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Register a new userScaling of geometric phase versus band structure in cluster-Ising models
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We study the phase diagram of a class of models in which a generalized cluster interaction can be quenched by Ising exchange interaction and external magnetic field. We characterize the various phases through winding numbers. They may be ordinary phases with local order parameter or exotic ones, known as symmetry protected topologically ordered phases. Quantum phase transitions with dynamical critical exponents z = 1 or z = 2 are found. Quantum phase transitions are analyzed through finite-size scaling of the geometric phase accumulated when the spins of the lattice perform an adiabatic precession. In particular, we quantify the scaling behavior of the geometric phase in relation with the topology and low energy properties of the band structure of the system.
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We present a fully analytically solvable family of models with many-body cluster interaction and Ising interaction. This family exhibits two phases, dubbed cluster and Ising phases, respectively. The critical point turns out to be independent of the cluster size $n+2$ and is reached exactly when both interactions are equally weighted. For even $n$ we prove that the cluster phase corresponds to a nematic ordered phase and in the case of odd $n$ to a symmetry protected topological ordered phase. Though complex, we are able to quantify the multi-particle entanglement content of neighboring spins. We prove that there exists no bipartite or, in more detail, no $n+1$-partite entanglement. This is possible since the non-trivial symmetries of the Hamiltonian restrict the state space. Indeed, only if the Ising interaction is strong enough (local) genuine $n+2$-partite entanglement is built up. Due to their analytically solvableness the $n$-cluster-Ising models serve as a prototype for studying non trivial-spin orderings and due to their peculiar entanglement properties they serve as a potential reference system for the performance of quantum information tasks.
Motivated by the recent success of tensor networks to calculate the residual entropy of spin ice and kagome Ising models, we develop a general framework to study frustrated Ising models in terms of infinite tensor networks %, i.e. tensor networks that can be contracted using standard algorithms for infinite systems. This is achieved by reformulating the problem as local rules for configurations on overlapping clusters chosen in such a way that they relieve the frustration, i.e. that the energy can be minimized independently on each cluster. We show that optimizing the choice of clusters, including the weight on shared bonds, is crucial for the contractibility of the tensor networks, and we derive some basic rules and a linear program to implement them. We illustrate the power of the method by computing the residual entropy of a frustrated Ising spin system on the kagome lattice with next-next-nearest neighbour interactions, vastly outperforming Monte Carlo methods in speed and accuracy. The extension to finite-temperature is briefly discussed.
For the one-dimensional Ising chain with spin-$1/2$ and exchange couple $J$ in a steady transverse field(TF), an analytical theory has well been developed in terms of some topological order parameters such as Berry phase(BP). For a TF Ising chain, the nonzero BP which depends on the exchange couple and the field strength characterizes the corresponding symmetry breaking of parity and time reversal(PT). However, there seems to exist a topological phase transition for the one-dimensional Ising chain in a longitudinal field(LF) with the reduced field strength $epsilon$. If the LF is added at zero temperature, researchers believe that the LF also could influence the PT-symmetry and there exists the discontinuous BP. But the theoretic characterization has not been well founded. This paper tries to aim at this problem. With the Jordan-Wigner transformation, we give the four-fermion interaction form of the Hamiltonian in the one-dimensional Ising chain with a LF. Further by the method of Wicks theorem and the mean-field theory, the four-fermion interaction is well dealt with. We solve the ground state energy and the ground wave function in the momentum space. We discuss the BP and suggest that there exist nonzero BPs when $epsilon=0$ in the paramagnetic case where $J<0$ and when $-1<epsilon<1$, in the diamagnetic case where $J>0$.
The competition between interactions and dissipative processes in a quantum many-body system can drive phase transitions of different order. Exploiting a combination of cluster methods and quantum trajectories, we show how the systematic inclusion of (classical and quantum) nonlocal correlations at increasing distances is crucial to determine the structure of the phase diagram, as well as the nature of the transitions in strongly interacting spin systems. In practice, we focus on the paradigmatic dissipative quantum Ising model: in contrast to the non-dissipative case, its phase diagram is still a matter of debate in the literature. When dissipation acts along the interaction direction, we predict important quantitative modifications of the position of the first-order transition boundary. In the case of incoherent relaxation in the field direction, our approach confirms the presence of a second-order transition, while does not support the possible existence of multicritical points. Potentially, these results can be tested in up-to date quantum simulators of Rydberg atoms.
Non-stoquastic drivers are known to improve the performance of quantum annealing by reducing first-order phase transitions into second-order ones in several mean-field-type model systems. Nevertheless, statistical-mechanical analysis shows that some target Hamiltonians still exhibit unavoidable first-order transitions even with non-stoquastic drivers, making them difficult for quantum annealing to solve. Recently, a mechanism called coherent catalysis was proposed by Durkin [Phys. Rev. A textbf{99}, 032315 (2019)], in which he showed the existence of a particular point on the line of first-order phase transitions where the energy gap scales polynomially as expected for a second-order transition. We show by extensive numerical computations that this phenomenon is observed in a few additional mean-field-type optimization problems where non-stoquastic drivers fail to change the order of phase transition in the thermodynamic limit. This opens up the possibility of using coherent catalysis to search for exponential speedups in systems previously thought to be exponentially slow for quantum annealing to solve.
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