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arXiv:1911.12353 · Physical review. D/Physical review. D.

Qubit regularized O(N) nonlinear sigma models

Hersh Singh

Published Jun 17, 2022• 22 citations
arXivHTMLPDF

Abstract

Motivated by the prospect of quantum simulation of quantum field theories, we formulate the $O(N)$ nonlinear sigma model as a ``qubit'' model with an ($N+1$)-dimensional local Hilbert space at each lattice site. Using an efficient worm algorithm in the worldline formulation, we demonstrate that the model has a second-order critical point in ($2+1$) dimensions, where the continuum physics of the nontrivial $O(N)$ Wilson-Fisher fixed point is reproduced. We compute the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\eta}$ for the $O(N)$ qubit models up to $N=8$, and find excellent agreement with known results in literature from various analytic and numerical techniques for the $O(N)$ Wilson-Fisher universality class. Our models are suited for studying $O(N)$ nonlinear sigma models on quantum computers up to $N=8$ in $d=2$, 3 spatial dimensions.

DOI
10.1103/physrevd.105.114509

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We provide strong evidence that the asymptotically free (1+1)-dimensional nonlinear O(3) sigma model can be regularized using a quantum lattice Hamiltonian, referred to as the "Heisenberg comb," that acts on a Hilbert space with only two qubits per spatial lattice site. The Heisenberg comb consists of a spin-half antiferromagnetic Heisenberg chain coupled antiferromagnetically to a second local spin-half particle at every lattice site. Using a world-line Monte Carlo method, we show that the model reproduces the universal step-scaling function of the traditional model up to correlation lengths of 200000 in lattice units and argue how the continuum limit could emerge. We provide a quantum circuit description of the time evolution of the model and argue that near-term quantum computers may suffice to demonstrate asymptotic freedom.
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