Bose-Einstein condensation and Berezinskii-Kosterlitz-Thouless transition in the two-dimensional nonlinear Schrödinger model

We analyze the Bose-Einstein condensation process and the Berezinskii-Kosterlitz-Thouless phase transition within the nonlinear Schrödinger model and their interplay with wave turbulence theory. By using numerical experiments we study how the condensate fraction and the first-order correlation function behave with respect to the mass, the energy, and the size of the system. By relating the free-particle energy to the temperature, we are able to estimate the Berezinskii-Kosterlitz-Thouless transition temperature. Below this transition we observe that for a fixed temperature the superfluid fraction appears to be size independent, leading to a power-law dependence of the condensate fraction with respect to the system size.

Figure 1

Dark regions indicate the dispersion relation positive branch measured in a simulation with a weak (negligible) condensate. Results refer to the simulation having $L=256$ and free-particle energy density of ${\epsilon }_2=5.51$. The red dashed line shows the expected free-particle dispersion relation (16) derived in the four-wave interaction regime with the nonlinear frequency shift (17).

Figure 2

Dark regions indicate the dispersion relation positive branch measured in a simulation with a strong condensate. Results refer to the simulation having $L=256$ and free-particle energy density of ${\epsilon }_2=0.02$. The blue dashed line shows the expected Bogoliubov dispersion relation (55) derived in the three-wave interaction regime.

Figure 3

Measured frequency shift as a function of the free-particle energy density ${\epsilon }_2$. Results taken from simulations with $L=256$. The frequency shift, estimated using the dispersion relation data, is expected to be closer to the pure condensate shift ${\rho }_0$ at low temperatures and to reach the limiting theoretical value ${\omega }_{\mathrm{NL}}=2\overline{\rho }$ (17) for high temperatures where condensate is negligible.

Figure 4

Interaction energy density over the free-particle energy density ${\epsilon }_4/{\epsilon }_2$ as a function of the free-particle energy density ${\epsilon }_2$ measured at final steady states in simulations having different system sizes. Results show independence of the ratio with respect to the size of the system.

Figure 5

Final steady-state condensate fraction $C={\rho }_0/\overline{\rho }$ as a function of the free-particle energy density ${\epsilon }_2$. Different colored symbols indicate different system sizes. The condensate transition becomes shaper for larger sizes.

Figure 6

Final steady-state condensate fraction $C={\rho }_0/\overline{\rho }$ as a function of the system size $L$ for different temperatures in a log-log plot. The straight dashed lines are the best fits obtained with the power law (46). Error bars indicate the time-averaged standard deviation of the condensate fraction fluctuation in time in the final steady states.

Figure 7

Real part of the first-order correlation function of the field $\psi$ for different temperatures as a function of an angle-averaged distance $r$ in a log-log plot. Results are taken from simulations having system size $L=256$. The change from exponential to power-law behavior characteristic to the BKT transition is clearly observed. The power law having slope $\alpha =-1/4$ at the transition temperature $T_{\mathrm{BKT}}$ is also shown.

Figure 8

Real part of the first-order correlation function of the field $\psi$ for different temperatures below the estimated $T_{\mathrm{BKT}}$ as a function of an angle-averaged distance $r$ in a log-log plot. Same-colored lines refer to simulations having the same temperature but different system sizes: Similar exponents in the power laws are observed for the same temperature.

Figure 9

Estimated superfluid fraction ${\rho }_s$ with respect to different values of temperature. The superfluid fraction is evaluated by fitting the power-law exponents in Fig. 8 and inverting Eq. (44). Error bars indicate the measured standard deviation of the estimated superfluid fraction.

Figure 10

Angle-averaged steady-state spectra (points) measured in different simulations having high free-particle energy densities and negligible condensate fraction. The spectra are fitted with curves corresponding to the Rayleigh-Jeans distribution (18) integrated over angles, that is, $N_k={\int }_0^{2\pi }{n}_{k}kd\theta$. Corresponding fitted values of temperature and chemical potential are shown in the labels. Results are taken from simulations with $L=256$.

Figure 11

Angle-averaged steady-state spectra (points) of Bogoliubov normal variables (23) measured in different simulations having low free-particle energy densities and strong condensate fraction. The spectra are fitted with curves corresponding to the Rayleigh-Jeans distribution (32) integrated over angles, that is, $N_k={\int }_0^{2\pi }{n}_{k}kd\theta$. Corresponding fitted values of temperature and condensate fraction are shown in the labels. Results are taken from simulations with $L=256$.

Figure 12

Temperatures (points) evaluated by fitting with the Rayleigh-Jeans distributions the final steady states with respect to the free-particle energy density ${\epsilon }_2$ (see Figs. 10 and 11 for the corresponding values). Two sets of simulations with same system size $L=256$ but different resolution ($\Delta x=\xi$ and $\Delta x=\xi /2)$ are shown. The straight lines correspond to prediction (19) where the term containing the chemical potential $\mu$ has been neglected. Points out of the lines correspond to thermodynamic states where $\mu \ne 0$ and thus have no condensate fraction. Results are taken from simulations with $L=256$.

Figure 13

Measured condensate fraction with respect to the estimated temperatures fitted from spectra for systems having same size $L=256\xi$ but different resolution $\Delta x=\xi$ and $\Delta x=\xi /2$. Results indicate that a better resolved system manifests a lower condensation temperature.

Figure 14

Numerically detected quantum vortices in the final steady states for four simulations with $L=256$ and different temperatures. Clockwise and counterclockwise vortices are shown by dark yellow and black colors, respectively. Vortex number increases when increasing the temperature and clear hydrodynamic dipoles are only observed at very low temperatures, well below the estimated $T_{\mathrm{BKT}}$.

Figure 15

Numerically measured detected number of vortices per density for different values of the free-particle energy density. Results are taken from simulations having a side equal to $L=256$.