Quench dynamics of the three-dimensional U(1) complex field theory: Geometric and scaling characterizations of the vortex tangle

We present a detailed study of the equilibrium properties and stochastic dynamic evolution of the U(1)-invariant relativistic complex field theory in three dimensions. This model has been used to describe, in various limits, properties of relativistic bosons at finite chemical potential, type II superconductors, magnetic materials, and aspects of cosmology. We characterize the thermodynamic second-order phase transition in different ways. We study the equilibrium vortex configurations and their statistical and geometrical properties in equilibrium at all temperatures. We show that at very high temperature the statistics of the filaments is the one of fully packed loop models. We identify the temperature, within the ordered phase, at which the number density of vortex lengths falls off algebraically and we associate it to a geometric percolation transition that we characterize in various ways. We measure the fractal properties of the vortex tangle at this threshold. Next, we perform infinite rate quenches from equilibrium in the disordered phase, across the thermodynamic critical point, and deep into the ordered phase. We show that three time regimes can be distinguished: a first approach toward a state that, within numerical accuracy, shares many features with the one at the percolation threshold; a later coarsening process that does not alter, at sufficiently low temperature, the fractal properties of the long vortex loops; and a final approach to equilibrium. These features are independent of the reconnection rule used to build the vortex lines. In each of these regimes we identify the various length scales of the vortices in the system. We also study the scaling properties of the ordering process and the progressive annihilation of topological defects and we prove that the time-dependence of the time-evolving vortex tangle can be described within the dynamic scaling framework.

Figure 1

Temperature dependence of the order parameter $m$. (a) Comparison between the estimate for $m\left(T\right)$ obtained with different stochastic dynamics: overdamped Langevin Eq. (15) (over), underdamped Langevin Eq. (20) (rel), ultrarelativistic limit of the underdamped Langevin Eq. (21a) (ultrarel), and nonrelativistic limit of the underdamped Langevin Eq. (21b) (nonrel). Systems with $L=100$ linear size are used in all cases. (b) Finite-size dependence of $m\left(T\right)$ as obtained from the underdamped Langevin Eq. (20). In both panels, we plot with a solid line the critical decay of the order parameter, $m\left(T\right)\propto {(T_c-T)}^{\beta }$, at temperatures $T<T_c$ with $T_c=2.26$ and $\beta =0.347$. For details on these values see Figs. 2 and 4, and the corresponding discussion. In the inserts: zoom over the critical region.

Figure 2

Temperature dependence of the Binder ratio, $U_1$, and the ratio of the correlation function at different distances, $U_2$, as obtained from the underdamped Langevin Eq. (20), for different system sizes $L$ given in the keys.

Figure 3

Temperature dependence of the susceptibility, $\chi$, the two definitions of the specific heat, $C_1$ and $C_2$, and the correlation length, ${\xi }_{\mathrm{eq}}$, for a system with linear size, $L=100$, and the four stochastic dynamic equations. Inserts: zoom over the critical region.

Figure 4

Finite-size scaling of the order parameter, $m$, the susceptibility, $\chi$, the specific heat, $C_1$, and the correlation length, ${\xi }_{\mathrm{eq}}$, all obtained from the underdamped Langevin Eq. (20). The system sizes and color code are given in the key. Inserts: zoom over the critical region.

Figure 5

Finite-size scaling of the correlation function $C\left(r\right)$ near the critical temperature $T_c=2.26$ obtained using the underdamped Langevin Eq. (20). The dashed line is the analytic prediction ${(r/L)}^{-1-\eta }$ with $\eta =0.038$ and the dotted line includes the exponential cutoff.

Figure 6

Temperature dependence of the helicity modulus $\mathrm{{\rm Y}}$. (a) Data for different Langevin equations with $L=100$. (b) Data for different system sizes $L$ as obtained from the underdamped Langevin Eq. (20). Inserts: zoom over the critical region.

Figure 7

Finite-size scaling of the helicity modulus $\mathrm{{\rm Y}}$ obtained from the underdamped Langevin Eq. (20). The sizes used are given in the key and the dashed line represents the critical behavior close to the transition; see the text for a discussion.

Figure 8

Temperature dependence of the relaxation time $\tau$ for different Langevin equations with $L=100$ (a) and for different system sizes $L$ obtained from the underdamped Langevin Eq. (20) (b).

Figure 9

Finite-size scaling of the relaxation time $\tau$ obtained from (a) the overdamped Langevin Eq. (15), (b) the underdamped Langevin Eq. (20), (c) the ultrarelativistic limit of the underdamped Langevin Eq. (21a), and (d) the nonrelativistic limit of the underdamped Langevin Eq. (21b). $z_{\mathrm{eq}}=2.1$ and $\nu =0.6703$. The system sizes are given in the keys. Inserts: zoom over the critical region.

Figure 10

Dependence of the averaged vortex density ${\rho }_{\mathrm{vortex}}$ on (a) temperature and (b) inverse temperature for different system sizes $L$. The critical temperature $T_c$ is indicated with a vertical dotted line in both panels. In the inset to panel (a) we zoom over a temperature interval around $T_c$. The meaning of the temperatures $T_{\mathrm{L}}^{\left(\mathrm{M}\right)}$ and $T_{\mathrm{L}}^{\left(\mathrm{S}\right)}$ will be explained below. The fitting parameter $\varepsilon \simeq 6.63$ for ${\rho }_{\mathrm{vortex}}\propto e^{-\varepsilon /T}$ is the thermal activation energy for small vortex rings that describes ${\rho }_{\mathrm{vortex}}$ at small $T$, shown with a dashed line in panel (b). The equilibrium configurations used in this and all other figures in this subsection are generated with the underdamped Langevin Eq. (20).

Figure 11

Example of a unit cube comprising four plaquettes with (shaded) and two plaquettes without (not shaded) nonzero flux, respectively. The vortex elements are shown with arrows and their directions indicate the sign of the fluxes. When four vortex elements pierce a unit cube (left), we face an ambiguity in the two ways of connecting them (mid and right images). Two ways of resolving this ambiguity are explained in the text and in Fig. 13.

Figure 12

The density of recombination as a function of temperature, in equilibrium.

Figure 13

Reconnection of vortex elements using the maximal criterium. Black and gray colored circles show the centers of the unit cubes with a pair of ingoing and outgoing vortex elements while the white circles show those with just one ingoing and one outgoing arrow. In the case of the black circles there are two possible ways of connecting the two ingoing and two outgoing vortex elements [see Fig. 11 and panel (c)], whereas in the case of the gray circles the connection is unique since crossing vortex lines is not allowed [see panel (d)]. In practice, we first draw all vortex line elements passing through the centres of the cubes [see panel (a)], and we then select the connection of elements at all black circles in such a way that the length of the total vortex loop is maximized [panel (b)]. At the gray vertices there is no choice [see panel (d)] and the connection may lead to the separation into two loops, as shown in the example in panels (a) and (b).

Figure 14

Equilibrium snapshots of the system configurations at temperatures (a) $T=0.6T_c$, (b) $T=0.8T_c$, (c) $T=T_c$, and (d) $T=1.2T_c$ in a system with size $L=40$. The vortex line elements are connected with the maximal criterion (upper panels) and the stochastic criterion (lower panels) and they are shown in gray (blue) in the black background. The longest vortex lines in each image are highlighted in light gray (yellow).

Figure 15

Equilibrium vortex length number at various temperatures below and above $T_c$ in (a) and ($\text{c}$) and approaching $T_c$ from below in (b) and (d), for a system with linear size $L=100$. Upper (lower) panels account for the maximal (stochastic) criteria for connecting vortex elements. We take a mean over 100 noise realizations and we further average over 1000 different times for each dynamical run. The dashed line in panels (b) and (d) represent the algebraic decay $l^{-2.17}$. The two power laws in (c) are $l^{-5/2}$ and $l^{-1}$ as indicated in the key.

Figure 16

Equilibrium vortex length distribution per unit volume $N\left(l\right)/L^3$ at $T=0.8T_c$ in (a) and (c), and $T=T_c$ in (b) and (d) for systems with linear sizes $L=40,60,80$, and 100. Upper (lower) panels account for the maximal (stochastic) criteria for connecting vortex elements. In panels (a) and (c), the dashed straight line represents the exponential decay $e^{-0.026l}$ and $e^{-0.052l}$, respectively. Although the data in panel (d) may suggest that the system is at its line percolation threshold, at $T_c$ it is already beyond it, see the text for a discussion.

Figure 17

Stochastically reconnected loop length number density at infinite temperature against $l$ (a) and $N^{\left(\mathrm{S}\right)}l$ against $l/L^2$ (b).

Figure 18

(a) Temperature dependence of the mass parameter $m_{\mathrm{L}}^{(\mathrm{M},\mathrm{S})}$ in the fit Eq. (66) to the vortex length number densities. The insets show $m_{\mathrm{L}}^{(\mathrm{M},\mathrm{S})}$ as a function of $T_{\mathrm{L}}^{(\mathrm{M},\mathrm{S})}-T$ with $T_{\mathrm{L}}^{\left(\mathrm{M}\right)}=0.94T_c$ and $T_{\mathrm{L}}^{\left(\mathrm{S}\right)}=0.98T_c$ in double logarithmic scale together with an algebraic dependence with ${\beta }_{\mathrm{L}}^{(\mathrm{M},\mathrm{S})}=1.7$. (b) Equilibrium vortex length number per unit volume $N^{(\mathrm{M},\mathrm{S})}\left(l\right)/L^3$ at the line-tension point $T_{\mathrm{L}}^{(\mathrm{M},\mathrm{S})}$. Upper (lower) panels account for the maximal (stochastic) criteria for connecting vortex elements.

Figure 19

Finite-size scaling of the bump structure in $N_{\mathrm{L}}^{(\mathrm{M},\mathrm{S})}$ at (a) the vortex line-tension point $T_{\mathrm{L}}^{(\mathrm{M},\mathrm{S})}$ and (b) the critical temperature $T_c$. Upper (lower) panels account for the maximal (stochastic) criterium for connecting vortex elements. The scaling variables are ${\tilde{l}}^{(\mathrm{M},\mathrm{S})}=l/L^{D_{\mathrm{L}}^{(\mathrm{M},\mathrm{S})}}$ and the fractal dimensions are fixed to $D_{\mathrm{L}}^{(\mathrm{M},\mathrm{S})}=2.56$; see the text for a discussion. Their is data collapse in (a) but not in (b).

Figure 20

Temperature dependence of the mean number of vortex loops connected with the maximal rule $N_{\mathrm{loop}}^{\left(\mathrm{M}\right)}$ (a) and with the stochastic rule $N_{\mathrm{loop}}^{\left(\mathrm{S}\right)}$ (b) in both cases normalized by the system size. Several system sizes were used and are given in the key. In the insets, zooms over the peaks.

Figure 21

The size of a vortex loop explained with an example. A vortex loop is shown with a broken solid line made of straight vortex line elements. The minimal rectangular parallelepiped that covers the loop in the $x,y$, and $z$ directions is also shown. The size of the vortex loop is defined as the maximal linear length of the faces of the covering parallelepiped. In the case in the figure, the three linear sizes of the rectangular parallelepiped are $3,2$, and 2, and the size of the vortex loop is $\max (3,2,2)=3$.

Figure 22

Temperature dependence of the number of vortices satisfying different conditions. (a) Ratio between the number of vortex loops larger than the system size $N_{\mathrm{system}\mathrm{size}}^{\left(\mathrm{M}\right)}$ and the total number of vortex loops $N_{\mathrm{loop}}^{\left(\mathrm{M}\right)}$. In the inset, a zoom over the peak. (b) Averaged number of noncontractible vortex loops $N_{\mathrm{noncontract}}^{\left(\mathrm{M}\right)}$. (c) Number of vortex loops larger than the system size $N_{\mathrm{system}\mathrm{size}}^{\left(\mathrm{S}\right)}$. (d) Number of noncontractible loops. In (a) and (b) the maximal criterium is used. In (c) and (d) the stochastic one is used. The system sizes are given in the keys.

Figure 23

Examples of contractible and noncontractible vortex loops. In panel (a) there is one contractible vortex loop, the size of which is larger than the system size. In panel (b) there are two noncontractible vortex loops. The two vortex configurations in panels (a) and (b) can be continuously transformed one into the other through the reconnection of two vortex elements.

Figure 24

Three equilibrium snapshots of the system at $T=T_{\mathrm{L}}^{\left(\mathrm{S}\right)}$. We show in light gray (yellow) the longest vortex loop obtained with the stochastic criterion for the connection of vortex elements. In panel (a) the size of the vortex loop is smaller than the system size $L$. In panel (b) the size of the vortex loop is longer than the system size $L$ both in the horizontal and vertical directions, but it is contractible. In panel (c) the vortex loop is noncontractible in the vertical direction.

Figure 25

In panels (a) and (b) the finite-size scaling of the data in Fig. 22 for the number of vortex loops larger than the system size $N_{\mathrm{system}\mathrm{size}}^{\left(\mathrm{S}\right)}$. In panels (c) and (d) the finite-size scaling of the data in Fig. 22 for the number of noncontractible vortex loops $N_{\mathrm{noncontract}}^{\left(\mathrm{S}\right)}$. The stochastic rule for connecting vortex elements was used. The system sizes are given in the keys. From the scaling analysis, we obtain the exponent ${\nu }_{\mathrm{L}}=0.76$ for $T<T_{\mathrm{L}}^{\left(\mathrm{S}\right)}$ in panels (a) and (c), and ${\nu }_{\mathrm{L}}^{\prime }=0.34$ for $T>T_{\mathrm{L}}^{\left(\mathrm{S}\right)}$ in panels (b) and (d).

Figure 26

Finite-size scaling of the mass parameter $m_{\mathrm{L}}^{\left(\mathrm{S}\right)}$. The sizes used are given in the key and the dotted (blue) line represents the critical behavior close to the transition.

Figure 27

Snapshots of the vortex configurations at (a) $t=2$, (b) $t=3$, (c) $t=4$, and (d) $t=5$, after an instantaneous quench at $t=0$ from equilibrium at $2T_c$. We plot all vortex line elements at the centers of plaquettes with nonzero flux (the total system linear size is $L=40$). The vortex line elements are shown in gray (blue) in the black background and the longest vortex lines in each image are highlighted in light gray (yellow). The configurations are generated with the underdamped Langevin Eq. (20) running at $T=0$. The maximal and stochastic line reconnection criteria were used in the upper and lower panels, respectively. We note that while they influence the vortex tangle at the initial state, the rule used to link the vortex elements becomes irrelevant after a very short time scale, $t\simeq 4$.

Figure 28

The dynamical correlation length ${\xi }_{\mathrm{d}}$ against time $t$ obtained from the overdamped Langevin Eq. (15) (a), the underdamped Langevin Eq. (20) (b), the ultrarelativistic limit of the underdamped Langevin Eq. (21a) (c), and the nonrelativistic limit of underdamped Langevin Eq. (21b) (d) in systems with different system sizes given in the key. The dashed line is $t^{1/2}$ and the dotted line in (a) is $t^{0.43}$ that executes an unexpected better fit to the data at the measuring times.

Figure 29

The time-dependent vortex density ${\rho }_{\mathrm{vortex}}\left(t\right)$. (a) Overdamped Langevin Eq. (15), (b) underdamped dynamics Eq. (20), (c) ultrarelativistic limit of the underdamped Langevin Eq. (21a), and (d) nonrelativistic limit of the underdamped Langevin Eq. (21b). The different curves in each panel are for different system and mesh sizes. The dashed lines are $t^{-1}$ and the dotted line in (a) is $t^{-0.86}$ that, consistently with what we obtained for ${\xi }_{\mathrm{d}}\left(t\right)$, provides a better fit to the data measured with overdamped dynamics.

Figure 30

Dynamic scaling regime. Snapshots of the vortex configurations at (a) $t=5$, (b) $t=10$, (c) $t=15$, and (d) $t=20$, after an instantaneous quench at $t=0$ from equilibrium at $2T_c$. We plot all vortex line elements at the centers of the plaquettes with nonzero flux (the total system linear size is $L=60$). The vortex line elements are shown in gray (blue) in the black background and the longest vortex lines in each image are highlighted in light gray (yellow). The configurations are generated with the underdamped Langevin Eq. (20) running at $T=0$. The reconnection criterium is not important at this time scale.

Figure 31

Very short time scale dynamics. Differences induced by the various microscopic dynamics in the vortex density and dynamic correlation length.

Figure 32

Time-dependent number of vortex loops that are larger than the system size $N_{\mathrm{system}\mathrm{size}}^{(\mathrm{M},\mathrm{S})}$ and time-dependent number of noncontractible vortex loops $N_{\mathrm{noncontractible}}^{(\mathrm{M},\mathrm{S})}$. (a) Overdamped Langevin dynamics Eq. (15), (b) underdamped dynamics Eq. (20), (c) ultrarelativistic limit of the underdamped Langevin Eq. (21a), and (d) nonrelativistic limit of the underdamped Langevin Eq. (21b). The interval over which the algebraic decay of ${\rho }_{\mathrm{vortex}}$ is apparent [see Figs. 29–29] is shown in each panel.

Figure 33

Late epochs. Snapshots of the vortex configurations at (a) $t=50$, (b) $t=100$, (c) $t=150$, and (d) $t=200$, after an instantaneous quench at $t=0$ from equilibrium at $2T_c$. We plot all vortex line elements at the centers of the plaquettes with nonzero flux. The system linear size is $L=100$. The vortex line elements are shown in gray (blue) in the black background and the longest vortex lines in each image are highlighted in light gray (yellow). The configurations are generated with the underdamped Langevin Eq. (20) running at $T=0$. At these times the reconnection rule is irrelevant.

Figure 34

Finite-size scaling plots for (a) the vortex density ${\rho }_{\mathrm{vortex}}\left(t\right)$ and (b) the dynamic correlation length with the dynamical exponent $z_d=2$ obtained from the underdamped Langevin Eq. (20) at $T=0$. The dashed lines show the power laws $t^{-1}$ (a) and $t^{1/2}$ (b).

Figure 35

Finite-size scaling plots of the vortex density ${\rho }_{\mathrm{vortex}}\left(t\right)$ [panels (a) and (b)] and the dynamic correlation length [panels (c) and (d)] obtained from the overdamped Langevin Eq. (20) at $T=0$. The dashed lines in panels (a), (c), and (d) are the predictions with the dynamical exponent $z_d=2$ while the dotted lines in panels (c) and (d) display the algebraic growth with exponent $1/z_d=0.43$. Note the different scaling with $L$ in the horizontal axis in the first and second column.

Figure 36

Early stages of evolution. Time-dependent length number density $N^{\left(\mathrm{M}\right)}(l,t)$ in the initial stage of evolution, $t=1,3,5,7$ of the (a) overdamped Langevin dynamics Eq. (15), (b) underdamped dynamics Eq. (20), (c) ultrarelativistic limit of the underdamped Langevin Eq. (21a), and (d) nonrelativistic limit of the underdamped Langevin Eq. (21b). The maximal reconnection rule was used here to identify the vortex loops. The linear system size is $L=100$. The dashed line is the power law $l^{-{\alpha }_{\mathrm{L}}^{\left(\mathrm{M}\right)}}$ with ${\alpha }_{\mathrm{L}}^{\left(\mathrm{M}\right)}=2.17$, the dotted line the power $l^{-1}$, and the almost vertical peaks at the far right of the plot correspond to length scales that diverge with the system size.

Figure 37

Early stages of evolution. Finite-size scaling of the number density $N^{\left(\mathrm{M}\right)}(l,t)$ at (a) $t=1$, (b) $t=4$, (c) $t=7$, and (d) $t=10$ of the underdamped dynamics Eq. (20), with the maximal reconnection rule to identify the vortex loops. The data are presented in a way that selects the weight at very long $l$. The scaling variable is ${\tilde{l}}^{\left(\mathrm{M}\right)}=l/L^{D_{\mathrm{L}}^{\left(\mathrm{M}\right)}}$.

Figure 38

Early stages of evolution. Time-dependent length number densities $N^{\left(\mathrm{S}\right)}(l,t)$ in the initial stage of evolution, $t=1,3,5,7$ with (a) overdamped Langevin dynamics Eq. (15), (b) underdamped dynamics Eq. (20), (c) the ultrarelativistic limit of the underdamped Langevin Eq. (21a), and (d) the nonrelativistic limit of the underdamped Langevin Eq. (21b). In all cases the stochastic reconnection rule was used to identify the vortex loops. The linear system size is $L=100$. The dark dashed line is the power law $l^{-{\alpha }_{\mathrm{L}}^{\left(\mathrm{S}\right)}}$ with ${\alpha }_{\mathrm{L}}^{\left(\mathrm{S}\right)}=2.17$, the light dotted line is the power law $l^{-1}$ characterizing the large-scale statistics at high temperatures, and the dashed-dotted line is the power law $l^{-5/2}$ of the Gaussian random walks that characterize the finite-size loops at high temperature.

Figure 39

Dynamic scaling regime. Time-dependent length number densities $N(l,t)$ in the early stages of the scaling regime, $t=10,15,20,30$ of the (a) overdamped Langevin dynamics Eq. (15), (b) underdamped dynamics Eq. (20), (c) ultrarelativistic limit of the underdamped Langevin Eq. (21a), and (d) nonrelativistic limit of the underdamped Langevin Eq. (21b). We use here the maximal reconnection rule to identify the vortex loops, although this choice is irrelevant in this regime. The dashed line is the power law $l^{-{\alpha }_{\mathrm{L}}}$ with ${\alpha }_{\mathrm{L}}=2.17$, the dotted line is the power law $l^{-1}$, and we also include a line proportional to $l$ for the statistics of the shortest loops.

Figure 40

Dynamic scaling regime. Time-dependent length number densities scaled as $t^{2}N(l,t)$ as a function of $l/t^{1/2}$ in the early stages of the scaling regime, $t=10,15,20,30$ of the (a) overdamped Langevin dynamics Eq. (15), (b) underdamped dynamics Eq. (20), (c) ultrarelativistic limit of the underdamped Langevin Eq. (21a), and (d) nonrelativistic limit of the underdamped Langevin Eq. (21b). (The maximal reconnection rule was used here although this choice is irrelevant at these times.)

Figure 41

Late epochs—saturation. Time-dependent length number densities $N(l,t)$ in the late stages of the scaling regimes $t=100,200,300,400$ of (a) overdamped Langevin dynamics Eq. (15), (b) underdamped dynamics Eq. (20), (c) ultrarelativistic limit of the underdamped Langevin Eq. (21a), and (d) nonrelativistic limit of the underdamped Langevin Eq. (21b), with the maximal reconnection rule to identify the vortex loops, although this choice is irrelevant at these times. In the insets the total number of loops and the number of noncontractible loops as functions of time. The linear system size is $L=100$.