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Emergence of curved light-cones in a class of inhomogeneous Luttinger liquids

by Jérôme Dubail, Jean-Marie Stéphan, Pasquale Calabrese

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Submission summary

Authors (as registered SciPost users): Pasquale Calabrese · Jerome Dubail
Submission information
Preprint Link:  (pdf)
Date submitted: 2017-05-18 02:00
Submitted by: Dubail, Jerome
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Quantum Physics
Approach: Theoretical


The light-cone spreading of entanglement and correlation is a fundamental and ubiquitous feature of homogeneous extended quantum systems. Here we point out that a class of inhomogenous Luttinger liquids (those with a uniform Luttinger parameter $K$) at low energy display the universal phenomenon of curved light cones: gapless excitations propagate along the geodesics of the metric $ds^2=dx^2+v(x)^2 d\tau^2$, with $v(x)$ being the calculable spatial dependent velocity induced by the inhomogeneity. We confirm our findings with explicit analytic and numerical calculations both in- and out-of-equilibrium for a Tonks-Girardeau gas in a harmonic potential and in lattice systems with artificially tuned hamiltonian density.

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Reports on this Submission

Anonymous Report 2 on 2017-6-26 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1705.00679v2, delivered 2017-06-26, doi: 10.21468/SciPost.Report.176


Discussion clear and to the point
Results simple and important
Comparison with numerics very compelling in most cases


Observed but not fully explained discrepancy between prediction and numerics for entanglement entropy
Discussion of some assumptions
Some references


In this paper, the authors develop further a theory for inhomogeneous critical models. The main point is that, based on very general arguments, models whose large scale physics is described by the Luttinger liquid theory (CFT with central charge $c=1$), when modified to account for external large scale, slowly varying inhomogeneities, should be described by CFT in curved space. This works in a certain class of inhomogeneities, in which the Luttinger liquid parameter is constant. There are various other assumptions that are made in this paper, and explicit models satisfying these assumptions are studied, with very compelling numerical comparison.

I find this paper very nice. It is well written, the discussion is clear, and the results interesting and very well justified, both analytically and numerically. There is numerical discrepancy for the entanglement entropy, which is not fully explained, but I think it is fine to leave this for future work. I just have small points (see "requested changes"), which are nevertheless important before the paper can be published.

Requested changes

1. Assumption 1 is probably quite crucial for the method, at least in its current form, as discussed. This is fine and can be left for future works. However I believe that assumptions 2 and 3 are not that crucial. Could the authors discuss what would happen if assumptions 2 and 3 do not hold? How would the method be modified? Are there natural models where this would occur (e.g. modifying the Hamiltonian density with a space-time dependent factor)?

2. The definition of the stretched coordinate is good, and of course the choice of the initial point $x_0/\ell$ (e.g. in (67)) is somewhat arbitrary. With appropriate choice of initial point, the problem of non-integrability of the integrand with the sine-square deformation could be averted quite easily I believe. Physically, it is just that with an non-integrable deformation, the time to reach the singular point is infinite, hence no excitation can ever reach such a point. This is natural as zeroes of $f$ give zero interaction, hence no propagation through. Thus the region is divided into different regions, between non-integrable points, each with their own stretched coordinate that goes to +/- infinity at the boundaries of the region. Perhaps the authors could discuss this a bit more.

3. This is a reference-related point. In discussing the entanglement entropy, the authors make use of a twist field. I understand this twist field was actually first introduced in the work [J. L. Cardy, O. A. Castro-Alvaredo and B Doyon J. Stat. Phys. 130, 129-168 (2008)] in the context of entanglement entropy, clearly crucially inspired by the first paper in [71]. For instance as far as I'm aware no literature on the entanglement entropy before the above work talks about twist field, while almost all after it refer to this concept. The twist field was also of course studied much earlier in different contexts [e.g. L. Dixon, D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B675, 13-73 (1987)]. Since the authors actually refer to a twist field and use the notation of the above work, why not cite this literature?

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

Anonymous Report 1 on 2017-6-24 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1705.00679v2, delivered 2017-06-24, doi: 10.21468/SciPost.Report.174


- Timely subject.
- Analytical predictions supported by numerical evidences.


- Hard to understand what is really new.
- The implications of the main results are not clear.


The authors study in more details the features of a CFT for Inhomogeneous one-dimensional quantum
systems, where the Luttinger parameter K is nonetheless constant in space. Despite this restriction they show that there are still some models with inhomogenous Hamiltonians (non translationary invariant) that can be described this way. The two examples they provide are free fermions in a harmonic trap (which is exactly solvable) and generic spin chains with space-dependent couplings. Some questions that I have are listed below.

a) Why there is no a general classification of such models with constant K in space? The authors provide two examples but it is not clear how to say a priori if a model fall in their category or not.
b) The authors should stress how different is (and why it is better) their description compared to a standard Luttinger liquid theory with a Local Density Approximation (LDA). Namely one could expect that the same curved light-cone velocities can be predicted by employing a LDA approximation on top of a uniform system.
c) What are the conditions that the function f they introduce in (62) has to obey in order for their approach to be valid? Any smooth (C^\infty) function is ok or there are corrections of order 1/ell^alpha with alpha some number? Could then the authors reproduce a disorder model by considering a function f such that its Fourier coefficients are chosen to be random numbers?
d)The authors show that a curved light cone emerges when observing dynamical correlations. What about a quantum quench or a inhomogenous quench as the ones studied in ref. [35]-[37] ?
e) Is it possible to construct inhomogenous Hamiltonians such that the light cone closes on itself, namely where the particle propagation is confined, as observed in Nature Physics 13 (2017) 246-249?
f) What are the connections of this work with the recent progresses that have been achieved on the hydrodynamic description of integrable model in presence of inhomogenous fields, see SciPost Phys. 2, 014 (2017)?
e) What are the consequences of a curved light cone on transport coefficients like the thermal and charge Drude weight?

Requested changes

I kindly ask the authors to comment about the questions above, mainly in the introduction or in the conclusions.

  • validity: high
  • significance: good
  • originality: high
  • clarity: high
  • formatting: perfect
  • grammar: perfect

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