Density of states approach for lattice field theory with topological terms^††thanks: Based on a parallel talk by C. Gattringer.

It is well known that using open boundary conditions lifts the quantization of the topological charge and in the thermodynamic and continuum limits describes the correct physics. Using open boundary conditions in lattice simulations was proposed in [24, 25] as a tool for reducing topological autocorrelation, which, due to the index theorem, is a problem that is related to our problem of singular densities of states when a topological term is added. In [23] it was demonstrated, that also the latter problem can be solved with open boundary conditions and that for topological quantities the correct continuum limit is found (see also [26]). In this contribution we review the new techniques, discuss the case of 2-d U(1) lattice gauge theory with a $\theta$-term as an example where analytic reference results can be used to assess the approach, and finally present first tests for 4-d SU(2) lattice gauge theory with a $\theta$-term.

For non-vanishing topological angle $\theta$ the Boltzmann factor $e^{\,-\,S[A]\;-\;i\theta\,Q[A]}$ is complex and thus cannot be used as a probability in a Monte Carlo simulation. For setting up the density of states approach we introduce (generalized) densities

$$\rho^{(J)}(x)\;=\,\int\!\!D[A]\;e^{\,-\,S[A]}\;J[A]\;\delta\Big{(}x-Q[A]\Big{)}\;,$$

(2)

where we allow for the insertion of some functional $J[A]$ of the gauge fields.

Obviously, when $Q[A]$ becomes integer in the continuum limit (or is integer also at finite lattice spacing if a geometrical or a fermionic definition of $Q[A]$ is used) the density will be the aforementioned superposition of Dirac deltas. However, we may use open boundary conditions to lift the quantization of the topological charge. Thus for open boundary conditions in at least one of the lattice directions the density remains smooth also in the continuum limit. This is a key step in the approach presented here.

Using the densities (2) we may write the vacuum expectation values as

$$\langle O\rangle_{\theta}\,=\,\frac{1}{Z_{\theta}}\!\int\!dx\;\rho^{(O)}(x)\;e^{\,-\,i\,\theta\,x}\qquad\mbox{with}\qquad Z_{\theta}\,=\int\!dx\;\rho^{(\mathds{1})}(x)\;e^{\,-\,i\,\theta\,x}\;.$$

(3)

For an actual determination of the densities we need to parameterize them in a suitable way. As a first step towards such a parameterization we truncate the range of the argument $x$ to the interval $[0,l_{max}]$. Such a truncation is justified since the densities are either even or odd and decay fast for large arguments $x$, a property that of course needs to be checked in the end. Next we divide the (truncated) $x$-range $[0,l_{max}]$ into $N$ intervals $I_{n},\;n=0,1\,...\,N-1$ of sizes $\Delta_{n}$. For the densities we now make the ansatz:

$$\rho(x)\;=\;e^{\,-\,L(x)}\quad\mbox{with}\;L(x)\;\mbox{continuous and piecewise linear on the intervals}\;\;I_{n}\;.$$

(4)

The parameterization of the exponent of the density with the piecewise linear and continuous function is illustrated in the lhs. plot of Fig. 1. Imposing the normalization $\rho(0)=1$ completely determines the densities $\rho(x)$ in terms of the slopes $k_{n}$ defined for each of the intervals $I_{n}$,

$$\rho(x)\;=\;A_{n}\;e^{\,-\,x\,k_{n}}\qquad\mbox{for}\quad x\in I_{n}\qquad\mbox{with}\quad A_{n}\;=\;e^{\,-\sum_{j=0}^{n-1}\big{[}k_{j}-k_{n}\big{]}\Delta_{j}}\;.$$

(5)

To determine the slopes we use so-called restricted vacuum expectation values [8]–[11] which are defined as

$$\langle Q\rangle_{n}(\lambda)\;=\;\frac{1}{Z_{n}(\lambda)}\int\!\!D[A]\,e^{\,-\,S[A]}\,J[A]\;e^{\,\lambda\,Q[A]}\;Q[A]\;\Theta_{n}\Big{(}Q[A]\Big{)}\quad\mbox{with}\quad\Theta_{n}(x)\;=\;\left\{\begin{array}[]{cc}1&\mbox{for}\;\;\;x\in I_{n}\\ 0&\mbox{for}\;\;\;x\notin I_{n}\end{array}\right..$$

(6)

Obviously the function $\Theta_{n}\Big{(}Q[A]\Big{)}$ restricts the path integral such that the values of the topological charge are required to be in the interval $I_{n}$. The restricted vacuum expectation values $\langle Q\rangle_{n}(\lambda)$ are free of the complex action problem and thus can be evaluated with standard Monte Carlo simulations. We will compute them as a function of the parameter $\lambda\in\mathds{R}\,$.

However, the $\langle Q\rangle_{n}(\lambda)$ may also be computed from the parameterized density $\rho(x)\;=\;A_{n}\;e^{\,-\,x\,k_{n}}$ of Eq. (5). A trivial calculation gives

$$\langle Q\rangle_{n}(\lambda)\;=\;\frac{d}{d\,\lambda}\ln\int_{x_{n}}^{x_{n+1}}\!\!dx\;\rho(x)\;e^{\,\lambda\,x}\;=\;g_{n}(\lambda-k_{n})\;,$$

(7)

with some function $g_{n}$ (see, e.g., [19] for details). After an additive and a multiplicative normalization of the restricted vacuum expectation values one finds the more convenient form

$$\frac{\langle\,Q\,\rangle_{n}(\lambda)-x_{n}}{\Delta_{n}}-\frac{1}{2}\,=\,h\Big{(}\Delta_{n}\big{[}\lambda-k_{n}\big{]}\Big{)}\;,$$

(8)

where the function $h(s)$ is given by

$$h(s)\,\equiv\,\frac{1}{1-e^{-s}}-\frac{1}{s}-\frac{1}{2}\;.$$

(9)

We thus may fit the Monte Carlo data for $(\langle Q\rangle_{n}(\lambda)-x_{n})/\Delta_{n}-1/2$ with the function $h(\Delta_{n}[\lambda-k_{n}])$ and in this way determine the slopes $k_{n}$ from simple stable 1-parameter fits (DoS FFA approach [16] – [20]). The rhs. plot of Fig. 1 provides an example of such fits, where we show the results for different intervals $I_{n}$ for $n=0,20$ and 40. The data are for the simulation of 2-d U(1) lattice gauge theory presented in [23].

Beyond the more fundamental question of the correct continuum limit, the exact reference results allow one to also address the question whether the DoS FFA techniques we use give rise to sufficient accuracy, such that the oscillating integrals (3) can be reliably evaluated. We begin our assessment of the accuracy with the results for the density $\rho(x)$ shown in the lhs. plot of Fig. 2. The data are from a simulation of 2-d U(1) lattice gauge theory with Wilson action and a simple field theoretical discretization of the topological charge (see [23] for the details of the simulation). We show results for $L\times L$ lattices with different sizes $L$. The inverse gauge coupling $\beta$ was chosen such that the ratio $R=L^{2}/\beta$ remains constant which means that we approach the continuum limit for a fixed physical volume. In the continuum the ratio is given by $R=Ve^{2}$ where $V$ is the physical volume and $e$ the electric charge. In Fig. 2 this ratio is chosen as $R=10$ and the symbols represent the results from the DoS FFA calculation. The dashed lines are the corresponding analytical results and for all volumes we find excellent agreement with the numerical data. For comparison we also show the analytical result for the Villain action in the continuum limit at $R=10$ and the sequence of volumes and couplings we consider for the Wilson action at $R=10$ approaches the corresponding Villain continuum result. It is important to note that the densities show a Gaussian-type of shape, i.e., they decay quickly for increasing $x$ such that the truncation of $x$ to a finite interval is justified.

In the rhs. plot of Fig. 2 we show the result for the topological charge density $q=Q/V$ as a function of the topological angle $\theta$. To obtain this result one only needs the density $\rho(x)$ of Eq. (2) for $J\equiv\mathds{1}$, which then is integrated over with $e^{-i\theta x}$ to obtain the partition function $Z_{\theta}$ and with $x\,e^{-i\theta x}$ to obtain the numerator of the vacuum expectation value $\langle q\rangle$. The corresponding results in the rhs. plot of Fig. 2 are for lattice size $L=24$, with again $R=10$. We find that the DoS FFA results (symbols) and the corresponding exact results (red full line) match very well for the full range of $\theta$-values. The insert shows the relative error and we find that it remains below 0.3 %.

The tests presented in [23] show that, at least for the case of 2-d U(1) lattice gauge theory with a topological term, the approach of using DoS FFA with open boundary conditions correctly reproduces the continuum physics at finite $\theta$. Based on analytic results it was shown that the open boundary conditions give rise to the correct continuum limit, and the numerical tests established that the parameterized density can be obtained with sufficient accuracy, such that the necessary integration with the oscillating factors reliably provides vacuum expectation values at non-zero $\theta$.

Our calculation is based on the Wilson action and we use the field theoretical definition of the topological charge $Q$ [27] (”clover discretization”). We work on lattices of size $16^{3}\times 4$, i.e., a finite temperature setting and we use mixed boundary conditions where one of the spatial directions remains open, while the other space directions and the time direction have periodic boundary conditions. Thus the topological charge does not become quantized to integers in the continuum limit and the density of states remains regular such that we can access it with the DoS FFA. The finite temperature setting, i.e., a short temporal direction, has been chosen such that changing the inverse gauge coupling $\beta$ drives the system through the deconfinement transition, which allows us to test the DoS FFA approach to in the two different phases of the theory. Besides this technical aspects, from a physical point of view it is a first attempt at studying the change of the $\theta$ dependence across the phase transition.

The details of the numerical simulation used for the results shown here will be presented in detail in an upcoming publication [28], and we here only address a new technical step that leads to a considerable improvement of the results. In Eq. (5) we have written the density in terms of the slopes $k_{n}$ describing the density in the corresponding discretization intervals $I_{n}$. Using the DoS FFA these $k_{n}$ are computed for each interval independently. Clearly the $k_{n}$ will have statistical errors that will lead to fluctuations around their true values. These fluctuations will introduce some roughness into the numerical results for the densities, which in turn will spoil the accuracy of the oscillating integrals (3) and thus the results for physical observables. However, this naive approach ignores the fact that (at least for finite lattices) $\rho(x)=e^{-L(x)}$ and thus also the function $L(x)$ in the exponent of the density must be smooth functions. In order to take this into account we may make an ansatz for $L(x)$ in the form of an even (or odd) finite polynomial, such as (note that we may normalize to $L(0)=0$, such that no constant term appears in $L(x)$)

$$L(x)\;\sim\;a_{2}\,x^{2}+a_{4}\,x^{4}+\,...\,a_{2k}\,x^{2k}\;\sim\;\sum_{j=0}^{n-1}[k_{j}-k_{n}]\,\Delta_{j}\,+\,k_{n}\,x\quad\mbox{for}\quad x\,\in\,I_{n}\;,$$

(10)

where in the first step we have represented $L(x)$ by the finite polynomial and in the second step exploited (5) to write $L(x)$ using our representation in terms of the slopes $k_{n}$. We may now use the numerical data for all slopes $k_{n}$ as input and then determine the values for the coefficients $a_{2j}$ from a global fit to all numerical data. Subsequently the new parameterized form $\rho(x)=\exp(-a_{2}\,x^{2}\,...\,-a_{2k}\,x^{2k})$ is used for evaluating the integrals (3) and thus the observables. Obviously this step, which corresponds to a reparameterization of the density now takes into account the smoothness discussed above.

In Fig. 3 we show the results for the density of states at different values of the inverse gauge coupling $\beta$. We remark that for a temporal extent of 4 the critical gauge coupling is $\beta_{c}\sim 2.35$ [29]. We observe that for $\beta$ above $\beta_{c}$, i.e., in the deconfined phase, the density is considerably more narrow than in the confined phase ($\beta<\beta_{c}$), which reflects the fact that in the deconfined phase the moments of $Q$ become suppressed quickly. For the consistency of our approach, in particular the truncation of the $x$-range, it is important to note that as in the 2-d U(1) case we find that for all values of $\beta$ the densities have a Gaussian-like shape, i.e., they decay fast, such that the truncation in our parameterization of the densities is justified.

We conclude our first discussion of the 4-d SU(2) results with an assessment of the accuracy we have achieved in our calculation of the density. Clearly no analytical reference results are available in this case, but we may use the density also for the evaluation of observables at $\theta=0$, which can then be compared to results of a conventional simulation. More specifically we look at the DoS results for $\langle Q^{2}\rangle$ and $\langle Q^{4}\rangle$ at $\theta=0$ and compare them with reference data from a conventional simulation. This is done for different inverse gauge couplings, i.e., different temperatures such that we can assess the accuracy for different physical situations.

The corresponding results are shown in Fig. 4 where $\langle Q^{2}\rangle$ (lhs. plot) and $\langle Q^{4}\rangle$ (rhs.) are shown as a function of $\beta$. The open circles represent the results from DoS FFA, while the filled triangles are the results from a conventional simulation. We find very good agreement between the Dos FFA results and the reference data from the conventional simulation for all values of $\beta$, which indicates that DoS FFA is ready for the analysis of the $\theta$-dependence of observables in 4-d SU(2) lattice gauge theory.

In a second part we present our new tests for 4-d SU(2) lattice gauge theory with a $\theta$-term. Here the focus is on achieving the necessary accuracy for the DoS approach in a theory that is considerably closer to the target theory of QCD. We present results for the case without topological term and show that the standard simulation results for the second and the fourth moment of the topological charge are precisely reproduced by the DoS method in a wide range of temperatures. The method thus is ready for computing observables at non-zero topological angle $\theta$ [28].