## Final Report Summary - HOLOLAND (Charting the holographic quantum landscape, and consequences on charge transport at strong coupling)

Strongly-coupled matter is notoriously hard to tackle: since elementary constituents interact strongly, no perturbative expansion in a small coupling constant can be set up, as is the case in Quantum Electro-Dynamics or Fermi liquid theory (which describes conventional metals such as lead and copper). Prime examples where perturbative field theory techniques fail due to strong coupling are Quantum ChromoDynamics at low energies, but also Graphene near the charge neutrality point or high Tc superconductors. The latter have been one of the central challenges in Condensed Matter Physics since their discovery thirty years ago. Despite intense research efforts, spurred on by potential applications to room temperature superconductivity, no theoretical consensus yet exists on the physical mechanism underpinning pairing in the superconducting phase or transport in their normal phase.

Thermoelectric transport in the normal phase has been thoroughly investigated, and much is known on the temperature, magnetic field and frequency dependence thermoelectric conductivities. The experiments suggest that a Quantum Critical Point underlies the phenomenology of the normal phase, albeit of a non-conventional kind. Indeed, the scaling properties measured experimentally depart from the expectations at a conventional Quantum Critical Point, which are completely fixed by scale invariance.

Non-scale invariant Quantum Critical Points are challenging to model in Condensed Matter. However they arise naturally in the low temperature limit of black holes in Anti de Sitter, which can be used to model strongly-coupled systems in general through Gauge/Gravity Duality (aka Holography). Gauge/Gravity duality was discovered in 1997 by Juan Maldacena, who showed that within string theory a certain gauge theory (N=4 super Yang Mills) was dual to a certain theory of gravity (Type IIb supergravity on AdS5xS5). This a weak/strong, classical/quantum duality. That is, when the field theory is quantum and strongly-coupled, it can be described by a weakly-coupled, classical theory of gravity. Essentially, Einstein’s gravity coupled to a few matter fields.

Part of the work accomplished in the Fellowship was precisely to study holographically these non conventional Quantum Critical Points. We have shown how in general the low temperature scalings of thermoelectric conductivities arise from the charge and entropy density operators acquiring anomalous dimensions in the effective long wavelength theory. These anomalous dimensions are directly responsible for the departure from strict scale invariant behaviour. These results partially inspired a theoretical proposal for the scaling of zero frequency conductivities in high Tc superconductors. It could be confirmed or disproved by future experiments providing missing transport data.

Another important aspect of the phenomenology of high Tc superconductors is the pattern of symmetry breaking as the temperature is lowered. Of course, spontaneous breaking of a U(1) symmetry and low temperature superconductivity have long been appreciated to play an important role in the phase diagram of these materials. However, over the last 20 years, spontaneous breaking of space translations and time reversal symmetry have also been observed experimentally, both as static and fluctuating orders. Understanding how the pattern of symmetry breaking affects transport is thus also a pressing question.

I have tackled these questions using hydrodynamics and the memory matrix approach. Hydrodynamics is an effective theory valid at late times and long wavelengths compared to the mean free path. Said otherwise, it captures small variations above local equilibrium. At strong coupling, individual constituents quickly decay. The effective degrees of freedom are thus conserved collective modes, such as the energy, charge and momentum densities, as well as Nambu-Goldstone bosons arising from spontaneously broken symmetries. This are the effective degrees of freedom which survive in the long wavelength approximation. Real materials often only have approximate symmetries, and the memory matrix provides a way of computing the relaxation rates associated to the slow decay of conserved quantities. Together with hydrodynamics, we can thus obtain expressions for the low frequency, long wavelength behaviour of thermoelectric conductivities.

As an example, we have proposed together with my collaborators that fluctuating translation symmetry breaking (stripes) could play an important role in explaining some of the transport properties of the normal, strange metallic phase of cuprate high Tc superconductors. With refined sensivity and probes sensitive to the spatial dependence of conductivities, this proposal can be tested in future experiments.

The Fellowship consisted of two phases: an Outgoing phase (2014-2016) at the Stanford Institute for Theoretical Physics (Stanford University, CA, USA) under the supervision of Pr. Sean Hartnoll; and a Return phase (2016-2017) at the Nordic Institute for Theoretical Physics (NORDITA, Stockholm, Sweden) under the supervision of Pr. Konstantin Zarembo. Overall, 10 articles have been published in peer-reviewed journals, one is currently under review and several are in preparation.

Dissemination activities have included seminars at 14 research institutes, oral presentations at 14 international workshops and conferences and 3 series of short lecture courses on topics related to the project.

To conclude, the Fellowship has been extremely successful. It has contributed significantly to the advancement of the field through the research conducted, with new, testable proposals. The training objectives were satisfactorily completed, as well as the goal of reaching a position of professional maturity. The Fellow was offered a five-year Assistant Professorship at NORDITA (2016-), and more recently a permanent faculty position at Ecole Polytechnique (Palaiseau, France) as well as an ERC Starting Grant (2017) on 'Hydrodynamics, holography and strongly-coupled quantum matter'.

Thermoelectric transport in the normal phase has been thoroughly investigated, and much is known on the temperature, magnetic field and frequency dependence thermoelectric conductivities. The experiments suggest that a Quantum Critical Point underlies the phenomenology of the normal phase, albeit of a non-conventional kind. Indeed, the scaling properties measured experimentally depart from the expectations at a conventional Quantum Critical Point, which are completely fixed by scale invariance.

Non-scale invariant Quantum Critical Points are challenging to model in Condensed Matter. However they arise naturally in the low temperature limit of black holes in Anti de Sitter, which can be used to model strongly-coupled systems in general through Gauge/Gravity Duality (aka Holography). Gauge/Gravity duality was discovered in 1997 by Juan Maldacena, who showed that within string theory a certain gauge theory (N=4 super Yang Mills) was dual to a certain theory of gravity (Type IIb supergravity on AdS5xS5). This a weak/strong, classical/quantum duality. That is, when the field theory is quantum and strongly-coupled, it can be described by a weakly-coupled, classical theory of gravity. Essentially, Einstein’s gravity coupled to a few matter fields.

Part of the work accomplished in the Fellowship was precisely to study holographically these non conventional Quantum Critical Points. We have shown how in general the low temperature scalings of thermoelectric conductivities arise from the charge and entropy density operators acquiring anomalous dimensions in the effective long wavelength theory. These anomalous dimensions are directly responsible for the departure from strict scale invariant behaviour. These results partially inspired a theoretical proposal for the scaling of zero frequency conductivities in high Tc superconductors. It could be confirmed or disproved by future experiments providing missing transport data.

Another important aspect of the phenomenology of high Tc superconductors is the pattern of symmetry breaking as the temperature is lowered. Of course, spontaneous breaking of a U(1) symmetry and low temperature superconductivity have long been appreciated to play an important role in the phase diagram of these materials. However, over the last 20 years, spontaneous breaking of space translations and time reversal symmetry have also been observed experimentally, both as static and fluctuating orders. Understanding how the pattern of symmetry breaking affects transport is thus also a pressing question.

I have tackled these questions using hydrodynamics and the memory matrix approach. Hydrodynamics is an effective theory valid at late times and long wavelengths compared to the mean free path. Said otherwise, it captures small variations above local equilibrium. At strong coupling, individual constituents quickly decay. The effective degrees of freedom are thus conserved collective modes, such as the energy, charge and momentum densities, as well as Nambu-Goldstone bosons arising from spontaneously broken symmetries. This are the effective degrees of freedom which survive in the long wavelength approximation. Real materials often only have approximate symmetries, and the memory matrix provides a way of computing the relaxation rates associated to the slow decay of conserved quantities. Together with hydrodynamics, we can thus obtain expressions for the low frequency, long wavelength behaviour of thermoelectric conductivities.

As an example, we have proposed together with my collaborators that fluctuating translation symmetry breaking (stripes) could play an important role in explaining some of the transport properties of the normal, strange metallic phase of cuprate high Tc superconductors. With refined sensivity and probes sensitive to the spatial dependence of conductivities, this proposal can be tested in future experiments.

The Fellowship consisted of two phases: an Outgoing phase (2014-2016) at the Stanford Institute for Theoretical Physics (Stanford University, CA, USA) under the supervision of Pr. Sean Hartnoll; and a Return phase (2016-2017) at the Nordic Institute for Theoretical Physics (NORDITA, Stockholm, Sweden) under the supervision of Pr. Konstantin Zarembo. Overall, 10 articles have been published in peer-reviewed journals, one is currently under review and several are in preparation.

Dissemination activities have included seminars at 14 research institutes, oral presentations at 14 international workshops and conferences and 3 series of short lecture courses on topics related to the project.

To conclude, the Fellowship has been extremely successful. It has contributed significantly to the advancement of the field through the research conducted, with new, testable proposals. The training objectives were satisfactorily completed, as well as the goal of reaching a position of professional maturity. The Fellow was offered a five-year Assistant Professorship at NORDITA (2016-), and more recently a permanent faculty position at Ecole Polytechnique (Palaiseau, France) as well as an ERC Starting Grant (2017) on 'Hydrodynamics, holography and strongly-coupled quantum matter'.