Periodic Reporting for period 1 - HATS (Holography and Topological Semimetals)
Reporting period: 2023-10-01 to 2025-09-30
Weyl semimetals are a recently-discovered class of materials, which have various unusual properties. For example, they conduct electricity in a manner different from ordinary metals, which may make them useful for building novel electronic devices. Theoretical models are an important tool for understanding the physics of Weyl semimetals and for finding applications of them. Weyl semimetals are an example of a broader class of material, called topological semimetals.
Some Weyl semimetals are “strongly correlated”, meaning that the electrons within the material interact very strongly with each another, with important consequences for the properties of the material. Strong correlations make theoretical modelling difficult, as they cause some of the techniques that are often used to approximately solve physical models to fail.
A useful tool for modelling strongly coupled materials is holography, also known as the AdS/CFT correspondence. The idea of holography is that some theoretical models of strongly correlated materials are equivalent to different models of totally different physical systems.
Concretely, some models of strongly correlated systems are mathematically equivalent to Einstein’s theory of general relativity, which describes gravity, combined with various types of particles interacting under gravity. Two models related by holography are said to be “holographically dual” to one another. What the equivalence between holographically dual models means is that predictions for one model may be made by performing calculations in the other. This is useful when the equations arising in one model are easier to solve than the equations arising in its holographic dual.
In this project, we used holography to build and study models of strongly correlated Weyl semimetals and related materials, such as other topological semimetals. The principle behind such a model is to construct a model of gravity (general relativity plus particles) that has the right particle content to be equivalent to a model of the desired type of material. Then, predictions for the strongly correlated material may be made by solving the equations arising in the gravity model.
One of the key objectives of the project was to construct models exhibiting various important aspects of Weyl semimetal physics that have not so far been modelled successfully using holography, such as by breaking symmetries that are present in existing models. Doing so makes the models constructed more realistic, at the cost of making their construction and analysis more complicated.
Some Weyl semimetals are “strongly correlated”, meaning that the electrons within the material interact very strongly with each another, with important consequences for the properties of the material. Strong correlations make theoretical modelling difficult, as they cause some of the techniques that are often used to approximately solve physical models to fail.
A useful tool for modelling strongly coupled materials is holography, also known as the AdS/CFT correspondence. The idea of holography is that some theoretical models of strongly correlated materials are equivalent to different models of totally different physical systems.
Concretely, some models of strongly correlated systems are mathematically equivalent to Einstein’s theory of general relativity, which describes gravity, combined with various types of particles interacting under gravity. Two models related by holography are said to be “holographically dual” to one another. What the equivalence between holographically dual models means is that predictions for one model may be made by performing calculations in the other. This is useful when the equations arising in one model are easier to solve than the equations arising in its holographic dual.
In this project, we used holography to build and study models of strongly correlated Weyl semimetals and related materials, such as other topological semimetals. The principle behind such a model is to construct a model of gravity (general relativity plus particles) that has the right particle content to be equivalent to a model of the desired type of material. Then, predictions for the strongly correlated material may be made by solving the equations arising in the gravity model.
One of the key objectives of the project was to construct models exhibiting various important aspects of Weyl semimetal physics that have not so far been modelled successfully using holography, such as by breaking symmetries that are present in existing models. Doing so makes the models constructed more realistic, at the cost of making their construction and analysis more complicated.
The laws of physics are, to the best of our knowledge, invariant under translations, meaning that they are the same wherever in the universe you are. However, physical objects themselves are not invariant under translations. Many physical models ignore this, as translational invariance makes their equations much simpler. One of the achievements of this project was the construction of a holographic model that breaks translational invariance in a relatively simple manner.
A Weyl semimetal also has to break one of two symmetries, either inversion symmetry (a type of reflection), or time-reversal symmetry. A holographic model of a family of topological semimetals with broken inversion symmetry was constructed. This represents a breakthrough towards making holographic models more useful, since previously existing models of Weyl semimetals and other topological semimetals were based on broken time-reversal symmetry instead. However, many of the known existing Weyl semimetals are of the type that breaks inversion symmetry.
An interface refers to the boundary between two types of material. In this project, a quantity called entanglement entropy, which measures the amount of quantum entanglement between different regions of space, was computed for an interface. This was used to quantify the number of degrees of freedom residing at the interface.
Other work focused on a mechanism for generating light states in holographic models, and further developments for the holographic computation of entanglement entropy.
A Weyl semimetal also has to break one of two symmetries, either inversion symmetry (a type of reflection), or time-reversal symmetry. A holographic model of a family of topological semimetals with broken inversion symmetry was constructed. This represents a breakthrough towards making holographic models more useful, since previously existing models of Weyl semimetals and other topological semimetals were based on broken time-reversal symmetry instead. However, many of the known existing Weyl semimetals are of the type that breaks inversion symmetry.
An interface refers to the boundary between two types of material. In this project, a quantity called entanglement entropy, which measures the amount of quantum entanglement between different regions of space, was computed for an interface. This was used to quantify the number of degrees of freedom residing at the interface.
Other work focused on a mechanism for generating light states in holographic models, and further developments for the holographic computation of entanglement entropy.
The holographic model of systems with broken translational invariance described above depends on a free function, which may be specified at will. Different choices of this function allow for the modelling of different types of system, including semimetals, topological insulators, and more, and also allows for study of various time-dependent processes. Thus, the construction of the model exceeds the aims of the project, allowing for the study of a wide range of systems. To exploit this fully it will be important to disseminate the model to researchers working on applications on holography. This will serve both to encourage other research groups to work on the model, and for the researcher to learn of potential new applications of the model.