Bound on the dynamical exponent of frustration-free Hamiltonians and Markov processes

          @misc{ pirsa_PIRSA:25040115,
            doi = {},
            url = {https://pirsa.org/25040115},
            author = {Soejima, Tomohiro},
            keywords = {Condensed Matter},
            language = {en},
            title = {Bound on the dynamical exponent of frustration-free Hamiltonians and Markov processes},
            publisher = {Perimeter Institute},
            year = {2025},
            month = {apr},
            note = {PIRSA:25040115 see, \url{https://pirsa.org}}
          }
          

Abstract

Exactly solvable models have tremendously helped our understanding of condensed matter systems. A notable number of them are "frustration-free" in the sense that all local terms of the Hamiltonian can be minimized simultaneously. It has been particularly successful at describing the physics of gapped phases of matter, such as symmetry protected topological phases and topologically ordered phases. On the other hand, relatively little has been understood about gapless frustration-free Hamiltonians, and their ability to teach us about more generic systems. In this talk, we derive a constraint on the spectrum of frustration-free Hamiltonians. Their dynamical exponent z, which captures the scaling of the energy gap versus the system size, is bounded from below to be z >= 2. This proves that frustration-free Hamiltonians are incapable of describing conformal critical points with z = 1. Further, by a well-known mapping from Markov processes to frustration-free Hamiltonians, we show that the relaxation time for many Markov processes also scale with z >=2. This improves the previously known bound on the relaxation time scaling of z >= 7/4. The talk is based on works with Rintaro Masaoka and Haruki Watanabe.