Polaron in a $P+Ip$ Fermi Topological Superfluid

PHYSICAL REVIEW A 99, 033613 (2019)

Polaroninap + ip Fermi topological superﬂuid

Fang Qin (),1,2,* Xiaoling Cui,3,4,† and Wei Yi1,2,‡ 1CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Chinese Academy of Sciences, Hefei, Anhui 230026, China 2CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China 3Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 4Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China

(Received 21 January 2019; published 18 March 2019)

We study polaron excitations induced by an impurity interacting with a two-dimensional p + ip Fermi superfluid. As the Fermi-Fermi pairing interaction is tuned, the background Fermi superfluid undergoes a topological phase transition. We show that such a transition is accompanied by a discontinuity in the second derivative of the polaron energy, regardless of the impurity-fermion interaction. We also identify a polaron to trimer crossover when the Fermi superfluid is in the strongly interacting, thus topologically trivial, regime. However, the trimer state is metastable against the molecular state where the impurity binds a Bogoliubov quasiparticle from the Fermi superfluid. By comparing the polaron to molecule transition in our system with that of an impurity in a noninteracting Fermi sea, we find that pairing interactions in the background Fermi superfluid effectively facilitate the impurity-fermion binding. Our results suggest the possibility of using the impurity as a probe for detecting topological phase transitions in the background; they also reveal interesting competitions between various many-body states in the system.

DOI: 10.1103/PhysRevA.99.033613

I. INTRODUCTION where μ is the chemical potential of fermions. In contrast, in the strong-coupling regime with μ<0, the superfluid is topo- The study of topological matter has attracted much at- logically trivial. A topological phase transition occurs at μ = tention in recent years [1–8]. Whereas topological bands 0, where the system becomes gapless [58,59]. Assuming a and phases in noninteracting systems are well understood tunable impurity-fermion interaction, we calculate the energy, by now, the interplay of interaction and topological bands the impurity residue, and the wave function of the polaron is still under extensive study [9–23]. A bottom-up approach state, where the impurity is dressed by a pair of Bogoliubov here concerns an impurity interacting with a topologically quasiparticles. We find that as the background Fermi super- nontrivial background. As the impurity is dressed by quasipar- fluid is tuned across the topological phase transition, a discon- ticle excitations of its environment, it can also acquire some tinuity emerges in the second derivative of the polaron energy, topological features through interaction. In previous studies, which is consistent with the order of the topological phase both immobile and mobile impurities have been considered transition. On the other hand, when the impurity-fermion in topological environments [24–26], with the latter giving interaction is tuned, the polaron state can cross over into a rise to polarons where the impurity moves around dragging trimer state, as the impurity residue decreases rapidly within particle-hole excitations along. Such a scenario is especially a finite range of interaction strength. Interestingly, such a relevant to ultracold atomic gases, where key elements such polaron to trimer crossover only occurs when the background as topological bands [27–30], tunable interactions [31], and Fermi superfluid is in the topologically trivial strong-coupling polaron excitations [32–41] are all experimentally accessi- regime. Further, as the impurity-fermion interaction increases, ble. While polarons serve as a bridge connecting few- and the polaron state can become metastable against a molecular many-body physics [42–57], they provide a unique angle state, where the impurity forms a local bound state with a sin- for understanding topological systems in the presence of gle Bogoliubov quasiparticle. In the strong-coupling regime, interaction. the polaron to molecular transition occurs before the polaron- In this work, we study an impurity immersed in a two- trimer crossover, such that the trimer state is metastable. In dimensional p + ip Fermi superfluid. As the p-wave Fermi- the weak-coupling limit, the polaron state remains the ground Fermi interaction is tuned, the superfluid becomes topolog- state. ically nontrivial in the weak-coupling regime with μ>0, The paper is organized as the following: In Sec. II,we describe the system configuration and give the model Hamil- tonian. In Sec. III, we characterize the polaron state using a *[email protected] variational approach. We characterize the molecular state as †[email protected] well as the polaron to molecule transition in Sec. IV. Finally, ‡[email protected] we summarize in Sec. V.

II. MODEL We consider an impurity interacting with a two- dimensional p + ip Fermi superfluid. The Hamiltonian is − μ = − μ † + b † H N ( k )akak kbkbk k k 1 † † ∗ + ( a a + a− a ) 2 k k −k k k k k gfi † † + a b b − a , (1) V k q−k q k k k,k,q where N is the total number of spinless fermions in background superfluid, μ is the chemical potential of the fermions in the background, ak and bk are, respectively, the annihilation operators for the background p-wave superfluid fermions and 2 the impurity atom, k = k /(2m) is the kinetic energy of the b = 2/ μ spinless fermions with mass m, k k (2mb) is the kinetic FIG. 1. Chemical potential of the background p-wave super- energy of the impurity atom with mass mb, and V is the fluid as a function of the pairing order parameter . two-dimensional volume. Here, the pairing order parameter is k = k−, with = [g /(2V )] k+a−k ak , k± = kx ± ff k Fig. 1, we show the chemical potential as a function of the iky, and gff is the p-wave interaction between fermions in the background. g is the s-wave interaction rate between pairing order parameter. For a given kc, the chemical potential fi || fermions and the impurity, which can be renormalized follow- monotonically decreases with increasing , as the system μ> ing the standard procedure in two dimensions [48–52]: changes from a topologically nontrivial phase with 0to μ< a topologically trivial phase with 0. In the following, 1 1 1 μ p =− , (2) we will use or to characterize the background -wave g V + b + E interaction strength. fi k k k b where Eb is the two-body bound-state energy. The natural III. POLARON STATE unitsh ¯ = kB = 1 will be used throughout the paper. The p-wave Fermi pairing superfluid at zero temperature We adopt the Chevy-type ansatz to characterize the polaron can be described by the Bardeen-Cooper-Schrieffer (BCS)- state [42,57]: type wave function [59] † 1 † † † | = + † † | , |PQ = ψQb + ψk,k b − − α α |BCSp, (8) BCS p (uk vka−kak ) 0 (3) Q Q k k k k 2 , k k k

E + ( − μ) ψ ψ ψ = iθk k k where Q and k,k are polaron wave functions, k,k uk = e , (4) −ψ 2Ek k ,k due to the p-wave symmetry, and Q indicates the center-of-mass momentum of the polaron. The second term − − μ Ek ( k ) in the brackets, which effectively describes impurity-induced vk = , (5) 2Ek pair breaking in the superfluid and is therefore trimerlike, † † includes contributions from excitations like ak a−k, a−k ak, θ = + = − μ 2 +| |2 | † † where k arg(kx iky ), Ek ( k ) k , and 0 a a ,ora− a− . Here, we only keep excitations to the lowest | k k k k is the vacuum state. The BCS-type wave function BCS p is order. The existence of a trimerlike term in the polaron wave α | = the vacuum for Bogoliubov quasiparticles, with k BCS p function is unique for a pairing-superfluid background. α = + † 0 and k ukak vka−k. The ground-state solution can be obtained by minimizing The parameters characterizing the background p-wave su- Ep = QP|(H − μN)|PQ − EBCS, where Ep is the polaron perfluid are related through the gap and number equations, energy and the BCS ground-state energy is

2 ||2 2 4 =−1 k , 1 k (6) EBCS = [(k − μ) − Ek] + . (9) gff V Ek 2 4V E k k k k 1 n = |v |2, (7) V k Following a similar derivation as in Ref. [57], we have k ∗ ∗ | |2 where the particle number density n is given by n = k2 /(4π ) V uk vk k vk Ak uk Ak F − Ak = − uk , with the Fermi wave vector k . For the summations here, g A , E − b A , F ﬁ k k k p Q k k k we adopt a high-momentum cutoff kc, which is related to the short-range scattering parameter of the p-wave interaction. In (10)

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FIG. 2. (a) The lowest branch of the polaron energy versus Eb. FIG. 3. Momentum distribution of the trimer probability (b) The lowest branch of the polaron energy versus μ. (c) The first 2 |ψk ,k | in the (k1, k2 ) plane, where the polar angles θ1 and θ2 of k1 derivative of the lowest branch of the polaron energy with respect 1 2 and k2 have been integrated for (a) Eb = 0.5EF with μ =−0.5EF , to μ. (d) Impurity residue. The parameters are Q = 0, mb = m,and (b) Eb = 10EF with μ =−0.5EF ,(c)Eb = 0.5EF with μ = 0.5EF , momentum cutoff kc = 20kF . In (a) and (d), the solid green line is for and (d) Eb = 10EF with μ = 0.5EF . The other parameters are Q = 0, μ = 0.5EF , the solid blue line is for μ = 0.2EF , the dashed-dotted mb = m, and momentum cutoff kc = 20kF . black line is for μ = 0, the dashed blue line is for μ =−0.2EF ,and the dashed green line is for μ =−0.5EF . In (b) and (c), the solid = . = . green line is for Eb 0 5EF , the dashed blue line is for Eb 0 75EF , topologically trivial strongly interacting regime (μ<0), the and the dashed-dotted black line is for E = E . b F impurity residue drops precipitously over a small range of Eb, such that the residue Z essentially vanishes at large impurity- where fermion interactions. This is a clear signature of polaron to trimer crossover, first studied in the impurity problem for a = − − − b , Ak,k Ep Ek Ek Q−k−k (11) background of s-wave Fermi superfluid [57]. However, the polaron to trimer crossover becomes much smoother in the ∗ topologically nontrivial weak-interacting regime (μ>0), as A = g v ψ + u ψ , . (12) k fi k Q k k k the residue monotonically decreases but remains finite even at k large Eb. The coefficients in the polaron ansatz (8) give the impurity The polaron-trimer crossover is more apparent in the residue [57], momentum-space probability distribution of the wave func- ψ |ψ |2 tion k1,k2 .InFigs.3(a) and 3(b), we show the angular- Q 2 Z = . (13) integrated momentum-space distribution dθ dθ |ψ , | 2 1 2 1 2 k1 k2 |ψQ| + , |ψk,k | 4 k k for μ<0, where θ1 and θ2 are the polar angles of k1 and The impurity residue signals the weight of bare impurity in k2, respectively. When the impurity-fermion interaction is the polaron excitation. From the ansatz wave function Eq. (8), small, the distribution is localized in momentum space [see it is clear that the polaron would become trimerlike when Z Fig. 3(a)]. This suggests an extended polaronlike spatial wave approaches zero. function. When the impurity-fermion interaction is large, the We numerically solve Eq. (10) and get the lowest-energy distribution becomes extended in momentum space, as shown polaron branch as shown in Figs. 2(a) and 2(b). The po- in Fig. 3(b). This suggests a spatially localized trimerlike wave μ> laron binding energy |Ep| increases with increasing impurity- function. In contrast, for 0, the momentum distribution fermion interaction strength or with increasing fermion- of the wave function is always localized, regardless of the fermion interaction. We then plot the first derivative of the Ep impurity-fermion interaction as shown in Figs. 3(c) and 3(d). with respect to μ. As shown in Fig. 2(c), kinks appear at μ = Further, we have numerically checked that the probability |ψ |2 θ = θ + π 0 regardless of the impurity-fermion interaction, which indi- k1k2dk1dk2 k1,k2 always peaks at 1 2 . cates a discontinuity in the second derivative of the polaron energy. This is consistent with the fact that the topological IV. MOLECULAR STATE phase transition in the background is a third-order phase transition. Thus the information of the background topological For the impurity problem in a noninteracting Fermi gas, phase transition is carried over to the polaron excitation, a polaron to molecule transition occurs when the impurity- which can serve as a probe for the phase transition. fermion interaction increases. In two dimensions, such a In Fig. 2(d), we show the evolution of the impurity residue transition can be accurately captured by considering polaron as a function of Eb. When the Fermi superfluid is in the and molecular wave functions each dressed by a single pair

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Therefore, the trimerlike state is metastable. In contrast, when the Fermi superfluid is topological, we find no polaron to molecule transitions. However, similar to the case of two- dimensional Fermi polaron, where the polaron to molecule transition only appears when one considers dressed molecules with higher-order particle-hole fluctuations [48,49], we expect that a polaron to molecule transition should exist against a background of topological superfluid when higher-order fluctuations in the molecular sector are taken into account. Specifically, one needs to go beyond Eq. (14) and consider more pairs of quasiparticle excitations. Nevertheless, the fact that we find polaron to molecule transitions in the strongly interacting regime by considering the bare molecular state alone suggests that pairing interaction in the background ac- tually destabilizes the polaron state and effectively facilitates the binding of impurity and fermion.

FIG. 4. Molecular and polaron energies in the Q = 0 sector with V. SUMMARY μ = . μ = . μ =− . μ =− . (a) 0 5EF ;(b) 0 2EF ;(c) 0 2EF ;(d) 0 5EF . By using the Chevy-type variational wave functions, we = = Here, we choose mb m and kc 20kF . The red solid curve is the study the impurity problem of a p + ip Fermi topological su- lowest branch of the polaron energy and the black dashed curve is perfluid across the topological phase transition. The topolog- the molecular energy. ical phase transition gives rise to kinks of the polaron-enegy derivatives, which can serve as an external probe for the phase of particle-hole excitation [48,49]. To determine the ground transition. We then discuss the interplay of fermion-fermion state of the impurity, we should also compare the energy of a and impurity-fermion interactions on the polaron-trimer molecular state with that of the polaron state. crossover as well as the polaron-molecule transition in the We write the molecular ansatz wave function as [50,57] system. Our results reveal interesting competitions between various many-body states in an interacting topological system. | = φ(M) † α†| , M Q k bQ−k k BCS p (14) k ACKNOWLEDGMENTS where Q is the center-of-mass momentum of the molecule. Physically, we are considering the binding of the impurity We thank Ming Gong, Lijun Yang, Tian-Shu Deng, and and a Bogoliubov quasiparticle into a tightly confined dimer. Jing-Bo Wang for useful discussions. W.Y. acknowledges In the weak-coupling limit → 0, the wave function above Jiali Lu for his early contributions. X.C. acknowledges sup- reduces to that of a bare molecular state. port from the National Key Research and Development The ground-state solution can be obtained by minimizing Program of China (Grants No. 2018YFA0307600 and No. 2016YFA0300603), and the National Natural Science Foun- EM = QM|(H − μN)|MQ − EBCS, where EM is the molecular energy. The closed equation reads dation of China (Grants No. 11622436, No. 11421092, and No. 11534014). W.Y.acknowledges support from the National 1 1 |u |2 Key Research and Development Program of China (Grants = k , (15) b No. 2016YFA0301700 and No. 2017YFA0304100), and the gfi V EM − Ek − − k Q k National Natural Science Foundation of China (Grant No. where gfi needs to be renormalized following Eq. (2). 11522545). F.Q. acknowledges support from the National Key In Fig. 4, we show the lowest branch of the polaron energy Research and Development Program of China (Grant No. as well as the molecular energy. In the topologically trivial 2017YFA0304800), the National Natural Science Foundation regime, a polaron to molecule transition occurs. By comparing of China (Grant No. 11404106), and the project funded with Fig. 2(d), we find that the polaron to molecule tran- by the China Postdoctoral Science Foundation (Grant No. sition happens prior to the polaron-trimer crossover region. 2016M602011).

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