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arXiv:1905.13204 · Physical review. D/Physical review. D.

Qubit regularization of the O(3) sigma model

Hersh Singh, Shailesh Chandrasekharan A

Published Sep 17, 2019• 54 citations
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Abstract

We construct a qubit regularization of the $O(3)$ nonlinear sigma model in two and three spatial dimensions using a quantum Hamiltonian with two qubits per lattice site. Using a worldline formulation and worm algorithms, we show that in two spatial dimensions our model has a quantum critical point where the well-known scale-invariant physics of the three-dimensional Wilson-Fisher fixed point is reproduced. In three spatial dimensions, we recover mean-field critical exponents at a similar quantum critical point. These results show that our qubit Hamiltonian is in the same universality class as the traditional classical lattice model close to the critical points. Simple modifications to our model also allow us to study the physics of traditional lattice models with $O(2)$ and ${Z}_{2}$ symmetries close to the corresponding critical points.

DOI
10.1103/physrevd.100.054505

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Motivated by the prospect of quantum simulation of quantum field theories, we formulate the O(N)O(N)O(N) nonlinear sigma model as a ``qubit'' model with an (N+1N+1N+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+12+12+1) dimensions, where the continuum physics of the nontrivial O(N)O(N)O(N) Wilson-Fisher fixed point is reproduced. We compute the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\eta}$ for the O(N)O(N)O(N) qubit models up to N=8N=8N=8, and find excellent agreement with known results in literature from various analytic and numerical techniques for the O(N)O(N)O(N) Wilson-Fisher universality class. Our models are suited for studying O(N)O(N)O(N) nonlinear sigma models on quantum computers up to N=8N=8N=8 in d=2d=2d=2, 3 spatial dimensions.
Qubit O(N)O(N)O(N) nonlinear sigma models
We reformulate the traditional lattice O(N)O(N)O(N) nonlinear sigma models as qubit models with a N+1N+1N+1 dimensional Hilbert space at each lattice site. Using an efficient worm algorithm in the worldline formulation, we demonstrate that the model has a quantum critical point in (2+1)(2+1)(2+1) dimensions. We compute the critical exponents ν\nuν and η\etaη up to N=8N=8N=8. By comparing these exponents with results from large-NNN and ϵ\epsilonϵ-expansions, we demonstrate that these quantum critical points lie in the usual O(N)O(N)O(N) Wilson-Fisher universality class. Our models are suited for studying O(N)O(N)O(N) Wilson-Fisher universality class on quantum computers up to N=8N=8N=8.
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.
Other than the commonly used Wilson's regularization of quantum field theories (QFTs), there is a growing interest in regularizations that explore lattice models with a strictly finite local Hilbert space, in anticipation of the upcoming era of quantum simulations of QFTs. A notable example is Euclidean qubit regularization, which provides a natural way to recover continuum QFTs that emerge via infrared fixed points of lattice theories. Can such regularizations also capture the physics of ultraviolet fixed points? We present a novel regularization of the asymptotically free massive continuum QFT that emerges at the Berezenski-Kosterlitz-Thouless (BKT) transition through a hard core loop-gas model, discussing the advantages this model provides compared to traditional regularizations. In particular, we demonstrate that without the need for fine-tuning, it can reproduce the universal step-scaling function of the classical lattice XY model in the massive phase as we approach the phase transition.
We provide strong evidence that the asymptotically free (1+1)-dimensional non-linear 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 anti-ferromagnetic Heisenberg-chain coupled anti-ferromagnetically 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 200,000 in lattice units and argue how the continuum limit could emerge. We provide a quantum circuit description of time-evolution of the model and argue that near-term quantum computers may suffice to demonstrate asymptotic freedom.
I study the properties of the quantum critical point of the transverse-field quantum Ising model on various fractal lattices such as the Sierpiński carpet, Sierpiński gasket, and Sierpiński tetrahedron. Using a continuous-time quantum Monte Carlo simulation method and finite-size scaling analysis, I identify the quantum critical point and investigate its scaling properties. Among others, I calculate the dynamic critical exponent and find that it is greater than one for all three structures. The fact that it deviates from one is a direct consequence of the fractal structures not being integer-dimensional regular lattices. Other critical exponents are also calculated. The exponents are different from those of the classical critical point and satisfy the quantum scaling relation, thus confirming that I have indeed found the quantum critical point. I find that the Sierpiński tetrahedron, of which the dimension is exactly 2, belongs to a different universality class than that of the two-dimensional square lattice. I conclude that the critical exponents depend on more details of the structure than just the dimension and the symmetry.