Periodic Reporting for period 4 - HHQM (Hydrodynamics, holography and strongly-coupled quantum matter)
Okres sprawozdawczy: 2023-03-01 do 2024-02-29
Describing states of matter starting from their microscopic constituents is often a very hard task. If the interactions between the microscopic constituents are strong, as is the case in Quantum ChromoDynamics at low energies or in strongly-correlated quantum materials, perturbation theory fails for lack of a small expansion parameter. If they are weak, exact solutions are out of reach in quantum many-body problems: while conceptually straightforward, diagonalizing Hamiltonians for a large number of particles is in practice impossible. Effective theories and dualities are two of the tools at our disposal which circumvent these problems, each in their own specific way.
Effective theories rely on a truncation of the microscopic degrees of freedom by coarse-graining them into collective variables. A famous example is hydrodynamics, where the dynamics is not described by tracking the motion of individual molecules but by evolving a set of equations for the density, momentum and energy which follow from the global symmetries of the system.
Gauge/gravity duality was discovered in the context of String Theory and allows to compute the dynamics of strongly-interacting particles (gauge theories) using a weakly-coupled theory of gravity, which is a close cousin of General Relativity.
The goal of the ERC project was to apply these methods, more traditionally used in high energy physics, to strongly-correlated condensed matter systems, such as the strange metallic phase of high Tc superconductors. In these phases, charge transport defies the predictions of Fermi liquid theory, which is an effective theory that describes the low-energy dynamics of conventional metals in terms of weakly-coupled quasiparticle excitations in the vicinity of the Fermi surface. In strange metals, these quasiparticles are short-lived, ie strongly-coupled, making them amenable to effective field theoretic approaches or dualities.
Establishing controlled theoretical frameworks is an essential stepping stone towards understand the dynamics of these states, which remain unresolved in spite of decades of intense research.
Effective theories rely on a truncation of the microscopic degrees of freedom by coarse-graining them into collective variables. A famous example is hydrodynamics, where the dynamics is not described by tracking the motion of individual molecules but by evolving a set of equations for the density, momentum and energy which follow from the global symmetries of the system.
Gauge/gravity duality was discovered in the context of String Theory and allows to compute the dynamics of strongly-interacting particles (gauge theories) using a weakly-coupled theory of gravity, which is a close cousin of General Relativity.
The goal of the ERC project was to apply these methods, more traditionally used in high energy physics, to strongly-correlated condensed matter systems, such as the strange metallic phase of high Tc superconductors. In these phases, charge transport defies the predictions of Fermi liquid theory, which is an effective theory that describes the low-energy dynamics of conventional metals in terms of weakly-coupled quasiparticle excitations in the vicinity of the Fermi surface. In strange metals, these quasiparticles are short-lived, ie strongly-coupled, making them amenable to effective field theoretic approaches or dualities.
Establishing controlled theoretical frameworks is an essential stepping stone towards understand the dynamics of these states, which remain unresolved in spite of decades of intense research.
In this ERC project, we have blended together the effective field theory and holographic approaches to study strongly-coupled quantum phases of matter. Doing so has provided several insights which were not obvious by other approaches. More specifically, we have made several discoveries in the context of hydrodynamics of electronic fluids, low-energy dynamics of pinned charge density waves, instabilities of superfluids, and dynamics in the vicinity of quantum critical points.
*Hydrodynamics with approximate momentum conservation:
We have revisited the effective theory of systems with only approximate momentum conservation, for instance the electronic fluid in Graphene, but also more speculatively, the strange metallic phase of high Tc superconductors. To leading order, it is well-known that the weak relaxation of momentum causes a sharp `Drude' peak centered at zero frequency to appear in the frequency dependent response to an electric field. Its spectral weight is usually thought to be given by a combination of thermodynamic parameters, that is, equilibrium properties of the state. Further, the width of the peak is given by viscous transport coefficients. We showed that in fact the spectral weight receives out-of-equilibrium corrections as well. Since spectral weights are often used to extract thermodynamic properties, our results imply that caution should be exercised when doing so. They may be relevant to interpret differing trends in effective masses of charge carriers reported in strange metals using either thermodynamics or linear response.
*Phases breaking translations spontaneously:
Charge density waves and other phases where translations are spontaneously broken abound in the phase diagram of strongly-correlated materials. The hydrodynamic description of these states is textbook material. Spontaneous symmetry breaking generates a new degree of freedom, the so-called Nambu-Goldstone boson, which couples to the other hydrodynamic variables. Here it represents the freedom to slide the density wave around in the material without energy cost. Impurities are known to `pin’ the Goldstone mode by giving it a small mass. By using gauge/gravity duality and field theory, we discovered that there is a very simple relation between the lifetime of the Goldstone mode, its mass, and its diffusivity in the clean limit.
*Superfluids:
In a superfluid, particle number conservation is spontaneously broken. We established that the vicinity of quantum critical point could spoil the properties that the system flows without dissipation at zero temperature, as has been reported in some high Tc superconductors. Further, we gave a universal, thermodynamic description of their instability at large supercurrent, irrespective of the existence of quasiparticles.
* Breakdown of hydrodynamics and fundamental bounds on transport coefficients :
The breakdown of hydrodynamics at scales of order the local equilibration scale is expected to depend on the microscopic details of the system in the ultra-violet. Using gauge/gravity duality, we have found that the breakdown of hydrodynamics near a quantum critical point is controlled by the quantum critical degrees of freedom of the universal infra-red theory. Specifically, the equilibration time τ=h/(4πΔkBT), where T is temperature, kB the Boltzmann constant, h the Planck constant and Δ the scaling dimension of the least irrelevant deformation away from the quantum critical point.
Bounds on transport coefficients are of great import, as these quantities are not always easy to calculate microscopically. Such bounds may originate from causality, from consistency with Quantum Mechanics, etc. The Kovtun-Son-Starinets bound on the ratio of the shear viscosity over entropy density has played a key rôle in our qualitative understanding of strongly-coupled quantum phases, such as the Quark-Gluon-Plasma. It can be reformulated as a Quantum bound on the diffusivity of transverse momentum. Similar bounds have been conjectured for other transport coefficients, such as the thermal diffusivity. These bounds usually involved a timescale and a distinguished velocity. In the holographic system we studied, we gave a precise identification of the thermal diffusivity and the diffusivity of transverse momentum in terms of the local equilibration time and length scales, which saturate the bounds mentioned above.
*Hydrodynamics with approximate momentum conservation:
We have revisited the effective theory of systems with only approximate momentum conservation, for instance the electronic fluid in Graphene, but also more speculatively, the strange metallic phase of high Tc superconductors. To leading order, it is well-known that the weak relaxation of momentum causes a sharp `Drude' peak centered at zero frequency to appear in the frequency dependent response to an electric field. Its spectral weight is usually thought to be given by a combination of thermodynamic parameters, that is, equilibrium properties of the state. Further, the width of the peak is given by viscous transport coefficients. We showed that in fact the spectral weight receives out-of-equilibrium corrections as well. Since spectral weights are often used to extract thermodynamic properties, our results imply that caution should be exercised when doing so. They may be relevant to interpret differing trends in effective masses of charge carriers reported in strange metals using either thermodynamics or linear response.
*Phases breaking translations spontaneously:
Charge density waves and other phases where translations are spontaneously broken abound in the phase diagram of strongly-correlated materials. The hydrodynamic description of these states is textbook material. Spontaneous symmetry breaking generates a new degree of freedom, the so-called Nambu-Goldstone boson, which couples to the other hydrodynamic variables. Here it represents the freedom to slide the density wave around in the material without energy cost. Impurities are known to `pin’ the Goldstone mode by giving it a small mass. By using gauge/gravity duality and field theory, we discovered that there is a very simple relation between the lifetime of the Goldstone mode, its mass, and its diffusivity in the clean limit.
*Superfluids:
In a superfluid, particle number conservation is spontaneously broken. We established that the vicinity of quantum critical point could spoil the properties that the system flows without dissipation at zero temperature, as has been reported in some high Tc superconductors. Further, we gave a universal, thermodynamic description of their instability at large supercurrent, irrespective of the existence of quasiparticles.
* Breakdown of hydrodynamics and fundamental bounds on transport coefficients :
The breakdown of hydrodynamics at scales of order the local equilibration scale is expected to depend on the microscopic details of the system in the ultra-violet. Using gauge/gravity duality, we have found that the breakdown of hydrodynamics near a quantum critical point is controlled by the quantum critical degrees of freedom of the universal infra-red theory. Specifically, the equilibration time τ=h/(4πΔkBT), where T is temperature, kB the Boltzmann constant, h the Planck constant and Δ the scaling dimension of the least irrelevant deformation away from the quantum critical point.
Bounds on transport coefficients are of great import, as these quantities are not always easy to calculate microscopically. Such bounds may originate from causality, from consistency with Quantum Mechanics, etc. The Kovtun-Son-Starinets bound on the ratio of the shear viscosity over entropy density has played a key rôle in our qualitative understanding of strongly-coupled quantum phases, such as the Quark-Gluon-Plasma. It can be reformulated as a Quantum bound on the diffusivity of transverse momentum. Similar bounds have been conjectured for other transport coefficients, such as the thermal diffusivity. These bounds usually involved a timescale and a distinguished velocity. In the holographic system we studied, we gave a precise identification of the thermal diffusivity and the diffusivity of transverse momentum in terms of the local equilibration time and length scales, which saturate the bounds mentioned above.
Our results go beyond the state of the art in several aspects:
- out-of-equilibrium corrections to spectral weights had not been explicitly discussed before;
- the pinning-diffusivity relation giving the damping of Goldstones in pinned charge density waves is a new result;
- the zero temperature density in superfluids was thought to be always equal to the superfluid density, ie that no normal degrees of freedom could survive to zero temperature;
- there was no framework to connect the Landau instability in weakly-coupled and strongly-coupled superfluids;
- there are very few calculable and controlled models of quantum critical phases besides holographic ones; we constructed and explored the properties of several families of them and used our results to derive more general insights.
- out-of-equilibrium corrections to spectral weights had not been explicitly discussed before;
- the pinning-diffusivity relation giving the damping of Goldstones in pinned charge density waves is a new result;
- the zero temperature density in superfluids was thought to be always equal to the superfluid density, ie that no normal degrees of freedom could survive to zero temperature;
- there was no framework to connect the Landau instability in weakly-coupled and strongly-coupled superfluids;
- there are very few calculable and controlled models of quantum critical phases besides holographic ones; we constructed and explored the properties of several families of them and used our results to derive more general insights.