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LagrangeEuler
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Why density matrix renormalization group theory works only for 1D systems?
atyy said:There are generalizations to higher dimensions using matrix product states (and other tensor networks like the MERA).
http://arxiv.org/abs/1008.3477v2
http://arxiv.org/abs/1308.3318
http://arxiv.org/abs/1306.2164
DMRG (Density Matrix Renormalization Group) theory is a computational method used in condensed matter physics to study the properties of quantum systems. It is based on the concept of renormalization, which involves breaking down a complex system into smaller, simpler pieces and then recombining them to understand the overall behavior of the system.
DMRG works by systematically reducing the dimensionality of a quantum system in order to accurately calculate its ground state properties. This is achieved by iteratively truncating the system's Hilbert space and calculating the density matrix of the resulting reduced system. By repeating this process, DMRG is able to approximate the ground state of the original system with high accuracy.
DMRG is primarily used to study one-dimensional quantum systems, such as spin chains or interacting lattice models. However, it can also be extended to study higher-dimensional systems by using a combination of DMRG and other techniques.
One of the main advantages of DMRG is its ability to accurately calculate the ground state properties of complex systems that would be too difficult to solve using traditional methods. It is also relatively efficient and can handle large systems with many degrees of freedom, making it a powerful tool for studying a wide range of quantum systems.
Although DMRG is a powerful and versatile method, it does have some limitations. It is most effective for studying systems with strong one-dimensional correlations and may struggle with systems that exhibit long-range correlations. Additionally, DMRG is limited to studying quantum systems at low temperatures and cannot be used to study systems at finite temperatures.