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Scaling at quantum phase transitions above the upper critical dimension
by A. Langheld, J. A. Koziol, P. Adelhardt, S. C. Kapfer, K. P. Schmidt
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Submission summary
| Authors (as registered SciPost users): | Patrick Adelhardt · Sebastian Kapfer · Jan Alexander Koziol · Anja Langheld · Kai Phillip Schmidt |
| Submission information | |
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| Preprint Link: | scipost_202207_00015v1 (pdf) |
| Data repository: | https://zenodo.org/record/6645107 |
| Date accepted: | 2022-07-26 |
| Date submitted: | 2022-07-12 15:38 |
| Submitted by: | Schmidt, Kai Phillip |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
The hyperscaling relation and standard finite-size scaling (FSS) are known to break down above the upper critical dimension due to dangerous irrelevant variables. We establish a coherent formalism for FSS at quantum phase transitions above the upper critical dimension following the recently introduced Q-FSS formalism for thermal phase transitions. Contrary to long-standing belief, the correlation sector is affected by dangerous irrelevant variables. The presented formalism recovers a generalized hyperscaling relation and FSS form. Using this new FSS formalism, we determine the full set of critical exponents for the long-range transverse-field Ising chain in all criticality regimes ranging from the nearest-neighbor to the long-range mean field regime. For the same model, we also explicitly confirm the effect of dangerous irrelevant variables on the characteristic length scale.
Author comments upon resubmission
Response: Anonymous Report 2 on 2022-5-11 (Invited)
We thank the referee for thoroughly examining our work and raising interesting points.
Below we address the points made by the referee:
a) We thank the referee for this point. We added a short note after the expression for the PCE (Eq (31)) for clarification. Demanding that the PCE of a quantum system is the same as the PCE of its classical analogue by considering the quantum-classical mapping, would lead to an expression for the PCE of $(d+z)/(d_{uc}+z)$ instead of Eq (31).
b) We thank the referee for this point. We added a short note after the expression for the PCE (Eq (31)) and around equation (35). For the classical case, one can also extend the definition to $d<d_{uc}$ with a PCE of unity. For the subtle difference see also our response in a) or the discussion in 2.2.4.
c) We thank the referee for this point. The result should also hold for small T as long as T is sufficiently small in comparison to the finite-size gap. We added this as a footnote in 2.2.4. For increasing T, we believe that there is a crossover from FSS in d dimensions at small T with the PCE of the quantum system to a FSS in d+z dimensions with the PCE of the classical analogue. However our numerical data was not able to resolve this subtle difference. We made this more transparent in and around Eqs (32) and (33).
d) We thank the referee for this suggestion. We agree that disorder and, in particular FSS for disordered systems, is an interesting problem to investigate. We added this to the outlook in Sec. 4.
e) We thank the referee for this question. The derivation is not based on the quantum-classical correspondence and the absence of a classical counterpart should not make any difference. However, for more exotic systems, other preconditions like the GHF structure might not be met. Section 2.2.4 was included to resolve the apparent contradiction of the different PCE under quantum-classical mapping but is not part of the derivation.
Response: Report 1 by Bertrand Berche on 2022-4-2 (Invited)
We thank Bertrand Berche for the thorough examination and report, his support and further insights on Q-FSS.
Below we address the point made by him: We thank Bertrand Berche for the suggestion to include the Q-FSS values. We added a short calculation at the end of section 3.1 and in the plots of the directly extracted critical exponents.
List of changes
- All figures except fig. 4 have been updated due to increased statistics of the underlying data.
- We improved fig. 4 by coloring and labeling the regime above the upper critical dimension.
- We added Ref. [71] which gives access to the raw data of our work. We cite this reference in the captions below all figures.
- On page 8 at the end of section 2.2.2, we added a new paragraph with a comparison to the classical case in order to clarify the non-trivial nature of our result as requested by the second referee.
- On page 9 in Eq (32) and (33) we renamed the scaling power of the temperature to z*.
- On page 9 above Eq (32) we added a footnote that puts the scaling power of the temperature in context and gives an argument about how this scaling power is related to other scaling powers.
- On page 9, we rephrased the last paragraph of section 2.2.3 to address a question raised by the second referee.
- On page 9 after the last paragraph, we added a footnote to address a question raised by the second referee.
- On page 11 before section 3.1, we added a sentence about our raw and processed data provided in Ref [71].
- On page 12 at the end of section 3.1, we added a calculation and table with the analytically known values in the two limiting regimes as suggested by the first referee.
- In Fig. 1 and Fig. 2, we added references to the table mentioned in the point above as suggested by the first referee.
- On page 18 in the middle of the second/last paragraph, we updated the deviations of the pCUT exponents from the analytical values according to the improved statistic of the data.
- On page 22 at the end of section 4, we added an outlook to disorder systems as suggested by the second referee.
Published as SciPost Phys. 13, 088 (2022)