{V}\sum\limits_{x,y}\biggl{(}\bigg{.}\sum\limits_{a,b}\mathopen{}\mathclose{{}\left\langle\phi^{a\,b}\mathopen{}\mathclose{{}\left(x}\right)\phi^{b\,a}\mathopen{}\mathclose{{}\left(y}\right)}\right\rangle-\frac{1}{N}\bigg{.}\biggr{)}\ ,

(3.2)

and the so-called second moment correlation length (see e.g. [8]),

[TABLE]

with those obtained from data presented in [16] for an over-heat-bath algorithm that directly simulates the (non-dual) system (1.4). As can be seen, for the given choice of parameters, all three algorithms show very similar dynamical critical exponents.

4 Conclusion

We have implemented and tested two worm algorithms which are suitable to simulate the lattice $\operatorname{\mathbb{C}P}^{N-1}$ model in two different dual "flux-variable" formulations, which both allow one to couple the model to \mathopen{}\mathclose{{}\left(N-1}\right) independent chemical potentials without causing a sign-problem.

For the simulation parameters used in this work ($N=10$, fixed), the worm algorithms do not seem to significantly reduce critical slowing down in comparison to traditional simulation techniques. It is however possible, that the worm algorithms might start to perform better if $N$ is increased, since the slowest dynamical modes are those that change topology, and topological degrees of freedom are integrated out in the dual formulation (see [1]).