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The quantum many body problem has been at the heart of much of theoretical and experimental physics over the past few decades. Even though we have understood the fundamental laws that govern the behavior of elementary particles for almost a century, the issue is that many interesting phenomena are the result of the complex collective behavior of many interacting quantum particles. In the words of condensed matter theorist Philip W. Anderson: "More is different."

Since simulating models with this many degrees of freedom exactly is entirely intractable computationally, approximations such as perturbation theory have been widely used to gain insight into their behavior. However, this approach requires that the theory is close to non-interacting, which renders it unusable in many cases of physical interest.

More recently, an approach based on insights from quantum information theory has shown great promise for tackling these non-perturbative regimes. It was understood that the low-energy quantum states of local models display relatively little entanglement compared to generic quantum states, a feature that is exploited in tensor network methods.

Researchers at University of Cambridge, Institut des Hautes Études Scientifiques and Ghent University recently introduced a new tensor network-based strategy that could advance the simulation of quantum many-body systems. Their proposed approach, outlined in a paper in Nature Â鶹ÒùÔºics, could enable the efficient simulation of quantum lattice models that are difficult to simulate using conventional tensor network-based methods.

"Recent works have cemented matrix product operators, a type of operator represented as a tensor network that explicitly encodes its entanglement structure, as the correct language for studying generalized global symmetries of one-dimensional quantum systems," Laurens Lootens, first author of the paper, told Â鶹ÒùÔº.

"Mathematically, such symmetries are encoded in structures that generalize ordinary groups called fusion categories, and it was understood that matrix product operators encode the different ways in which such a symmetry can act on a chain of quantum spins."

Tensor network-based approaches, particularly those leveraging networks known as matrix product states, are renowned for their numerical power. By efficiently representing many-body systems, they can help to overcome the limitations of standard computational methods, enabling the simulation of low-energy behavior in strongly interacting quantum systems.

"In our recent paper, we were looking to unite the more recent theoretical component concerning representation theory of generalized symmetries with well-established variational methods for optimizing matrix product states," explained Lootens.

"By leveraging the matrix product operator representation theory for generalized symmetries, we were able to prove that any one-dimensional quantum Hamiltonian with symmetry can be mapped to an equivalent dual Hamiltonian with the exact same spectrum, but whose ground state spontaneously breaks the complete dual symmetry."

The variational approach employed by researchers allowed them to obtain symmetry-breaking ground states of quantum many-body systems far more efficiently than symmetric approaches. This is because symmetric approaches generally enforce redundancies in entanglement patterns that are costly to compute.

"Mapping to a symmetry-breaking model removes this redundancy and reveals the mathematical structure that underpins the ground state as well as its quasiparticle excitation spectrum," said Lootens. "In doing so, we significantly enlarge the scope of traditional symmetric tensor network methods, which only perform well in the completely symmetric phase."

The method devised by Lootens and his colleagues lies at the intersection between mathematical and computational strategies. By combining the two, it could outperform conventional tensor network-based methods in the efficient representation of quantum many-body systems and their ground states.

"On the one hand, our work provides the necessary mathematical framework for describing the low-energy behavior of a general quantum spin chain," said Lootens. "On the other hand, it provides a completely new way of leveraging symmetries in quantum spin systems; it is both simpler and more efficient than current methods and additionally extends their applicability to all possible gapped phases."

As part of their recent study, the team applied their method to the study of one-dimensional (1D) quantum systems, which are simpler to approach both mathematically and computationally. In their future studies, however, they hope to apply it to higher-dimensional and more complex many-body systems.

"Tensor network methods have been widely applied to higher-dimensional problems as well, but these are notoriously challenging and their computational complexity is significantly worse than the one-dimensional case," added Lootens. "For this reason, it is all the more important to exploit all possible symmetries present in these models, which requires the generalization of our approach to the higher dimensional case.

"Fortunately, much progress has been made in the mathematical understanding of higher-dimensional generalized symmetries in the past few years, and we are convinced that this will have strong repercussions on the numerical tractability of the higher-dimensional quantum many-body problem."

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