Topological transitions in the euclidean 2d U(1)-Higgs model

TL;DR

This paper investigates topological transitions in the 2D Euclidean U(1)-Higgs model, analyzing sphaleron transition rates through lattice simulations and their relation to vortex formation and Chern-Simons number behavior.

Contribution

It provides a novel analysis of sphaleron transitions in the 2D Euclidean U(1)-Higgs model using lattice simulations at finite temperature.

Findings

Sphaleron transition rates can be inferred from vortex formation.

Chern-Simons number behavior correlates with topological transitions.

High-temperature Euclidean simulations reveal insights into topological dynamics.

Abstract

The two-dimensional U(1)-gauged Higgs model is studied on an euclidean lattice of size $L_1\times L_2$, where the temperature $T=L_2^{-1}$ is of the order of the sphaleron mass. The simulation parameters are taken from zero temperature results. By comparison with classical and semiclassical results I discuss, whether the sphaleron transition rate can be extracted from the behavior of the Chern-Simons number and from the formation of vortices in an euclidean simulation at high temperatures.

Equations

$${\chi }_{\text{mbox}top}=\frac{1}{\beta V}<[N_{\text{mbox}CS}(\beta )-N_{\text{mbox}CS}(0){]}^2>\to \frac{\Gamma }{V},$$

$$\tilde{\rho }\equiv \frac{1}{\beta Z}\int \mathcal{D}[x(t)]\chi [x(t)]e^{-S[x(t)]},$$

$$\chi[x(t)]=\left\{\begin{array}[]{ll}1&\mbox{if }\exists\ t\in[0,\beta]\mbox{ with }x(t)=x_{m}\\ 0&\mbox{else}\end{array}\right.\!\!.$$

$$\tilde{\rho }=(\pi /2){\Gamma }_{\text{mbox}cl},$$

$${\Gamma }_{\text{mbox}cl}=<|p|\delta (x-x_m)>=\sqrt{\frac{2}{\pi \beta }}\frac{e^{-\beta V_{\text{mbox}max}}}{{\int }_0^{2\pi }dx{e}^{-\beta V(x)}}.$$

$$\rho \equiv -\ln (1-\tilde{\rho }\beta )/\beta .$$

$${\Gamma }_{\infty }=\frac{1}{4{\pi }^2\beta }<[x(\beta )-x(0){]}^2>.$$

$$B_x^{\mu }=-{\omega }_x+A_x^{\mu }+{\omega }_{x+\hat{\mu }}\in [-\pi ,\pi ],$$

$$N_{\text{mbox}CS}(0)=\sum _{x_1}\frac{A_x^1}{2\pi },{\partial }_t^{+}{N}_{\text{mbox}CS}(t)=\sum _{x_1}\frac{F_x}{2\pi },$$

$$B_x^1+B_{x+e_1}^2-B_x^2-B_{x+e_2}^1\notin [-\pi ,\pi ].$$

$$N_B(t)\equiv \frac{1}{2\pi }\sum _{x_1}{B}_{(x_1,t)}^1=N_{\text{mbox}CS}(t)(\text{mbox}mod1).$$

$$S_{\text{mbox}eff}[N_B]=\frac{2{\pi }^2}{e^2{L}_1}\int dt\left({\dot{N}}_B^2+v^2{e}^2{N}_B^2\right).$$

$$P({\bar{N}}_{\text{mbox}CS})\sim \sum _k{e}^{-\frac{d\beta }{L_1}({\bar{N}}_{\text{mbox}CS}-k{)}^2},d=2{\pi }^2{v}^2.$$

$${\rho }_V=<N_{\text{mbox}vortices}>/(\beta L_1).$$

Full text

1 Motivation

In the 2d U(1)-Higgs model, as well as in the Standard Model, the transition between gauge equivalent vacua with integer values of the Chern-Simons number $N_{\mbox{\scriptsize\it CS}}$ is related to the anomalous violation of the axial U(1)-symmetry. The transition rate $\Gamma$ is usually evaluated by semiclassical methods [2] and classical real-time simulations [3]. It is desirable that an evaluation of $\Gamma$ by euclidean simulations gives control over the full quantum corrections to these calculations.

At low temperatures (inverse temperature $\beta\rightarrow\infty$) $\Gamma$ is given by the topological susceptibility

[TABLE]

$V$ is the spatial volume. This relation makes use of the random walk of $N_{\mbox{\scriptsize\it CS}}$ for large $t$.

At high temperatures $\beta$ is too small to see this long-time behavior. In fact, since the topological charge $N_{\mbox{\scriptsize\it CS}}(\beta)-N_{\mbox{\scriptsize\it CS}}(0)$ must be an integer, a configuration contributing to $\chi_{\mbox{\scriptsize\it top}}$ is forced to change $N_{\mbox{\scriptsize\it CS}}$ by at least 1 in the short time interval given by $\beta$. This leads to an exponential suppression $\chi_{\mbox{\scriptsize\it top}}\sim e^{-c/\beta}$, whereas $\Gamma$ should be enhanced by high temperatures [4]. Can other observables do better?

2 The high temperature transition rate of the quantum pendulum

I shall address this question at first for the quantum pendulum, a particle of unit mass in the potential $V(x)=[1+\cos(x)]$. An observable asking for paths from vacuum to vacuum will be suppressed for $\beta\rightarrow 0$, since $\Delta x=2\pi$ in a time $\Delta t=\beta$ is required. I thus consider paths, which only need to cross one of the barrier tops $V(x)=V_{\mbox{\scriptsize\it max}}$ at $x_{m}=(2n\!+\!1)\pi$. I define

[TABLE]

[TABLE]

For any periodic potential I find in the limit $\beta\!\rightarrow\!0$

[TABLE]

the classical transition rate $\Gamma_{\mbox{\scriptsize\it cl}}$ is [5]

[TABLE]

$\tilde{\rho}$ counts repeated fluctuations over the barrier only once. Otherwise it would not be well-defined due to small time fluctuations of any quantum path. However, for small $\beta$ one expects at most one physical transition to occur, thus $\tilde{\rho}$ has the meaning of a transition density per unit euclidean time. For large $\beta$ and periodic boundary conditions $x(0)\!=\!x(\beta)\!+\!2\pi n$ the non-transition probability $p\!=\!1\!-\!\tilde{\rho}\beta$ decays as $p\!=\!e^{-\beta\rho}$. So, for the interpretation as a transition density I consider

[TABLE]

For large $\beta$ the behavior of $\rho$ can be compared with the $T=0$ transition rate given by the analogue of the topological susceptibility

[TABLE]

In Figure LABEL:pd_rho the diamonds give the value of $2\rho/\pi$, the squares give the values of $\Gamma_{\infty}$, the dashed line shows the classical rate $\Gamma_{\mbox{\scriptsize\it cl}}$, the full line shows the rate $\Gamma_{\mbox{\scriptsize\it Sph}}$ in sphaleron approximation [5].

There is a qualitative agreement of $2\rho/\pi$ and $\Gamma_{\infty}$ at $\beta\rightarrow\infty$. For higher temperatures $\Gamma_{\infty}$ becomes suppressed as expected, whereas $2\rho/\pi$ approaches the classical behavior of $\Gamma$. However, I yet didn’t manage to improve $\rho$ such that there is an exact correspondence to $\Gamma$ on all temperature scales. This question shall be addressed in a forthcoming paper.

3 Topological transitions in the 2d U(1)-Higgs model

Can a similar observable be defined in the 2d U(1)-Higgs model? Lattice formulation and scaling behavior of this model at $T=0$ are described in [1]. Here the quantities of interest are the link fields

[TABLE]

$A^{\mu}_{x}$ is the gauge field, $\omega_{x}$ is the phase of the scalar field. The difficulty is the analogue of the sector boundaries. A naive guess is the requirement $N_{\mbox{\scriptsize\it CS}}=n+1/2$ with the lattice definition

[TABLE]

$F_{x}/(2\pi)$ is the topological density, see [1]. In addition a topological transition shows a vortex

[TABLE]

However, the above condition $N_{\mbox{\scriptsize\it CS}}=n+1/2$ is in general not related to the formation of a vortex. Consider the related quantity

[TABLE]

In the continuum model with fixed scalar field length $\rho(x)=v$, $N_{B}(t)$ decouples from all other degrees of freedom. Its effective action reads

[TABLE]

This leads to a probability distribution of the constant mode $\bar{N}_{\mbox{\scriptsize\it CS}}=\int_{0}^{\beta}dtN_{\mbox{\scriptsize\it CS}}(t)/\beta$

[TABLE]

With growing $L_{1}$ it becomes constant. Thus, even with fixed $\rho(x)$, i.e. with parameters which do not allow for the standard instanton or sphaleron solutions, configurations with $N_{\mbox{\scriptsize\it CS}}\simeq n+1/2$ are not suppressed. I found this $L_{1}$-dependence of $P(\bar{N}_{\mbox{\scriptsize\it CS}})$ also in lattice simulations with variable scalar field length, completely dominating the effects induced by vortices. Only the parameter $d$ had to be matched. So the crossing of $N_{\mbox{\scriptsize\it CS}}(t)$ through the points $N_{\mbox{\scriptsize\it CS}}=n+1/2$ is no good condition for topological transitions, see also [6].

Another possibility is to consider the density of vortices per unit time and spatial volume

[TABLE]

Again it is important not to count nearby vortices and antivortices separately, which tend to occur in small clusters. I evaluated $\rho_{V}(\beta)$ in the ‘Higgs region’ of parameter space with the MC algorithm described in [1]. The $T=0$ results $am_{H}=0.442(19)$, $am_{V}=0.258(6)$, $v=1.849(1)$ lead to a sphaleron energy $aE_{\mbox{\scriptsize\it Sph}}=1.01(4)$.

A comparison of $\rho_{V}(\beta)$ (diamonds) with the transition rate per volume $\Gamma_{\mbox{\scriptsize\it Sph}}/L_{1}$ in sphaleron approximation [2] (full line with dashed error) shows a plateau of $\rho_{V}(\beta)$ for large $\beta$, see Figure LABEL:u1h_red. It exceeds the topological susceptibility at large $\beta$ (horizontal line) by far. This plateau is related to dislocation-like vortex pairs (DVPs). These objects have a size of a few lattice spacings, but fixed $\delta S$ for $a\rightarrow 0$ compared to a vacuum configuration. A typical example is shown in Figure 3.

For this example I find $\delta S=4\kappa\bar{\rho}^{2}$, compared with a vacuum configuration scalar field length $\rho_{x}\!=\!\bar{\rho},\,B^{\mu}_{x}\!=\!0$. The DVPs contribute to the expectation value $\rho_{V}$ and therefore destroy its scaling behavior in the continuum limit. Thus it is crucial to separate the true topological transitions from these effects on the cut-off scale.

Under cooling dislocation-like objects should loose their energy density faster than physical objects of the size of the correlation length. In fact, there is a correlation between vortices and lumps in the energy density $\epsilon(x)$ after some cooling sweeps. Figure 4 shows the distribution $p(\epsilon_{\mbox{\scriptsize\it max}})$ of the local maxima of the energy density near a vortex. I define the minimal required energy density for a true topological transition in the valley between the two peaks, thus throwing away a part of the vortices, identified as the dislocation-like ones. The cooling parameters are fixed for all $\beta$-values, a detailed study of the behavior under cooling shall be given in a future publication.

The such reduced vortex density $\rho_{\mbox{\scriptsize\it red}}$ (squares in Figure LABEL:u1h_red) fits better to the sphaleron rate at small $\beta$ and to the topological susceptibility at large $\beta$. However, this is a rough estimate, far from giving quantitative results. Rather it gives a hint which quantities should be better understood even in this simple toy model for the high temperature physics of the Standard Model.