Final Report Summary - CMADS (Condensed matter applications of the AdS / CFT correspondence)
The project concerned the application of holographic techniques for the study of condensed matter problems. The first aim of the project was to investigate the interactions of 'defects' by finding and studying (mostly finite temperature) dual gravity-solutions with back-reacting D-branes. The 'easiest' cases to be studied involve D-branes carrying no world-sheet electromagnetic field (i.e. no charge density in the dual system) in dual relativistic theories. The most relevant extension that has been pursued in the project includes the world-sheet gauge fields.
At the same time, the study in the first period of the fellowship had to allow for the identification of other problems in condensed matter physics which are suitable for the investigation in String Theory. The second aim of the project was to model some of these strong coupling systems in dual String backgrounds and study their physical properties holographycally.
The project has faced these tasks in two main directions. The first one has started from a full-fledged String Theory model of a system at finite temperature and chemical potential, the D3-D7 system (top-down approach). The second direction has adopted holographic techniques in the so-called bottom-up approach, in which a gravity action is considered containing the minimal field content for the description of a dual-condensed matter system, regardless the possible embedding of this gravity theory in String Theory.
The D3-D7 system is the benchmark model for holographic dualities including matter in the fundamental representation of the gauge group ('defects' coming from the D7-branes in String Theory) on top of the ubiquitous matter in the adjoint representation (given by the D3-branes). As such, it constitutes the optimal system for the study of strongly interacting (quasi-)particles, charged under an Abelian gauge symmetry (the dual of the electromagnetic symmetry), immersed in an uncharged medium. The former particles come from the D7-branes and are charged under the world-volume U(1) symmetry. The uncharged environment is provided by the D3 degrees of freedom.
While the system has been known from a long time, it was analysed only in the so-called probe approximation, where the influence of the D7 degrees of freedom on the whole D3 system was dropped. This approach loses a significant fraction of the physics of the system, e.g. its complete thermodynamics or relevant transport properties.
The first task of the project has been to consider the charged D3-D7 system in its completeness. To begin with, the backreaction of the D7-branes at finite temperature and charge density has been calculated. This has been possible due to a simplifying mechanism known as the smearing technique. Then, the complete thermodynamics of the system has been calculated, including the matrix of susceptibilities which allowed to check the thermodynamic stability. The calculation has been performed in the gravity theory, but the results are valid for the dual field theory at finite density, a model for e.g. strongly correlated electrons in a metal. As a second application, the energy loss of a (quasi-)particles traveling in the system has been analysed. To conclude with the direct practical applications, some non-standard optical properties, such as negative refraction and the presence of Additional Light Waves, have been considered. These studies have enlightened the usefulness of the D3-D7 charged system has a versatile and powerful model where to study a number of physical processes at strong quasi-particle correlation. In order to have the results in a comprehensive description, a complete review on the subject has been published on top of the original papers.
Further progress has been made, from a technical point of view, by deriving the consistent truncation to five dimensions of the whole ten-dimensional D3-D7 system. This far-from elementary step is necessary in order to have access, in the near future, to the study of the dynamical stability of the D3-D7 system, e.g. at small temperature towards non-homogeneous configurations, and to many transport properties, e.g. conductivities or hydrodynamics.
The second main line of the project has considered systems of unbalanced holographic superconductors and spintronics. The main motivation was to study the simplest system of holographic s-wave superconductors in the realistic regime where the condensing electrons have unbalanced Fermi-surfaces, which is the generic situation in experimental settings. The imbalance can in fact be created by impurities, doping or external fields coupling in different ways to the electron spins. The result at weak coupling is a rich phase diagram including limiting values of the imbalance, beyond which the system loses superconductivity, and a region of parameter space where the ground state relax to a non-homogeneous phase, known as Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) phase. The interest was to analyse this system in the regime of strong coupling, relevant to many unconventional superconductors and strange metals which represent one of the main unsolved study area of modern condensed matter physics.
The results have been very interesting. The phase diagram at strong coupling is quite different from the weak coupling one. There is only a second order phase transition between the superconducting phase and the normal, non-superconducting phase. There is no sign of inhomogeneous LOFF phase, consistently with previous phenomenological results.
On top of this results, the model for unbalanced superconductor has been shown to encode in a natural way spintronic effects, i.e. effects of mixed transport properties of spin and electric charge. Spintronics is a very active field of research, due to its important technological outcomes, e.g. in ordinary computer devices. The mixing in the transport of spin and charge is crucial to enhance the transport and storage properties of devices.
The holographic model analysed in this project has shown that this mixing spintronic effects are generic in the strongly coupled regime, being due to the simple and generic mixing of the energy-momentum tensor operator with the spin and charge current operators. This is a crucial observation, which paves the road for further future studies of spintronic effects in holographic models. In particular, as it has been shown in the simple model analysed in the project, all the conductivities can be straightforwardly computed, both in the normal and the superconducting phase of the system. This allowed for the understanding of the interplay of the spin and charge degrees of freedom at strong coupling.
Finally, as an aside minor direction, during the development of the project there has been the possibility for the calculation of meson spectra in a holographic model of a quantum chromodynamic (QCD)-like theory, with a surprisingly high precision for what concerns the rho-meson spectrum and an interesting mass-inversion behavior for the mesons made by light or heavy quarks.
In conclusion, the project has shown how holographic techniques are suitable and relevant for the study of strongly coupled Condensed Matter systems. Both the construction and the study of properties of strongly correlated systems has been performed with success, both in the top-down and bottom up approach. In the first case, it has been shown how the charged D3-D7 system constitutes the best candidate for a String Theory dual to systems with (quasi-)particles in the fundamental representation. In the second case, the simplest model for holographic s-wave superconductors and spintronics has been analysed.
The results of this project can be (and are actually being) employed directly for further developments in the field. In particular, studies are being performed on further properties of the D3-D7 charged systems, most importantly in the small temperature limit relevant for the study of the dual Fermi surface. Analogously, studies are being performed on holographic models of s-wave superconductors, with modified parameters and / or interaction terms to explore the possible variation of the phase diagram and the occurrence of non-homogeneous phases.
At the same time, the study in the first period of the fellowship had to allow for the identification of other problems in condensed matter physics which are suitable for the investigation in String Theory. The second aim of the project was to model some of these strong coupling systems in dual String backgrounds and study their physical properties holographycally.
The project has faced these tasks in two main directions. The first one has started from a full-fledged String Theory model of a system at finite temperature and chemical potential, the D3-D7 system (top-down approach). The second direction has adopted holographic techniques in the so-called bottom-up approach, in which a gravity action is considered containing the minimal field content for the description of a dual-condensed matter system, regardless the possible embedding of this gravity theory in String Theory.
The D3-D7 system is the benchmark model for holographic dualities including matter in the fundamental representation of the gauge group ('defects' coming from the D7-branes in String Theory) on top of the ubiquitous matter in the adjoint representation (given by the D3-branes). As such, it constitutes the optimal system for the study of strongly interacting (quasi-)particles, charged under an Abelian gauge symmetry (the dual of the electromagnetic symmetry), immersed in an uncharged medium. The former particles come from the D7-branes and are charged under the world-volume U(1) symmetry. The uncharged environment is provided by the D3 degrees of freedom.
While the system has been known from a long time, it was analysed only in the so-called probe approximation, where the influence of the D7 degrees of freedom on the whole D3 system was dropped. This approach loses a significant fraction of the physics of the system, e.g. its complete thermodynamics or relevant transport properties.
The first task of the project has been to consider the charged D3-D7 system in its completeness. To begin with, the backreaction of the D7-branes at finite temperature and charge density has been calculated. This has been possible due to a simplifying mechanism known as the smearing technique. Then, the complete thermodynamics of the system has been calculated, including the matrix of susceptibilities which allowed to check the thermodynamic stability. The calculation has been performed in the gravity theory, but the results are valid for the dual field theory at finite density, a model for e.g. strongly correlated electrons in a metal. As a second application, the energy loss of a (quasi-)particles traveling in the system has been analysed. To conclude with the direct practical applications, some non-standard optical properties, such as negative refraction and the presence of Additional Light Waves, have been considered. These studies have enlightened the usefulness of the D3-D7 charged system has a versatile and powerful model where to study a number of physical processes at strong quasi-particle correlation. In order to have the results in a comprehensive description, a complete review on the subject has been published on top of the original papers.
Further progress has been made, from a technical point of view, by deriving the consistent truncation to five dimensions of the whole ten-dimensional D3-D7 system. This far-from elementary step is necessary in order to have access, in the near future, to the study of the dynamical stability of the D3-D7 system, e.g. at small temperature towards non-homogeneous configurations, and to many transport properties, e.g. conductivities or hydrodynamics.
The second main line of the project has considered systems of unbalanced holographic superconductors and spintronics. The main motivation was to study the simplest system of holographic s-wave superconductors in the realistic regime where the condensing electrons have unbalanced Fermi-surfaces, which is the generic situation in experimental settings. The imbalance can in fact be created by impurities, doping or external fields coupling in different ways to the electron spins. The result at weak coupling is a rich phase diagram including limiting values of the imbalance, beyond which the system loses superconductivity, and a region of parameter space where the ground state relax to a non-homogeneous phase, known as Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) phase. The interest was to analyse this system in the regime of strong coupling, relevant to many unconventional superconductors and strange metals which represent one of the main unsolved study area of modern condensed matter physics.
The results have been very interesting. The phase diagram at strong coupling is quite different from the weak coupling one. There is only a second order phase transition between the superconducting phase and the normal, non-superconducting phase. There is no sign of inhomogeneous LOFF phase, consistently with previous phenomenological results.
On top of this results, the model for unbalanced superconductor has been shown to encode in a natural way spintronic effects, i.e. effects of mixed transport properties of spin and electric charge. Spintronics is a very active field of research, due to its important technological outcomes, e.g. in ordinary computer devices. The mixing in the transport of spin and charge is crucial to enhance the transport and storage properties of devices.
The holographic model analysed in this project has shown that this mixing spintronic effects are generic in the strongly coupled regime, being due to the simple and generic mixing of the energy-momentum tensor operator with the spin and charge current operators. This is a crucial observation, which paves the road for further future studies of spintronic effects in holographic models. In particular, as it has been shown in the simple model analysed in the project, all the conductivities can be straightforwardly computed, both in the normal and the superconducting phase of the system. This allowed for the understanding of the interplay of the spin and charge degrees of freedom at strong coupling.
Finally, as an aside minor direction, during the development of the project there has been the possibility for the calculation of meson spectra in a holographic model of a quantum chromodynamic (QCD)-like theory, with a surprisingly high precision for what concerns the rho-meson spectrum and an interesting mass-inversion behavior for the mesons made by light or heavy quarks.
In conclusion, the project has shown how holographic techniques are suitable and relevant for the study of strongly coupled Condensed Matter systems. Both the construction and the study of properties of strongly correlated systems has been performed with success, both in the top-down and bottom up approach. In the first case, it has been shown how the charged D3-D7 system constitutes the best candidate for a String Theory dual to systems with (quasi-)particles in the fundamental representation. In the second case, the simplest model for holographic s-wave superconductors and spintronics has been analysed.
The results of this project can be (and are actually being) employed directly for further developments in the field. In particular, studies are being performed on further properties of the D3-D7 charged systems, most importantly in the small temperature limit relevant for the study of the dual Fermi surface. Analogously, studies are being performed on holographic models of s-wave superconductors, with modified parameters and / or interaction terms to explore the possible variation of the phase diagram and the occurrence of non-homogeneous phases.