Quantum Phase Transition In Multiconnected-Jaynes-Cummings Model And Rabi-Hubbard Model

Strong-correlated many body quantum physics has been the key question of the con-densed matter physics for decades.The entanglement between particles makes some methods like mean-field method fail to study these questions and the exponential wall is so hard to be conquered.Although some advanced methods such as quantum Monte-Carlo and density matrix renormalization group shed light on us,but we are still too far to solve this problem.The concept of quantum computer has injected a powerful agent to solve the problem of strong correlation.If a universal quantum computer can be implemented,quantum many body correlation will no longer be difficult,and overcoming the exponential wall is no longer a dream.However,the current technology is difficult to meet the needs of universal quantum computers.Instead,scientists construct specific quantum simulation experiments on specific systems to achieve the purpose of simulation research.Current quantum simulation experiments are based on the technology of ultra-cold atom or superconducting quantum circuit,while the Jaynes-Cummings model and the Rabi model are models describing the basic components of cold atomic system or super-conducting quantum circuit system.Through the theoretical analysis of those two models,we can understand the application range of quantum simulation technology at this stage.Based on the Jaynes-Cummings model and the Rabi model,we constructed the corresponding condensed matter model,namely the multiconnected-Jaynes-Cummings?MCJC?model and the anisotropic-Rabi-Hubbard?ARH?model.This paper mainly dis-cusses the phase transition and related physical properties of these two models.The condensed matter model constructed by the Jaynes-Cummings model,due to the different connection modes,two main models can be obtained.The first one is Jaynes-Cummings-Hubbard?JCH?model,which is obtained by coupling between the cavities on lattice points.The MCJC model can be obtained by coupling between the two-level system?TLS?on one site and the cavity on adjacent site,which is the second one.The quantum phase transition and related properties of JCH model can be analyzed by the mean field method.The MCJC model is more complex,and larger quantum fluctuations invalidate the mean field method.We used the density matrix renormalization group the-ory to numerically simulate MCJC.We theoretically analyzed the physical properties of the MCJC model by the polariton picture.It was found that the Jaynes-Cummings cou-pling at the same site in the MCJC model would provide an effective on-site repelling interaction.Jaynes-Cummings coupling between sites would provide an effective jump coefficient.Effective exclusion and effective hopping respectively make the system more inclined to local or non-local.It is the competition between them that leads to a Mott-superfluid phase transition in the MCJC system.Due to the unique connection way,the jump term of the MCJC system and the coupling term on site are symmetric,so the phase diagram of MCJC should be symmetric.We also studied the MCJC model by DMRG method.We determine the phase diagram by calculating the ground state energy and chemical potentials.Through the analysis of the correlation function,the correlation properties of the particles in different phases are determined.The particles in the Mott phase are localized,the correlation function decays rapidly with the increase of the lattice distance,and the particles in the superfluid phase are non-local,the function decays slowly.Similarly,we studied the anisotropic-Rabi-Hubbard?ARH?model,which is a con-densed matter model constructed by the anisotropic Rabi model.As a generalization of the Rabi model and the Jaynes-Cummings model,the existence of a non-rotating term in anisotropic-Rabi breaks the U?1?symmetry,making its energy spectrum difficult to be solved.However,the model is an integrable model whose energy spectrum is hid-den in a transcendental function and there is no analytical form till now.Based on the anisotropic-Rabi model,we studied the finite ARH model and its related properties.We found that when the ratio of TLS energy split and the difference between the cavity frequency and the system jump coefficient tends to infinity,the ARH system will have an analytical solution.We calculated the analytical form of anisotropic-Rabi-dimer?ARD?and found that the system has two solutions under different conditions?the crit-ical conditions?,one solution is the non-degenerate ground state,and the other is double degenerate ground state.The double degenerated ground state consists both of the parity states,which means that this phase transition is along with the spontaneous Z₂symmetry breaking.Then we developed a method to deal with the analytical solution of the finite ARH model under this extreme condition.We obtained the analytical energy spectrum of the finite system,as well as the ground state.There is also a quantum phase transition in the finite ARH.On both sides of the critical condition,there are two sets of energy spectrum and two kinds of ground states,among which one is a local state,and the other is a non-local state.In the local state,the ground state of the system is non-degenerate,and there is an energy gap between the ground state and the excited states.As the system approaches critical conditions,the gap is closing while the ground state is degenerate at critical condition.In the non-local state,the ground states of the system are double de-generated.There is also an energy gap between the ground states and the excited states.As the system approaches the critical condition,the gap will also be closing.The corre-lation function of the local state decays quickly while the non-local states stay the same.We calculated the relevant physical properties of the system by DMRG.The numerical results support our theoretical results.In addition to this,we also calculated the relevant properties of the ARH system when the two-level system splitting tends to zero.We gave the ground state of the Rabi-Hubbard model in the case of TLS with zero splitting,and find that the ground states of the system are double degenerated.Near this limit,we used the perturbation method to deal with the finite ARH model based on this double-degenerate ground state.It is found that the double degeneracy is stable at this time,and the perturbation could not destroy it.That is,under this condition,there is no quantum phase transition in the ARH system.Our research provides theoretical support for the corresponding quantum simulation experiments,lays a foundation for the subsequent research of related models.