# Viscosity Bound Violation in Viscoelastic Fermi Liquids

###### Abstract

The anti-de Sitter/conformal field theory correspondence (AdS/CFT) has been used to determine the lower bound of the shear viscosity to entropy density ratio for strongly-coupled field theories with a gravity dual. The universal lower bound, given as , is a measure of interaction strength in a quantum fluid where equality indicates a perfect quantum fluid. In this letter we study in a Fermi gas in the unitary limit. We show that while there is a local minimum for the lower bound at , a more interesting result exists in the violation of the bound due to the superfluid fluctuations above . Subsequently, we examine the viscoelastic properties of the unitary Fermi gas. Previous work has brought to light the connection between violation of the bound and a viscoelastic response in the context of holographic solids. We ultimately find that, in addition to holographic solids, all Fermi liquids with a viscoelastic response produced by superfluid fluctuations can violate the universal lower bound.

###### pacs:

67.85.Lm, 67.10.Jn, 51.20.+d^{†}

^{†}preprint: APS/123-QED

## I Introduction

Using the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, strongly interacting quantum field theories can be described in terms of weakly interacting gravitational systems. This has led to the conjecture that there exists a universal lower bound —the KSS bound— for of a strongly coupled field theory given by Cremonini (2011); Kovtun et al. (2005, 2003); Iqbal and Liu (2009); Policastro et al. (2001)

(1) |

Quantum fluids of varying density, such as the quark gluon plasma and the unitary Fermi gas, that obey eqn. (1), are called nearly perfect quantum liquids where equality denotes a perfect quantum liquid Müller et al. (2009); Thomas (2010). The AdS/CFT correspondence additionally creates a bridge between gravitational physics and condensed matter and allows one to be studied in terms of the other as in Alberte et al. (2018, ). It’s been shown that if the unitary Fermi gas undergoes a superfluid phase transition, that superfluid fluctuations above the phase transition temperature, , have significant affects on the spin transport Li et al. (2015). Recent experiments on the unitary Fermi gas show a normal/superfluid phase transition at a transition temperature, Sommer et al. (2011), where stands for the Fermi temperature of the unitary Fermi gas. The ratio was also measured experimentally in this unitary Fermi gas where a minimum close to was found at temperatures to the superfluid transition temperature Cao et al. (2011).

To better understand from the context of strongly correlated systems, we develop a simple theoretical model to calculate the quasiparticle scattering rates of a strongly correlated quantum liquid above . Such a model differs from past calculations Rupak and Schäfer (2007); How and LeClair (2010); Salasnich and Toigo (2011); Pakhira and McKenzie (2015) in that we include the effects of superfluid fluctuations as . The model separates the quasiparticle scattering amplitude for the strongly correlated quantum fluid into two components: the superfluid fluctuations term coming from the particle-particle pairing fluctuations in the singlet scattering channel above , and a normal Fermi liquid scattering term calculated from the local version of the induced interaction model Engelbrecht and Bedell (1995); Jackiewicz and Bedell (2005). Applying our theory to the unitary Fermi gas, we calculate for the unitary Fermi gas about following the methods used in the transport studies of Landau Fermi-liquid theory Baym and Pethick (1991). We find a local minimum as of which agrees with the experimentally measured lower bound Cao et al. (2011). However, our result shows eventually drops to zero at due to the superfluid fluctuations effect, thus violating the universal lower bound. While our result appears to be in contradiction to the work done by Cao et. al. Cao et al. (2011), we find good qualitative agreement with the more recent analysis done by Joseph et. al. Joseph et al. (2015). We believe this is because the measurements were done over a wide temperature range while the violation of the bound happens in a small window around . Additionally, due to the breakdown of the quasiparticle picture, numerous other methods have been employed such as those performed by Enss et. al. Enss et al. (2011) to determine the viscosity. While we don’t necessarily disagree with these results, we feel our method is valid due to the experimental support of the quasiparticle picture near (as shown in Fig. 4 and will be discussed later). Previous work by Alberte, Baggioli, and Pujolàs Alberte et al. (2016); Baggioli (2016) presents the idea that the viscoelastic nature of holographic solids violates the bound. The purpose of this paper is to expand on their work and provide insight into this high-energy problem from the viewpoint of condensed matter.

## Ii Superfluid Fluctuations in the Unitary Fermi Gas

The high transition temperature, , of the unitary Fermi gas allows for the experimental measurement of at temperatures close to , where superfluid fluctuations could play a role Sommer et al. (2011); Cao et al. (2011). Previous study of the spin transport found the superfluid fluctuations play a significant role in the spin diffusion Li et al. (2015). As such, we aim to understand how the superfluid fluctuations may affect . The superfluid fluctuations come from the particle-particle pairing fluctuations in the spin singlet quasiparticle scattering channel closely above . Due to the pairing fluctuations, the quasiparticle scattering amplitudes for small total momentum scattering diverge at . Considering here only the s-wave (spin singlet) pairing mechanism for the Cooper pairs, we reveal the superfluid fluctuations in the scattering amplitudes by evaluating the temperature vertex function of particle-particle type in the spin singlet channel for small total momentum scattering using standard quantum field theory methods Abrikosov et al. (1975). The spin singlet temperature vertex function is generated from the diagram shown in Fig. 1, leading to the following integral equation,

(2) | |||||

where, , are the four momenta of the scattering particles, and, , stands for the total momentum of the incident particles. depends only on the total momentum , , when for and . Solving eqn. (2), we can express in the small limit as

(3) |

where , is the Euler’s constant, is the Fermi momentum and is the cutoff frequency Gorkov and Melik-Barkhudarov (1961). is the zero temperature irreducible particle-particle vertex function, which is approximately equal to the spin singlet normal Fermi-liquid scattering amplitude a given diagramatically in Fig. 2b Baym and Pethick (1991). In order to calculate the viscosity of the Fermi system, we need the normal Fermi-liquid scattering amplitude. The total quasiparticle scattering probability, , is obtained by averaging the quasiparticle scattering amplitudes of different K’s over the phase space Baym and Pethick (1991). For the unitary Fermi gas, is separated into a superfluid fluctuations term, , and a normal Fermi-liquid scattering term, :

(4) | |||||

stands for the critical value of the total momentum of the incident particles, beyond which Cooper pairs start to break down and the particles scatter off of each other as in the normal Fermi liquid state. It is given by , where , from regular quantum field theory analysis Abrikosov et al. (1975). It’s important to note that the angular averages in eqn. (4) are different due to the different angular dependencies in and Ainsworth and Bedell (1987); Li et al. (2015).

The Landau parameters needed for computing the quasiparticle scattering amplitudes are determined from the local induced interaction model, shown diagramtically in Fig. 2. First developed to study the quasiparticle interactions in liquid He, it has seen success in its application to other Fermi systems and been further generalized to account for the momentum dependence in the scattering amplitudes Babu and Brown (1973); Ainsworth et al. (1983); Bedell and Quader (1985); Ainsworth and Bedell (1987). According to the model, the quasiparticle interaction parameter, , is generated from a direct term, , which is equivalent to a model dependent effective quasiparticle potential, and an induced term coming from the coupling of collective excitations to the quasiparticles. The mechanism is shown diagrammatically in Fig. 1 in Li et. al. Li et al. (2015)

In this work we use a local, momentum independent, version of the induced interaction model where only the Landau parameters, , are nonzero Li et al. (2015); Engelbrecht and Bedell (1995); Jackiewicz and Bedell (2005); Gaudio et al. (2009) and given as

(5) |

(6) |

where, . In the unitary limit, the Landau parameters take on the following values: and . These parameters capture the strong interactions and successfully explain various universal thermodynamic properties of the unitary Fermi gas Li et al. (2015); Giorgini et al. (2008).

Following the approach of Landau Fermi-liquid theory Baym and Pethick (1991), along with the local induced interaction model, we calculate the quasiparticle scattering amplitudes and :

(7) |

(8) |

where in the unitary limit. The total scattering probability becomes

To calculate the viscosity, , of the unitary Fermi gas using Landau Fermi-liquid theory, we need the viscous lifetime in addition to the scattering probabilities eqn. (LABEL:eq:Total_Scattering). In the low temperature limit, the viscous lifetime, , is Baym and Pethick (1991)

(10) | |||||

where without any index is the quasiparticle lifetime and the factor of 0.205 is from the different angular average of the scattering amplitude in the unitary limit. A finite temperature correction is added to to give Dy and Pethick (1969)

(11) | |||||

The viscosity is then given by for . For , we fit the classical lifetime Brunn (2011) to data of the viscosity coefficient Cao et al. (2011) and obtain for . This leads to the following for viscosity:

(12) |

A natural concern is our use of the Landau Kinetic equation (LKE) to calculate the viscosity since the quasiparticle lifetime is short. Typically, one resorts to use of the Kubo formalism to calculate transport quantities when the quasiparticle picture is insufficient. Bruun and Smith performed a calculation Bruun and Smith (2007) and show that corrections to the LKE result are small compared to those using the Kubo formalism. Additionally, the entropy from Ku et. al. Ku et al. (2012), shown in Fig. 4, exhibits Fermi liquid like behavior with minor corrections thus providing validity of our approach in using the quasiparticle picture. To calculate the ratio , we also need the entropy density of the unitary Fermi gas. According to Fermi liquid theory Baym and Pethick (1991), the low temperature entropy density is given by

(13) |

where is a cutoff temperature much less than and , for a local Fermi liquid in the unitary limit, where the logarithmic term stands for the finite temperature correction to the zero temperature result. In the high temperature limit, the entropy density takes the form of a classical Fermi gas Pathria (1996)

(14) |

where is the thermal wave length, and for a two component Fermi gases.

The ratio is plotted over the entire temperature regime in Fig. 3. The experimental data of from Joseph et al. (2015), shown in the inset of Fig. 3, is measured with respect to the energy scale . A ratio of corresponds roughly to a temperature ratio of Luo and Thomas (2009), therefore the low temperature portions of the calculated and measured ratio of are plotted in the same temperature window. A local minimum, with value , is found in the calculated ratio at (shown by the red curve in Fig. 3) agrees roughly with the experimental saturation value of for a nearly perfect Fermi gas Thomas (2010); Cao et al. (2011) (in the inset of Fig. 3) and is not far from the holographic prediction Kovtun et al. (2005), . However, as can be seen from Fig. 3 and eqns. (LABEL:eq:Total_Scattering) and (12), the ratio is not bounded by this local minimum as it clearly drops to zero at due to superfluid fluctuations as

(15) |

which qualitatively agrees with the behavior in Fig.3. The conjectured universal lower bound for is therefore violated in our theory. A natural concern is if potential hidden behavior of the entropy density, not captured by eqn. (13), is causing . While eqn. (13) may not be the complete low temperature behavior, the data given by Fig. 4 suggests that although a kink is present, there is no divergence or singularity. Neither theoretical, eqn. (13) nor experimental result diverges and therefore the entropy is well behaved and cannot be driving the ratio to zero. Violation of the conjectured bound on begs the following question: how do superfluid fluctuations in the unitary Fermi gas violate the KSS bound?

## Iii Viscoelasticity of the Unitary Fermi Gas

Previous work Bedell and Pethick (1982); Alberte et al. (2016) has led us to look at the viscoelastic behavior of the unitary Fermi gas. Alberte et. al have shown that holographic solids, solid massive gravity black branes with nonzero graviton mass, violate the KSS bound Alberte et al. (2016). Their work ultimately found that holographic solids with a non-zero bulk modulus, specifically, finite shear modulus, violate the KSS bound, with strong evidence for extension to real solids. Our work aims to go a step further and provide a real system, the unitary Fermi gas, that has viscoelasic behavior and that violates the KSS bound with the hope that this furthers investigation into how viscoelastic behavior violates the bound.

We must first ask if the viscoelastic model is suitable to describe the unitary Fermi gas. The conditions for viscoelasticity are (i) , and/or (ii) as where and are the speeds of zero and first sound respectively, and l is the viscous mean free path. Although (i) is violated for the unitary Fermi gas since , (ii) is satisfied since the electron mean free path goes to zero as and cooper pairs form. Additionally, the unitary Fermi gas has both elastic and viscous response, therefore the model is appropriate to describe the unitary Fermi gas.

We start with the general form for the stress tensor for a viscoelastic model, different from those found in Baym and Pethick (1991); Bedell and Pethick (1982); Landau and Lifshitz (1986):

(16) |

where

is the strain tensor for small displacements and is the flow velocity. is the bulk viscosity and may be ignored since at low temperature for a Normal Fermi Liquid Baym and Pethick (1991); Son (2007). is the shear modulus which contains the viscous (viscosity) and elastic (elasticity) behavior. In the viscoelastic model, due to the short lifetime, i.e. , and the the elasticity is no different from the viscosity. Using the LKE we get

(17) |

(18) |

where the real and imaginary parts of (17) are analyzed separately. Letting , we obtain the following expression for the coefficient of sound attenuation Pethick (1969) in the unitary Fermi gas

(19) |

The real part of (17) gives

(20) | |||||

Eqns. (19) and (20) provide experimentally attainable quantities relating to the viscoelasticity of unitary Fermi gases. As the temperature of unitary Fermi gas approaches , two things happen: (i) and (ii) . From Landau and Lifshitz (1987) we interpret as the penetration depth of being infinite. As the unitary Fermi gas approaches its transition temperature, the zero sound mode predicted by Landau Fermi Liquid Theory is damped and a typical sound mode, first sound, propagates through the entire system. The propagating sound mode being first sound is another trademark of viscoelatic behavior. Continuing with the Landau Kinetic equation, we can use conservation laws (momentum and number) to obtain a hydrodynamic equation of motion for the mass density

(21) |

where if , we obtain

(22) |

a standard wave equation for a sound wave propagating at velocity which is in agreement with our analysis and interpretation of eqn. (20).

## Iv Conclusion

Based on our theory, superfluid phase transitions have significant affects on for the unitary Fermi gas resulting in clear violation of the proposed universal lower bound. Strongly coupled systems often exhibit phase transitions, leading us to believe that similar conclusions could be drawn about other strongly correlated fermionic quantum fluids such as dense nuclear matter produced in heavy ion collisions, where Cremonini (2011); Jacak and Steinberg (2010). While agreement with eqn. (1) is possible, we argue that the small ratio for in dense nuclear matter is related to fluctuations that would arise from the strongly interacting quark gluon plasma (QGP) phase Chen and Nakano (2007). The transition temperature for the QGP phase is predicted from lattice QCD computations Karsch et al. (2001) to be, , and from the experiments below this temperature, , is close to the KSS bound Jacak and Steinberg (2010). Therefore, we raise the question: Is the minimum found in the transport coefficients of the nearly perfect quantum liquid due to universal quantum behavior predicted by the AdS/CFT correspondence or is it a local minimum in the ratio caused by the interplay between correlated liquid effects that want the ratio to grow and the fluctuations of a nearby phase that want to drive them to zero at or near the phase transition?

We have developed a new model, taking into consideration of the amplitude fluctuations, to study the ratio of strongly correlated quantum fluids. Our calculations have shown that fluctuations from the nearby superfluid phase can drive the ratio to very low values, even to zero at the phase boundary, thus violating the conjectured universal lower bound. In the case of low values found for the ratio in heavy ion collisions at , it appears that fluctuation effects coming from the nearby QGP phase could be the driver for this low value. More precise measurements of near the phase boundaries, in tighter temperature windows around , are needed to establish if the KSS bound is a truly “universal lower bound”. Additionally, we bring to light the connection between viscoelastic responses and violation of the conjectured bound. Both systems, the unitary Fermi gas and the holographic solids Alberte et al. (2016), share a combination of viscous and elastic responses. It should be of interest if, in addition to phase fluctuations, complicated viscoelastic responses also violate the bound. In conclusion, our theory provides an alternative way of studying in a strongly correlated quantum fluid by considering the effects of pairing instabilities in the quasiparticle scattering amplitude. We hope this work will shed light on the connection between condensed matter and high energy problems through (bottom up) AdS/CFT.

## Acknowledgements

The authors M. Gochan, H. Li, and K. Bedell would like to thank Joshuah Heath for his valuable discussions on AdS/CFT, and Mark Ku for experimental data for the entropy in the unitary Fermi gas. This work is supported by the John H. Rourke Boston College endowment fund.

## References

- Cremonini (2011) S. Cremonini, Mod. Phys. Lett. B 25, 1867 (2011).
- Kovtun et al. (2005) P. K. Kovtun, D. T. Son, and A. O. Starinets, Phys. Rev. Lett. 94, 111601 (2005).
- Kovtun et al. (2003) P. Kovtun, D. T. Son, and A. O. Starinets, J. High Energy Physics 2003, 064 (2003).
- Iqbal and Liu (2009) N. Iqbal and H. Liu, Phys. Rev. D 79, 025023 (2009).
- Policastro et al. (2001) G. Policastro, D. Son, and A. Starinets, Phy. Rev. Lett. 87, 081601 (2001).
- Müller et al. (2009) M. Müller, J. Schmalian, and L. Fritz, Phys. Rev. Lett. 103, 025301 (2009).
- Thomas (2010) J. Thomas, Physics Today 63, 34 (2010).
- Alberte et al. (2018) L. Alberte, M. Ammon, M. Baggioli, A. Jiménez, and O. Pujulàs, J. High Energy Phys. 2018, 129 (2018).
- (9) L. Alberte, M. Ammon, M. Baggioli, A. Jiménez, and O. Pujulàs, arXiv:1711.03100 [hep-th] .
- Li et al. (2015) H. Li, J. Jackiewicz, and K. S. Bedell, Phys. Rev. B 91, 075107 (2015).
- Sommer et al. (2011) A. Sommer, M. Ku, G. Roati, and M. W. Zwierlein, Nature 472, 201 (2011).
- Cao et al. (2011) C. Cao, E. Elliott, J. Joseph, H. Wu, J. Petricka, T. Schäfer, and J. Thomas, Science 331, 58 (2011).
- Rupak and Schäfer (2007) G. Rupak and T. Schäfer, Phys. Rev. A 76, 053607 (2007).
- How and LeClair (2010) P. How and A. LeClair, J. Stat. Mech. 2010, 1 (2010).
- Salasnich and Toigo (2011) L. Salasnich and F. Toigo, J. Low Temp. Phys. 165, 239 (2011).
- Pakhira and McKenzie (2015) N. Pakhira and R. McKenzie, Phys. Rev. B 92, 125103 (2015).
- Engelbrecht and Bedell (1995) J. Engelbrecht and K. Bedell, Phys. Rev. Lett. 74, 4265 (1995).
- Jackiewicz and Bedell (2005) J. A. Jackiewicz and K. Bedell, Phil. Mag. 85, 1755 (2005).
- Baym and Pethick (1991) G. Baym and C. Pethick, Landau Fermi-Liquid Theory, 1st ed. (Wiley-VCH, 1991).
- Joseph et al. (2015) J. A. Joseph, E. Elliott, and J. E. Thomas, Phys. Rev. Lett. 115, 020401 (2015).
- Enss et al. (2011) T. Enss, R. Haussmann, and W. Zwerger, Ann. Phys. 326, 770 (2011).
- Alberte et al. (2016) L. Alberte, M. Baggioli, and O. Pujolàs, J. High Energy Physics 74, 1 (2016).
- Baggioli (2016) M. Baggioli, Gravity, holography and applications to condensed matter, Ph.D. thesis, Universitat Autònoma de Barcelona, The address of the publisher (2016).
- Abrikosov et al. (1975) A. Abrikosov, L. Gorkov, and I. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, 1st ed. (Dover Publications, 1975).
- Gorkov and Melik-Barkhudarov (1961) L. Gorkov and T. Melik-Barkhudarov, J. Exp. Theo. Phys. 13, 1018 (1961).
- Ainsworth and Bedell (1987) T. L. Ainsworth and K. S. Bedell, Phys. Rev. B 35, 8425 (1987).
- Babu and Brown (1973) S. Babu and G. Brown, Ann. of Phys. 78, 1 (1973).
- Ainsworth et al. (1983) T. L. Ainsworth, K. S. Bedell, G. Brown, and K. Quader, J. Low Temp. Phys. 50, 319 (1983).
- Bedell and Quader (1985) K. S. Bedell and K. F. Quader, Phys. Rev. B 32, 3296 (1985).
- Gaudio et al. (2009) S. Gaudio, J. Jackiewicz, and K. Bedell, Phil. Mag. 89, 1823 (2009).
- Giorgini et al. (2008) S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 80, 1215 (2008).
- Dy and Pethick (1969) K. S. Dy and C. J. Pethick, Phys. Rev. 185, 373 (1969).
- Brunn (2011) G. Brunn, New J. Phys. 13, 035005 (2011).
- Bruun and Smith (2007) G. Bruun and H. Smith, Phys. Rev. A 75, 043612 (2007).
- Ku et al. (2012) M. J. H. Ku, A. T. Sommer, L. W. Cheuk, and M. W. Zwierlein, Science 335, 563 (2012).
- Pathria (1996) R. Pathria, Statistical Mechanics, 2nd ed. (Butterworth-Heinemann, 1996).
- Luo and Thomas (2009) L. Luo and J. Thomas, J. Low Temp. Phys. 154, 1 (2009).
- Bedell and Pethick (1982) K. Bedell and C. Pethick, J. Low Temp. Phys. 49, 213 (1982).
- Landau and Lifshitz (1986) L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 3rd ed. (Butterworth-Heinemann, 1986).
- Son (2007) D. T. Son, Phys. Rev. Lett. 98, 020604 (2007).
- Pethick (1969) C. J. Pethick, Phys. Rev. 185, 384 (1969).
- Landau and Lifshitz (1987) L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. (Butterworth-Heinemann, 1987).
- Jacak and Steinberg (2010) B. Jacak and P. Steinberg, Physics Today 63, 39 (2010).
- Chen and Nakano (2007) J. Chen and E. Nakano, Phys. Lett. B 647, 371 (2007).
- Karsch et al. (2001) F. Karsch, E. Laermann, and A. Perkert, Nucl. Phys. B 605, 579 (2001).