Lattice simulations of real-time quantum fields

We investigate lattice simulations of scalar and non-Abelian gauge fields in Minkowski space-time. For $SU(2)$ gauge-theory expectation values of link variables in $3+1$ dimensions are constructed by a stochastic process in an additional (5th) “Langevin-time.” A sufficiently small Langevin step size and the use of a tilted real-time contour leads to converging results in general. All fixed point solutions are shown to fulfil the infinite hierarchy of Dyson-Schwinger identities, however, they are not unique without further constraints. For the non-Abelian gauge theory the thermal equilibrium fixed point is only approached at intermediate Langevin-times. It becomes more stable if the complex time path is deformed towards Euclidean space-time. We analyze this behavior further using the real-time evolution of a quantum anharmonic oscillator, which is alternatively solved by diagonalizing its Hamiltonian. Without further optimization stochastic quantization can give accurate descriptions if the real-time extent of the lattice is small on the scale of the inverse temperature.

Figure 1

Graphical representation of the Dyson-Schwinger equation for a Wilson loop.

Figure 2

Complex time-contours (left) and the corresponding distribution of eigenvalues (right) characterizing the Langevin dynamics for a free scalar theory as described in the main text.

Figure 3

Real and imaginary part of the unequal-time two-point correlation function for the quantum anharmonic oscillator as a function of real time $t$. The results obtained from stochastic quantization agree to very good accuracy to those obtained from directly solving the Schrödinger equation.

Figure 4

Similar evolution as in Fig. 3 but for much stronger coupling $\lambda =96$ and $\beta =1$.

Figure 5

Similar evolution as in Fig. 3 but for smaller temperature ($\beta =8$ and $\lambda =6$) on an isosceles triangle contour.

Figure 6

Shown is the Langevin-time evolution of the two-point function and four-point function given by the LHS and RHS of the Dyson-Schwinger identity (66). The curves are for fixed real-time values. At sufficiently late Langevin-time they agree to very good accuracy, thus becoming related by the Dyson-Schwinger equation.

Figure 7

Real and imaginary part of the equal-time correlator as a function of time (index of lattice site along the time contour) for the parameters ($\lambda =24$ and $\beta =1$) as well as the contour of Fig. 3. Compared are two simulations where the real-time extent of the lattice differs by a factor of 2. The larger lattice leads to a qualitatively different, nonunitary behavior.

Figure 8

Similar evolution as in Fig. 7, but on an isosceles triangle contour. While the thermal solution obtained for $t_{\mathrm{final}}=1$ does not depend on the contour details (see also Fig. 6), the properties of the nonunitary fixed point obtained for $t_{\mathrm{final}}=2$ are contour dependent (see also Figs. 9, 10).

Figure 9

The left-hand side (LHS) and right-hand side (RHS) of the Dyson-Schwinger equation (67) for the nonunitary fixed point, which agree well at late Langevin-time. Contrary to the properties of the thermal fixed point (see Fig. 6), the (0, 0) and $(t,t)$ components are not equal, which reflects the loss of time-translation invariance.

Figure 10

The LHS and RHS of Eq. (66), showing a discrepancy beyond the statistical error.

Figure 11

Nonequilibrium time evolution of the field expectation value as a function of time. Shown are results for different real-time extents of the lattice. Shorter $t_{\mathrm{final}}$ lead to improved agreement with results of the Schrödinger equation on the employed complex contour.

Figure 12

The spatial plaquette average as a function of Langevin-time. Shown are results for different complex contours. Here $\alpha =0$ corresponds to a contour with an infinite extent along the real-time axis, while $\alpha =\pi /2$ denotes the Euclidean contour. The longer the real-time component of the contour the less accurately the thermal solution is reproduced. The imaginary part is zero to a good approximation.

Figure 13

The Langevin-time ${\vartheta }_{\mathrm{crossover}}$ for the crossover away from the approximate thermal solution to another fixed point. (See Fig. 12.) It is displayed as a function of $\tan \alpha$ for various time contours. Contours with different real-time extent, but with the same $\tau +$ are connected with a line.

Figure 14

The spatial plaquette averages as a function of the index of lattice sites along the time contour. One observes that it is time-translation invariant to rather good accuracy for the approximate thermal fixed point, while it is not for the nonunitary fixed point.

Figure 15

Numerical check of the Dyson-Schwinger equation for a spatial plaquette variable. Displayed are separately the LHS and the RHS of (57).