Luttinger liquid on a lattice

Zakharov, V.

Understanding interactions in quantum many-body systems remains one of the most profound and difficult challenges in condensed matter physics. While free particle systems can be solved through single-particle techniques, the introduction of interactions turns the problem essentially many-body and quickly intractable. The Hilbert space grows exponentially, and traditional approaches become useless.
To deal with this, one typically turns to numerical methods by discretizing the system on a lattice. This approach generally works well for many 1D problems, including those described by Schrödinger equation. However, when it comes to massless Dirac fermions with protected chirality, the situation becomes more complicated. Any naive discretization inevitably runs into the fermion doubling problem, formalized by the Nielsen–Ninomiya theorem, which forbids a straightforward lattice realization of a single chiral mode without either breaking symmetry or introducing unphysical degrees of...Show moreUnderstanding interactions in quantum many-body systems remains one of the most profound and difficult challenges in condensed matter physics. While free particle systems can be solved through single-particle techniques, the introduction of interactions turns the problem essentially many-body and quickly intractable. The Hilbert space grows exponentially, and traditional approaches become useless.
To deal with this, one typically turns to numerical methods by discretizing the system on a lattice. This approach generally works well for many 1D problems, including those described by Schrödinger equation. However, when it comes to massless Dirac fermions with protected chirality, the situation becomes more complicated. Any naive discretization inevitably runs into the fermion doubling problem, formalized by the Nielsen–Ninomiya theorem, which forbids a straightforward lattice realization of a single chiral mode without either breaking symmetry or introducing unphysical degrees of freedom.
My thesis is motivated by this problem: the need to describe 1D interacting systems, and the difficulty of doing so numerically due to these fundamental obstacles. The main focus of this work is to develop and explore lattice-based numerical methods for strongly correlated chiral fermions in one dimension, with the Luttinger liquid as a central case study.
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Luttinger liquid on a lattice