# Specific heat and sound velocity at the relevant competing phase of high-temperature superconductors

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Edited by Zachary Fisk, University of California, Irvine, CA, and approved April 6, 2015 (received for review September 4, 2014)

## Significance

A variety of experiments have supported the idea that high-temperature superconductivity is caused by the interaction of quantum mechanical zero-point fluctuations of a competing order with electrons. One of the strangest features of this order is that a transition to it appears to occur without a noticeable specific heat anomaly. We show unambiguously that a proposed order, in which currents spontaneously flow in close loops within each unit cell of the compound, is such that an unusual signature of transition to it is found in the sound velocity experiments but that it is unobservable in the specific heat measurements. Stringent conditions are provided by the analysis for other possible orders to be relevant to the fundamental physics of high-temperature superconductivity.

## Abstract

Recent highly accurate sound velocity measurements reveal a phase transition to a competing phase in YBa_{2}Cu_{3}O_{6+δ} that is not identified in available specific heat measurements. We show that this signature is consistent with the universality class of the loop current-ordered state when the free-energy reduction is similar to the superconducting condensation energy, due to the anomalous fluctuation region of such a transition. We also compare the measured specific heat with some usual types of transitions, which are observed at lower temperatures in some cuprates, and find that the upper limit of the energy reduction due to them is about 1/40th the superconducting condensation energy.

- high-temperature superconductor
- pseudogap
- thermodynamics
- resonant ultrasound spectroscopy
- loop current order

A significant step forward toward understanding high-temperature superconductivity is the variety of experimental results that have led to the widespread acceptance of the idea (1) that a phase with a broken symmetry competes with superconductivity in the underdoped region, often called the pseudogap region. There are a plethora of suggested phases (2⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–15). However, experimental results (16) consistent with transition to only one of them are observed at the pseudogap temperature _{2}Cu_{3}O_{6+δ} (18⇓–20) to show that phase transitions to the universality class of the loop current-ordered state with free-energy reduction similar to the measured superconducting condensation are consistent with the sound velocity and with lack of identifiable observation in the specific heat.

Sound velocity changes near a phase transition, as shown below, are proportional to

The free energy due to a phase transition at temperature *λ* associated with the phase transition, normalized to the background smoothly varying sound velocity *ρ* is the density. In mean field phase transitions, such as the superconducting transition, this reduces to the relation commonly used. Noting that the second contribution above is much smoother than the first and typically

## Results

### Transitions to the Loop-Ordered State.

Changes in sound velocity, consistent in their location in the phase diagram with the previous observations both of _{2}Cu_{3}O_{6+δ}. The results for the larger doping, **1** is amplified near the quantum critical point. Fig. 1 gives the measured change in relative frequency for three different modes, or, equivalently, the relative change in the sound velocity as a function of temperature (17) showing both the superconducting transition and a transition at about 68 K consistent with the continuation of the loop current order transition seen through polarized neutron scattering (16). The Fig. 1 *Insets* give, on a different scale, the signature of the superconducting transition and the pseudogap transition after subtracting the background signal. It is noteworthy that the width of the transition at the *d*-wave superconducting transition at

Consider the thermodynamic properties of the loop current order transition. This broken symmetry, with which many varieties of experimental results (16, 17, 21⇓⇓⇓⇓–26) are consistent, is in the statistical mechanical class of the Ashkin−Teller (AT) model that does not have a specific heat divergence, but singularities exist in the order parameter as a function of temperature. The asymptotically exact critical exponents were derived by Baxter (27); over the range of the parameters for the AT model consistent with the symmetry of the observed phase, the specific heat exponent varies from 0 to −2. The specific heat as a function of temperature (and the order parameter) was calculated (28) by Monte Carlo methods on asymptotically large lattices and is given for two sets of parameters in Fig. 2. In the *Lower Inset* of Fig. 1, the sharpest of the observed sound velocity changes near **1**, a sound velocity signature similar to those observed. Note that the two theoretical curves shown bound the region of parameters that reproduce the sound velocity variations, and note the striking similarity between the measurements and the calculations with regard to the wide fluctuation regime for transitions of this universality class.

Having thus approximately fixed the parameters of the AT model, we calculate the specific heat quantitatively to compare with the observations. The free-energy reduction due to the transition(s) is calculated from the measured specific heat (18⇓–20) (see *Calculations of the Condensation Energy* for details). The value obtained is 52.7 joules per mole. The calculated specific heat for the AT model (28) shown in Fig. 2 is for the four discrete states of the AT model with one classical spin 1 per unit cell, so that the asymptotic entropy at high temperatures in every case is *Calculations of the Condensation Energy*. With this constraint,

The electronic specific heat for YBa_{2}Cu_{3}O_{6+δ}, at various *δ*, has been deduced by subtracting the measured value from that of samples with partial substitution of Cu by Zn (18⇓–20). The Zn-doped samples are not superconducting and have almost the same lattice-specific heat. Unfortunately, as is now known, independently, from Knight shift (29), optical conductivity (30), and neutron scattering (31) measurements, the pseudogap properties are observed in the Zn-doped samples starting at the same

We note that the peaks of ^{2}) and 4.1 millijoules per (mol K^{2}), a factor of 6 and 9 below the deduced value, respectively. The width of the peak at half height is about 40 K. Besides the point made in the last paragraph, we can try to estimate what error is allowed in an absolute measurement of the specific heat as a function of temperature to see this specific heat feature. Given the width of the specific heat feature in the pure limit, one finds that an uncertainty in determining ^{2}) would be the limiting error for the smoother of the two sets of parameters to be decipherable.

It should be mentioned that a broad bump in the specific heat of the magnitude expected from the AT model is directly observed (32) (without making any subtractions) for La_{2−x} Sr_{x}CuO_{4}, systematically decreasing to lower temperature as doping is increased and invisible above *x* = 0.22. The error bars in these measurements were not specified. Ultrasound anomalies, besides those at the superconducting transition temperature, in this compound, as well as those in YBa_{2}Cu_{3}O_{6+δ} with a

Let us return now to Eq. **2** to estimate the relative factors of variation of the transition temperatures with strain. From Fig. 3, the peak height of **2**, one concludes that *δ* and that near the quantum critical point

### Transitions to Other Ordered States.

Let us now consider some of the other transitions that have been reported in the cuprates (6⇓–8, 34, 35) in relation to the specific heat expected due to them for a given free-energy reduction. Some of the transitions are being reported in underdoped cuprates at various temperatures through high-quality measurements in NMR (34), neutron scattering (35), elastic and inelastic X-ray scattering (6⇓–8), and ultrasound measurements as a function of a magnetic field (36). Let us confine ourselves to those at zero magnetic field. The first important point is that no phase transition other than the time reversal breaking order, consistent in its symmetry with the loop current order, has ever been reported near the onset of the pseudogap. Nevertheless, the upper limits on the free-energy reduction due to transitions to other ordered states may be placed from the symmetry class of the phase transition and the lack of observation of a specific heat signature. Detailed sound velocity measurements, which have been made so far only for a small number of samples, would be even more effective in discerning different proposed orders. Let us specifically consider transitions of the incommensurate charge density wave type and compare the observations to those in the well-studied electronically highly anisotropic material 2H-TaSe_{2}. The energy reduction due to the second-order charge density wave transition at about 120 K is measured to be about 67 joules per mole (37), only about 15% larger than the energy reduction due to superconductivity in YBa_{2}Cu_{3}O_{6.9}. We plot the measured _{2}Cu_{3}O_{6+δ} also in Fig. 4 at the transition temperature of 2H-TaSe_{2}. It is clear that a structural transition of this type with about (1/40) of the condensation energy of the superconducting transition would be easily visible in the specific heat measurements if it were to occur at 120 K; a smaller fraction of condensation energy would be detectable at lower temperatures. Sound velocity features of a transition, presumably a CDW, are clearly seen in YBa_{2}Cu_{3}O_{6.67} (36), but only for fields larger than about 20 teslas and temperature greater than about 40 K. A CDW has been observed through both hard and soft X-rays at _{2}Cu_{3}O_{6.67}. In YBa_{2}Cu_{4}O_{8}, which also shows quantum oscillations above about 50 teslas, no CDW distortions are visible at zero field, using the same methods as those that show them in YBa_{2}Cu_{3}O_{6.67}.

If the energy gain due to other transitions is much less than the superconducting condensation energy, they may be regarded as phenomena incidental to the principal remarkable features of the phase diagram of cuprates, which are all extravagant in spending the free energy. As mentioned, a variety of charge stripe/checkerboard phases (2⇓⇓⇓⇓⇓⇓⇓⇓⇓–12, 14, 15) with varying support in experiments have been discussed without their magnitudes from which the energy reduction may be estimated. They are all of the Ising/CDW variety discussed in relation to Fig. 3. An exceptional case is the intraunit cell **Q** = 0 charge order (14, 15) of

## Discussion

To summarize, we have shown that the sound velocity and specific heat measurements can be used to discern the broken symmetry phase in high-temperature superconductors. We have found the universality class of the loop current-ordered phase is consistent with the experimental observation that the reduction of free energy of the transition captured from the sound velocity measurements is not identifiable in specific heat measurements. We have also analyzed other types of transitions and shown that the reduction of free energy for these transitions can only be up to about 1/40 the superconducting condensation energy to be in consistent with the experiments.

We discuss now whether it is possible that the relative width of the ultrasound signatures at the superconducting transition and at the pseudogap transition shown in Fig. 1 may be due to disorder, rather than due to the intrinsic difference between the universality class of the superconducting transition and the loop order transition in the pure limit. This is a legitimate question since, as mentioned, the variation of *δ* is much larger than that of *δ* of the sample for which the data are shown in Fig. 1. Therefore, sample inhomogeneities will have a much bigger effect on the variations in *d*-wave transition, for which nonmagnetic impurities are also pair breaking. Knowing that the superconducting correlation length **1**, this would give orders of magnitude larger sound velocity anomaly at its peak than observed for the same total free-energy reduction as superconductivity.

We should mention that consistency of specific heat and sound velocity has not been proven here for the specific model of loop current order, only for a statistical model to which the loop current order belongs. We have ruled out classes of order of the Ising kind, irrespective of the details of the microscopic model underlying it. Commensurate CDWs belong to this class. For the transition at

Finally, the difficulty of the loop current model—that in the pure limit, it does not give a gap—should be mentioned. This issue has been addressed by showing that small angle scattering due to pinned domain walls between different orientations of loop current order, due to defects, gives the observed features of a pseudogap (42). Some experiments have been suggested to test this conclusion and to see if the ideas and calculations are valid.

## Acknowledgments

We thank Albert Migliori, Cyril Proust, Brad Ramshaw, and Arkady Shekhter for discussions of the sound velocity measurements. Discussions with Nevin Barišić, Philippe Bourges, Martin Greven, Marc-Henri Jullien, and Suchitra Sebatian were very helpful. This research was supported by National Science Foundation Grant DMR 1206298.

## Footnotes

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^{1}To whom correspondence may be addressed. Email: chandra.varma{at}ucr.edu or lijun.zhu{at}ucr.edu.

Author contributions: C.M.V. designed research; C.M.V. and L.Z. performed research; L.Z. analyzed data; and C.M.V. and L.Z. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1417150112/-/DCSupplemental.

## References

- ↵
- ↵.
- Comin R, et al.

- ↵
- ↵
- ↵
- ↵.
- Ghiringhelli G, et al.

_{2}Cu_{3}O_{6+x}. Science 337(6096):821–825 - ↵
- ↵.
- da Silva Neto EH, et al.

- ↵
- ↵
- ↵.
- Hoffman JE, et al.

_{2}Sr_{2}CaCu_{2}O_{8+}_{δ}. Science 295(5554):466–469 - ↵
- ↵
- ↵.
- Fujita K, et al.

- ↵.
- Fujita K, et al.

*d*-form factor density wave in underdoped cuprates. Proc Natl Acad Sci USA 111(30):E3026–E3032 - ↵
- ↵
- ↵
- ↵
- ↵.
- Cooper JR,
- Loram JW,
- Kokanović I,
- Storey JG,
- Tallon JL

_{2}Cu_{3}O_{6+δ}is not bounded by a line of phase transitions: Thermodynamic evidence.*Phys Rev B*89(20):201104(R) - ↵
- ↵
- ↵.
- He RH, et al.

- ↵
- ↵
- ↵
- ↵.
- Baxter RJ

- ↵
- ↵
- ↵.
- Uykur E,
- Tanaka K,
- Masui T,
- Miyasaka S,
- Tajima S

_{2}(Cu,Zn)_{3}O(y) revealed by c-axis optical conductivity measurements for several Zn concentrations and carrier doping levels. Phys Rev Lett 112(12):127003 - ↵
- ↵
- ↵
- ↵
- ↵.
- Hücker M, et al.

_{2-x}Ba_{x}CuO_{4}in a magnetic field. Phys Rev B 87(1):014501 - ↵.
- LeBouef D, et al.

_{2}Cu_{3}O_{y}. Nat Phys 9(2):79–83 - ↵
- ↵
- ↵
- ↵
- ↵.
- Blackburn E

- ↵

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