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Numerical signatures of ultralocal criticality in a one dimensional Kondo lattice model
by Alexander Nikolaenko, YaHui Zhang
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Authors (as registered SciPost users):  Alexander Nikolaenko 
Submission information  

Preprint Link:  https://arxiv.org/abs/2306.09402v1 (pdf) 
Date submitted:  20230720 15:40 
Submitted by:  Nikolaenko, Alexander 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
Heavy fermion criticality has been a longstanding problem in condensed matter physics. Here we study a onedimensional Kondo lattice model through numerical simulation and observe signatures of local criticality. We vary the Kondo coupling $J_K$ at fixed doping $x$. At large positive $J_K$, we confirm the expected conventional Luttinger liquid phase with $2k_F=\frac{1+x}{2}$ (in units of $2\pi$), an analogue of the heavy Fermi liquid (HFL) in the higher dimension. In the $J_K \leq 0$ side, our simulation finds the existence of a fractional Luttinger liquid (LL*) phase with $2k_F=\frac{x}{2}$, accompanied by a gapless spin mode originating from localized spin moments, which serves as an analogue of the fractional Fermi liquid (FL*) phase in higher dimensions. The LL* phase becomes unstable and transitions to a spingapped LutherEmery (LE) liquid phase at small positive $J_K$. Then we mainly focus on the `critical regime' between the LE phase and the LL phase. Approaching the critical point from the spingapped LE phase, we often find that the spin gap vanishes continuously, while the spinspin correlation length in real space stays finite and small. For a certain range of doping, in a point (or narrow region) of $J_K$, the dynamical spin structure factor obtained through the timeevolving block decimation (TEBD) simulation shows dispersionless spin fluctuations in a finite range of momentum space above a small energy scale (around $0.035 J$) that is limited by the TEBD accuracy. All of these results are unexpected for a regular gapless phase (or critical point) described by conformal field theory (CFT). Instead, they are more consistent with exotic ultralocal criticality with an infinite dynamical exponent $z=+\infty$. Lastly, we propose to simulate the model in a bilayer optical lattice with a potential difference.
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Strengths
In this paper, the authors carry out an extensive DMRG study of a one dimensional Kondo lattice model derived from a twochain Hubbard model that potentially, may be simulated in an optical lattice. The authors have discovered two novel phases that appear to develop between the LutherEmery insulator and the large Fermi surface Luttinger Liquid; moreover, the associated quantum phase transition displays the characteristic
of a local quantum phase transition. This is a fascinating result.
Weaknesses
I do however have a number of questions for the authors that require addressing before the paper is published. In particular, there is a minor error in the Schrieffer Wolff transformation that allows passage from the twochain model to the Kondo lattice that omits cotunneling terms. The authors results need to be revised to take these terms into account.
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Review of "Numerical signatures of ultralocal criticality in a one dimensional Kondo
lattice model" by Alexander Nikolaenko and YaHui Zhang.
arXiv:2306.09402v1
In this paper, the authors carry out an extensive DMRG study of a one
dimensional Kondo lattice model derived from a twochain Hubbard model.
The authors have discovered two novel phases that appear to develop between
the LutherEmery insulator and the large Fermi surface Luttinger Liquid;
moreover, the associated quantum phase transition displays the characteristic
of a local quantum phase transition. This is a fascinating result that
deserves publication in scipost. I do however have a number of questions
for the authors that require addressing before the paper is pubished. Question
(1) below is crucial.
Requested changes
(1) The derived Kondo lattice model in equation (2) is missing the cotunneling
terms that describe the simultaneous "hop and spinflip" processes that
develop when an electron on layer 1 undergoes a virtual transition into
layer 2 and back again. In the authors work, they have assumed that when
this process occurs, the electron always returns to the same site on chain
1. However, it can return to its neighboring, or next neighboring
site. The authors need to carefully repeat their Schrieffer Wolff
transformation to include this effect. These terms should be included in
(2). (See eg Phys. Rev. B 57, 12757 (1998)).
(2) The model Eq (2) is interesting in its own right, but its motivation and
potential link with an optical lattice is slightly blurred without the
link with the twochain model of equation (1). Can we be sure that the
misssing cotunneling terms in equation (2) are restored, the phase diagram
is unchanged? This is a crucial question for the paper. I recognize that
this would require more computation, but without it, the results of the
paper are only valid for the two chain model with t_12=0.
(3) If the authors do not wish to do the extra runs, then can we at least
be sure that the physics is the same for J_cs = 0, i.e t_12=0? I note
that region II was studied at J_cs =0, so this may be fine, but region
II was studied at finite J_cs, so it is not clear whether these results
will hold at J_cs =0, or with the extra cotunneling.
(4) In region I, how do you rule out the possibility that the spin gap is just
too small to be resolved  i.e can you rule out the possibility that the
c=3 region is just a crossover that preceeds the transition to the LE
liquid?
(5) In region II, I was not clear what the authors meant by a ferromagnetic
moment. Normally we reserve this description for a system with a finite
moment per unit length, yet the authors claim on p9, column 1, that the
total moment S_z is 1% of (1/L). I suspect this is a typo  do the authors
mean M = (Sz/L) = 1% (1/L), i.e S_z ~ 1%. On page page 10, column 1
says M ~ 1%. This is confusing! Which is the right claim? Is there a bulk
finite M, or is M~ 1% (1/L)?
(6) What do the authors mean by "ultralocal"?
The correlation lengths appear to be of order 15. Surely, "local" is
sufficient?
nd to the the questions (1)(6) above, returning the revised manuscript and appropriate response to the referee.