Kinetic theory of nonthermal fixed points in a Bose gas

We outline a kinetic theory of nonthermal fixed points for the example of a dilute Bose gas, partially reviewing results obtained earlier, thereby extending, complementing, generalizing, and straightening them out. We study universal dynamics after a cooling quench, focusing on situations where the time evolution represents a pure rescaling of spatial correlations, with time defining the scale parameter. The nonequilibrium initial condition set by the quench induces a redistribution of particles in momentum space. Depending on conservation laws, this can take the form of a wave-turbulent flux or of a more general self-similar evolution, signaling the critically slowed approach to a nonthermal fixed point. We identify such fixed points using a nonperturbative kinetic theory of collective scattering between highly occupied long-wavelength modes. In contrast, a wave-turbulent flux, possible in the perturbative Boltzmann regime, builds up in a critically accelerated self-similar manner. A key result is the simple analytical universal scaling form of the nonperturbative many-body scattering matrix, for which we lay out the concrete conditions under which it applies. We derive the scaling exponents for the time evolution as well as for the power-law tail of the momentum distribution function, for a general dynamical critical exponent $z$ and an anomalous scaling dimension $\eta$. The approach of the nonthermal fixed point is, in particular, found to involve a rescaling of momenta in time $t$ by $t^{\beta }$, with $\beta =1/z$, within our kinetic approach independent of $\eta$. We confirm our analytical predictions by numerically evaluating the kinetic scattering integral as well as the nonperturbative many-body coupling function. As a side result, we obtain a possible finite-size interpretation of wave-turbulent scaling recently measured by Navon et al.

Figure 1

Universal dynamics of a dilute Bose gas, as induced by cooling quenches of different strength. (a) Weak cooling quench. Sketch of the time evolution of the single-particle radial number distribution $n(p,t)$ as a function of momentum $p$ at three different times $t$ (gray short-dashed, solid, and long-dashed lines). Starting from the initial distribution $n(p,t_0)$ (blue dashed line) after a weak quench, removing part of the thermal tail (gray shaded area), an inverse wave-turbulent cascade develops, transporting particles toward lower momenta. See text for more details. (b) Strong cooling quench. The initial distribution $n(p,t_0)$ (red dashed line) is created by a strong cooling quench which removes the thermal tail (gray shaded area). In the aftermath, a strong bidirectional redistribution of particles in momentum space (arrows) occurs. This eventually builds up a quasicondensate in the infrared while refilling the thermal tail at large momenta. In (a) and (b), the particle transport toward zero as well as large momenta is characterized by self-similar scaling evolution in space and time, $n(p,t)=t^{\alpha }f\left(t^{\beta }p\right)$, with characteristic scaling exponents $\alpha$, $\beta$, in general different for the two directions. The infrared transport moves particles to low-$p$ modes (gray arrow) while their energy is deposited by the scattering of much fewer particles to higher momenta (green arrow), conserving total energy and particle number. Note the double-logarithmic scale. (c) Schematic sketch of the cooling quenches in the equilibrium phase diagram of the dilute Bose gas, leading to the same final $\mu =g\rho$, for illustrating the initial and long-time states of the system, before the quench and after reequilibration, respectively (cf. Appendix pp2). We emphasize that the dynamics ensuing the quenches can not be described within this equilibrium plane.

Figure 2

Sketch of the self-similar evolution of the scaling form (41) for $n_Q(p,t)$ according to Eq. (39). Note the double-logarithmic scale. The IR cutoff scale $p_{\mathrm{\Lambda }}$ and the UV scale $p_{\lambda }$, as well as the amplitude $f_1$ rescale with time $t$ such that the total quasiparticle density remains invariant. The sketch shows the case of an inverse particle transport following a strong cooling quench. See Table 2 for our predictions for the scaling exponents (first row, NTFP).

Figure 3

Sketch of the buildup of the inverse quasiparticle cascade, defined as a scaling evolution of the form (49) for $n_Q(p,t)$ according to Eq. (50), with $\alpha =\beta \kappa$ and, to leading order, ${\alpha }^{\prime }={\beta }^{\prime }=0$. Note the double-logarithmic scale. The IR cutoff scale $p_{\mathrm{\Lambda }}$ shifts without significantly changing the total quasiparticle density. Only a small non-leading-order rescaling of the UV scale $p_{\lambda }$ is required to satisfy number conservation. The sketch shows the case of an inverse particle transport following a weak cooling quench. See Table 2 for our predictions for the scaling exponents. In the case of a weak-wave-turbulence quasiparticle cascade, $p_{\mathrm{\Lambda }}$ shifts in an accelerated way, with $\tau$ replaced by ${\tau }^{*}$ [see Eq. (51)].

Figure 4

Effective coupling $g_{\mathrm{eff}}(p_0,p)$ in $d=3$ dimensions as a function of the spatial momentum $p=\left|\mathbf{p}\right|$, on a double-logarithmic scale. The figure shows cuts in the $p_0$–$p$ plane, with $p_0=0.5{\varepsilon }_{\mathbf{p}}$ (dark lines) and $p_0=1.5{\varepsilon }_{\mathbf{p}}$ (transparent bright lines). Different colors (line styles) refer to different infrared cutoff scales $p_{\mathrm{\Lambda }}$ as listed in the legend. $p_{\mathrm{\Lambda }}$ is set by the scaling form of the occupation-number distribution entering the nonperturbative coupling function (see, e.g., Fig. 2). All momenta are measured in units of the “healing”-length wave number $p_{\mathrm{\Xi }}={\left(2g{\rho }_{\mathrm{nc}}m\right)}^{1/2}$ which is set by the noncondensed particle density ${\rho }_{\mathrm{nc}}$. Note that $p_{\mathrm{\Xi }}$ sets the scale separating the perturbative region at large $p$ from the nonperturbative collective-scattering region within which the coupling assumes the form given in Eq. (74). See Fig. 5 for the dependence in the full $p_0\text{−}p$ plane.

Figure 5

Contour plot of the effective coupling function $g_{\mathrm{eff}}(E,p)/g$ of free particles, for $d=3$, as a function of $E=2m{p}^0$ and $p=\left|\mathbf{p}\right|$, for $p_{\mathrm{\Lambda }}={10}^{-3}{p}_{\mathrm{\Xi }}$. This function enters the $T$ matrix as defined in Eq. (71). Cuts through this function, for $E=1.5p^2$ and $E=0.5p^2$ (black dashed lines), are shown in Fig. 4. The quasiparticle distribution $n_Q$ was chosen to scale with $\kappa =3.5$. The coupling function in the collective-scattering region (73), is well described by the universal form (74) which does not depend on $\kappa$. The scattering resonance marked by the dark red “ridge” is centered around the Bogoliubov-type energy-momentum transfer $E={[2p_{\mathrm{\Xi }}^2{p}^2+p^4]}^{1/2}$.

Figure 6

Left panel: dependence of the perturbative Boltzmann scattering integral $I\left[n_Q\right]\left(p\right)$ for free particles ($z=2$, $\eta =0$; $n=n_Q$) in $d=3$ spatial dimensions, at the momentum $p=1.5p_{\mathrm{\Xi }}$, on the momentum scaling exponent $\kappa$ characterizing the occupation-number distribution $n\left(p\right)\sim p^{-\kappa }$. The vertical dashed lines mark, from the left, the thermal zero at ${\kappa }_T=2$, the particle-cascade exponent ${\kappa }_Q=\frac{7}{3}$, and the energy-cascade exponent ${\kappa }_P=3$. In the figure, the rescaled integral $\tilde{I}\left[n_Q\right](p/p_{\mathrm{\Xi }})={\left(2\pi \right)}^3{p}_{\mathrm{\Xi }}^{3\kappa -4}{\left(2m{g}^2{\mathrm{\Lambda }}^{3\kappa }\right)}^{-1}I\left[n_Q\right]\left(p\right)$ is shown [see Eq. (E8) for the definition of $\mathrm{\Lambda }]$. The different colors (line styles) correspond to different values of the IR cutoff $p_{\mathrm{\Lambda }}$, as indicated in the legend. As the cutoff is lowered, the zeros approach the predicted values. The sign of the slope $\partial I\left[n_Q\right]/\partial \kappa$ at the zeros determines the direction of the cascade. Right panel: scaling exponents $\kappa$ of occupation-number distribution $n\left(p\right)\sim p^{-\kappa }$ for which the perturbative Boltzmann scattering integral $I\left[n\right]\left(p\right)$ for free particles ($z=2$, $\eta =0$; $n=n_Q$) in $d=3$ spatial dimensions has a zero, for different momenta $p$ in units of the IR cutoff scale $p_{\mathrm{\Lambda }}$. Lower line: particle cascade for which $\kappa$ approaches ${\kappa }_Q=\frac{7}{3}$ for $p_{\mathrm{\Lambda }}\to 0$. Upper line: zeros of the energy cascade, approaching ${\kappa }_P=3$. We emphasize that the two lines represent different physical situations and, as ${\kappa }_Q=\frac{7}{3}<d=3$, due to constraints from conservation laws, only the inverse quasiparticle cascade is relevant in an isolated system (cf. Sec. 2c1).

Figure 7

Left panel: dependence of the nonperturbative scattering integral $I\left[n\right]\left(p\right)$ in the collective-scattering regime, for free particles ($z=2$, $\eta =0$; $n=n_Q$) in $d=3$ spatial dimensions, at the momentum $p=0.001p_{\mathrm{\Xi }}$, on the momentum scaling exponent $\kappa$ characterizing the occupation-number distribution $n\left(p\right)\sim p^{-\kappa }$. The vertical dashed lines mark, from the left, the inverse particle-cascade exponent ${\kappa }_Q=\frac{11}{3}$ and the direct energy-cascade exponent ${\kappa }_P=\frac{13}{3}$. In the figure, the rescaled integral $\tilde{I}\left[n_Q\right](p/p_{\mathrm{\Xi }})={\left(2\pi \right)}^3{p}_{\mathrm{\Xi }}^{3\kappa -4}{\left(2m{g}^2{\mathrm{\Lambda }}^{3\kappa }\right)}^{-1}I\left[n_Q\right]\left(p\right)$ is shown [see Eq. (E8) for the definition of $\mathrm{\Lambda }]$. The different colors (line styles) correspond to different values of the IR cutoff $p_{\mathrm{\Lambda }}$, as indicated in the legend. As the cutoff is lowered, the zeros approach the predicted values. The sign of the slope $\partial I\left[n\right]/\partial \kappa$ at the zeros determines the direction of the cascade. Note that the slope at ${\kappa }_P\simeq \frac{13}{3}$ is finite and positive. Right panel: scaling exponents $\kappa$ of occupation-number distribution $n\left(p\right)\sim p^{-\kappa }$ for which the nonperturbative Boltzmann scattering integral $I\left[n\right]\left(p\right)$ for free particles ($z=2$, $\eta =0$; $n=n_Q$) in $d=3$ spatial dimensions has a zero. The figure applies to the IR region of large occupation numbers where the effective many-body coupling describing collective scattering scales with ${\gamma }_{\kappa }=2$ and modifies the scaling properties. The colors (line styles) mark different choices of the IR cutoff scale $p_{\mathrm{\Lambda }}$. The upper line corresponds to an energy cascade and approaches ${\kappa }_P=d+4/3$ for $p_{\mathrm{\Lambda }}\to 0$, as obtained from Eq. (F8) with ${\gamma }_{\kappa }=2$, while the lower line approaches the particle-cascade exponent ${\kappa }_Q=d+2/3$ [cf. Eq. (F7)]. The exponent ${\kappa }_S=4$ [cf. Eq. (140)], characterizing the scaling function of the self-similar evolution which is the only one relevant in the isolated system after a quench is marked by the middle dashed line.

Figure 8

Momentum dependence of the scattering integral (54) multiplied by ${p^{\prime }}^4={(p/p_{\mathrm{\Lambda }})}^4$, for free particles ($z=2$, $\eta =0$, $n=n_Q$) in $d=3$ dimensions. The integral is evaluated in the collective-scattering regime, with effective many-body $T$-matrix, Eqs. (71), (80), (E6), for different values of the IR cutoff scale $p_{\mathrm{\Lambda }}$ (distinguished by color and line style), and the results are scaled on top of each other by showing $p^{\prime 4}{\tilde{I}}_S\left[n_Q\right]\left(p^{\prime }\right)={(p/p_{\mathrm{\Xi }})}^4{\left(2\pi \right)}^3{p}_{\mathrm{\Xi }}^{3\kappa -4}{\left(2m{g}^2{\mathrm{\Lambda }}^{3\kappa }\right)}^{-1}I\left[n_Q\right]\left(p\right)$. The horizontal plateau demonstrates the power-law dependence $I\left[n_Q\right]\left(p\right)\sim p^{-4}$ predicted by ${\kappa }_S=4$, cf. Eq. (140).

Figure 9

Exemplary momentum distribution at a nonthermal fixed point [$n_{\mathrm{NTFP}}\left(p\right)$, solid line] vs prequench thermal Bose-Einstein distribution in the Rayleigh-Jeans regime $\left|\mu \right|\ll k_{\mathrm{B}}{T}_{\mathrm{c}}$ (short-dashed line). All particles to the right of the red long-dashed vertical line at $p=p_{\mathrm{q}}$ are removed in the strong cooling quench $p_{\mathrm{q}}<\zeta p_{T_{\mathrm{c}}}$ (shaded area, see main text). In $d=3$ dimensions, according to our kinetic theory, the NTFP distribution of modes with frequency $\omega \left(p\right)\sim p^2$ shows power-law scaling as $n_{\mathrm{NTFP}}\left(p\right)\sim p^{-4}$ and evolves self-similarly in time [see Fig. 1]. The figure shows the special, transient case of such a distribution with a single momentum power law (black solid line). As the evolution conserves the total density $\rho ={\int }_{\mathbf{k}}n\left(\mathbf{p}\right)$, the point $[p_{\mathrm{\Lambda }}\left(t\right),n_{\mathrm{NTFP}}(p_{\mathrm{\Lambda }}\left(t\right))]$ moves along the gray fine-dotted arrow $n_{\mathrm{NTFP}}(p_{\mathrm{\Lambda }}\left(t\right))\sim p_{\mathrm{\Lambda }}{\left(t\right)}^{-3}$. The momentum scale where the prequench distribution is cut off in the UV, corresponding approximately to the prequench critical temperature $p_{T_{\mathrm{c}}}\sim {\left(m{T}_{\mathrm{c}}\right)}^{1/2}$, is outside the graph's frame. The gray vertical arrows mark the momenta corresponding to the quench scale $p_{\mathrm{q}}\sim \epsilon \zeta p_{T_{\mathrm{c}}}$, with prequench diluteness parameter $\zeta$ and some positive number $\epsilon <1$, the postquench chemical potential scale $p_{\mathrm{\Xi }}={\left(g{\rho }_q\right)}^{1/2}={\zeta }_q^{1/2}{p}_{T_{\mathrm{c},\mathrm{q}}}$, with postquench diluteness parameter ${\zeta }_{\mathrm{q}}$ and critical temperature scale $p_{T_{\mathrm{c},\mathrm{q}}}$, and the IR and UV cutoff scales $p_{\mathrm{\Lambda }}\sim {\epsilon }^{3/4}{p}_{\mathrm{\Xi }}$ and $p_{\lambda }\sim {\epsilon }^{1/4}{p}_{\mathrm{\Xi }}$, at the moment when the scaling function is given by the solid line and still entirely within the nonperturbative regime. The regime where the perturbative wave-Boltzmann equation is applicable [$n\left(p\right)\ll {\left(ap\right)}^{-1}$, cf. Eq. (B4)] and [98], is marked by the red dashed area. See Appendices pp2-s2 and pp2-s3 for details.

Figure 10

Diagrams which are contributing to ${\mathrm{\Gamma }}_2\left[G\right]$ and are relevant in this work. (a) The two lowest-order diagrams of the loop expansion which lead to the quantum-Boltzmann equation and thus to the coupling $g_{\mathrm{eff}}=g$ in the perturbative region. Black dots represent the bare vertex $\sim g\delta (x-y)$, solid lines the propagator $G(x,y)$. (b) Diagram representing the resummation approximation which, in the IR, replaces the diagrams in (a) and gives rise to the modified scaling of the $T$ matrix. (c) The wiggly line is the effective coupling function which is represented as a sum of bubble-chain diagrams. Summation of the geometric series gives the expression in Eq. (E1).

Figure 11

Effective coupling $g_{\mathrm{eff}}(E,p)$ as a function of momentum $p$. Shown are cuts in the $p\text{−}E$ plane, with $E=0.99p^2$ (red solid, green dotted-dashed, and blue double-dotted-dashed lines), $E=0.999p^2$ (blue dashed), $E=0.9999p^2$ (blue dotted). Different colors refer to different infrared cutoff scales $p_{\mathrm{\Lambda }}$ as listed in the legend.

Figure 12

Effective coupling $g_{\mathrm{eff}}(E,p)$ for $E=1.5p^2$, $p_{\mathrm{\Lambda }}={10}^{-3}{p}_{\mathrm{\Xi }}$. The panels show the coupling as a function of (a) $p$, for different $2\le \kappa \le 4$ (see legend), $p_{\lambda }\to \infty$; (b) of $\kappa$, for $p={10}^{-2}{p}_{\mathrm{\Lambda }}$ and $p_{\lambda }={10}^5{p}_{\mathrm{\Xi }}$ [dashed line: approximate scaling (E33)]; (c) of $p_{\lambda }$, for $p={10}^{-2}{p}_{\mathrm{\Lambda }}$, $\kappa =0$, 1, and 2 (see legend); (d) of $p$, for $\kappa =0$ and different $p_{\lambda }$ as labeled; (e) of $p$, for $\kappa =2$ and different $p_{\lambda }$ as labeled.

Figure 13

Contour plot of $g_{\mathrm{eff}}(E,p)/g$ defined in Eq. (E1), for interactions of Bogoliubov quasiparticles, with ${\mathrm{\Pi }}^R(E,p)$ given in Eq. (E37), for $\kappa =3.5$ and $p_{\mathrm{\Lambda }}={10}^{-3}{p}_{\mathrm{\Xi }}$. In leading-order approximation the coupling function does not depend on $\kappa$.

Figure 14

Effective coupling $g_{\mathrm{eff}}^2(E,p)$ as a function of momentum $p$. Shown are different cuts in the $p\text{−}E$ plane, with (a) $E=0.5p$ and (b) $E=1.5p$. Different colors (line styles) refer to different infrared cutoff scales $p_{\mathrm{\Lambda }}$ as listed in the legends.

Figure 15

Left panel: dependence of the perturbative wave-Boltzmann scattering integral $I\left[n_Q\right]\left(p\right)$ for Bogoliubov sound $(z=1,\eta =0)$ in $d=3$ spatial dimensions, at the momentum $p=0.1p_{\mathrm{\Xi }}$, on the momentum scaling exponent $\kappa$ characterizing the occupation-number distribution $n_Q\left(p\right)\sim p^{-\kappa }$. The vertical dashed lines mark, from the left, the thermal zero at ${\kappa }_T=1$, the inverse quasiparticle-cascade exponent ${\kappa }_Q=\frac{4}{3}$, and the direct energy-cascade exponent ${\kappa }_P=\frac{5}{3}$. In the figure, $\tilde{I}\left[n_Q\right](p/p_{\mathrm{\Xi }})=2^{3/2}{\left(2\pi \right)}^3{p}_{\mathrm{\Xi }}^{3\kappa -1}{\left[m{g}^2{\left(p_{\xi }{\mathrm{\Lambda }}^{\kappa }\right)}^3\right]}^{-1}I\left[n_Q\right]\left(p\right)$ is shown. The different colors (line styles) correspond to different values of the IR cutoff $p_{\mathrm{\Lambda }}$, as indicated in the legend. As the cutoff is lowered, the zeros approach the predicted values. The sign of the slope $\partial I\left[n_Q\right]/\partial \kappa$ at the zeros determines the direction of the cascade. Note that the slope at ${\kappa }_P\simeq \frac{5}{3}$ is finite and positive. Right panel: scaling exponents $\kappa$ of the quasiparticle distribution $n_Q\left(p\right)\sim p^{-\kappa }$ for which the perturbative wave-Boltzmann scattering integral $I\left[n_Q\right]\left(p\right)$ in $d=3$ spatial dimensions has a zero, for different momenta $p$ in units of the IR cutoff scale $p_{\mathrm{\Lambda }}$. As in Fig. 6, the lower data correspond to the particle cascade, with the zeros approaching ${\kappa }_Q=\frac{4}{3}$, the upper data to the energy cascade, approaching ${\kappa }_P=\frac{5}{3}$ as labeled.

Figure 16

Left panel: dependence of the nonperturbative scattering integral $I\left[n_Q\right]\left(p\right)$ for Bogoliubov sound $(z=1,\eta =0)$ in $d=3$ spatial dimensions, at the momentum $p=0.002p_{\mathrm{\Xi }}$, on the momentum scaling exponent $\kappa$ characterizing the occupation-number distribution $n_Q\left(p\right)\sim p^{-\kappa }$. The vertical dashed lines mark, from the left, the quasiparticle-cascade exponent ${\kappa }_Q=\frac{8}{3}$ and the energy-cascade exponent ${\kappa }_P=3$. In the figure, $\tilde{I}\left[n_Q\right](p/p_{\mathrm{\Xi }})=2^{3/2}{\left(2\pi \right)}^3{p}_{\mathrm{\Xi }}^{3\kappa -1}{\left[m{g}^2{\left(p_{\xi }{\mathrm{\Lambda }}^{\kappa }\right)}^3\right]}^{-1}I\left[n_Q\right]\left(p\right)$ is shown. The different colors (line styles) correspond to different values of the IR cutoff $p_{\mathrm{\Lambda }}$, as indicated in the legend. As the cutoff is lowered, the zeros approach the predicted values. The sign of the slope $\partial I\left[n_Q\right]/\partial \kappa$ at the zeros determines the direction of the cascade. Right panel: scaling exponents $\kappa$ of the quasiparticle distribution $n_Q\left(p\right)\sim p^{-\kappa }$ for which the nonperturbative wave-Boltzmann scattering integral $I\left[n_Q\right]\left(p\right)$ in the IR collective-scattering region, in $d=3$, has a zero, for different momenta $p$ in units of the IR cutoff scale $p_{\mathrm{\Lambda }}$, for three different $p_{\mathrm{\Lambda }}$. As in Fig. 15, the lower data mark the quasiparticle cascade, with the zeros approaching ${\kappa }_Q=\frac{8}{3}$, while the upper data mark the energy cascade, approaching ${\kappa }_P=3$ as labeled.