You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
author:
- |
    \
    Institut für Theoretische Physik\
    Auf der Morgenstelle 14\
    72076 Tübingen\
    Germany\
    E-mail:
title: '’t Hooft loop and the phases of SU(2) LGT'
---

Introduction
============

Most of the ideas aimed at solving confinement in $SU(N)$ Yang-Mills theories involve topological degrees of freedom of some sort. Among these, $\mathbb{Z}_N$ (i.e. center) vortices [@'tHooft:1979uj] have received much attention in the literature in general and in lattice investigations in particular [@Del; @Debbio:1996mh; @Langfeld:1997jx].

The gauge group of pure Yang-Mills $SU(N)/\mathbb{Z}_N$ possesses a non trivial first homotopy class, corresponding to its center, $\pi_1(SU(N)/\mathbb{Z}_N) = \mathbb{Z}_N$. A super-selection rule will thus arise for the physical Hilbert space of gauge invariant states [@Burgio:1999tg], with sectors labeled by a center vortex topological index $n \in \mathbb{Z}_N$. According to ’t Hooft’s original idea [@'tHooft:1979uj], the low temperature confinement phase should then correspond to a superposition of all topological sectors, while above the deconfinement transition vortex symmetry gets broken to the trivial sector $n = 0$; the ’t Hooft loop $H$, dual to the Wilson loop $W$, is the natural observable to describe the transition, “counting” the number of topological vortices piercing it. From $H$ one can reconstruct the free energy for vortex creation, $F = \Delta U -T \Delta S$, which should jump at $T_c$; the monitoring of such behaviour across the deconfinement transition has received broad attention in the literature [@Kovacs:2000sy; @deForcrand:2000fi; @deForcrand:2001nd; @Burgio:2006dc; @Burgio:2006xj].

The natural choice to investigate $F$ upon lattice discretization of Yang-Mills theories would be to define the partition function through the adjoint Wilson action $Z \sim e^{\beta_A {\mathsf{Tr}}_A(U)}$, transforming under $SU(N)/\mathbb{Z}_N$; in this case all topological sectors are dynamically included [@deForcrand:2002vs]. Universality should of course allow the equivalent use of the standard Wilson plaquette action $S\sim{\mathsf{Tr}}_F(U)$. In this case topology must however be introduced “by hand” summing over all twisted boundary conditions[^1]. The “proper” partition function $\tilde{Z}$ can then be defined through the weighted sum of all partition functions with fixed twisted b.c.[^2]. Since each of them must be determined by independent simulations, their relative weights can only be calculated through indirect means [@deForcrand:2000fi; @deForcrand:2001nd; @vonSmekal:2012vx].

There is however a loophole in the argument given above. The two partition functions $Z$ and $\tilde{Z}$ can only be shown to be equivalent when $\mathbb{Z}_N$ magnetic monopoles are absent [@deForcrand:2002vs; @Mack:1978rq]. Taking the explicit case of $SU(2)/\mathbb{Z}_2 = SO(3)$, this translates into the constraint: $$\sigma_c=\prod_{\overline{P}\in\partial c}\mathrm{sign}
({\mathsf{Tr}}_{F} \,U_{\overline{P}})=1
\label{eq:mon}$$ being satisfied for every elementary 3-cube $c$, where $U_{\overline{P}}$ denotes the plaquettes belonging to the cube surface $\partial c$. This ensures that endpoints of open center vortices, ${\mathbb{Z}}_2$ magnetic monopoles, are suppressed and only [*closed*]{}, i.e. topological ${\mathbb{Z}}_2$ vortices winding around the boundaries can form.

The above condition is usually quoted when claiming that the bulk transition separating the strong and weak coupling regime along $\beta_A$ [@Greensite:1981hw; @Bhanot:1981eb; @Halliday:1981te; @Halliday:1981tm; @Baig:1987qa] constitutes an obstacle in defining the continuum limit for the adjoint Wilson action. This however assumes that topological sectors along the fundamental coupling $\beta$ are always well defined. We will show this [*not*]{} to be the case. Together with a set of established results demonstrating that above the adjoint bulk transition topological sectors are well defined and a physical continuum limit of the theory exists [@Burgio:2006dc; @Burgio:2006xj; @Barresi:2001dt; @Barresi:2002un; @Barresi:2003jq; @Barresi:2003yb; @Barresi:2004qa; @Barresi:2004gk; @Burgio:2005xe; @Barresi:2006gq], we can turn the argument around, casting doubts that investigations of vortex topology for the fundamental Wilson are well defined. Preliminary results had been presented in Ref. [@Burgio:2007np].

Setup
=====

We will investigate as a test case the $SU(2)$ fundamental Wilson action with periodic b.c.: $$S = \beta \sum_{x,\mu>\nu}[1-{\mathsf{Tr}}_F(U_{\mu\nu}(x)]$$ in $D=2,3,4$ dimensions. Different groups or b.c. can be considered as well and won’t change the main results given below.

The twist operator, measuring the number of topological vortices piercing the ’t Hooft loop in the $\mu,\nu$ planes, can be constructed via [@Kovacs:2000sy]: $$z_{\mu\nu} = \frac{1}{L^{D-2}}\,\sum_{\vec{y}\bot{\mu\nu}{\rm plane}}\; \prod_{\vec{x} \in {\mu\nu} {\rm plane}} 
{\sf sign}({{\mathsf{Tr}}}_{F}U_{\mu\nu}(\vec{x},\vec{y}))\,.
\label{eq:twist}$$ When topological sectors are well defined $z_{\mu\nu}$ takes values $\pm 1$ for all fixed $\mu$ and $\nu$; e.g. in the case at hand, i.e. periodic b.c., the topological sector must be trivial and one should always have $z_{\mu\nu}=1$ $\forall \mu, \nu$. It is now easy to define an order parameter $z$ such that $z=1$ if, whatever the b.c., the vortex topology takes the correct value expected in the continuum theory, while $z=0$ when $\mathbb{Z}_2$ monopoles are still present and open $\mathbb{Z}_2$ center vortices dominate the vacuum, making the identification of topological sectors ill defined at best: $$\begin{aligned}
z &=& 1-| z_{12} - \langle z_{12} \rangle|\qquad\qquad\qquad  D=2\\
z &=&\frac{2}{D(D-1)} \sum_{\mu>\nu=1}^{D} \langle |z_{\mu\nu}|\rangle \quad\qquad D \geq 3\,.\end{aligned}$$ In the following we will monitor $z$ as a function of $\beta$ and the volume $L^D$. The continuum limit of our lattice discretization is of course defined by taking the thermodynamic limit $L\to\infty$ first and then the weak coupling limit $\beta\to\infty$.

Results
=======

The $D =2$ case offers an interesting cross-check of the numerical results in higher dimensions, since here everything can be calculated analytically. For the order parameter $z$ and its susceptibility $\chi$ we have, for fixed volume $L^2$: $$\begin{aligned}
\langle {z} \rangle_L &=& e^{-4 L^2 p(\beta)}\,;\qquad
\chi_L = L^2 \left[e^{-4 L^2 p(\beta)}-e^{-8 L^2 p(\beta)}\right]\\
p(\beta) &=& \frac{1}{2}\left[1-\frac{L_1(\beta)}{I_1(\beta)}\right] 
=
\sqrt{\frac{2 \beta}{\pi}}e^{-\beta}(1+{\cal{O}}(\frac{1}{\beta}))\,,\end{aligned}$$ where $L$ and $I$ denote the modified Struve and Bessel functions, respectively. Plotting the above functions (see Fig. (\[fig\_2\])) we can clearly distinguish a “strong” coupling regime, where the topology is ill defined, and a “weak” coupling one, where $z$ takes the correct value it should have in the continuum theory.

![Order parameter $z$ in $D=2$ (left) and its susceptibility $\chi$ (right).[]{data-label="fig_2"}](z2 "fig:"){width="49.00000%"} ![Order parameter $z$ in $D=2$ (left) and its susceptibility $\chi$ (right).[]{data-label="fig_2"}](x2 "fig:"){width="49.00000%"}

The finite size scaling analysis can be performed exactly, giving for the susceptibility peaks and the corresponding pseudo-critical coupling: $$\beta_c(L) = \ln{L^2} + \frac{1}{2}\ln{\ln{L^2}} + {\cal{O}}(1)\,;\qquad
\chi_L(\beta_c(L)) = \frac{L^2}{4}\,.
\label{eq_fss_2}$$ From the above equation it is easy to extract the critical behaviour of the correlation length around the critical coupling $\beta_c = \infty$, including logarithmic corrections: $$\xi \sim \sqrt[4]{\frac{\pi \ln^2{2}}{2 \beta}}\;\cdot\, e^{\frac{1}{2}\beta}
\label{eq_ess}$$ i.e. an essential scaling[^3] with critical exponents $\nu = 1$, $\eta = 0$ and $r=0$.

Summarizing, although for any fixed volume $L^2$ one can always find a coupling above which the topology corresponds to that dictated by the boundary conditions, taking the thermodynamic limit first, as one should, the “strong” coupling regime extends to $\beta = \infty$ and the system is always in the disordered phase. A vortex topology cannot be defined.

Turning now to the $D =3$ case and using Monte-Carlo simulations to calculate $z$ we basically get the same picture. In Fig. (\[fig\_3\]) we show the susceptibility $\chi$ and its FSS with an essential scaling Ansatz, again with $\beta_c = \infty$:

![Left: susceptibility $\chi$ of the order parameter $z$ in $D=3$. Right: same with FSS as in Eq. (\[eq\_fss\]).[]{data-label="fig_3"}](3d "fig:"){width="49.00000%"} ![Left: susceptibility $\chi$ of the order parameter $z$ in $D=3$. Right: same with FSS as in Eq. (\[eq\_fss\]).[]{data-label="fig_3"}](fss "fig:"){width="49.00000%"}

$$\beta_c(L)\simeq A \ln{L^2} + B\ln{\ln{L^2}}\,;\qquad\chi_L(\beta_c(L)) \simeq C {L^2} \ln^{-2 r}{L}
\label{eq_fss}$$

The result is the same, i.e. in the continuum limit the theory is always in disordered phase and no vortex topology can be defined. The values obtained for $\beta_c$ are well within the scaling region and for fixed volumes $L^3$ they are always lower than the pseudo-critical coupling at which the “finite temperature” transition for time length $L$ would be measured.

The situation is inverted at $D=4$. Here we still get the same result as above for our order parameter $z$, i.e. an essential scaling as in Eq. (\[eq\_fss\]). The values of the pseudo-critical coupling $\beta_c$ are however higher than the values measured for the deconfinement transition at time length $L$, cfr. Fig. (\[fig\_4\]), making the scale at which topological vortices stabilize way above any physical scale involved in the process.

![Left: susceptibility $\chi$ of the order parameter $z$ in $D=4$. Right: same with FSS as in Eq. (\[eq\_fss\]).[]{data-label="fig_4"}](x4 "fig:"){width="49.00000%"} ![Left: susceptibility $\chi$ of the order parameter $z$ in $D=4$. Right: same with FSS as in Eq. (\[eq\_fss\]).[]{data-label="fig_4"}](fss4 "fig:"){width="49.00000%"}

Conclusions
===========

We have shown that $\forall\; D \leq 4$ the vortex topology for the standard Wilson action is always ill defined in the continuum limit. This is driven by a too slow fall-off of discretization artefacts density, i.e. ${\mathbb{Z}}_2$ magnetic monopoles Eq. (\[eq:mon\]). This means that for any fixed $\beta$ there always exists a lattice size $L$ for which enough open center vortices can form, spoiling the identification of topological sectors and making a measurement of the conjectured super-selection rule in the thermodynamic limit ill-defined.

The details of such bulk effect will of course strongly depend on the discretization chosen. For example, the separation among the regimes in $D=3$ and $D=4$ are substantially different. While in $D=3$ the deconfinement transition for fixed length $L$ always lies in the spurious phase above the pseudo-critical coupling $\beta_c(L)$ Eq. (\[eq\_fss\]), for $D=4$ the latter is always way above the physical scale, making the physical volumes quite small.

Other choices for the discretization would of course change the picture. For example, in $D=4$ one could define the theory through the adjoint Wilson action, for which, above the the bulk transition [@Greensite:1981hw; @Bhanot:1981eb; @Halliday:1981te; @Halliday:1981tm], topology is well defined Ref. [@Burgio:2006dc; @Burgio:2006xj]. There it was however found that $F\neq0$ in the confined phase, calling for more investigations of the standard center vortices symmetry breaking argument as a model for confinement. Simulations in this case are however technically quite demanding.

Another choice, which would in principle work in any dimensions, would be to define the discretized theory through a Positive Plaquette Model [@Fingberg:1994ut]. In this case the ${\mathbb{Z}}_2$ magnetic monopole constraint is always satisfied while the order parameter $z \equiv 1$ by construction. As was proven in Ref. [@Barresi:2006gq], this is indeed equivalent to simulating the adjoint Wilson action in a fixed topological sector. This model should then be the discretization of choice if one is interested in investigating the role of center vortices.

We would like to stress that none of the above results contradicts universality. First, universality hold as long as no discretization artifacts obstacle the continuum limit. In our case, center monopoles and the associated open center vortices spoil the equivalence among different discretizations. Second, all physical properties measurable in “experiments”, like glueball masses and the critical exponents at the transition, should be reflected by physical observables which can be defined [*irrespective*]{} of the discretization chosen; this however does not mean that everything that can be defined in a given discretization should acquire a physical meaning.

Aknowledgements {#aknowledgements .unnumbered}
===============

We wish to thank F. Bursa for useful discussions on the $D=2$ case. This work was partially supported the DFG under the contract DFG-Re856/6-3.

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[^1]: Such topological boundary conditions also play a rôle in lattice investigations of the string spectrum or large $N$ reduction.

[^2]: See e.g. Ref. [@vonSmekal:2012vx], Chapt. 3.

[^3]: Compare with the critical behaviour of the XY model, $\xi \sim \exp\left({b t^{-\nu}}\right)$ and $\chi \sim \xi^{2-\eta} 
    \ln^{-2 r}{\xi}$, with $t = \left|T/T_c - 1\right|$ the reduced temperature and $\nu = 1/2$, $\eta = 1/4$ $r = -1/16$ the critical exponents [@Kenna:1996bs].