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arXiv:1609.06554 · Physical review. B./Physical review. B

Class of topological phase transitions of Rashba spin-orbit coupled fermions on a square lattice

Yu Yi-Xiang, Fadi Sun, Jinwu Ye

Published Nov 19, 2018• 4 citations
arXivHTMLPDF

Abstract

We study the system of attractively interacting fermions hopping in a square lattice with any linear combinations of Rashba or Dresselhaus spin-orbit coupling in a normal Zeeman field. By imposing self-consistence equations at half filling with zero chemical potential, we find that there are three phases: band insulator (BI), superfluid (SF), and topological superfluid (TSF) with a Chern number $C=2$. The $C=2$ TSF happens in small Zeeman fields and weak interactions which is in the experimentally most easily accessible regime. The transition from the BI to the SF is a first-order one due to the multiminima structure of the ground state energy landscape. There is a class of topological phase transitions (TPTs) from the SF to the $C=2$ TSF at the low critical field ${h}_{c1}$, then another one from the $C=2$ TSF to the BI at the upper critical field ${h}_{c2}$. We derive effective actions to describe these two classes of topological phase transitions, then use them to study the Majorana edge modes and the zero modes inside the vortex core of the $C=2$ TSF near both ${h}_{c1}$ and ${h}_{c2}$, especially exploring their spatial and spin structures. We find that the edge modes decay into the bulk with oscillating behaviors and determine both the decay and oscillating lengths. We compute the bulk spectra and map out the Berry curvature distribution in momentum space near both ${h}_{c1}$ and ${h}_{c2}$. We elaborate some intriguing bulk-Berry curvature-edge-vortex correspondences. We also discuss the competitions between SFs and charge density wave states in more general cases. A possible classification scheme of all the TPTs in a square lattice is outlined. Comparisons with previous works on related systems are discussed. Possible experimental implications in cold atoms in an optical lattice are given.

DOI
10.1103/physrevb.98.174506

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We study the system of attractively interacting fermions hopping in a square lattice with any linear combinations of Rashba or Dresselhaus spin-orbit coupling in a normal Zeeman field. By imposing self-consistence equations at half filling with zero chemical potential, we find that there are three phases: band insulator (BI), superfluid (SF), and topological superfluid (TSF) with a Chern number C=2C=2C=2. The C=2C=2C=2 TSF happens in small Zeeman fields and weak interactions which is in the experimentally most easily accessible regime. The transition from the BI to the SF is a first-order one due to the multiminima structure of the ground state energy landscape. There is a class of topological phase transitions (TPTs) from the SF to the C=2C=2C=2 TSF at the low critical field hc1{h}_{c1}hc1​, then another one from the C=2C=2C=2 TSF to the BI at the upper critical field hc2{h}_{c2}hc2​. We derive effective actions to describe these two classes of topological phase transitions, then use them to study the Majorana edge modes and the zero modes inside the vortex core of the C=2C=2C=2 TSF near both hc1{h}_{c1}hc1​ and hc2{h}_{c2}hc2​, especially exploring their spatial and spin structures. We find that the edge modes decay into the bulk with oscillating behaviors and determine both the decay and oscillating lengths. We compute the bulk spectra and map out the Berry curvature distribution in momentum space near both hc1{h}_{c1}hc1​ and hc2{h}_{c2}hc2​. We elaborate some intriguing bulk-Berry curvature-edge-vortex correspondences. We also discuss the competitions between SFs and charge density wave states in more general cases. A possible classification scheme of all the TPTs in a square lattice is outlined. Comparisons with previous works on related systems are discussed. Possible experimental implications in cold atoms in an optical lattice are given.
We investigate the effects of attractive Hubbard interaction on two-component fermionic atoms in narrow two-dimensional (2D) energy bands that exhibit Rashba spin-orbit coupling (SOC) in the presence of an applied Zeeman field. This narrow-band 2D spin-orbit-coupled attractive Fermi-Hubbard model can potentially be realized in cold-atom systems in optical lattices with artificially engineered Rashba SOC and Zeeman field. Employing a self-consistent mean-field approximation for the pairing potential, we uncover a complex phase diagram featuring various topological superfluid (TS) phases, dependent on the chemical potential and the Zeeman field. We focus on the pairing potential and the corresponding quasiparticle gap characterizing the TS phases, which are notably small for a wide-band model with quadratic dispersion near the $\mathrm{\ensuremath{\Gamma}}$ point, as found in earlier work. We identify the parameter regimes that maximize the gap. We find that, while generally the value of the pairing potential increases with the reduction of the fermionic bandwidth, as expected for narrow- or flat-band systems, the magnitude of the topological gap characterizing the TS phases reaches a significantly improved maximum of about 10−−12.510--12.5%10−−12.5 of the interaction strength at finite values of the hopping amplitude, Rashba coupling, and Zeeman field. Our work establishes a robust operating regime for experimenting with topological superfluids in cold-atom optical lattices.
We consider a two-component Fermi gas with attractive interactions on a square optical lattice, and study the interplay of Zeeman field, spin-orbit coupling, and next-nearest-neighbor hopping on the ground-state phase diagrams in the entire BCS-BEC evolution. In particular, we first classify and distinguish all possible superfluid phases by the momentum-space topology of their zero-energy quasiparticle-quasihole excitations, and then numerically establish a plethora of quantum phase transitions in between. These transitions are further signaled and evidenced by the changes in the corresponding topological invariant of the system, i.e., its Chern number. Lastly, we find that the superfluid phase exhibits a reentrant structure, separated by a fingering normal phase, the origin of which is traced back to the changes in the single-particle density of states.
Abstract The equal Rashba-Dresselhaus (ERD) spin-orbit coupling, as the first type to be implemented in cold atom experiments, provides a unique platform for exploring topological quantum phases. This work investigates the ground state of a two-dimensional Fermi gas featuring both ERD coupling and a Zeeman field, mapping out its topological phase diagram throughout the BCS-BEC crossover. We classify the homogeneous superfluid phases into topologically trivial gapped or nontrivial gapless phase based on the nodal structure of the quasiparticle excitation spectrum. We reveal the evolution of topological superfluid states under various conditions by systematically characterizing the phase distribution across key parameters including the spin-orbit coupling strength, two-body binding energy, Zeeman field, and chemical potential. Our findings indicate that spin-orbit coupling enhances triplet pairing and stabilizes the gapless superfluid phase. Moreover, small binding energy, and strong Zeeman field are found to promote the emergence of the gapless phase, particularly the US-II phase identified in this work. These results elucidate the phase transition mechanisms in low-dimensional quantum systems.
Abstract Based on the two-dimensional (2D) attractive Fermi-Hubbard model with Rashba spin–orbit coupling (SOC), the SOC strength and Zeeman field dependences of the phase diagram are investigated by calculating the pairing gap self-consistently. The results reveal that the phase transition from the BCS superfluid to the topological superfluid happens under proper Zeeman field strength and SOC strength. In particular, in contrast to the BCS superfluid decreasing monotonically as the SOC strength increasing, the topological superfluid region shows a dome with the SOC strength increasing. An optimal region in the phase diagram to find the topological superfluid can be found, which is important to realize the topological superfluid in optical lattice experimentally. Then we obtain the change of both band structure and density of states (DOSs) during the topological phase transition, and explain the four peaks of DOS in the topological superfluid by the topology change of the low-energy branch of quasiparticle energy spectra. Moreover, the topological superfluid can be suppressed by the doping concentration.
We study possible many body phenomena in the Quantum Anomalous Hall system of weakly interacting spinor bosons in a square lattice. There are various novel spin-bond correlated superfluids (SF) and quantum or topological phase transitions among these SF phases. One transition is a first order one driven by roton droppings ( but with non-zero gaps $ Δ_R $ ) tuned by the Zeeman field $ h $. Another is a second order bosonic Lifshitz transition with the dynamic exponents $ z_x=z_y=2 $ and an accompanying $ [C_4 \times C_4]_D $ symmetry breaking. It is driven by the softening of the superfluid Goldstone mode tuned by the ratio of spin-orbit coupled (SOC) strength over the hopping strength. The two phase boundaries meet at a topological tri-critical (TT) point which separates the $ h=0 $ line into two SF phases with $ N=2 $ and $ N=4 $ condensation momenta respectively. At the $ h=0 $ line where the system has an anti-unitary $ Z_2 $ Reflection symmetry, there are infinite number of classically degenerate family of states on both sides. We perform a systematic order from quantum disorder analysis to find the quantum ground states, also calculate the roton gaps $ Δ_R $ generated by the order from disorder mechanism on both sides of the TT point. The $ N=2 $ and $ N=4 $ SF phases have the same spin-orbital XY-AFM spin structure, respect the anti-unitary symmetry and break the $ [C_4 \times C_4]_D $ symmetry, so they be distinguished only by the different topology of the BEC condensation momenta instead of by any differences in the symmetry breaking patterns. All these novel quantum or topological phenomena can be probed in the recent experimentally realized weakly interacting Quantum Anomalous Hall (QAH) model of $ ^87 Rb $ by Wu, {\sl et.al}, Science 354, 83-88 (2016).
The twodimensional chiral p-wave superfluid is known to be topological, with gapless Majorana states expected to exist at its edge. The possibility to utilise such Majorana excitations as fault-tolerant quantum bits has spurred interest in robust experimental realisation of such systems. One proposed pathway is to use spin-orbit coupled and spin-imbalanced ultracold gases of fermionic atoms subject to conventional s-wave pairing, as inducing a large-enough Zeeman energy splitting between pseudo-spin-1/2 states drives the system into a topological superfluid (TSF) phase with chiral-p-wave characteristics. In this thesis, we apply and extend advanced theoretical methods to study this topological phase transition in detail. Specifically, our work elucidates what happens when the s-wave interaction strength between atoms is increased so that the system undergoes the BCS-to-BEC crossover. While previous studies have largely focused on the weakly interacting BCS regime, studying the full crossover reveals new properties of the TSF phase that may aid its experimental realisation in the future. In particular, we obtain phase diagrams in the parameter space of the s-wave interaction strength and Zeeman energy. As long as the system is in the nontopological superfluid (NSF) phase, it shows the ordinary features expected during the BCS-to-BEC crossover, such as the shrinking and disappearance of the Fermi surfaces as the s-wave interaction strength is increased. The TSF phase, however, exhibits strikingly different behavior: it retains the BCS-limit characteristics of a TSF even as the s-wave interaction strength reaches extremely large values. The phase diagrams we obtain also show that increasing the s-wave interaction strength changes the nature of the topological transition from second-order continuous to first-order. During the first-order topological transition, the system splits into coexisting regions of NSF and TSF phases. We study this phase coexistence theoretically and obtain analytical formulae that capture the behaviour of all relevant thermodynamic quantities during the first-order topological transition. In that, our work extends current fundamental knowledge about spin-imbalanced Fermi gases to situations with finite spin-orbit coupling. We also investigate the gapless edge states emerging at the system’s boundaries, as these are crucial for experiments that aim to observe topological phases and realise novel quantum-information-processing architectures. The principle of bulk-boundary correspondence guarantees that gapless Majorana excitations should be localised to the interface between coexisting NSF and TSF regions. We develop a detailed theoretical description of this novel topological interface state and compare its physical features with those exhibited by the Majorana states expected to exist at the system’s boundary with vacuum.