Correlation Functions of the Quantum Sine-Gordon Model in and out of Equilibrium

Complete information on the equilibrium behavior and dynamics of quantum field theory (QFT) is provided by multipoint correlation functions. However, their theoretical calculation is a challenging problem, even for exactly solvable models. This has recently become an experimentally relevant problem, due to progress in cold-atom experiments simulating QFT models and directly measuring higher order correlations. Here we compute correlation functions of the quantum sine-Gordon model, a prototype integrable model of central interest from both theoretical and experimental points of view. Building upon the so-called truncated conformal space approach, we numerically construct higher order correlations in a system of finite size in various physical states of experimental relevance, both in and out of equilibrium. We measure deviations from Gaussianity due to the presence of interaction and analyze their dependence on temperature, explaining the experimentally observed crossover between Gaussian and non-Gaussian regimes. We find that correlations of excited states are markedly different from the thermal case, which can be explained by the integrability of the system. We also study dynamics after a quench, observing the effects of the interaction on the time evolution of correlation functions, their spatial dependence, and their non-Gaussianity as measured by the kurtosis.

Figure 1

Density plots of 2P correlations $G^{(2)}(x_1,x_2)$ on the ground state of the SGM in a box (right) in comparison with those of the massless (left) and massive (middle) free boson (FB) case [interaction $\mathrm{\Delta }=1/18$, system size $L=25$ (in units of $M$), mass of the free case chosen equal to first breather mass of the SGM].

Figure 2

Density plots of 2P correlations $G^{(2)}(x_1,x_2)$ and 4P full and connected correlations $G_{(\mathrm{con})}^{(4)}(x_1,x_2,L/4,3L/4)$ for the ground and two thermal states ($\mathrm{\Delta }=1/18$, $L=25$, $T=0$, 0.25, 1 in units $M$). For better comparison, the last column shows plots of the 4P full (solid line), connected (dashed), and disconnected (dotted) correlations along the antidiagonal section $x_2=L-x_1$.

Figure 3

Kurtosis (measure of non-Gaussianity) in thermal states of the quantum sine-Gordon model, as a function of the inverse temperature $T^{-1}$ for interactions $\mathrm{\Delta }\equiv {\beta }^2/(8\pi )=1/100$ (red line) and $1/18$ (green line). Deviations from Gaussianity increase with $T$ up to a maximum at $T\sim 1$ (in units of soliton mass $M$) and decrease at higher $T$ since the high-energy states are essentially free. Our method describes accurately the low and intermediate temperature regime $T\lesssim 1$. Insets illustrate the excitation level relative to the height of the cosine potential (see Supplemental Material [33]).

Figure 4

Density plots of 2P and 4P full and connected correlations for some excited states. Note the strong qualitative differences in the patterns, even at nearby energy levels (top and middle rows). The last column shows comparative plots of 4P antidiagonal correlations.

Figure 5

Time evolution of 2P correlations $G^{(2)}(L/3,2L/3;t)$ (top row), kurtosis (second row), and spatial density plots of 2P correlations $G^{(2)}(x_1,x_2;t)$ and 4P correlations $G_{\mathrm{con}}^{(4)}(x_1,x_2,L/4,3L/4;t)$ at various times $t$ (bottom four rows) and time averaged (last column) after a quench starting from an excited state of the SGM (prequench interaction ${\mathrm{\Delta }}_0=1/18$, postquench interaction $\mathrm{\Delta }=1/8$, initial state energy $\sim 0.73M$ above ground state, corresponding to 50% of the height of the cosine potential, $L=30/M$). Correlations under massless and massive free boson dynamics (with mass matched to that of the first breather of the SGM) are plotted together for comparison. (Density plot axes correspond to $x_1/L$ and $x_2/L$ as in Figs. 2 and 4. For an animation of these data and technical details about the computation of the dynamics of the kurtosis, see the Supplemental Material [33].)