Functional renormalization group study of a dissipative Bose--Hubbard model

TL;DR

This work studies a one-dimensional dissipative Bose–Hubbard model where each site couples to an independent bath, producing non-Markovian dissipation. Using a nonperturbative functional renormalization group, it uncovers a complete low-energy phase diagram featuring a line of Luttinger-liquid fixed points and a dissipative fixed point, separated by a bath-dependent BKT transition. The dissipative fixed point exhibits finite compressibility and vanishing superfluid stiffness, with universal scaling exponents that depend on the bath exponent $s$, and subleading corrections are characterized. The results provide a unified, systematically improvable framework for dissipative quantum phases in one dimension and demonstrate how FRG can interpolate between weak and strong dissipation from a single microscopic action.

Abstract

We investigate the phase diagram of a one-dimensional dissipative Bose-Hubbard model using the nonperturbative functional renormalization group (FRG). Each lattice site is coupled to an independent bath, generating long-range temporal interactions that encode non-Markovian dissipation. For a broad class of bath spectra -- ohmic, sub-ohmic, and super-ohmic -- we identify two competing low-energy regimes: a Luttinger-liquid line of fixed points and a dissipative fixed point characterized by finite compressibility, vanishing superfluid stiffness, and universal scaling exponents, separated by a Berezinskii-Kosterlitz-Thouless transition. The FRG framework is essential here, as it provides access to the complete renormalization group flow and all fixed points from a single microscopic action, beyond the reach of perturbative or variational methods. This work establishes a unified and systematically improvable framework for describing dissipative quantum phases in one dimension.

Figures (5)

• Figure 1: Schematic picture of the dissipative bosonic system considered in this work. Bosons are constrained to live on a one-dimensional lattice with a nearest-neighbour hopping amplitude $t$ and an on-site repulsion $U$. Each site is coupled to an independent bath characterized by a coupling $\alpha$ and a spectral exponent $s$, which characterize the low-energy spectrum of the bath.
• Figure 2: RG flow for ohmic ($s=1$) dissipation. The trajectories flow either toward the continuum of LL fixed points (blue) or toward the DFP (yellow). The flow around $(K_c,\tilde{y}_k)=(1/2,0)$ is that of a BKT transition with two (incoming and outgoing) separatrices drawn in black. Near the DFP, all trajectories flow into a single "large river" Bagnuls_2001aBagnuls_2001b.
• Figure 3: Left panels (top to bottom): RG flow of the running dynamical exponent $z_k$ and exponents $\eta_{\tau,k}$, $\eta_{0,k}$, and $\eta_{2,k}$ for trajectories flowing to the DFP. The RG time is $-t=\ln (\Lambda/k)$. The different lines correspond to $s=0.35$ (dashed), $s=0.4$ (dotted), $s=1$ (solid), and $s=1.5$ (dash-dotted). The rapid evolution of $\eta_{\tau,k}$ for $s=0.4$ at $-t \simeq 11$ is continuous; it gets more peaked as $s \to 0.5^-$. Right panels: asymptotic values of $z_k$, $\eta_{\tau,k}$, $\eta_{0,k}$, and $\eta_{2,k}$ as a function of $s$. The black curves are the analytical predictions obtained from the expansion about the DFP described in the main text.
• Figure 4: RG flow for super-ohmic ($s=1.5$) dissipation. The RG trajectories either flow to a continuum of LL fixed points in blue, or to the DFP in yellow. The BKT transition is at $K_c=1-s/2$ and $\tilde{y}_k=0$.
• Figure 5: RG flow for sub-ohmic ($s=0.35$) dissipation in the three-dimensional space $(K_k, \tilde{y}_{0,k}, \tilde{y}_{2,k})$. After a transient regime where $\tilde{y}_{2,k}$ increases rapidly, the RG trajectories either flow to a continuum of LL fixed points in blue, or to the DFP in yellow. The BKT transition occurs at $K_c=1- s/2$ and $\tilde{y}_{0,k}=\tilde{y}_{2,k}=0$. The DFP attracts all trajectories through a “large river” flow. Because the LL fixed points form a continuum, they are reached in a "large surface" which, here, is asymptotically defined by $\tilde{y}_{2,k}=0$. The initial values $\tilde{y}_{2,\Lambda}$ were taken small but finite, as the numerical flow equations become ill-behaved at $\tilde{y}_{2,k}=0$.