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Bad Metals from Fluctuating Density Waves
by Luca V. Delacrétaz, Blaise Goutéraux, Sean A. Hartnoll, Anna Karlsson
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Luca Delacrétaz · Blaise Goutéraux · Sean Hartnoll |
| Submission information | |
|---|---|
| Preprint Link: | http://arxiv.org/abs/1612.04381v5 (pdf) |
| Date accepted: | 2017-09-07 |
| Date submitted: | 2017-09-04 02:00 |
| Submitted by: | Hartnoll, Sean |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Bad metals have a large resistivity without being strongly disordered. In many bad metals the Drude peak moves away from zero frequency as the resistivity becomes large at increasing temperatures. We catalogue the position and width of the `displaced Drude peak' in the observed optical conductivity of several families of bad metals, showing that $\omega_\text{peak} \sim \Delta \omega \sim k_BT/\hbar$. This is the same quantum critical timescale that underpins the $T$-linear dc resistivity of many of these materials. We provide a unified theoretical description of the optical and dc transport properties of bad metals in terms of the hydrodynamics of short range quantum critical fluctuations of incommensurate density wave order. Within hydrodynamics, pinned translational order is essential to obtain the nonzero frequency peak.
Author comments upon resubmission
We thank both of the referees for their encouraging positive comments and constructive criticism.
In response to referee 2 we have moved all of appendix C and most of appendix B into the main text. We agree that this improves readability. We kept the plot in appendix B as an appendix because we feel this plot is not essential for a first reading of the paper. We kept the derivations in appendix A as an appendix because we believe that keeping the main text relatively decluttered from equations will make it more accessible to experimentalists who may wish to use our formulae for fitting data.
We would like to make the following comments in reply to the very reasonable questions raised by referee 1. Here we have only made two minor additions to the text, relating to the points (3) and (5) below. The remaining points are all explicitly addressed already to varying extents in the main text. We hope that having these replies available online together with the referee’s comments will be sufficiently useful to readers for whom the same questions may arise.
(1) The first concern is that transport is an insufficient foundation to build a theory of the complex physics at work (despite the fact that regimes we are studying — bad metals — are precisely defined by their transport behavior). In fact, as the referee surely knows well and as we noted in the text, especially the final two paragraphs, fluctuating charge density waves have been argued to be important across the phase diagram of e.g. cuprates for reasons that have nothing to do with transport. See e.g. figure 1 of the review by Kilveson et al. There have been explicit recent observations of CDWs in overdoped, as well as underdoped cuprates. Therefore the invocation of fluctuating CDWs as underpinning the bad metal regime is not completely without context. Further, various of the other materials in our list show very compelling evidence for SDW order close to the bad metal regime. While our theory as developed does not directly apply to SDWs, it is the case that SDWs also involve spontaneous breaking of translation invariance and so we expect that some of the same physics will be at work. But, indeed, the main point of our paper is that there is a forceful logic for translational order that operates purely in the realm of sigma(w) — it is nontrivial to displace the Drude peak!
(2) The second concern is that one should be able to fit the data in detail using our functional form, rather than just extract the peak locations and widths. We have preferred to go for a “low-tech” extraction of the peak characteristics in order to minimize human interference with the interpretation of the data. Careful fitting of the peak will likely involve subtracting out other non-hydrodynamic components in sigma(w), and this can be an involved task. At the time of writing this reply we are aware of at least one experimental group that is successfully using our form of sigma(w) to fit their data and we hope that there will be more in the future.
(3) The third concern is that (a) one should look for CDWs in X-rays etc. before transport and there is no evidence of an X-ray signal in bad metal regimes and (b) in regimes where there is an X-ray signal of CDWs our expressions do not match the transport. Regarding the first of these concerns, in fact, one of the points we emphasized in the text (in the final paragraph) is that when the magnitude of the CDW condensate becomes small, sigma(w) becomes a more powerful diagnostic than X-rays! This is because the weight of the displaced Drude peak in sigma(w) is set by the Drude weight and, unlike the peaks in X-ray data, is not proportional to the condensate. This allows the peak to be pronounced even with a small condensate. We have suggested in the text that looking at sigma(w,k) experimentally would amount to the best of both worlds — a stronger peak than in X-rays together with an explicit connection to spatial dependence. Regarding the second concern, in this paper we have preferred to focus on bad metal regimes only as these are the most similar between different materials and hence are good candidates for “universal” hydrodynamic reasoning. However, in our longer and more technical paper (ref 40) we have noted that aspects of underdoped cuprate transport, such as resistivity upturns, may possibly admit an explanation in terms of density wave dynamics. Furthermore, some optical data in the underdoped regime of cuprates (eg. our ref [21]) show peaks that continue into the bad metal regime. These peaks will fit within our theory, so long as the frequencies are low enough to admit a hydrodynamic interpretation. Of course this regime is not quantum critical and so the temperature dependence of the various hydrodynamic parameters will be different. There are many non-hydrodynamic peaks seen in the optical conductivity of underdoped samples, but because these are at non-hydrodynamic energy scales our approach cannot tie them CDW physics as opposed to other pseudogap related phenomena.
—> We added the sentence, in the second to last paragraph, “In some cases the peak in the optical conductivity of bad metals continues to exist in underdoped and overdoped samples [21]. This may allow a quantitative comparison with results from direct imaging techniques.”
(4) The fourth concern is that the pinning frequency is proportional to the condensate strength and this should not increase with temperature. This is a fair point, and we agree that at present we know of no microscopic theory in which the pinning frequency increases with temperature. However, as we note in the text (in the paragraph above equation 6), if these peaks are indeed within quantum critical fans, then pinning due to disorder is not the only dynamics at work. There will also be temperature-dependent “thermal masses” generated in the quantum critical theory. A true microscopic computation will need to take into account the interplay between this quantum critical effect and pinning. I.e. the question is whether the quantum critical Planckian mass scale can “drag” the Drude peak out to it. This will require the inclusion of disorder effects at the quantum critical point for e.g. charge density waves in a metal. One of us is currently investigating this question -- it is a well-defined technical question -- but it is beyond the scope of the present paper.
(5) The fifth concern is that bad metallicity occurs only at high temperatures wheres, at least in cuprates, the dc transport behavior is remarkably continuous all the way down to low temperatures. This is a fair point and indeed the persistence of the T-linear resistivity from the lowest to highest temperatures in bad metals is perhaps *the* central mystery of transport in the cuprates. Even if our theory of bad metals is experimentally corroborated, we will not necessarily have solved that more general problem. That said, in principle our expressions (4) and (5) for the conductivity can hold down to arbitrarily low temperatures in the quantum critical fan. While fluctuating CDWs offer an avenue to bad metallicity, by displacing the Drude peak, they are also compatible, in principle, with good metallicity if quantum fluctuations are strong enough.
—> We added the following sentence at the end of the paragraph containing equation (5): “Within a quantum critical fan, the behavior (5) can furthermore continue down to low temperatures where the resistivity will no longer be large.”
In response to referee 2 we have moved all of appendix C and most of appendix B into the main text. We agree that this improves readability. We kept the plot in appendix B as an appendix because we feel this plot is not essential for a first reading of the paper. We kept the derivations in appendix A as an appendix because we believe that keeping the main text relatively decluttered from equations will make it more accessible to experimentalists who may wish to use our formulae for fitting data.
We would like to make the following comments in reply to the very reasonable questions raised by referee 1. Here we have only made two minor additions to the text, relating to the points (3) and (5) below. The remaining points are all explicitly addressed already to varying extents in the main text. We hope that having these replies available online together with the referee’s comments will be sufficiently useful to readers for whom the same questions may arise.
(1) The first concern is that transport is an insufficient foundation to build a theory of the complex physics at work (despite the fact that regimes we are studying — bad metals — are precisely defined by their transport behavior). In fact, as the referee surely knows well and as we noted in the text, especially the final two paragraphs, fluctuating charge density waves have been argued to be important across the phase diagram of e.g. cuprates for reasons that have nothing to do with transport. See e.g. figure 1 of the review by Kilveson et al. There have been explicit recent observations of CDWs in overdoped, as well as underdoped cuprates. Therefore the invocation of fluctuating CDWs as underpinning the bad metal regime is not completely without context. Further, various of the other materials in our list show very compelling evidence for SDW order close to the bad metal regime. While our theory as developed does not directly apply to SDWs, it is the case that SDWs also involve spontaneous breaking of translation invariance and so we expect that some of the same physics will be at work. But, indeed, the main point of our paper is that there is a forceful logic for translational order that operates purely in the realm of sigma(w) — it is nontrivial to displace the Drude peak!
(2) The second concern is that one should be able to fit the data in detail using our functional form, rather than just extract the peak locations and widths. We have preferred to go for a “low-tech” extraction of the peak characteristics in order to minimize human interference with the interpretation of the data. Careful fitting of the peak will likely involve subtracting out other non-hydrodynamic components in sigma(w), and this can be an involved task. At the time of writing this reply we are aware of at least one experimental group that is successfully using our form of sigma(w) to fit their data and we hope that there will be more in the future.
(3) The third concern is that (a) one should look for CDWs in X-rays etc. before transport and there is no evidence of an X-ray signal in bad metal regimes and (b) in regimes where there is an X-ray signal of CDWs our expressions do not match the transport. Regarding the first of these concerns, in fact, one of the points we emphasized in the text (in the final paragraph) is that when the magnitude of the CDW condensate becomes small, sigma(w) becomes a more powerful diagnostic than X-rays! This is because the weight of the displaced Drude peak in sigma(w) is set by the Drude weight and, unlike the peaks in X-ray data, is not proportional to the condensate. This allows the peak to be pronounced even with a small condensate. We have suggested in the text that looking at sigma(w,k) experimentally would amount to the best of both worlds — a stronger peak than in X-rays together with an explicit connection to spatial dependence. Regarding the second concern, in this paper we have preferred to focus on bad metal regimes only as these are the most similar between different materials and hence are good candidates for “universal” hydrodynamic reasoning. However, in our longer and more technical paper (ref 40) we have noted that aspects of underdoped cuprate transport, such as resistivity upturns, may possibly admit an explanation in terms of density wave dynamics. Furthermore, some optical data in the underdoped regime of cuprates (eg. our ref [21]) show peaks that continue into the bad metal regime. These peaks will fit within our theory, so long as the frequencies are low enough to admit a hydrodynamic interpretation. Of course this regime is not quantum critical and so the temperature dependence of the various hydrodynamic parameters will be different. There are many non-hydrodynamic peaks seen in the optical conductivity of underdoped samples, but because these are at non-hydrodynamic energy scales our approach cannot tie them CDW physics as opposed to other pseudogap related phenomena.
—> We added the sentence, in the second to last paragraph, “In some cases the peak in the optical conductivity of bad metals continues to exist in underdoped and overdoped samples [21]. This may allow a quantitative comparison with results from direct imaging techniques.”
(4) The fourth concern is that the pinning frequency is proportional to the condensate strength and this should not increase with temperature. This is a fair point, and we agree that at present we know of no microscopic theory in which the pinning frequency increases with temperature. However, as we note in the text (in the paragraph above equation 6), if these peaks are indeed within quantum critical fans, then pinning due to disorder is not the only dynamics at work. There will also be temperature-dependent “thermal masses” generated in the quantum critical theory. A true microscopic computation will need to take into account the interplay between this quantum critical effect and pinning. I.e. the question is whether the quantum critical Planckian mass scale can “drag” the Drude peak out to it. This will require the inclusion of disorder effects at the quantum critical point for e.g. charge density waves in a metal. One of us is currently investigating this question -- it is a well-defined technical question -- but it is beyond the scope of the present paper.
(5) The fifth concern is that bad metallicity occurs only at high temperatures wheres, at least in cuprates, the dc transport behavior is remarkably continuous all the way down to low temperatures. This is a fair point and indeed the persistence of the T-linear resistivity from the lowest to highest temperatures in bad metals is perhaps *the* central mystery of transport in the cuprates. Even if our theory of bad metals is experimentally corroborated, we will not necessarily have solved that more general problem. That said, in principle our expressions (4) and (5) for the conductivity can hold down to arbitrarily low temperatures in the quantum critical fan. While fluctuating CDWs offer an avenue to bad metallicity, by displacing the Drude peak, they are also compatible, in principle, with good metallicity if quantum fluctuations are strong enough.
—> We added the following sentence at the end of the paragraph containing equation (5): “Within a quantum critical fan, the behavior (5) can furthermore continue down to low temperatures where the resistivity will no longer be large.”
List of changes
(a) Appendix C and most of Appendix B moved to main text.
(b) We added the sentence, in the second to last paragraph, “In some cases the peak in the optical conductivity of bad metals continues to exist in underdoped and overdoped samples [21]. This may allow a quantitative comparison with results from direct imaging techniques.”
(c) We added the following sentence at the end of the paragraph containing equation (5): “Within a quantum critical fan, the behavior (5) can furthermore continue down to low temperatures where the resistivity will no longer be large.”
Published as SciPost Phys. 3, 025 (2017)