In this paper, we convert the lattice configurations into networks with different modes of links and consider models on networks with arbitrary numbers of interacting particle-pairs. We solve the Heisenberg model by revealing the relation between the Casimir operator of the unitary group and the conjugacy-class operator of the permutation group. We generalize the Heisenberg model by this relation and give a series of exactly solvable models. Moreover, by numerically calculating the eigenvalue of Heisenberg models and random walks on network with different numbers of links, we show that a system on lattice configurations with interactions between more particle-pairs have higher degeneracy of eigenstates. The highest degeneracy of eigenstates of a lattice model is discussed.In this paper, we convert the lattice configurations into networks with different modes of links and consider models on networks with arbitrary numbers of interacting particle-pairs. We solve the Heisenberg model by revealing the relation between the Casimir operator of the unitary group and the conjugacy-class operator of the permutation group. We generalize the Heisenberg model by this relation and give a series of exactly solvable models. Moreover, by numerically calculating the eigenvalue of Heisenberg models and random walks on network with different numbers of links, we show that a system on lattice configurations with interactions between more particle-pairs have higher degeneracy of eigenstates. The highest degeneracy of eigenstates of a lattice model is discussed.

arXiv:2012.12074Dec 22, 20201 citation

Quantum Stochastic Walks on the Permutation Group

Feng He, Arthur Hutsalyuk, Giuseppe Mussardo +1 more

How rapidly does order give way to randomness, and can quantum coherence accelerate this process? We address these questions through the paradigmatic problem of card shuffling, formulated as a random walk on the symmetric group $S_n$ . We first recast the random-transposition walk studied by Diaconis and Shahshahani, as well as more general walks generated by conjugacy classes of $S_n$ , in continuous time. We then identify the transition matrix of each classical walk with a permutation Hamiltonian generating a corresponding unitary quantum walk. Purely unitary evolution, however, does not generically converge to the uniform distribution in the classical sense of mixing: coherence preserves information rather than erasing it. We therefore embed the problem into a quantum stochastic walk, where coherent dynamics competes with the dissipative process responsible for classical mixing. In this setting, quantum coherence assists randomization. We prove that it can only decrease the distance from the uniform distribution in the computational basis and can therefore accelerate mixing. An analysis of the slowest mode yields a criterion for the coupling strength required to produce an appreciable speedup. Finally, numerical results reveal a scaling collapse of the ratio between quantum and classical mixing times onto a simple one-parameter form. Our results illustrate how coherence and dissipation can cooperate in the emergence of randomness in walks on permutation groups.How rapidly does order give way to randomness, and can quantum coherence accelerate this process? We address these questions through the paradigmatic problem of card shuffling, formulated as a random walk on the symmetric group $S_n$. We first recast the random-transposition walk studied by Diaconis and Shahshahani, as well as more general walks generated by conjugacy classes of $S_n$, in continuous time. We then identify the transition matrix of each classical walk with a permutation Hamiltonian generating a corresponding unitary quantum walk. Purely unitary evolution, however, does not generically converge to the uniform distribution in the classical sense of mixing: coherence preserves information rather than erasing it. We therefore embed the problem into a quantum stochastic walk, where coherent dynamics competes with the dissipative process responsible for classical mixing. In this setting, quantum coherence assists randomization. We prove that it can only decrease the distance from the uniform distribution in the computational basis and can therefore accelerate mixing. An analysis of the slowest mode yields a criterion for the coupling strength required to produce an appreciable speedup. Finally, numerical results reveal a scaling collapse of the ratio between quantum and classical mixing times onto a simple one-parameter form. Our results illustrate how coherence and dissipation can cooperate in the emergence of randomness in walks on permutation groups.

cond-mat.stat-mechCross-listedarXiv:2609.13387Sep 15, 20260 citations

S_n

. We first recast the random-transposition walk studied by Diaconis and Shahshahani, as well as more general walks generated by conjugacy classes of

S_n

, in continuous time. We then identify the transition matrix of each classical walk with a permutation Hamiltonian generating a corresponding unitary quantum walk. Purely unitary evolution, however, does not generically converge to the uniform distribution in the classical sense of mixing: coherence preserves information rather than erasing it. We therefore embed the problem into a quantum stochastic walk, where coherent dynamics competes with the dissipative process responsible for classical mixing. In this setting, quantum coherence assists randomization. We prove that it can only decrease the distance from the uniform distribution in the computational basis and can therefore accelerate mixing. An analysis of the slowest mode yields a criterion for the coupling strength required to produce an appreciable speedup. Finally, numerical results reveal a scaling collapse of the ratio between quantum and classical mixing times onto a simple one-parameter form. Our results illustrate how coherence and dissipation can cooperate in the emergence of randomness in walks on permutation groups.