SciPost Submission Page
Duality in quantum transport models
by Rouven Frassek, Cristian Giardinà, Jorge Kurchan
Submission summary
As Contributors:  Rouven Frassek 
Arxiv Link:  https://arxiv.org/abs/2008.03476v2 (pdf) 
Date submitted:  20210105 14:49 
Submitted by:  Frassek, Rouven 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We develop the `duality approach', that has been extensively studied for classical models of transport, for quantum systems in contact with a thermal `Lindbladian' bath. The method provides (a) a mapping of the original model to a simpler one, containing only a few particles and (b) shows that any dynamic process of this kind with generic baths may be mapped onto one with equilibrium baths. We exemplify this through the study of a particular model: the quantum symmetric exclusion process introduced in [D. Bernard, T. Jin, Phys. Rev. Lett. 123, 080601 (2019)]. As in the classical case, the whole construction becomes intelligible by considering the dynamical symmetries of the problem.
Current status:
Editorincharge assigned
Author comments upon resubmission
Dear Editor,
we carefully read the very interesting reports of the referees and thank them for their work. We modified our manuscript accordingly. Please find the changes below.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
REFEREE 1:
(0) The abstract appears in a funny way in SciPost.
It has been now fixed.
(1) "In all systems where duality has been introduced, it happens that the dynamics may be written in terms of the generators of a group" :
By definition in quantum physics, the Hamiltonian belongs to the lie algebra and generates the evolution. Can the authors be a bit more precise about what they mean here?
We changed “the dynamics” > “Hamiltonian generating the dynamics”
(2) Eq (10) :
The † symbol both stands for the adjoint of endomorphism on the Hilbert space and for the adjoint of superoperators which leads to some confusion.
To avoid the confusion, this is now explicitly mentioned in the text
(3) Eq(20) :
\tilde{λ}_i is not defined. Also I think this operation should be detailed a bit more. For instance, as far as my understanding goes, i should be the index of the vertex belonging to the system and K+ and D− should be superoperators acting on the vertices belonging to the bath (as in eq(15)).
Thanks. We added some text.
(4) Inline equation before (21) : It seems that the part in the exponential that are just numbers should cancel with each other, what's the point in writing them? Also the cancellation used after equation (27) should rather be with D− instead of D− − 1?
Thank you, we corrected a mistake, now it is ok.
(5) From this point on, the authors invert the role of operators and density matrices. However one can not be trivially taken for the other since the density matrix obviously fulfills more constraint such as preservation of the trace. It would be nice if this exchange was explained a bit more.
We added the following comment:
Note that if $\tilde O$ has not the requirements of unit trace and positivity of a density matrix,
it can always be brought into one that has those properties, by addition of a term proportional to the invariant
measure, and multiplication by a constant (to normalize).
(6) First paragraph in VII :
The position in the tensor product of the bath and the bulk have been inverted with respect to the previous convention which can lead to confusion.
Thanks. Now it has been restored.
(7) Eq (29) :
The step from (28) to (29) is a bit abrupt, especially since it is one of the important results of the paper. It would be good to give a more detailed explanation on how the coefficients c[O]{n_i} are related to O.
We have clarified this point with the example that follows (29).
(8) In the same manner, it would be great to have more details on how the results (30), (31), (41) have been derived, as they constitute important illustrations on how the formalism works.
See (27) (and we added in the text).
We also improved the notation in (42).
(9) Paragraph before (42) : What is the result for λL ≠λR?
We added a text explaining that the result was given in ref [4].
(10) In general it is not clear from the results presented how this formalism helps getting more results than those which were presented in [4]. Can the authors comment on that?
The duality formalism helps here as much as it does in the classical setting.
Namely, it allows to describe correlation functions by just following dual
particles, whose dynamics is qualitatively immediately guessed.
As far as the stationary state is concerned, as the referee remarks,
the solution of the open quantum exclusion process is given in [4].
We remark that, should one be able to follow the dual dynamics at any time t, then
one could compute timedependent correlation functions.
Furthermore, in the macroscopic hydrodynamic limit, it is conceivable that particles will be effectively independent,
thus simplifying the analysis.
What we really learn from duality applied to the open quantum exclusion process
is discussed in section IX, i.e. the mapping between equilibrium and nonequilibrium
systems.
(11) “the generalization to other models (e.g. the quantum Kac model[8, 20] or quantum KMP [21]) should be immediate –just changing group."
In the conclusion, the authors state that the formalism presented can be easily applied to other type of models. It would be nice to have a more precise statement on the general requirements for a given quantum model to have a dual structure. For instance, from the construction, it seems to me that the algebra describing the generators must fulfill some strict conditions.
We agree with the referee. It should be pointed out that this
has been an open problem for many years.
This will be discussed in a forthcoming paper by the authors.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
REFEREE 2:
(1) In the long introduction, background and motivations are given for
duality. However, some extrareferences could be useful. In the
context of classical stochastic processes, duality has been discussed
by G. M. Sch\"utz 1997, Duality Relations for Asymmetric Exclusion
Processes, J. Stat. Phys., 86, 1265–1287 and more recently by
T. Imamura and T. Sasamoto 2011, Current moments of 1D ASEP by
duality, J. Stat. Phys 142, 91993. It has then become an essential
tool in the field of Integrable Probabilities, see for example
A. Borodin, I. Corwin, and T. Sasamoto 2014, From duality to
determinants for qTASEP and ASEP, Ann. Prob., 42, 2314–2382 (or any
review paper or lecture notes by these authors or their
collaborators).
All the suggested references have been added.
(2) Similarly, in the introduction, the authors could give more
examples of exactly solvable quantum Liouvillians (or Lindbladians),
beyond the quantum symmetric exclusion. (Maybe, one could refer
to some works by the groups of Essler, Prosen, Caux, Calabrese
...).
Done! We added the references and new sentence at the end of
the introduction.
(3) It would be helpful to give some details or references for the
derivation of Eq. (6).
We added a sentence explaining that Eq. (6) (which is Eq. (8) is the new version)
is just a consequence of the the fact that the external noise is Gaussian.
We also cited reference [4].
(4) Similarly, a rationale for finding the su(2) algebra (8) and (9)
could be given.
We have added a phrase about applying two superoperators in succession.
(5) Section V, below eq. (14), c_{1,m} should probably be replaced
by \sqrt{c_{1,m}}.
Done!
(6) The authors obtain a Lindbladian in Eq. (17). Usually this form
requires a Markovian assumption, weak coupling and a small time
expansion. Would it be possible to trace down these assumptions in
the present context?
The Markovian assumption comes from the assumption that the noise is white.
(7) Between Eq. (19) and Eq. (20) the symbol
\tilde{λ}_i \to 0 is used. I have not been able to find
the definition of this quantity in
the text before. In fact, I do not understand what the authors mean
in Eq. (20) ; some details will be useful (the deduction of the
duality Eq. (27) then follows from pure algebra).
We added some extra text to explain what is meant.
(8) In Eq. (27), \rho_0 should be replaced by
\rho(0).
Done!
(9) I would suggest to move section IV (real replicas) just before
section VII.
We prefer to keep the current structure: section 4 is introductory and
section 8 contains the explicit computations.
(10) When applying duality to the quantum exclusion process (Eq. (43))
the authors may describe in more details that model.
The process was defined at the very beginning of the paper (Eq.(1)).
The results of Eq.(43) refer to the onedimensional lattice geometry,
which is now recalled in the paper.
(11) Section IX contains a very important result and is deceptively
short. It may even be overlooked. I would advise that the authors
discuss in more details this mapping to an equilibrium model. This
was already remarkable in the classical case !
We have added a paragraph.
(12) The authors have considered a model of hopping free fermions. Is
it possible to construct a dual model for an interacting system? It
is known that the classical symmetric exclusion which is selfdual
is equivalent to the (Wick rotated) XX spin chain: does the quantum
XXX model also satisfy some kind of duality?
As far as we know the answer is no.
we carefully read the very interesting reports of the referees and thank them for their work. We modified our manuscript accordingly. Please find the changes below.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
REFEREE 1:
(0) The abstract appears in a funny way in SciPost.
It has been now fixed.
(1) "In all systems where duality has been introduced, it happens that the dynamics may be written in terms of the generators of a group" :
By definition in quantum physics, the Hamiltonian belongs to the lie algebra and generates the evolution. Can the authors be a bit more precise about what they mean here?
We changed “the dynamics” > “Hamiltonian generating the dynamics”
(2) Eq (10) :
The † symbol both stands for the adjoint of endomorphism on the Hilbert space and for the adjoint of superoperators which leads to some confusion.
To avoid the confusion, this is now explicitly mentioned in the text
(3) Eq(20) :
\tilde{λ}_i is not defined. Also I think this operation should be detailed a bit more. For instance, as far as my understanding goes, i should be the index of the vertex belonging to the system and K+ and D− should be superoperators acting on the vertices belonging to the bath (as in eq(15)).
Thanks. We added some text.
(4) Inline equation before (21) : It seems that the part in the exponential that are just numbers should cancel with each other, what's the point in writing them? Also the cancellation used after equation (27) should rather be with D− instead of D− − 1?
Thank you, we corrected a mistake, now it is ok.
(5) From this point on, the authors invert the role of operators and density matrices. However one can not be trivially taken for the other since the density matrix obviously fulfills more constraint such as preservation of the trace. It would be nice if this exchange was explained a bit more.
We added the following comment:
Note that if $\tilde O$ has not the requirements of unit trace and positivity of a density matrix,
it can always be brought into one that has those properties, by addition of a term proportional to the invariant
measure, and multiplication by a constant (to normalize).
(6) First paragraph in VII :
The position in the tensor product of the bath and the bulk have been inverted with respect to the previous convention which can lead to confusion.
Thanks. Now it has been restored.
(7) Eq (29) :
The step from (28) to (29) is a bit abrupt, especially since it is one of the important results of the paper. It would be good to give a more detailed explanation on how the coefficients c[O]{n_i} are related to O.
We have clarified this point with the example that follows (29).
(8) In the same manner, it would be great to have more details on how the results (30), (31), (41) have been derived, as they constitute important illustrations on how the formalism works.
See (27) (and we added in the text).
We also improved the notation in (42).
(9) Paragraph before (42) : What is the result for λL ≠λR?
We added a text explaining that the result was given in ref [4].
(10) In general it is not clear from the results presented how this formalism helps getting more results than those which were presented in [4]. Can the authors comment on that?
The duality formalism helps here as much as it does in the classical setting.
Namely, it allows to describe correlation functions by just following dual
particles, whose dynamics is qualitatively immediately guessed.
As far as the stationary state is concerned, as the referee remarks,
the solution of the open quantum exclusion process is given in [4].
We remark that, should one be able to follow the dual dynamics at any time t, then
one could compute timedependent correlation functions.
Furthermore, in the macroscopic hydrodynamic limit, it is conceivable that particles will be effectively independent,
thus simplifying the analysis.
What we really learn from duality applied to the open quantum exclusion process
is discussed in section IX, i.e. the mapping between equilibrium and nonequilibrium
systems.
(11) “the generalization to other models (e.g. the quantum Kac model[8, 20] or quantum KMP [21]) should be immediate –just changing group."
In the conclusion, the authors state that the formalism presented can be easily applied to other type of models. It would be nice to have a more precise statement on the general requirements for a given quantum model to have a dual structure. For instance, from the construction, it seems to me that the algebra describing the generators must fulfill some strict conditions.
We agree with the referee. It should be pointed out that this
has been an open problem for many years.
This will be discussed in a forthcoming paper by the authors.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
REFEREE 2:
(1) In the long introduction, background and motivations are given for
duality. However, some extrareferences could be useful. In the
context of classical stochastic processes, duality has been discussed
by G. M. Sch\"utz 1997, Duality Relations for Asymmetric Exclusion
Processes, J. Stat. Phys., 86, 1265–1287 and more recently by
T. Imamura and T. Sasamoto 2011, Current moments of 1D ASEP by
duality, J. Stat. Phys 142, 91993. It has then become an essential
tool in the field of Integrable Probabilities, see for example
A. Borodin, I. Corwin, and T. Sasamoto 2014, From duality to
determinants for qTASEP and ASEP, Ann. Prob., 42, 2314–2382 (or any
review paper or lecture notes by these authors or their
collaborators).
All the suggested references have been added.
(2) Similarly, in the introduction, the authors could give more
examples of exactly solvable quantum Liouvillians (or Lindbladians),
beyond the quantum symmetric exclusion. (Maybe, one could refer
to some works by the groups of Essler, Prosen, Caux, Calabrese
...).
Done! We added the references and new sentence at the end of
the introduction.
(3) It would be helpful to give some details or references for the
derivation of Eq. (6).
We added a sentence explaining that Eq. (6) (which is Eq. (8) is the new version)
is just a consequence of the the fact that the external noise is Gaussian.
We also cited reference [4].
(4) Similarly, a rationale for finding the su(2) algebra (8) and (9)
could be given.
We have added a phrase about applying two superoperators in succession.
(5) Section V, below eq. (14), c_{1,m} should probably be replaced
by \sqrt{c_{1,m}}.
Done!
(6) The authors obtain a Lindbladian in Eq. (17). Usually this form
requires a Markovian assumption, weak coupling and a small time
expansion. Would it be possible to trace down these assumptions in
the present context?
The Markovian assumption comes from the assumption that the noise is white.
(7) Between Eq. (19) and Eq. (20) the symbol
\tilde{λ}_i \to 0 is used. I have not been able to find
the definition of this quantity in
the text before. In fact, I do not understand what the authors mean
in Eq. (20) ; some details will be useful (the deduction of the
duality Eq. (27) then follows from pure algebra).
We added some extra text to explain what is meant.
(8) In Eq. (27), \rho_0 should be replaced by
\rho(0).
Done!
(9) I would suggest to move section IV (real replicas) just before
section VII.
We prefer to keep the current structure: section 4 is introductory and
section 8 contains the explicit computations.
(10) When applying duality to the quantum exclusion process (Eq. (43))
the authors may describe in more details that model.
The process was defined at the very beginning of the paper (Eq.(1)).
The results of Eq.(43) refer to the onedimensional lattice geometry,
which is now recalled in the paper.
(11) Section IX contains a very important result and is deceptively
short. It may even be overlooked. I would advise that the authors
discuss in more details this mapping to an equilibrium model. This
was already remarkable in the classical case !
We have added a paragraph.
(12) The authors have considered a model of hopping free fermions. Is
it possible to construct a dual model for an interacting system? It
is known that the classical symmetric exclusion which is selfdual
is equivalent to the (Wick rotated) XX spin chain: does the quantum
XXX model also satisfy some kind of duality?
As far as we know the answer is no.
Submission & Refereeing History
Submission 2008.03476v1 on 22 October 2020