Periodic Reporting for period 1 - SCLoTHiFi (Numerically exact theory of transport in strongly correlated systems at low temperature and under magnetic fields)
Okres sprawozdawczy: 2023-01-01 do 2025-06-30
Superconductivity is a striking phenomenon, occurring naturally at low temperature in various crystalline materials. It presents an absence of electrical resistance to the flow of a direct current. This exceptional property makes superconductors suitable for technological applications. Yet, many applications are still limited by the necessity to cool these materials down, below their critical temperature - the temperature at which the resistivity drops to zero and a material actually becomes superconducting.
For all the known materials (at atmospheric pressure), the critical temperature is below about 135 Kelvin. The family of materials with the highest critical temperatures are the cuprates, also known as "the high temperature superconductors". Finding ways to build materials with a higher critical temperature (ideally, as high as the room temperature) has motivated intense research in the last 4 decades. At present, there is no clear understanding of what property of the crystal lattice determines the critical temperature. For more than two decades, the highest recorded critical temperature has not increased.
The problem is that there is no simple theory of the superconductivity in the cuprates. As the current understanding goes, cuprate superconductivity is a true quantum many-body phenomenon, where interactions between individual electrons cannot be considered at any simplified level. This makes the description of the cuprates a quantum many-body problem, unsolvable in general, and one of the hardest in all of physics. In addition, it is not clear that there should be a single property that dominates the critical temperature; There may not be a simple model of the cuprate superconductors, and the critical temperature might be a detail, something that is a product of many different and equally important factors.
In our project we are working on the two main fronts related to theory of cuprate superconductors. First, we are developing numerical methods for the solution of the quantum many-body problem that arises in the description of cuprate superconductors. Second, we are searching for the minimal theoretical models able to capture the mechanisms of cuprate superconductivity, and look at how the parameters of those models correlate with the critical temperature. The latter is of utmost importance, as it might eventually allow for resverse engeneering of superconducting materials with desired properties.
For all the known materials (at atmospheric pressure), the critical temperature is below about 135 Kelvin. The family of materials with the highest critical temperatures are the cuprates, also known as "the high temperature superconductors". Finding ways to build materials with a higher critical temperature (ideally, as high as the room temperature) has motivated intense research in the last 4 decades. At present, there is no clear understanding of what property of the crystal lattice determines the critical temperature. For more than two decades, the highest recorded critical temperature has not increased.
The problem is that there is no simple theory of the superconductivity in the cuprates. As the current understanding goes, cuprate superconductivity is a true quantum many-body phenomenon, where interactions between individual electrons cannot be considered at any simplified level. This makes the description of the cuprates a quantum many-body problem, unsolvable in general, and one of the hardest in all of physics. In addition, it is not clear that there should be a single property that dominates the critical temperature; There may not be a simple model of the cuprate superconductors, and the critical temperature might be a detail, something that is a product of many different and equally important factors.
In our project we are working on the two main fronts related to theory of cuprate superconductors. First, we are developing numerical methods for the solution of the quantum many-body problem that arises in the description of cuprate superconductors. Second, we are searching for the minimal theoretical models able to capture the mechanisms of cuprate superconductivity, and look at how the parameters of those models correlate with the critical temperature. The latter is of utmost importance, as it might eventually allow for resverse engeneering of superconducting materials with desired properties.
In the first two years of the project, we have worked on developing numerical methods for the computation of electrical conductivity in interacting lattice models. We have focused on the Hubbard model, shown previously to qualitatively describe the normal-phase resistivity of the cuprates (i.e. above the critical temperature, where the resistivity is non-zero). The computation of conductivity is a difficult, long standing problem. In our work we have found a way to overcome the two main obstacles, and we are now able to compute numerically exact results, in parameter regime where none have been previously computed.
Our approach to this problem relies on diagrammatic Monte Carlo, a numerical method which breaks up a physical quantity into contributions from all possible scattering processes (Feynman diagrams). Using our new algorithm that does a part of the calculation analytically, we are able to avoid analytical continuatation and work with a formally infinite-size lattice. We are further able to cross-check the obtained results by an entirely different numerical method, with similar desirable properties.
This line of work should allow us to investigate the signatures of quantum criticality in the Hubbard model. Quantum phase transitions and the associated critical behavior have been recently shown (including in our own work) to be ubiquitous in the Hubbard model, in and outside of the parameter regime relevant for the cuprates. This is important because, in the cuprates, one of the standing hypotheses is that the large magnitude of the critical temperature is somehow related to an underlying quantum criticality. However, the Hubbard model might not capture all the physics relevant for the critical temperature.
For this reason, we are also studying various generalizations of the Hubbard model - we are looking for the minimal model that captures the physical mechanisms that majorly contribute to the critical temperature. We are focusing on the Emery model and the relation of its parameters with the experimentally measured critical temperature in various cuprates. Our results indicate that the geometry of the lattice is not the only important aspect of a crystal structure; rather, the strength of the effective coupling between the electrons must play an important role. We have made great progress in computing the parameters of the Emery model for more than 40 different cuprate compounds.
Our approach to this problem relies on diagrammatic Monte Carlo, a numerical method which breaks up a physical quantity into contributions from all possible scattering processes (Feynman diagrams). Using our new algorithm that does a part of the calculation analytically, we are able to avoid analytical continuatation and work with a formally infinite-size lattice. We are further able to cross-check the obtained results by an entirely different numerical method, with similar desirable properties.
This line of work should allow us to investigate the signatures of quantum criticality in the Hubbard model. Quantum phase transitions and the associated critical behavior have been recently shown (including in our own work) to be ubiquitous in the Hubbard model, in and outside of the parameter regime relevant for the cuprates. This is important because, in the cuprates, one of the standing hypotheses is that the large magnitude of the critical temperature is somehow related to an underlying quantum criticality. However, the Hubbard model might not capture all the physics relevant for the critical temperature.
For this reason, we are also studying various generalizations of the Hubbard model - we are looking for the minimal model that captures the physical mechanisms that majorly contribute to the critical temperature. We are focusing on the Emery model and the relation of its parameters with the experimentally measured critical temperature in various cuprates. Our results indicate that the geometry of the lattice is not the only important aspect of a crystal structure; rather, the strength of the effective coupling between the electrons must play an important role. We have made great progress in computing the parameters of the Emery model for more than 40 different cuprate compounds.
Our work on developing numerical methods has moved forward our ability to compute conductivity in interacting lattice models. We are obtaining results at the level of accuracy that was not possible in previous works. We have made some of the commonly used approximations no longer necessary. Furthermore, our implementation of real-frequency diagrammatic Monte Carlo algorithm is fully general and very flexible: it can be applied to various models, and for computation of different dynamical response functions. Our progress with method development in the last two years will be of great use for the foreseeable future.
Our work on correlating the parameters of the Emery model with the experimentally measured critical temperatures has produced better quality results than have the previous studies of the same kind. The step forward in our work is that we consider an exact transformation of the Emery Hamiltonian. This step is apparently necessary to get physically relevant model parameters, which is indicative of the underlying physics, rarely considered in previous works. The continuation of this work, where we compute systematically the model parameters for a large number of cuprate compounds is currently under way. We believe this work will set a new standard for the level of systematicness of a theoretical study of the cuprates, might reveal some hidden trends in what properties of these materials correlate with the critical temperature, and perhaps change the thinking in the field.
Our work on correlating the parameters of the Emery model with the experimentally measured critical temperatures has produced better quality results than have the previous studies of the same kind. The step forward in our work is that we consider an exact transformation of the Emery Hamiltonian. This step is apparently necessary to get physically relevant model parameters, which is indicative of the underlying physics, rarely considered in previous works. The continuation of this work, where we compute systematically the model parameters for a large number of cuprate compounds is currently under way. We believe this work will set a new standard for the level of systematicness of a theoretical study of the cuprates, might reveal some hidden trends in what properties of these materials correlate with the critical temperature, and perhaps change the thinking in the field.