Symmetry is a guiding principle in physics that allows us to generalize conclusions between many physical systems. In the ongoing search for new topological phases of matter, symmetry plays a crucial role by protecting topological phases. We address two converse questions relevant to the symmetry classification of systems: is it possible to generate all possible single-body Hamiltonians compatible with a given symmetry group? Is it possible to find all the symmetries of a given family of Hamiltonians? We present numerically stable, deterministic polynomial time algorithms to solve both of these problems. Our treatment extends to all continuous or discrete symmetries of non-interacting lattice or continuum Hamiltonians. We implement the algorithms in the Qsymm Python package, and demonstrate their usefulness through applications in active research areas of condensed matter physics, including Majorana wires and Kekule graphene.
Symmetry is a guiding principle in physics that allows us to generalize conclusions between many physical systems. In the ongoing search for new topological phases of matter, symmetry plays a crucial role by protecting topological phases. We address two converse questions relevant to the symmetry classification of systems: is it possible to generate all possible single-body Hamiltonians compatible with a given symmetry group? Is it possible to find all the symmetries of a given family of Hamiltonians? We present numerically stable, deterministic polynomial time algorithms to solve both of these problems. Our treatment extends to all continuous or discrete symmetries of non-interacting lattice or continuum Hamiltonians. We implement the algorithms in the Qsymm Python package, and demonstrate their usefulness through applications in active research areas of condensed matter physics, including Majorana wires and Kekule graphene.
The study of topological band structures is an active area of research in condensed matter physics and beyond. Here, we combine recent progress in this field with developments in machine learning, another rising topic of interest. Specifically, we introduce an unsupervised machine learning approach that searches for and retrieves paths of adiabatic deformations between Hamiltonians, thereby clustering them according to their topological properties. The algorithm is general, as it does not rely on a specific parametrization of the Hamiltonian and is readily applicable to any symmetry class. We demonstrate the approach using several different models in both one and two spatial dimensions and for different symmetry classes with and without crystalline symmetries. Accordingly, it is also shown how trivial and topological phases can be diagnosed upon comparing with a generally designated set of trivial atomic insulators.
Symmetry plays a central role in modern physics, from classifying quantum states to characterizing phases of matter through spontaneous symmetry breaking. In interacting fermionic systems with multiple internal degrees of freedom, however, determining the full continuous symmetry group and classifying possible order parameters remain challenging. In this work, we present a systematic framework for analyzing continuous symmetries and identifying candidate order parameters in such systems. By mapping the Hamiltonian to a Majorana representation, we obtain the generators of continuous symmetries from the Lie algebra of operators that commute with the Hamiltonian. We then identify the structure of this Lie algebra using the theory of semisimple Lie algebras. Building on representation theory, we further develop a systematic method for exhaustively enumerating candidate order parameters. By decomposing the exterior-power representations induced by the symmetry algebra on the Majorana space and incorporating discrete lattice symmetries, we classify these order parameters according to the symmetries they break. (Abridged. Please see the PDF manuscript for the complete abstract and specific model applications.)
Topological phase transitions represent a new paradigm beyond conventional Landau-Ginzburg symmetry breaking. Even in the absence of symmetry breaking, the symmetries of a Hamiltonian play an important role in stabilizing topological signatures. In this thesis, we will study the interplay between symmetry and topology in many-body systems, with three distinct physical setups in mind. First, we investigate a two-dimensional system of ultracold bosons which is condensed in the p-band of a triangular lattice. Certain smooth interaction profiles result in non-Abelian symmetry generators of the order parameter manifold, leading to anomalous physical consequences. These include a lack of Berezinskii-Kosterlitz-Thouless transition due to unstable vortex configurations, additional Goldstone modes, and a marginally divergent normal fluid density which implies lack of superfluidity in the thermodynamic limit. Our results suggest possible connections to solid state experiments on 4He films. We then show that disorder which keeps a Hamiltonian in the same Altland-Zirnbauer symmetry class is capable of inducing a topological transition. This is achieved by considering the normalizability of Majorana edge modes in a topological superconductor, and corroborated with entanglement metrics. While degeneracies of the many-body spectrum are robust with respect to weak perturbations, strong disorder can both promote and destroy a topological index. Lastly, we study topological edge modes and degeneracies in atomic, molecular, and optical systems which evolve non-unitarily in time or space due to non-equilibrium effects, including: condensate instabilities after a quench, decoherence channels from an external environment, and dissipation into a medium. Mathematically, the non-unitary evolution is generated by effective Hamiltonians which break Hermiticity. These systems can possess uniquely non-Hermitian symmetries which are absent from equilibrium counterparts. We present a symmetry-based classification, and investigate several systems which can be solved exactly.
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.
Symmetry plays a central role throughout physics, from Fourier analysis on discrete lattices to the classification of elementary particles in the standard model. In finite quantum systems, symmetries are often discrete, such as reflection and rotational symmetries. Here, we present the SIM-GRAPH method (Symmetric Ising Models - Graph Reduction And Projected Hamiltonians), which uses such symmetries to efficiently calculate ground-state observables of interacting quantum systems. By projecting all interactions onto a symmetric subset of the system, the method effectively reduces the number of sites, thus providing an exponential speedup over standard exact diagonalization. Furthermore, we introduce two extensions that broaden the applicability of the method and allow for a controlled trade-off between computational cost and accuracy.
Discrete translational symmetry plays a fundamental role in condensed matter physics and lattice gauge theories, enabling the analysis of systems that would otherwise be intractable. Despite this, many open problems remain. Quantum simulation promises to offer new insights, but progress is often limited by device connectivity constraints, which lead to prohibitively long computation times. We extend the use of spatial symmetry from the systems to be simulated to the quantum circuits simulating them. One application is that it becomes possible to efficiently and optimally alleviate device connectivity constraints algorithmically. This leads to reductions in quantum computational time by several orders of magnitude even for moderate system sizes, making such simulations feasible, with even greater relative gains for larger systems. This substantially enhances the capabilities of quantum computers in the simulation of condensed matter systems and lattice gauge theories, even before hardware improvements. Our Letter forms the basis for using spatial symmetry of quantum circuits in other areas of quantum computation, such as in the design and implementation of quantum error correcting codes.