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arXiv:2002.09158 · Physical Review A

Non-Gaussian variational approach to Fermi polarons in one- and two-dimensional lattices

Ruijin Liu, Yue-Ran Shi, Wei Zhang

Published Sep 2, 2020• 7 citations
arXivHTMLPDF

Abstract

We study the Fermi polaron problem of one mobile spin-up impurity immersed atop the bath consisting of spin-down fermions in one- and two-dimensional square lattices. We solve this problem by applying a variational approach with non-Gaussian states after separating the impurity and the background via the Lee-Low-Pines transformation. The ground state with a fixed total momentum can be obtained via imaginary-time evolution. For the one-dimensional case, the variational ground-state energy is compared with exact Bethe ansatz solutions and numerical density matrix renormalization-group results with excellent agreement. In two-dimensional lattices, we focus on the dilute limit, and find a polaron-molecule evolution consistent with previous results obtained by variational and quantum Monte Carlo methods for models in continuum space. Compared to previous works, our method provides the lowest ground-state energy in the entire parameter region considered, and has an apparent advantage as it does not need to assume a priori any specific form of the variational wave function.

DOI
10.1103/physreva.102.033305

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arXiv:1709.07408•Physical Review A•Sep 21, 2017•15 citations
We study the Fermi polaron problem of one mobile spin-up impurity immersed atop the bath consisting of spin-down fermions in one- and two-dimensional square lattices. We solve this problem by applying a variational approach with non-Gaussian states after separating the impurity and the background via the Lee-Low-Pines transformation. The ground state with a fixed total momentum can be obtained via imaginary-time evolution. For the one-dimensional case, the variational ground-state energy is compared with exact Bethe ansatz solutions and numerical density matrix renormalization-group results with excellent agreement. In two-dimensional lattices, we focus on the dilute limit, and find a polaron-molecule evolution consistent with previous results obtained by variational and quantum Monte Carlo methods for models in continuum space. Compared to previous works, our method provides the lowest ground-state energy in the entire parameter region considered, and has an apparent advantage as it does not need to assume a priori any specific form of the variational wave function.
We investigate the Fermi-polaron system with one mobile impurity immersed atop a two-component interacting fermion bath in one- and two-dimensional square lattices, where the impurity atom interacts with only one species (the spin-up component) of the background. The ground state with a given total momentum can be approximated by an extended Gaussian state, which is constructed from the combination of Gaussian states and a non-Gaussian polaronic transformation (Lee-Low-Pines transformation). In the few-body limit, the variational energies of two- and three-body systems show good agreement with exact results, which indicates the validity of the non-Gaussian variational method. We then move on to the many-body limit, i.e., the polaron problem with an interacting background. We choose two representative fillings of the background fermions, $\ensuremath{\rho}=1/4,1/2$, and obtain the corresponding ground states. Our results show that the system will undergo a smooth crossover from a Fermi-polaron to a Bose-polaron system as the interaction between background fermions increases. We further analyze the double occupancy and momentum distribution of the background fermions and find that the impurity will not significantly affect the background pairing.
We develop a non-Gaussian variational approach that enables us to study both equilibrium and far-from-equilibrium physics of the two-dimensional Fermi polaron. This method provides an unbiased analysis of the polaron-to-molecule phase transition without relying on truncations in the total number of particle-hole excitations. Our results—which include the ground-state energy and quasiparticle residue—are in qualitative agreement with the known Monte Carlo calculations. The main advantage of the non-Gaussian states compared to conventional numerical methods is that they enable us to explore long-time polaron evolution and, in particular, study various spectral properties accessible to both solid-state and ultracold atom experiments. We design two types of radiofrequency spectroscopies to measure polaronic and molecular spectral functions. Depending on the parameter regime, we find that these spectral functions and fermionic density profiles near the impurity either display long-lived oscillations between the repulsive and attractive polaron branches or exhibit fast relaxational dynamics to the molecular state.
We consider an impurity problem in a quasi-two-dimensional Fermi gas, where a spin-down impurity is immersed in a Fermi sea of N spin-up atoms. Using a variational approach and an effective two-channel model, we obtain the energy for a wide range of interaction strength and for various different mass ratios between the impurity and the background fermion in the context of heteronuclear mixture. We demonstrate that in a quasi-two-dimensional Fermi gas there exists a transition of the ground state from polaron in the weakly interacting region to molecule in the strongly interacting region. The critical interaction strength of the polaron–molecule transition is non-universal and depends on the particle density of the background Fermi sea. We also investigate the excited repulsive polaron state, and find similar non-universal behavior.
A two-dimensional (2D) spin-up ideal Fermi gas interacting attractively with a spin-down impurity in the continuum undergoes, at zero temperature, a first-order phase transition from a polaron to a dimeron state. Here, we study a similar system on a square lattice, by considering the attractive 2D Fermi-Hubbard model with a single spin-down and a finite filling fraction of spin-up fermions. We study polaron and dimeron quasiparticle properties via variational ansatz up to one particle-hole excitation. Moreover, we develop a determinant diagrammatic Monte Carlo algorithm for this problem based on expansion in bare on-site coupling U . This algorithm turns out to be sign-problem-free at any filling of spin-up fermions, allowing one to sample very high diagram order (larger than 200 in our study) and to do simulations for large U / t (we go up to U / t = − 20 with t the hopping strength). Both methods give qualitatively consistent results. With variational ansatz we go to even larger on-site attraction. At very low spin-up filling fraction, we observe the polaron-to-dimeron transition, in agreement with the continuum case. Upon increasing the filling fraction, however, the transition shifts to higher values of | U | / t and the transition disappears beyond a filling fraction of about 20 % . In this region, the polaron state always gives a lower energy and has a finite quasiparticle residue. Our findings are directly relevant to cold atom experiments with 2D optical lattices: A small and finite density of spin-down impurities in the ground state will form a superfluid at strong coupling at low spin-up filling fraction. Above some critical spin-up filling fraction, the system is expected to remain a normal Fermi liquid, even in the strong-coupling limit.
We address the problem of a single ``spin-down'' impurity atom interacting attractively with a spin-up Fermi gas in two dimensions (2D). We consider the case where the mass of the impurity is greater than or equal to the mass of a spin-up fermion. Using a variational approach, we resolve the questions raised by previous studies and show that there is, in fact, a transition between polaron and molecule (dimer) ground states in 2D. For the molecule state, we use a variational wave function with a single particle-hole excitation on the Fermi sea and we find that its energy matches that of the exact solution in the limit of infinite impurity mass. Thus, we expect the variational approach to provide a reliable tool for investigating 2D systems.
We consider the Fermi polaron problem at zero temperature, where a single impurity interacts with noninteracting host fermions. We approach the problem starting with a Fr\"ohlich-like Hamiltonian where the impurity is described with canonical position and momentum operators. We apply the Lee-Low-Pine (LLP) transformation to change the fermionic Fr\"ohlich Hamiltonian into the fermionic LLP Hamiltonian, which describes a many-body system containing host fermions only. We adapt the self-consistent Hartree-Fock (HF) approach, first proposed by Edwards, to the fermionic LLP Hamiltonian in which a pair of host fermions with momenta k\mathbf{k}k and ${\mathbf{k}}^{\ensuremath{'}}$ interact with a potential proportional to $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{k}}^{\ensuremath{'}}$. We apply the HF theory, which has the advantage of not restricting the number of particle-hole pairs, to repulsive Fermi polarons in one dimension. When the impurity and host fermion masses are equal our variational ansatz, where HF orbitals are expanded in terms of free-particle states, produces results in excellent agreement with McGuire's exact analytical results based on the Bethe ansatz. This work raises the prospect of using the HF ansatz and its time-dependent generalization as building blocks for developing all-coupling theories for both equilibrium and nonequilibrium Fermi polarons in higher dimensions.