Hierarchical entanglement shells of multichannel Kondo clouds

Impurities or boundaries often impose nontrivial boundary conditions on a gapless bulk, resulting in distinct boundary universality classes for a given bulk, phase transitions, and non-Fermi liquids in diverse systems. The underlying boundary states however remain largely unexplored. This is related with a fundamental issue how a Kondo cloud spatially forms to screen a magnetic impurity in a metal. Here we predict the quantum-coherent spatial and energy structure of multichannel Kondo clouds, representative boundary states involving competing non-Fermi liquids, by studying quantum entanglement between the impurity and the channels. Entanglement shells of distinct non-Fermi liquids coexist in the structure, depending on the channels. As temperature increases, the shells become suppressed one by one from the outside, and the remaining outermost shell determines the thermal phase of each channel. Detection of the entanglement shells is experimentally feasible. Our findings suggest a guide to studying other boundary states and boundary-bulk entanglement.


Report on manuscript NCOMMS-22-39861-T
Using the tools of numerical renormalization, bosonization, and boundary conformal field theory, the authors exploit the entanglement measure of negativity to uncover the spatial structure of screening clouds of the multichannel Kondo effect. When the channel symmetry is broken, shells exhibiting distinct scaling properties are found to coexist in a cloud, with the particulars depending on temperature and the number of channels k. An experiment to test the predictions by the authors is proposed, using a quantum dot coupled to k quantum Hall edge channels where the two degenerate charge states of the dot emulate the s=1/2 spin states of a quantum impurity. This is a topical and very interesting paper, addressing the important issue of how to understand the spatial structure of Kondo clouds. In fact, considering the innovative and careful analysis of the authors, I think this paper may serve as a new benchmark for theoretical discussions of quantum impurity screening clouds. The paper is well written and laid out, and the results are presented in a transparent way. Provided that the points that I raise below are prudently taken into account by the authors, I will recommend publication in Nature Communications.
1) The authors' effort (in the introduction to the paper) to embed their work in the wider context of boundary-bulk entanglement problems (of which Kondo physics is but one example) is commendable. However, a drawback is that a reader not familiar with the Affleck-Ludwig BCFT approach to the multichannel Kondo effect may get lost. Given the title and the core of the paper, I think the authors owe it to the reader to add a paragraph early on where the Affleck-Ludwig approach is briefly summarized ("… reduction to 1D, folding the line to a half-line with a boundary, representing the impurity by a renormalized boundary condition…", etc., etc.), including references. In this context, I was surprised not to find the seminal paper by Andrei and Destri (Phys. Rev. Lett. 52, 364 (1984)) among the background references.
2) I think the statement "… presence of a boundary causes various critical behaviors in the bulk…" (p1) may be misunderstood. While it is true that e.g. the tail of the Kondo cloud exhibits critical scaling in the bulk, a boundary cannot cause bulk criticality per se (a singularity in the extensive term of the free energy/ground state energy, symmetry breaking, etc). While the statement refers to "critical behaviors in the bulk", not "critical behavior of the bulk" (which would be outright wrong), I would still suggest a reformulation.
3) Eq. (2) is key to the interpretation of the entanglement cloud as a screening cloud. However, I can find neither the formula nor its derivation in [31] (which is referenced to). Is "[31]" misprinted? In any event, considering the importance of Eq. (2) for the validity of the results in the paper, and also considering its nontriviality, a published proof must be available to the reader. (Actually, not having seen Eq. (2) before, I find it quite remarkable!) 4) I cannot find the proof of Eq. (4) in the Supplemental Material (as referenced to). Is it somehow hidden or implied in the BCFT discussion of the magnetization in the Supplemental material? If so, I suggest that it is lifted out of this context and be given a careful separate treatment.

5) Ref.
[7] ought to be cited after the sentence "…which signifies the non-Fermi liquid of the kCK." (p3) 6) How should one understand the statement "… \rho_n(L,T) quantifies the part of the entanglement distribution robust against thermal fluctuations with the help of the LSB." (p4) ? This is a cryptic statement. The only interpretation that comes to my mind is that since the LSB perturbation has already wiped out much of the entanglement, thermal fluctuations can do no worse. However, this can't be quite right! Please clarify and/or rewrite.
7) The formula for \rho_n applied to the charge-Kondo circuit is the same as the formula for {\cal N} in Eq. (2) (with M^2 replaced by <\Delta Q/e>^2. How can{\cal N} be replaced by \rho_n (as defined in Eq. (1))? 8) In the treatment of the channel anisotropic 2CK effects (p14, Supplemental material), the authors use the bosonization approach pioneered by Emery and Kivelson (reference missing!). However, the refermionization in the Emery-Kivelson approach works only for a special value of the z-component of the spin exchange (analogous to the Toulouse limit of the ordinary Kondo problem). I fail to see how the authors have implemented this in their analysis.
Reviewer #2 (Remarks to the Author): The Kondo effect with many-channels forms a screening cloud with a power-law decay at large distance. The exponent of the power law is governed by the dimension of the leading irrelevant operator acting at low energy.
The present manuscript explores this long-distance decay by evaluating the entanglement negativity -one particular marker of entanglement -between the quantum Kondo impurity and the bath electrons, with a distant local magnetic field breaking SU(2).
For an anisotropic Kondo interaction, a shell structure is nicely discovered with different exponents depending on the distance to the impurity and depending also on the channel where the local magnetic field is applied. This shell structure in position and channel space mirrors energydependent crossovers with different characteristics temperatures occurring in the anisotropic multichannel Kondo model.
Eventually, the way the shell structure gradually disappears with temperature is determined and an experimental setup to measure the shell structure is examined.
The paper proposes a very intuitive and simple methodology (also novel) to study impurity-bulk entanglement and the cloud structure of Kondo screening. It allows the authors to use both analytical (bosonization) and numerical (NRG) methods to solve the problem, and the approach could be readily applied to other impurity problems.
I would recommend publication of this paper but I have remarks which I believe could improve the quality of the paper. the bosonization procedure seems to obtain, in addition to the power-law decay, Friedel oscillations in the entanglement negativity. Such oscillations are absent in the numerics, if I am not mistaken. It would be informative to state why this is the case.
The discussion on the shell of anisotropic multichannel Kondo clouds (page 3) is not easy to read and follow. I think the authors should first discuss the different possible scalings : non-Fermi liquid with a certain exponent, Fermi liquid with a Kondo screening and what they call non-Kondo Fermi liquid (it is not straightforward to understand what this last phase is). After defining these different possibilities, a table that shows which scaling applies at which distance -and in which channelwould be more easy to read than the current text.
Still on the non-Kondo Fermi liquid, it would be useful to discuss more clearly what this phase/behavior is. In particular, the authors mention an absence of Pi (Pi/2 ?) phase shift but, as far as I can read, they don't show explicitly this absence.
The authors study theoretically the real-space entanglement structure of the multi-channel Kondo model, using a combination of numerical techniques (NRG) and analytical methods (BCFT). The entanglement negativity between the impurity and the rest of the conduction electron baths is computed. Real-space properties are deduced by comparing the negativity with and without a local perturbation applied to the bath at a distance L away from the impurity. This is done at zero temperature as well as finite temperature, in the k=1,2,3 channel Kondo models. Characteristic scaling behaviors are observed, which distinguish the different models. An experiment is proposed and briefly discussed, which is argued could probe the negativity scaling. The main conclusion is that for multi-channel Kondo models, there are a set of real-space 'shells' surrounding the impurity, each characterized by distinct behavior, and that this is picked up by the particular entanglement measure considered in this paper.
I do not believe that this work contains sufficient new material to be published in Nature Communications --neither in terms of the methodology nor the physical insights drawn from them.
The main message drawn from the study is the 'hierarchical shell' structure of multichannel Kondo models in real-space. However, this is known already from Ref [41], which is cited in the introduction of the present work, but then not mentioned again despite is obvious relevance. Ref [41] does not consider entanglement, as in the present paper, and so there are certainly differences, but the notion of real-space shells around the impurity is not new.
To conclude the introduction setting out the main finding: "This shows that different non-Fermi and Fermi liquids hierarchically coexist around the boundary with spatial and energetical separation, reflecting the renormalization of the quantum coherent impurity screening in the presence of the channel competition." However, this is already known.
Furthermore, deeper insights are provided in Ref [41]: the shells correspond to real-space regions described by the RG fixed points, with RG flow to lower energies corresponding to real-space flow away from the impurity. In particular, this leads to the conclusion that the shell referred to in this work as the 'core' is actually the local moment shell. This explains the observation in the present work that the core behavior is very similar for multichannel models with different numbers of channels, independent of critical behavior at larger distances ("the bulk does not show any characteristics of the zero-temperature bulk criticality, strongly "binding" with the impurity"). This is because the local moment fixed point is common to all the models considered. Note also that Ref [41] does consider the k=2 two-channel Kondo model and discusses the 2CK Kondo cloud, as well as the crossover in real space to Kondo Fermi liquid and non-Kondo shells due to symmetry breaking perturbations. Ref [41] also considered the "thermal evaporation" of the shells at finite-T.
The physical picture is therefore already known. This should be acknowledged and discussed in the paper. I accept that the present work provides more details on the shells in terms of their entanglement structure, as probed by the quite specific measurement setup described. This is surely interesting to specialists and deserves to be published in a more specialized journal.
In terms of methodology, again the present work does not really do something very new. The NRG technique used is sophisticated, but was developed in Ref [30] already, and used several times since to apprehend entanglement negativity properties of Kondo problems. Eq. 2 relates the T=0 negativity to magnetization, which is a very nice result --but this was derived already in Ref [31]. In Ref [31], finite-T negativity between the impurity and the channels for multi-channel Kondo models was already studied. The innovation here is to study this same quantity with the same methods, but now with a local perturbation at a distance L away from the impurity. This gives the possibility to discuss negativity changes as a function of the lengthscale L. Something similar was proposed in Refs [19,20] to probe the Kondo cloud (there the focus was measureable conductance, here the authors discuss the negativity).
With rho_n(L,T) the negativity difference with and without the perturbation at site L, the authors state: "Larger rho_n implies that at the distance L there exist more electrons participating in the entanglement." This seems crucial to the conclusions drawn, but is unsubstantiated. The negativity calculated is that of the impurity with the entire baths; adding the perturbation at position L can change the entanglement profile of the channels in real-space, which is then traced out. It is not clear to me that a perturbation at position L tells us about entanglement at lengthscale L. Philosophically, we can incorporate the perturbation at position L into the definition of a new non-interacting conduction electron bath, which will then have an energy-dependent hybridization function coupling to the impurity. The perturbation at L will lead (in 1d, which I guess is the setup used) to a feature at an energy scale omega_p~1/L in the hybridization function. If this perturbation scale omega_p is much smaller that the Kondo scale T_K, then there will be very little effect. If omega_p>>T_K then the Kondo effect will be affected. It seems that one can argue the results based only the energystructure of the hybridization function, and don't need to invoke a length-scale at all! For example, one would get the same rho_n and deduced entanglement structure in a different system that just happens to have the same energy-dependent hybridization function, where the length scale L has no meaning or analogue.
Finally, the authors propose and experiment to measure the entanglement signatures discussed in the paper. However, this potentially interesting section is not elaborated on much and is hard to follow, especially for non-experts. The main message however, is that a *charge-Kondo* setup allows to get an estimate on the negativity via dot charge changes, with and without a perturbation in the leads. This is an interesting idea, but unfortunately I do not think is practical, because the charge differences in the regime of interest are extremely small (that is because the interesting power-laws in rho_n set in at large L, which means very small rho_n in practice). This would be well below the charge detection noise level in an actual experiment. Furthermore, the scaling requires changing L, but a real device would have a fixed quantum point contact at a specific L, which is a physical property of the constructed device. One could make several different devices with different L, but the other inevitable differences between different devices would confound any comparison -again especially when considering the very small numbers involved. Therefore, the statements by the authors that "suppression of the entanglement can be ... experimentally detected" and "The boundary-bulk entanglement will be experimentally accessible" are wildly overstated. This would in practice be very difficult or even impossible, and there is certainly no evidence that this is doable. Some smaller points: 1) the language and presentation needs to be improved throughout 2) In the intro: "While bulk quantities have been understood, boundary states are yet to be explored." This is not true, they have been studied in the existing literature, as mentioned above, and in other works.
3) In the intro: "The properties of the cloud, such as its channel-resolved spatial distribution ... are yet to be studied." This is wrong, it has been studied before, with many of the same conclusions! See e.g. [41].
4) The form of the conduction electron Hamiltonians should be specified. In particular, are the authors explicitly considering 1d systems? This should be mentioned if so! 5) The LSB consists of a field B in the bath. What is the strength of B, and how do the results depend on B? If B is supposed to be infinitesimal, how is this dealt with in NRG? This should be mentioned in the main text. 6) "In the core, rho_n rapidly decays with L, showing that most electrons forming the cloud are in the core." I am not convinced this is implied. And furthermore, the decay of rho_n is more rapid in the tails, as shown in Fig 1! The powerlaw involves a more negative power in the tail than core, so surely its decay is more rapid *outside*. I would say the region inside the core is the "local moment cloud". 7) Original references should be given to published literature where the crossovers in 3CK to either 1CK or 2CK depending on the sign of delta J are discussed. This is not a new finding. 8) In the conclusion, it is stated that "In spin-1/2 boundary criticalities, [the entanglement structure] is obtained using the boundary magnetization, Eq 2." Referee #3 comment-1: The main message drawn from the study is the 'hierarchical shell' structure of multichannel Kondo models in real-space. However, this is known already from Ref [41], which is cited in the introduction of the present work, but then not mentioned again despite is obvious relevance. Ref [41] does not consider entanglement, as in the present paper, and so there are certainly differences, but the notion of real-space shells around the impurity is not new. To conclude the introduction setting out the main finding: "This shows that different non-Fermi and Fermi liquids hierarchically coexist around the boundary with spatial and energetical separation, reflecting the renormalization of the quantum coherent impurity screening in the presence of the channel competition." However, this is already known. Furthermore, deeper insights are provided in Ref [41]: the shells correspond to real-space regions described by the RG fixed points, with RG flow to lower energies corresponding to real-space flow away from the impurity. In particular, this leads to the conclusion that the shell referred to in this work as the 'core' is actually the local moment shell. This explains the observation in the present work that the core behavior is very similar for multichannel models with different numbers of channels, independent of critical behavior at larger distances ("the bulk does not show any characteristics of the zero-temperature bulk criticality, strongly "binding" with the impurity"). This is because the local moment fixed point is common to all the models considered. Note also that Ref [41] does consider the k=2 two-channel Kondo model and discusses the 2CK Kondo cloud, as well as the crossover in real space to Kondo Fermi liquid and non-Kondo shells due to symmetry breaking perturbations.

Response:
As aforesaid, the impurity-bulk entanglement [our achievement (i)] was never considered in Ref. [41]. Quantification of the spatial distribution of Kondo clouds by using the entanglement and the LSB [our achievement (ii)] was also not addressed in Ref.
Our impurity-bulk entanglement provides a direct measure of Kondo screening, as it is a key feature of the Kondo singlet states. Hence its spatial distribution quantifies the spatial distribution of Kondo clouds and reveals universal entanglement shells. By contrast, in Ref. [41], Kondo clouds were analyzed by a single-particle quantity, the excess charge density due to the impurity. Figs. 2 and 4 of Ref. [41] showed that the charge density is negative at certain distance from the impurity and that the absolute value of the charge density even increases with the distance. This position dependence cannot be interpreted as the spatial distribution of a Kondo cloud, while it indicates existence of the "local moment" and "strong coupling" spatial regions, providing valuable motivations for further studies.