Interacting chiral fermions on the lattice with matrix product operator norms (2405.10285v2)
Abstract: We develop a Hamiltonian formalism for simulating interacting chiral fermions on the lattice while preserving unitarity and locality and without breaking the chiral symmetry. The fermion doubling problem is circumvented by constructing a Fock space endowed with a semi-definite norm. When projecting our theory on the the single-particle sector, we recover the framework of Stacey fermions, and we demonstrate that the scaling limit of the free model recovers the chiral fermion field. Technically, we make use of a matrix product operator norm to mimick the boundary of a higher dimensional topological theory. As a proof of principle, we consider a single Weyl fermion on a periodic ring with Hubbard-type nearest-neighbor interactions and construct a variational generalized DMRG code to demonstrate that the ground state for large system sizes can be determined efficiently. As our tensor network approach does not exhibit any sign problem, we can add a chemical potential and study real-time evolution.
- H. B. Nielsen and M. Ninomiya, Absence of neutrinos on a lattice:(i). proof by homotopy theory, Nuclear Physics B 185, 20 (1981a).
- H. B. Nielsen and M. Ninomiya, Absence of neutrinos on a lattice:(ii). intuitive topological proof, Nuclear Physics B 193, 173 (1981b).
- K. G. Wilson, Confinement of quarks, Physical review D 10, 2445 (1974).
- P. H. Ginsparg and K. G. Wilson, A remnant of chiral symmetry on the lattice, Physical Review D 25, 2649 (1982).
- J. Kogut and L. Susskind, Hamiltonian formulation of wilson’s lattice gauge theories, Physical Review D 11, 395 (1975).
- W. Bietenholz and U.-J. Wiese, Perfect lattice actions for quarks and gluons, Nuclear Physics B 464, 319 (1996).
- S. D. Drell, M. Weinstein, and S. Yankielowicz, Strong-coupling field theories. ii. fermions and gauge fields on a lattice, Physical Review D 14, 1627 (1976).
- D. B. Kaplan, A method for simulating chiral fermions on the lattice, Physics Letters B 288, 342 (1992).
- D. B. Kaplan, Chiral gauge theory at the boundary between topological phases, Phys. Rev. Lett. 132, 141603 (2024).
- D. B. Kaplan and S. Sen, Weyl fermions on a finite lattice, Phys. Rev. Lett. 132, 141604 (2024).
- F. Verstraete, J. J. Garcia-Ripoll, and J. I. Cirac, Matrix product density operators: Simulation of finite-temperature and dissipative systems, Physical review letters 93, 207204 (2004a).
- R. Stacey, Eliminating lattice fermion doubling, Physical Review D 26, 468 (1982).
- F. Verstraete and J. I. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimensions, arXiv preprint cond-mat/0407066 (2004).
- L. Lootens, C. Delcamp, and F. Verstraete, Dualities in one-dimensional quantum lattice models: topological sectors, PRX Quantum 5, 010338 (2024).
- T. J. Osborne and A. Stottmeister, Conformal field theory from lattice fermions, Communications in Mathematical Physics 398, 219 (2023a).
- T. J. Osborne and A. Stottmeister, On the renormalization group fixed-point of the two-dimensional Ising model at criticality, Scientific Reports 13, 14859 (2023b).
- K. G. Wilson, The renormalization group: Critical phenomena and the Kondo problem, Reviews of Modern Physics 47, 773 (1975).
- E. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain, Annals of Physics 16, 407 (1961).
- O. O’Brien, L. Lootens, and F. Verstraete, Local jordan-wigner transformations on the torus, arXiv preprint arXiv:2404.07727 (2024).
- S. R. White, Density matrix formulation for quantum renormalization groups, Physical review letters 69, 2863 (1992).
- F. Verstraete, D. Porras, and J. I. Cirac, Dmrg and periodic boundary conditions: a quantum information perspective, arXiv preprint cond-mat/0404706 (2004b).
- N. E. Sherman, A. Avdoshkin, and J. E. Moore, Universality of critical dynamics with finite entanglement (2023), arXiv:2301.09681 [quant-ph] .
- A. Ueda and M. Oshikawa, Finite-size and finite bond dimension effects of tensor network renormalization, Phys. Rev. B 108, 024413 (2023).
- G. H. Golub and Q. Ye, An inverse free preconditioned krylov subspace method for symmetric generalized eigenvalue problems, SIAM Journal on Scientific Computing 24, 312 (2002), https://doi.org/10.1137/S1064827500382579 .
- B. Pirvu, J. Haegeman, and F. Verstraete, Matrix product state based algorithm for determining dispersion relations of quantum spin chains with periodic boundary conditions, Physical Review B 85, 035130 (2012b).
- L. Vanderstraeten, J. Haegeman, and F. Verstraete, Tangent-space methods for uniform matrix product states, SciPost Physics Lecture Notes , 007 (2019).
- F. Verstraete and J. I. Cirac, Matrix product states represent ground states faithfully, Physical review b 73, 094423 (2006).
- M. B. Hastings, An area law for one-dimensional quantum systems, Journal of statistical mechanics: theory and experiment 2007, P08024 (2007).
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