Dephasing Catastrophe in $4-\epsilon$ Dimensions: A Possible Instability of the Ergodic (Many-Body-Delocalized) Phase

In two dimensions, dephasing by a bath cuts off Anderson localization that would otherwise occur at any energy density for fermions with disorder. For an isolated system with short-range interactions, the system can be its own bath, exhibiting diffusive (non-Markovian) thermal density fluctuations. We recast the dephasing of weak localization due to a diffusive bath as a self-interacting polymer loop. We investigate the critical behavior of the loop in $d=4-\epsilon$ dimensions, and find a nontrivial fixed point corresponding to a temperature $T^{*}\sim \epsilon >0$ where the dephasing time diverges. Assuming that this fixed point survives to $\epsilon =2$, we associate it with a possible instability of the ergodic phase. Our approach may open a new line of attack against the problem of the ergodic to many-body-localized phase transition in $d>1$ spatial dimensions.

Figure 1

Cooperon as a self-interacting polymer loop: The Cooperon $C_{\eta ,-\eta }^t(\mathbf{r},\mathbf{r})$ entering the expression for the weak localization correction [Eq. (1)] is given by the return probability of a diffusing particle interacting with a fluctuating density field bath ${\rho }_{\mathrm{cl}}(\mathbf{r},t)$ [10, 11, 12, 35, 36, 39]. The diffusion process starts and ends at the same point $\mathbf{r}$ from time $-\eta$ to $\eta$. For short-range interactions, the density fluctuations are diffusive and, similar to the virtual diffusion process, controlled by the diffusion constant $D$ [see Eq. (4)]. The fluctuation-averaged Cooperon (return probability) in Eq. (3) can be interpreted as the path integral of a self-interacting polymer loop. The action is given by Eqs. (3b) and (3c), where $S_0$ is the action for the unperturbed random walk, and $S_I$ is the self-interaction term. The first term in $S_I$ describes the repulsive (causal) interaction between $\mathbf{r}({\tau }_1)$ and $\mathbf{r}({\tau }_2)$ of range $\sqrt{D_c|{\tau }_1-{\tau }_2|}$, and is indicated by the red dotted line in the figure; the second term, denoted by the blue dashed line, represents the attractive (anticausal) interaction between $\mathbf{r}({\tau }_1)$ and $\mathbf{r}(-{\tau }_2)$ of the same range. The intrinsic length scale of the polymer loop is determined by the dephasing length $L_{\varphi }=\sqrt{D{\tau }_{\varphi }}$, which should diverge as the system approaches the MBL transition point from the ergodic phase [1].

Figure 2

One-loop RG flow in spatial dimensions $d=4-\epsilon$. Panel (a) depicts the RG flow in the $g–r$ plane with $h=0$, $\xi =1$, and $\epsilon =0.1$. Here, the coupling constant $g$ is proportional to $k_{B}T/D$ (in $d=2$). The nontrivial fixed point denoted by the red dot has one relevant and two irrelevant directions. (The relevant direction and one irrelevant direction lie in the $g–r$ plane and are indicated by the red dashed lines.) The dephasing rate ${\tau }_{\varphi }^{-1}=2(r-r^{*})$ vanishes at the critical fixed point corresponding to temperature $T^{*}>0$. Above $T^{*}$, the system flows towards the ergodic metal phase where the dephasing rate ${\tau }_{\varphi }^{-1}$ is nonzero. The RG analysis suggests that the system undergoes a temperature-tuned transition that can be considered as a possible instability of the ergodic phase (in $d=2$). Panel (b) shows the RG flow in the $g–h$ plane.