Mapping a dissipative quantum spin chain onto a generalized Coulomb gas

1- exact mapping between a dissipative spin chain and a generalized Coulomb gas
2- RG equations are solved for sub- and superohmic baths
3- well written and well explained
4- an intuitive mechanism for the dissipation-driven quantum phase transition is developed

Quantum many-body systems coupled to an environment have gained increasing interest in the last years and the coupling to a bath can drive interesting quantum phase transitions. In this work, the author considers a one-dimensional XXZ chain where each local spin operator $S^{z}_i$ is coupled to an independent dissipative bosonic bath. The paper describes a mapping from a bosonized action for the dissipative chain to a generalized Coulomb gas. In particular, the Coulomb-gas approach provides an intuitive picture for the dissipation-induced unbinding transition, which includes a new set of particles originating from a bath-induced long-range interaction. One of the main results is the difference between sub- and superohmic baths which become relevant at different Luttinger parameters.

This work provides a link between a quantum many-body problem that is of current interest and a stat-mech problem that has an intuitive understanding. The paper includes sufficient details to follow the derivation, but also puts a lot of emphasis in describing the physical picture in simple words. I enjoyed reading the paper and learned a lot from the discussion. All in all, I recommend publication in SciPost Physics.

1- I am wondering about the decay of the dissipation kernel $D(x,\tau)$ in Eq. (14) and its dependence on the bath exponent $s$. For $\nu=0$ it becomes $D(x,\tau)\propto \delta(x)/(\tau/\tau_c)^{2+s}$. In particular, for an ohmic bath with $s=1$ this leads to a retarded interaction that decays as $1/\tau^3$. I always thought that an ohmic bath corresponds to a temporal decay $\propto 1/\tau^2$, as it is also the case in Ref. [33] (a closely related paper by the author). It would be good to clarify this difference to the author's previous work (or correct it if it is a mistake).

2- On page 8 [below Eq. (20)], the author writes $D(r-r)$. Isn't it just $D(r)$?

3- First line on page 11 [above Eq. (24)]: The formatting of the in-line equation $U=\dots$ is not clear because of the minus sign in front of $V_c(r)$. Can one just put it in front of the integral?

4- On page 12 [the line below Eq. (27)]: I assume it is $K_R$ instead of $K_r$, right?

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- derivation of a mapping of the Caldeira-Leggett action of a quantum sine-Gordon model coupled to a harmonic oscillator bath to a classical generalized Coulomb gas.
- renormalization group study of the Berezinskii-Kosterlitz-Thouless transition of the generalized Coulomb gas

The article reports on a mapping of a XXZ spin-1/2 chain interacting with a phonon bath on a generalized classical Coulomb gas. The mapping allows to identify a Berezinskii-Kosterlitz-Thouless transition between a Tomonaga-Luttinger liquid phase and a dissipative phase in the case of superohmic dissipation using renormalization group equations. With ohmic dissipation, the BKT transition occurs at the same point as the transition to the Ising antiferromagnet phase, and in the subhomic case, it is preempted by the formation of the Ising antiferromagnet. The article is very clearly written, with all the necessary details of derivations described in Appendices A-E.
The abstract and main results section describe the main findings which are also exposed in the first paragraph of the conclusions. The relevant previous work is covered in Refs. [21-34] and Ref. [38].
The article establishes an interesting link between quantum phase transitions in the presence of dissipation induced by a harmonic bath and classical statistical mechanics of generalized Coulomb gases. Phase transitions in dissipative quantum mechanics have undergone a recent regain of attention from theoretician, and a connection with a well studied subject such as Coulomb gas is likely to yield quick progress. An aspect which is not touched in the paper, but considered in Refs. [30,31] for a different model is the characterization of the phases in which dissipation dominates. The present article is likely to stimulate such studies. I thus recommend publication in SciPost.

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