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One-particle density matrix of trapped one-dimensional impenetrable bosons from conformal invariance
by Yannis Brun, Jérôme Dubail
This is not the current version.
|As Contributors:||Yannis Brun · Jerome Dubail|
|Submitted by:||Brun, Yannis|
|Submitted to:||SciPost Physics|
|Subject area:||Condensed Matter Physics - Theory|
The one-particle density matrix of the one-dimensional Tonks-Girardeau gas with inhomogeneous density profile is calculated, thanks to a recent observation that relates this system to a two-dimensional conformal field theory in curved space. The result is asymptotically exact in the limit of large particle density and small density variation, and holds for arbitrary trapping potentials. In the particular case of a harmonic trap, we recover a formula obtained by Forrester et al. [Phys. Rev. A 67, 043607 (2003)] from a different method.
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Submission & Refereeing History
Reports on this Submission
Anonymous Report 2 on 2017-2-21 Invited Report
- Cite as: Anonymous, Report on arXiv:1701.02248v1, delivered 2017-02-21, doi: 10.21468/SciPost.Report.85
1. Very well written, with the aim to reach a broad audience.
In this paper the authors analyse the equilibrium properties of a one-dimensional
Bose gas in the strong repulsive interacting regime. In particular, they focus on the
two-point bosonic correlation function, solving the problem of finding its shape
in the presence of an arbitrary confining potential by exploiting a very useful relation with
a two-dimensional conformal field theory in curved space. Such approach was recently
introduced by one of the author (see Ref. 34 and 35) in the context of noninteracting fermions,
and here it is further developed so as to obtain a very attractive result for a Tonks gas.
I really appreciated this work and I think it absolutely deserves to be published on SciPost.
In spite of that, I have a curiosity that the author may address:
1. In the present work the author consider hard-core bosons. In order to evaluate
the two-point function, they first refer to a Luttinger Liquid gaussian action (Eq.15) with depends
on the Luttinger parameter K (which for hard-core bosons is identically 1);
thereafter they identify the vertex operators which mostly contribute to the original
boson creation operator. Then they use the gaussian action with a properly deformed new metric.
Now, I was wondering if this approach can be extended to a generic critical system whose
low energy description is given by the Luttigner Liquid theory.
For example, in the XXZ spin-1/2 chain with space-dependent coupling
\Delta(x) in [-1,1] and space-dependent potential V(x) coupled to Sz (nothing but the density).
In this case I may imagine to have a Luttinger parameter K(\rho(x), \Delta(x)) depending
on space via the filling and the interaction. Is the author approach effective in this situation
or maybe there are some subtleties which make that such approach fails?
Anonymous Report 1 on 2017-1-31 Invited Report
- Cite as: Anonymous, Report on arXiv:1701.02248v1, delivered 2017-01-31, doi: 10.21468/SciPost.Report.74
1, Interesting and new results relevant to experiments.
2. Clear and pedagogical style.
I would appreciate a discussion of the applicability domain of the results, see report
The manuscripts presents derivation of one-body density matrix of impenetrable
(Tonks-Girardeau) bosons in a system with inhomogeneous density. The method
used is Conformal Field Theory adapted for inhomogeneous backgrounds. The
result is anticipated and reduces to known expression in all particular cases
considered. The manuscript is very clearly written and provides sufficient
details which can be followed easily. Overall I recommend the manuscript for
publication in SciPost. I would appreciate if the authors can clarify the
1. The use of Conformal Field Theory for interacting systems of bosons can be
used for any interaction strength (in a homogeneous setup). The author should
state why their results cannot be applied to any value of interactions. I
guess this is related to the density dependence of critical exponents
depending on Luttinger parameter K.
2. The density vanishing at the boundaries of the interval makes the
Conformal Field approach invalid there. The domain of validity of the derived
results should be stated carefully.