integrate out modes, the couplings themselves are renormalized. For each given coupling constant $\gamma_{i}$, which appears in the Hamiltonian as $H=H_0+\cdots + \int\!  dx \, \gamma_{i}
\mathcal{O}_i$, we can define a [*dimensionless*]{} coupling $\breve{\gamma}_i = \gamma_i/(v\Lambda_\ell^2)$, which is measured relative to the typical magnitude of the terms in the free bosonic field theory. Equivalently, division by both a factor of $v\Lambda_\ell$ (the typical energy at this scale) and an additional factor of $\Lambda_\ell$ (a typical inverse length) are needed to render $\gamma_i$, which is an energy density, dimensionless. To linear order, each dimensionless coupling “flows” according to the RG equation $$\label{eq:RGgeneral}
  \partial_\ell \breve{\gamma}_{i} = (2-\Delta_{i})\breve{\gamma}_{i}.$$ Note that the factor of $2$ in this equation, which comes from the normalization by $1/\Lambda_\ell^2$, is equivalent to the space-time dimensionality of the (1+1)-dimensional field theory of the spin chains. In RG schemes in which space-time is rescaled to keep the cutoff fixed, this factor arises directly from that rescaling. We prefer to formulate the RG without rescaling in this paper, so that all lengths, times, energies, etc. are explicit.

Many of the above operators may be [*relevant*]{} in the RG sense. This means that, with increasing $\ell$, the dimensionless coupling constants [*increase*]{}, which 