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arXiv:1801.04183 · Journal of High Energy Physics

Tensor network formulation for two-dimensional lattice $$ \mathcal{N} $$ = 1 Wess-Zumino model

Daisuke Kadoh, Yoshinobu Kuramashi, Y. Nakamura, Ryo Sakai, Shinji Takeda, Yusuke Yoshimura

Published Mar 1, 2018• 52 citations
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Abstract

Abstract Supersymmetric models with spontaneous supersymmetry breaking suffer from the notorious sign problem in stochastic approaches. By contrast, the tensor network approaches do not have such a problem since they are based on deterministic procedures. In this work, we present a tensor network formulation of the two-dimensional lattice $$ \mathcal{N} $$ N = 1 Wess-Zumino model while showing that numerical results agree with the exact solutions for the free case.

DOI
10.1007/jhep03(2018)141

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Tensor network formulation for two-dimensional lattice N \mathcal{N} N = 1 Wess-Zumino model
Abstract Supersymmetric models with spontaneous supersymmetry breaking suffer from the notorious sign problem in stochastic approaches. By contrast, the tensor network approaches do not have such a problem since they are based on deterministic procedures. In this work, we present a tensor network formulation of the two-dimensional lattice N\mathcal{N}N N = 1 Wess-Zumino model while showing that numerical results agree with the exact solutions for the free case.
We study a tensor network formulation of the two dimensional lattice N = 1 Wess–Zumino model with Wilson derivatives for both fermions and bosons. The tensor renormalization group allows us to compute the partition function without the sign problem, and basic ideas to obtain a tensor network for both fermion and scalar boson systems were already given in previous works. In addition to improving the methods, we have constructed a tensor network representation of the model including the Yukawatype interaction of Majorana fermions and real scalar bosons. We present some numerical results.
Future quantum computers will enable novel sign-problem-free studies of dynamical phenomena in non-perturbative quantum field theories, including real-time evolution and spontaneous supersymmetry breaking. We are investigating applications of quantum computing to low-dimensional supersymmetric lattice systems that can serve as testbeds for existing and near-future quantum devices. Here we present initial results for the N=1\mathcal{N} = 1N=1 Wess--Zumino model in 1+1~dimensions, building on our prior analyses of 0+1-dimensional supersymmetric quantum mechanics. In addition to exploring supersymmetry breaking using the variational quantum eigensolver, we consider the prospects for real-time evolution.
Lattice studies of spontaneous supersymmetry breaking suffer from a sign problem that in principle can be evaded through novel methods enabled by quantum computing. Focusing on lower-dimensional lattice systems with more modest resource requirements, in particular the N=1\mathcal N = 1N=1 Wess--Zumino model in 1+1 dimensions, we are exploring ways quantum computing could be used to study spontaneous supersymmetry breaking. A particularly promising recent development is to apply the variational quantum deflation algorithm, which generalizes the variational quantum eigensolver so as to resolve multiple low-energy states.
The variational tensor network renormalization approach to two-dimensional (2D) quantum systems at finite temperature is applied to a model suffering the notorious quantum Monte Carlo sign problem---the orbital eg{e}_{g}eg​ model with spatially highly anisotropic orbital interactions. Coarse graining of the tensor network along the inverse temperature $\ensuremath{\beta}$ yields a numerically tractable 2D tensor network representing the Gibbs state. Its bond dimension DDD---limiting the amount of entanglement---is a natural refinement parameter. Increasing DDD we obtain a converged order parameter and its linear susceptibility close to the critical point. They confirm the existence of finite order parameter below the critical temperature Tc{T}_{c}Tc​, provide a numerically exact estimate of Tc{T}_{c}Tc​, and give the critical exponents within 11%1 of the 2D Ising universality class.