Finite Time Path Field Theory and a New Type of Universal Quantum Spin Chain Quench Behavior

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors investigate the universal behavior of Loschmidt echo (LE) in quantum spin chains subjected to both sudden and non-sudden local quenches. They present several notable findings, including: (i) the long-time behavior of LE becomes insensitive to the details of the quench protocol, provided the switching function satisfies certain analyticity conditions; (ii) a universal LE response emerges even for disorder and global quenches, extending the results beyond local perturbations. While the study is thought-provoking, I have a few concerns and suggestions that should be addressed before publication:

• The derivation of the universal long-time behavior of the Loschmidt echo is mathematically detailed.However, the physical mechanism behind why the system becomes insensitive to quench details in the long-time limit remains abstract. Could the authors provide more intuitive or qualitative explanations for this universality?
• In the sentence “This function has two important limits; ‘adiabatic’ limit η→0 +ε...”, it would be more appropriate to use a colon rather than a semicolon for introducing the two limiting cases, e.g., “This function has two important limits: the adiabatic limit η→0+  and the sudden quench limit η→∞.” Additionally, the symbol “ε” appears without any definition or context. Could the authors clarify whether it is necessary here?  Otherwise, it is recommended to remove it for clarity.
• The derivation leading to E(18) is not fully detailed in the manuscript, which may affect the reproducibility of the results. It is recommended that the authors provide a more comprehensive derivation, possibly in an appendix, to enhance clarity and transparency.
• The potential applicability of the results to quantum information and energy storage technologies is mentioned but not elaborated. Providing a brief discussion of how these theoretical insights could be tested experimentally—e.g., using cold atoms, trapped ions, or superconducting qubits—would broaden the paper’s relevance.

Author Response

The authors would like to thank the reviewers for their helpful suggestions aimed at improving the manuscript. Since both reviewers organized their comments/suggestions/questions by points, our response to them is organized in the same way.

Reviewer 1

The authors investigate the universal behavior of Loschmidt echo (LE) in quantum spin chains subjected to both sudden and non-sudden local quenches. They present several notable findings, including: (i) the long-time behavior of LE becomes insensitive to the details of the quench protocol, provided the switching function satisfies certain analyticity conditions; (ii) a universal LE response emerges even for disorder and global quenches, extending the results beyond local perturbations. While the study is thought-provoking, I have a few concerns and suggestions that should be addressed before publication:

• The derivation of the universal long-time behavior of the Loschmidt echo is mathematically detailed.However, the physical mechanism behind why the system becomes insensitive to quench details in the long-time limit remains abstract. Could the authors provide more intuitive or qualitative explanations for this universality?
• In the sentence “This function has two important limits; ‘adiabatic’ limit η→0 +ε...”, it would be more appropriate to use a colon rather than a semicolon for introducing the two limiting cases, e.g., “This function has two important limits: the adiabatic limit η→0+  and the sudden quench limit η→∞.” Additionally, the symbol “ε” appears without any definition or context. Could the authors clarify whether it is necessary here?  Otherwise, it is recommended to remove it for clarity.
• The derivation leading to E(18) is not fully detailed in the manuscript, which may affect the reproducibility of the results. It is recommended that the authors provide a more comprehensive derivation, possibly in an appendix, to enhance clarity and transparency.
• The potential applicability of the results to quantum information and energy storage technologies is mentioned but not elaborated. Providing a brief discussion of how these theoretical insights could be tested experimentally—e.g., using cold atoms, trapped ions, or superconducting qubits—would broaden the paper’s relevance.

Response to Reviewer 1

• To shed light on the physical mechanism behind this universal behavior, we have added in Section 7. Conclusions what we see as the key physical reason for this, which, in our understanding, has three components: properties (1) and (2) of the Fourier transform of perturbation strength function, causality and a nature of the perturbation of a spin chain (lines 495 - 519).
• Symbol ε for infinitesimal quantity has no special significance, and we have replaced 0 + ε with 0⁺ , which is more appropriate in this context.
• We have reorganized Section 3. Perturbative expansion, in order to give a better explanation of the derivation of the equation for n-th order term in the perturbative expansion. To do this, we added also additional equation (Eq. 15), as an important step in the derivation.
• In the last paragraph of Section 7. Conclusions we have commented on this in the context of atoms in optical lattices and added an additional reference [59].

Reviewer 2 Report

Comments and Suggestions for Authors

The paper studies time-dependent perturbations of a one-dimensional spin-chain, building on methodology developed by the same authors in a previous paper. First a Jordan-Wigner transformation is applied to convert to a model of fermions, permitting application of real-time thermal field theory methods. The key result expressed in equation (33)  is that a previously derived result for the Loschmidt echo obtained in the sudden approximation is found to be universal once transient effects have died away in an adiabatic treatment, assuming a perturbation obeying reasonable analyticity conditions and characterised by a well-defined timescale. The results, which are readily generalisable to spatially-random perturbations, should inform attempts to develop robust quantum technologies.

The calculations are exhaustively presented and will permit a motivated reader to reproduce the result. Overall I think this is a useful addition to the literature and recommend publication, provided the authors attend to the following points in the interests of clarity.

(i) below eqn. (8) it is mentioned in passing that the Loschmidt echo is related to the "work probability distribution" P(W). Non-specialist readers will appreciate a definition of P(W).

(ii) the distinction between direct and twisted diagrams not well-explained - the two diagrams in figure 1 are topologically the same, so the vertical direction must imply some physical ordering principle - could this be explained further?

(iii) it is unfortunate that the same symbol k is used both as a momentum integration variable in eg. (18) and as an integer defined below (22) 

(iv) the key point of the paper is that summations over the poles in the lower half plane imply that only positive values of k need be considered, so that the e^-ηk terms are always decaying - yet k can equally well take negative values in (22). It would be very helpful if this technical point in the passage from (22) to (26) could be clarified - I guess this is equivalent to justifying the choice of integration contours for p_n and p_0,n

(v) it would also be nice if the origin of the 2^θ terms in (25) onwards could be commented on - ie how do Heaviside functions end up in an index?

 

Author Response

The authors would like to thank the reviewers for their helpful suggestions aimed at improving the manuscript. Since both reviewers organized their comments/suggestions/questions by points, our response to them is organized in the same way.

Reviewer 2

The paper studies time-dependent perturbations of a one-dimensional spin-chain, building on methodology developed by the same authors in a previous paper. First a Jordan-Wigner transformation is applied to convert to a model of fermions, permitting application of real-time thermal field theory methods. The key result expressed in equation (33)  is that a previously derived result for the Loschmidt echo obtained in the sudden approximation is found to be universal once transient effects have died away in an adiabatic treatment, assuming a perturbation obeying reasonable analyticity conditions and characterised by a well-defined timescale. The results, which are readily generalisable to spatially-random perturbations, should inform attempts to develop robust quantum technologies.

The calculations are exhaustively presented and will permit a motivated reader to reproduce the result. Overall I think this is a useful addition to the literature and recommend publication, provided the authors attend to the following points in the interests of clarity.

(i) below eqn. (8) it is mentioned in passing that the Loschmidt echo is related to the "work probability distribution" P(W). Non-specialist readers will appreciate a definition of P(W).

(ii) the distinction between direct and twisted diagrams not well-explained - the two diagrams in figure 1 are topologically the same, so the vertical direction must imply some physical ordering principle - could this be explained further?

(iii) it is unfortunate that the same symbol k is used both as a momentum integration variable in eg. (18) and as an integer defined below (22) 

(iv) the key point of the paper is that summations over the poles in the lower half plane imply that only positive values of k need be considered, so that the e^-ηk terms are always decaying - yet k can equally well take negative values in (22). It would be very helpful if this technical point in the passage from (22) to (26) could be clarified - I guess this is equivalent to justifying the choice of integration contours for p_n and p_0,n

(v) it would also be nice if the origin of the 2^θ terms in (25) onwards could be commented on - ie how do Heaviside functions end up in an index?

Response to Reviewer 2

1. We have added a definition of the work probability distribution P(W) (Eq. 9), its characteristic function (Eq. 10), and explained its relation with Loschmidt echo complex amplitude. This is taken from Ref. [19].
2. Indeed, direct and twisted diagrams are not topologically different. A “twist” only means a different order of contractions of other fermion operators with the fermion operators in the least time vertex. We added an explanation of this below Eq. (14).  Additional explanations for the direction of time and of propagation of fermions is given in the caption of Fig. 1.
3. We have changed this integer symbol “k” to “r” in all equations in the manuscript.
4. Exactly, this choice of integration contours for p_n and p_0,n is the only possible one. We have now elaborated this in detail below Eq. (28)
5. Explanation of this is now given in the second paragraph below Eq. (28). With that Heaviside function in the exponent, 2^θ can take values 1, corresponding to a contribution from a simple pole at p=0, and 2, corresponding to contributions of all other simple poles in the lower semiplane.