Structure of Many-Body Hamiltonians in Different Symmetry Classes

Calvin M. Johnson

Vijay B. Shenoy, Adhip Agarwala, Arijit Haldar

We provide a systematic treatment of the tenfold way of classifying fermionic systems that naturally allows for the study of those with arbitrary N-body interactions. We identify four types of symmetries such systems can possess, which consist of one ordinary type (usual unitary symmetries), and three non-ordinary symmetries (such as time reversal, charge conjugation and sublattice). Focusing on systems that possess no non-trivial ordinary symmetries, we demonstrate that the non-ordinary symmetries are strongly constrained. This approach not only leads very naturally to the tenfold classes, but also obtains the canonical representations of these symmetries in each of the ten classes. We also provide a group cohomological perspective of our results in terms of projective representations. We then use the canonical representations of the symmetries to obtain the structure of Hamiltonians with arbitrary N-body interactions in each of the ten classes. We show that the space of N-body Hamiltonians has an affine subspace (of a vector space) structure in classes which have either or both charge conjugation and sublattice symmetries. Our results can help address many open questions including the topological classification of interacting fermionic systems. (C) 2017 Elsevier Inc. All rights reserved.

Nicolas Regnault, Rahul Nandkishore

We investigate the role of symmetries in determining the random matrix class describing quantum thermalization in a periodically driven many-body quantum system. Using a combination of analytical arguments and numerical exact diagonalization, we establish that a periodically driven ``Floquet'' system can be in a different random matrix class from the instantaneous Hamiltonian. A periodically driven system can thermalize even when the instantaneous Hamiltonian is integrable. A Floquet system that thermalizes in general can display integrable behavior at commensurate driving frequencies. When the instantaneous Hamiltonian and the Floquet operator both thermalize, the Floquet problem can be in the unitary class while the instantaneous Hamiltonian is always in the orthogonal class, and vice versa. We extract general principles regarding when a Floquet problem can thermalize to a different symmetry class from the instantaneous Hamiltonian. A (finite-sized) Floquet system can even display crossovers between different random matrix classes as a function of driving frequency.

Saumya Shah, Patrick Rebentrost

The computational cost of simulating quantum many-body systems can often be reduced by taking advantage of physical symmetries. While methods exist for specific symmetry classes, a general algorithm to find the full permutation symmetry group of an arbitrary Pauli Hamiltonian is notably lacking. This paper introduces a new method that identifies this symmetry group by establishing an isomorphism between the Hamiltonian's permutation symmetry group and the automorphism group of a coloured bipartite graph constructed from the Hamiltonian. We formally prove this isomorphism and show that for physical Hamiltonians with bounded locality and interaction degree, the resulting graph has a bounded degree, reducing the computational problem of finding the automorphism group to polynomial time. The algorithm's validity is empirically confirmed on various physical models with known symmetries. We further show that the problem of deciding whether two Hamiltonians are permutation-equivalent is polynomial-time reducible to the graph isomorphism problem using our graph representation. This work provides a general, structurally exact tool for algorithmic symmetry finding, enabling the scalable application of these symmetries to Hamiltonian simulation problems.

Vasilios Alexiades, George Siopsis

H. Stern

An investigation is made of the relationship between long-wavelength, low-frequency normal modes and broken symmetry in nonrelativistic many-body systems. In particular, the relationship between broken symmetry as manifested through the so-called Goldstone pole and the normal-mode structure is examined. Through the study of various models, we show that the structure of the normal modes is correlated to the Goldstone pole either completely, partially, or not at all, according to the class of symmetry of the Hamiltonian of the system. For example, in the neutral superconductor, the Hamiltonian has such low symmetry that although the Anderson modes restore the symmetry of the ground state, they have no relationship at all to the Goldstone pole. It is observed that as the symmetry decreases, a dynamical sum rule takes the place of such a correlation and in all systems a sum rule gives the normal-mode frequency $\ensuremath{\omega}$. These sum rules also give the $\ensuremath{\omega}$ distribution of states through the Huang-Klein dispersion relation.