# Renormalization group for open quantum systems using environment temperature as flow parameter

### Submission summary

 As Contributors: Konstantin Nestmann · Maarten Wegewijs Arxiv Link: https://arxiv.org/abs/2111.07320v1 (pdf) Date submitted: 2021-11-16 09:11 Submitted by: Nestmann, Konstantin Submitted to: SciPost Physics Academic field: Physics Specialties: Quantum Physics Approach: Theoretical

### Abstract

We present the $T$-flow renormalization group method, which computes the memory kernel for the density-operator evolution of an open quantum system by lowering the physical temperature $T$ of its environment. This has the key advantage that it can be formulated directly in real time, making it particularly suitable for transient dynamics, while automatically accumulating the full temperature dependence of transport quantities. We solve the $T$-flow equations numerically for the example of the single impurity Anderson model. In the stationary limit, readily accessible in real-time for voltages on the order of the coupling or larger, we benchmark using results obtained by the functional renormalization group, density-matrix renormalization group and the quantum Monte Carlo method. Here we find quantitative agreement even in the worst case of strong interactions and low temperatures, indicating the reliability of the method. For transient charge currents we find good agreement with results obtained by the 2PI Green's function approach. Furthermore, we analytically show that the short-time dynamics of both local and non-local observables follow a "universal" temperature-independent behavior when the metallic reservoirs have a flat wide band.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission 2111.07320v1 on 16 November 2021

## Reports on this Submission

### Strengths

1 - Interesting generalization of fRG ideas, allowing one to study transient phenomena
2 - Clear presentation of the method and thorough analysis of the model
3 - Synergy of analytics and numerics

### Weaknesses

1 - A connection of the proposed scheme to other RG approaches could be described in a more extensive way
2 - Consideration of the paper appears too formal; a discussion in more physical (qualitative) terms should be extended
3 - The method is initially formulated for the general case of distinct temperatures of reservoirs; however, all the results are given for the case of equal temperatures

### Report

This is a nice attempt to formulate a renormalization-group approach to interacting systems by explicitly using the physical temperature as a flowing scale. The authors applied the developed machinery to the Anderson model of an interacting quantum dot and demonstrated a quantitative agreement between their results and those obtained by other numerical methods.

Let me comment on the points that could be improved in the manuscript.

1. A connection of the proposed scheme to other RG approaches could be described in a more extensive way. Indeed, usually temperature serves as an infrared cutoff for the running energy scale, e.g., in a poor-man scaling approach. It would be interesting to see this connection on a formal level, comparing the "conventional" RG approach with temperature serving only as a cutoff with the framework developed here. For this type of comparison, it would be nice to see a comparison of the approaches at an analytical level; for example, a derivation of the Luttinger-liquid renormalization of the finite-temperature conductance through a barrier (either in the limit of a weak barrier, or in the limit of weak interaction) in a quantum wire would be very instructive.

2. Consideration of the paper appears too formal; a discussion in more physical (qualitative) terms should be extended. It is well known that, in interacting systems, temperature separates the energy domains where the physics is dominated by real processes (with energy transfers smaller than T) or by virtual processes (energy transfers larger than T). The latter usually yield renormalization of the quantities entering the effective kinetic equation. In the present framework, it is not clear whether only the latter processes are accounted for, or the real inelastic processes are also effectively captured by the developed method.
Further, it is not quite clear whether the procedure depends on the initial density matrix of the system and how equilibration processes that lead to thermalization of the system state are described in this approach. I strongly suggest the authors discussing such points in a qualitative manner and simple physical terms.

3 . The method is initially formulated for the general case of distinct temperatures of reservoirs; however, all the results are given for the case of equal temperatures. It is extremely interesting to see how the method is applied to a situation with different temperatures of the reservoirs, which would correspond to a non-equilibrium steady state of the system. In particular, whether the notion of "non-equilibrium dephasing" (see, e.g., papers by Gutman et al. on non-equilibrium bosonization) would naturally emerge in this setting. But even without performing the numerical analysis of the flow equations in this intriguing case, it would be nice to present the set of such equations explicitly.

### Requested changes

1 - Extend the discussion of a relation between the proposed approach and other RG approaches with temperature serving as an infrared cutoff; present an analytical solution of the derived flow equations in some tractable well-known model.

2 - Add a discussion of the approach in more qualitative terms (see report).

3 - Present the flow equations for the general case of non-equal temperatures of the reservoirs.

• validity: high
• significance: high
• originality: high
• clarity: high
• formatting: excellent
• grammar: excellent