PHYSICAL REVIEW RESEARCH 2, 043264 (2020)

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Rényi entropy singularities as signatures of topological criticality in coupled - systems

F. P. M. Méndez-Córdoba ,1,* J. J. Mendoza-Arenas ,1 F. J. Gómez-Ruiz ,2,1 F. J. Rodríguez ,1 C. Tejedor ,3 and L. Quiroga 1 1Departamento de Física, Universidad de Los Andes, A.A. 4976, Bogotá, Colombia 2Donostia International Center, E-20018 San Sebastián, Spain 3Departamento de Física Teórica de la Materia Condensada and Condensed Physics Center (IFIMAC), Universidad Autónoma de Madrid, Madrid 28049, Spain

(Received 20 July 2020; accepted 13 October 2020; published 20 November 2020)

We show that the topological transition for a Kitaev chain embedded in a cavity can be identified by measuring experimentally accessible photon observables such as the Fano factor and the cavity quadrature amplitudes. Moreover, based on density matrix renormalization group numerical calculations, endorsed by an analytical Gaussian approximation for the cavity state, we propose a direct link between those observables and entropy singularities. We study two bipartite entanglement measures, the von Neumann and Rényi entanglement entropies, between and matter subsystems. Even though both display singularities at the topological points, remarkably only the Rényi entropy can be analytically connected to the measurable Fano factor. Consequently, we show a method to recover the bipartite entanglement of the system from a cavity observable. Thus, we put forward a path to experimentally access the control and detection of a topological quantum phase transition via the Rényi entropy, which can be measured by standard low noise linear amplification techniques in superconducting circuits. In this way, the main quantum information features of Majorana polaritons in photon-fermion systems can be addressed in feasible experimental setups.

DOI: 10.1103/PhysRevResearch.2.043264

I. INTRODUCTION Motivated by these remarkable advances, we are encour- aged to establishing new feasible hybrid cavity scenarios The understanding of correlated matter strongly coupled for the detection and control of nonlocal correlated features to quantum light has been an intense area of research both in -state setups such as topological materials [18–20]. theoretically and experimentally in the last few years. Hybrid A great deal of attention has been recently devoted to as- photonic technologies for control of complex systems have sessing nonlocal in chains been constantly improving, now acting as cornerstones for with strong spin-orbit coupling disposed over an s-wave quantum simulations in cutting-edge platforms such as optical superconductor [21–24]. Majorana , as topological lattices. Namely, trapped ions are subjected to high control quasiparticles in solid-state environments, have been widely by laser beams allowing the manipulation of the main system searched due to their unconventional properties against local parameters [1–5]. Strong light-matter couplings have been decoherence and hence for possible technological solutions to generated in superfluid and Bose-Einstein embedded fault-tolerant protocols [25–28]. in cavities now available to study systems with exquisitely Since the seminal work by Kitaev [29] where a one- tailored properties [6–10]. Furthermore, the analysis of light- dimensional spinless fermion chain was shown to fea- controlled condensed matter systems has led to predictions of ture Majorana physics, topological properties of hybrid a rich variety of phenomena, including the enhancement of -superconductor systems [21–24] have been -photon by cavity mediated fields explored looking for the presence of the so-called zero energy [11–15]. Experimentally, new physical features as well as con- modes (ZEM), corresponding to quasiparticles localized at trol opportunities in the ultrastrong and deep-strong-coupling the boundaries of the chain. The fact that these quasiparti- regimes, where coupling strengths are comparable to or larger cles have zero energy makes them potential candidates for than subsystem energies, have been observed recently using the implementation of non-Abelian gate operations within circuit quantum electrodynamics microwave cavities [16,17]. two-dimensional (2D) arrangements [30–34]. However, some open questions still remain about the experimental occurrence of these modes since the reported phenomena observed in *[email protected] those experiments could be caused by a variety of alter- native competing effects [35]. Therefore, new experimental Published by the American Physical Society under the terms of the frames are highly desirable to find unambiguous signs of such Creative Commons Attribution 4.0 International license. Further quasiparticles. distribution of this work must maintain attribution to the author(s) An important question in this context is whether the topo- and the published article’s title, journal citation, and DOI. logical phase transition of Majorana polaritons, for instance

2643-1564/2020/2(4)/043264(12) 043264-1 Published by the American Physical Society F. P. M. MÉNDEZ-CÓRDOBA et al. PHYSICAL REVIEW RESEARCH 2, 043264 (2020)

† Here Hˆ C = ωaˆ aˆ is the Hamiltonian describing the mi- crowave single-mode cavity, witha ˆ(ˆa†) the annihilation (creation) microwave photon operator, and ω is the energy of the cavity; we set the energy scale by taking ω = 1. The isolated open-end Kitaev chain Hamiltonian Hˆ K is given by

μ L L−1 † † † Hˆ =− [2ˆc cˆ − 1]ˆ − t [ˆc cˆ + + cˆ cˆ ] K 2 j j j j 1 j+1 j j=1 j=1 L−1 +  + † † . [ˆc jcˆ j+1 cˆ j+1cˆ j ] (2) j=1

FIG. 1. Schematic illustration of a Kitaev chain embedded in a † Herec ˆ j (ˆc j ) is the annihilation (creation) operator of spinless single-microwave cavity. The blue curve denotes the profile of the fermions at site j = 1,...,L, μ is the chemical potential, t fundamental mode of the cavity. Majorana fermion quasiparticles is the hopping amplitude between nearest-neighbor sites (we are depicted as blue and red spheres (bulk) and as green spheres for assume t  0 without loss of generality), and  is the nearest- the edge unpaired quasiparticles for an isolated Kitaev chain in the neighbor superconducting induced pairing interaction. The topological phase. The light red regions illustrate the hybridization effect yielding to Majorana polaritons. Kitaev model features two phases: a topological and a trivial phase. In the former the Majorana ZEM emerge, which occurs whenever |μ| < ±2 for the symmetric hopping-pairing Ki- in a fermion chain embedded in a microwave cavity [36–38], taev Hamiltonian, i.e., t = , the case we restrict ourselves can be detected by accessing observables such as the mean from now on [29,31]. Additionally, the general interaction number of , field quadratures or cavity Fano factor Hamiltonian is given by [36] (FF). In this paper, we report on an information-theoretic L L−1 approach based on the analysis of the Rényi entropy (SR) † + λ Hˆ = aˆ aˆ λ † + 1 † + † . of order two between light and matter subsystems, for con- Int √ 0 cˆ cˆ j cˆ cˆ j+1 cˆ + cˆ j L j 2 j j 1 necting its singular behavior, resulting from the topological j=1 j=1 transition, with the FF. Consequently, we show a path to char- (3) acterize the bipartite entanglement of the light-matter system and how to use it as a witness to identify quantum phase Thus, for the light-matter coupling, we shall consider a gen- λ transitions. Additionally, we show that in a wide parameter eral case which incorporates both on-site ( 0)aswellas λ coupling regime the cavity state is faithfully represented by hoppinglike ( 1) terms (without loss of generality we will λ λ > a Gaussian state (GS). Within this description, measurements assume 0, 1 0).InRef.[36], a typical value of the on-site λ  . ω of the Fano parameter and single-mode quadrature amplitudes chain-cavity coupling, 0 0 1 was estimated for a fermion yield directly to assessing the Rényi entropy. This approach chain length of L = 100 sites. Note that the whole chain is allows us to link directly accessible microwave observables assumed to be coupled to the same cavity field. to quantum light-matter correlations [39–41], and clarifies the role of topological phases hosted by cavity-fermion coupled III. MEAN-FIELD APPROACH systems. Our paper is organized as follows. Section II gives the In order to gain physical insights on how the original topo- description of the Kitaev model embedded in a microwave logical phase of the Kitaev chain is modified by its coupling cavity. In Sec. III, we present the mean-field approach of the to a cavity, we start by performing a mean-field (MF) treat- system, which is useful to predict the response of the cavity. ment. Although we develop the MF analysis for a chain with In Sec. IV, we present the phase diagram of the composite periodic boundary conditions, the relations we will discuss system obtained numerically. In Sec. V we show that the in this section are indeed useful guides for interpreting the composite system signals the phase transitions in the von quasiexact results obtained by density matrix renormalization Neumann entropy and that the state of the cavity can be group (DMRG) numerical simulations in chains with open approximated by a single-mode Gaussian state . In Sec. VI we boundary conditions, as illustrated below. show the connection between the Fano factor and the Rényi We start by separating the cavity and the chain subsys- entropy. Finally, in Sec. VII we present a summary of our tems by describing their interaction as the mean effect of work. one subsystem over the other. Applying the traditional MF approximation to the interaction Hamiltonian Hˆ Int,weset quantum fluctuations of products of bosonic and fermionic II. PHOTON-FERMION MODEL operators to 0, therefore We consider a Kitaev chain embedded in a single-mode mi- † + − † +  † − †  = . crowave cavity as schematically shown in Fig. 1. The system (ˆa aˆ aˆ aˆ )(ˆc j cˆ j cˆ j cˆ j ) 0 (4) is described by the Hamiltonian Following a similar procedure for the hoppinglike light-matter Hˆ = Hˆ C + Hˆ K + Hˆ Int. (1) interaction term and setting periodic boundary conditions, the

043264-2 RÉNYI ENTROPY SINGULARITIES AS SIGNATURES … PHYSICAL REVIEW RESEARCH 2, 043264 (2020) new interaction Hamiltonian is given by MF Hˆ ≈ L[λ1D + λ0(1 − Sz )][Xˆ − x] Int ⎡ ⎤ L λ L ⎣ † 1 † † ⎦ + 2x λ cˆ cˆ + (ˆc cˆ + + cˆ cˆ ) . (5) 0 j j 2 j j 1 j+1 j j=1 j=1 Here we define √ Xˆ = (ˆa + aˆ†)/2 L, x =Xˆ , 2 S = 1 − cˆ†cˆ , FIG. 2. Photon-fermion phase diagrams. NP, normal phase; TP, z L j j j topological phase; and SP, asymptotically super-radiant phase. (a) Chemical potential-like coupling. (b) Hoppinglike coupling. The 1 † † D = cˆ cˆ + + cˆ cˆ , (6) Kitaev-cavity parameters are L = 100 and  = 0.6ω. L j j 1 j+1 j j where expectation values are taken with respect to the photon- provided by a topological invariant, namely, the Majorana fermion ground state. The resulting Hamiltonian is that of a number [29]. displaced harmonic oscillator, with photon number aˆ†aˆ≡ For both types of cavity couplings, second-order phase nˆ=Lx2, and a Kitaev chain with effective chemical poten- transitions arise in the composite light-matter model (an ex- tial μeff ≡ μ − 2λ0x and hopping interaction teff ≡  − λ1x ample is shown in Appendix D), a result for which DMRG (see Appendix B for more details on this MF approach). and MF are in full agreement for a wide range of coupling The minimization of the MF Hamiltonian expected value, values. The phase diagram for the on-site coupling (λ0 = 0 ∂Hˆ MF/∂x = 0, yields and λ1 = 0) is presented in Fig. 2(a), whereas that for the hop- λ = λ = λ = λ + λ + ω , pinglike coupling ( 0 0 and 1 0) is depicted in Fig. 2(b). 0Sz 0 1D 2 x (7) For the on-site coupling, the critical points and the max- which shows the interdependence of the cavity and chain imum of correlations move asymmetrically to lower values λ +λ ∈ − 2 0 1 , of the chemical potential as the coupling strength increases states parameters. Since x [ 2ω 0], the effective MF renormalized Kitaev parameters turns out to be μeff  μ and (see Appendix C). The boundary between the topological teff  . By choosing λ1 = 0, it is easy to see that x will be phase (TP) and the asymptotically super-radiant phase (SP), related to the magnetization in the equivalent transverse Ising in which the number of photons approaches the maximum ob- chain [42–45], while when choosing λ = 0, x will be asso- tained by MF [cf. Fig. 7(a), see Appendix C], is affected more 0 λ /ω = ciated to the occupancy of first neighbor nonlocal Majorana dramatically causing the TP to disappear beyond 0 . ± . λ fermions in the Kitaev chain [25]. 1 39 0 01. For larger values of 0, there will only be one second-order phase transition between the normal phase (NP), which is topologically trivial and does not present radiation, IV. PHASE DIAGRAM and SP, holding only a trivial ordering of the chain. The ground state of the system has been obtained by For the hoppinglike photon-chain coupling case, the phase performing DMRG simulations in a matrix product state de- transition points are symmetrical with respect to the trans- scription [46,47], using the open-source TNT library [48,49]. formation μ →−μ (see Appendix D). Whenever the cavity Notably, matrix product algorithms have been successfully resides in a super-radiant phase, the chain is in the topological applied to correlated systems embedded in a cavity [50,51], as phase (see Appendix C); thus the mean number of photons well as to different interacting systems in starlike geometries acts as an orderlike parameter that correlates well with the [52–55]. In the following analysis, we consider separately quantum state of the chain. It is evident that this type of each kind of cavity-chain coupling term and we sweep cavity-chain coupling widens the topological phase allowed over μ. region. However, as the TP gets wider the maximum value The topological phase of the chain will be assessed through of Q decreases, indicating the degrading of nonlocal chain ≡  † + † the two-end correlations Q, defined as Q 2 cˆ1cˆL cˆLcˆ1 . correlations at high coupling values. This value is an indicator of the locality of edge modes, which is connected to the topology of the system [56,57]. For an V. VON NEUMANN ENTROPY, CRITICALITY, infinite isolated Kitaev chain, its value is 1 in the topological AND GAUSSIAN STATES phase while it goes to 0 in the trivial one. However, for finite sizes the value of Q takes on continuous values in between, A result well beyond the MF analysis for this photon- leaving a value of 1 at the point of maximum correlations [cf. fermion system is that phase transitions are associated with Figs. 7(e) and 7(f), see Appendix C]. Whenever Q > QTrigger singularities in the light-matter quantum von Neumann en- the phase is said to be topological, where QTrigger was defined tropy, SN [2,49,58], as shown in Fig. 3. Critical lines, as as the lowest Q that allows for ZEM to emerge in an isolated obtained from the nonlocal Q-correlation behavior, are fully Kitaev chain with the same  and L as in the simulated consistent with results extracted from the second derivative light-coupled case. In Appendix C, we show the agreement of the energy and SN behavior (see Appendix D). More- of this definition of the topological phase with the description over, the maximum nonlocal edge correlation Q = 1 coincides

043264-3 F. P. M. MÉNDEZ-CÓRDOBA et al. PHYSICAL REVIEW RESEARCH 2, 043264 (2020)

ances and it is simply expressed as [60,61,63]

SN = (N + 1) ln [N + 1] − N ln [N]. (11) In order to get the α, N, r, and φ Gaussian parameters, the covariance matrix and quadratures are numerically extracted from the corresponding expected values using ground-state DMRG calculations. The imaginary part of α and φ must be 0 to reach the ground state. Therefore, the relations that endorse us with the Gaussian parameters are the following [59]: FIG. 3. von Neumann entropy SN as a function of the chemi- √ cal potential of the chain μ/2 for any subsystem in the bipartite qˆ= 2Re[α], (12a) cavity-chain system. Symbols (lines) indicate DMRG (Gaussian) √ pˆ= 2Im[α], (12b) results. (a) Local photon-fermion couplings λ0 = 0.1ω (weak cou- λ = . ω + pling, black symbols and line) and 0 0 4 (moderate coupling, 2 2 1 2N λ = qˆ −qˆ = (cosh [2r] + sinh [2r] cos [φ]), red symbols and line). (b) Nonlocal photon-fermion coupling 1 2 . ω λ = . ω 0 07 (weak coupling, black symbols and line) and 1 0 4 (mod- (12c) erate coupling, red symbols and line). Other parameters are L = 100, ω =  = . ω 1 + 2N 1, and 0 6 pˆ2−pˆ2 = (cosh [2r] − sinh [2r] cos [φ]), 2 (12d) with the minimum of SN [compare Figs. 7(e) and 7(f) with Fig. 3(a)]. Consistency with S singularities is also found for 1 + 2N N qˆpˆ+pˆqˆ−qˆpˆ= (sinh [2r] sin [φ]). (12e) the hoppinglike coupling, as shown in Fig. 3(b). In any case, 2 singularities in SN are intimately connected to the phase tran- sitions for an ample domain of coupling strength parameters Results for SN obtained from DMRG and analytical GS cal- (see also weak-coupling behaviors in Fig. 3 for λ = 0.1ω and culations of Eq. (11) are in excellent agreement for different 0 coupling types and strengths, as shown in Fig. 3, thus confirm- λ1 = 0.07ω). The MF analytical description, which involves a single ing the adequacy of a GS photon description for the present coherent state for the cavity, provides an accurate description photon-fermion system (see also Appendix E). of the bulk expectation values in the chain, the mean number of cavity photons, the cavity quadratures, and the energy of VI. RÉNYI ENTROPY AND FANO FACTOR the whole system (see Appendix D). However, this effective The Rényi entropies, defined as theory is unable to account for entanglement properties be- −1 η tween subsystems and higher interaction terms such as the Sη(ρˆ ) = (1 − η) ln [tr[ρˆ ]] (13) FF. Many of the properties of GS have been broadly studied [59–62], being one of the most outstanding the fact that it for a stateρ ˆ , have been identified as powerful indicators of is fully characterized by its 2 × 2 covariance matrix σ and quantum correlations in multipartite systems [64]. The von first moments of the√ field-quadrature canonical√ variables given Neumann entropy SN is retrieved as the Rényi entropy in byq ˆ = (ˆa† + aˆ)/ 2 andp ˆ = i(ˆa† − aˆ)/ 2. The covariance the limit η → 1. It has also been established that the Rényi matrix for a single-mode GS is simply entropy of order η = 2 is well adapted for extracting corre-  2− 2  + −   lation information from GS. Thus, from now on we restrict qˆ qˆ qˆpˆ pˆqˆ qˆ pˆ ρ =− ρ2 σ = . (8) ourselves to consider only S2( ) ln [tr( )] which we  + −  2−  qˆpˆ pˆqˆ qˆ pˆ pˆ pˆ will simply denote as SR [2,3,27]. Specifically, SR for a GS can be simply expressed in terms of the GS covariance matrix Remarkably, an accurate description of the reduced photon 1 σ as SR = ln [det(σ )] [60]. system density matrix is possible by means of a single mode 2 = GS. Any single-mode GS can be expressed in terms of a fic- We also consider the photon FF, which is defined as FF Var (n ˆ )/nˆ, with Var(ˆn) =nˆ2−nˆ2. For further reference, tional thermal state on which squeezed (Sˆξ ) and displacement FF = 1 for a single coherent state (MF result) while it denotes (Dˆ α) operators act in the form: either a sub- (FF < 1) or super- (FF > 1) Poissonian photon aˆ†aˆ N † † state. We now argue that the GS approximation allows us ρˆGS = Dˆ αSˆξ Sξ Dα, (9) (1 + N)a†a to analytically work out a relation between the FF and the entanglement entropy SR, raising them as both reliable and with the operators defined as accessible indicators of phase transitions in composed photon- † ∗ fermion systems. For a cavity GS, the FF and the S can be Dˆ α = exp[αaˆ − α aˆ], R analytically expressed as [59,60,63]: ∗ 2 † 2 Sˆξ = exp[(ξ (ˆa) − ξ (ˆa ) )/2], (10) (N + 1/2)2 cosh [4r] + (1 + 2N)e2rα2 − 1/2 α ∈ ξ = iφ FF = , (14) where C, re is an arbitrary complex number with (N + 1/2) cosh [2r] + α2 − 1/2 modulus r and argument φ, and N is the thermal state pa- N 2 rameter [59]. Furthermore, a well-known property is that SN SR = 2ln[1 + N] + ln 1 − . (15) is maximized for a single-mode GS at given quadrature vari- 1 + N

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FIG. 5. Scaling of FF (black symbols and left scale) and SR (red symbols and right scale) with the lattice size at the critical point between the TP and SP for the on-site coupling. The correspondent colored solid lines are the result of a linear fit, which is the typical FIG. 4. [(a) and (c)] Rényi entropy S and [(b) and (d)] Fano form of the growth of entanglement at criticality for one-dimensional R  = . ω λ = . ω factor (FF −1) of the cavity state as a function of the chemical systems. The parameters are 0 6 and 0 0 49 . potential of the chain μ/2. Symbols (lines) indicate DMRG (GS) results. [(a) and (b)] Results for local photon-fermion coupling, λ0 = . ω λ = . ω 0 1 (black) and 0 0 4 (red). [(c) and (d)] Results for nonlocal between GS and DMRG results at the topological phase tran- photon-fermion coupling, λ1 = 0.07ω (black) and λ1 = 0.4ω (red). = ω =  = . ω sition are observed for the stronger coupling value. However, Other parameters are L 100, 1, and 0 6 . the locations of the singularities predicted by the analytical and numerical results coincide. Similarly, Figs. 4(c) and 4(d) For both kinds of photon-fermion couplings, results ob- display respective calculations for a hoppinglike coupled sys- tained from these analytical expressions fit exactly the tem (λ1 = 0.07 and 0.4), showing that GS results seem to numerical ones extracted from full DMRG calculations. As- slightly drift apart from the numerically exact DMRG ones suming a GS, the inequalities N, r |α| and N, r 1, for the highest coupling. which allow to clearly see the connection between both quan- We observe that the squeezing gets larger as the light- tities, are reliable and well justified for the range of parameters matter subsystems more entangled at the critical point (note of experimental interest (see Appendix E). Keeping first or- the behavior of the r parameter comparing the different curves der terms in r and N,inEqs.(14) and (15), we finally get in Fig. 4, see also Appendix E). In order to measure the FF = 1 + 2(r + N) (i.e., a super-Poissonian photon state) and squeezing parameter r, one can resort to a homodyne de- SR = 2N, from which a simple relation between SR, FF and tection technique which has been recently extended to the the squeezing parameter r immediately follows as microwave spectral region [65–67]. On the other hand, the FF can be assessed from measurements of the second-order S = FF − 2r − 1. (16) R correlation function g(2)(for the relation between both see It must be stressed that this last relation between the Rényi Refs. [59,63]), which has successfully been measured in ex- entropy and cavity observables does not rely on a MF analysis periments [68]. Moreover, in Fig. 5 we show a scaling analysis or equivalently on the assumption of a single coherent state on the value of both, FF and SR, at criticality. The results for the cavity, but rather it comes exclusively from numer- reveals a logarithmic growth with the size of the chain, which ical DMRG calculations and their analytical backing by a is reminiscent of the behavior of entanglement at criticality Gaussian-state approximation. in 1D systems [69]. From the results of Fig. 5 together with The validity of this important result is illustrated in Fig. 4 Eq. (16) it is straightforward to see that the squeezing at criti- regardless of the photon-fermion coupling type. In spite of cality depends logarithmically with the size of the chain. This the similar behavior through a topological phase transition behavior is not unique to the chemical potential-like coupling, (and corresponding analytical expressions for a GS) of von but we found this logarithmic response of the cavity for the Neumann and Rényi entropies, it is important to note that an hoppinglike interaction as well. Thus, the FF behavior and its equivalent relation to that in Eq. (16) but involving SN instead very close relation to SR turn out to be measurable, reliable and of SR is hardly workable. Therefore, we stress the relevance accurate indicators of entanglement for this light-matter inter- of this connection between a theoretical quantum informa- acting system. Aside from the fact that it is always interesting tion entropy, SR, and measurable photon field observables, FF to establish the connections between different approaches, and r. our main result in Eq. (16) raises the question of whether a Figures 4(a) and 4(b) exhibit the behavior of different terms GS approximation remains valid for quantum open systems involved in Eq. (16) for the local photon-fermion coupling and/or stronger light-matter coupling strengths. For example, (λ0 = 0.1ω and 0.4ω), and show an excellent agreement be- photon loss from the cavity is a ubiquitous deleterious effect tween the results directly obtained from DMRG and those in experimental setups, but key to measure the state of the assuming a cavity GS. This validates Eq. (16), according to cavity field. These subjects merit considerably further studies, which SR + 2r and FF − 1 coincide. Very small deviations motivated by our work.

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VII. CONCLUSIONS in the following way: In this work, we have developed a direct link between α − β − accessible microwave observables and quantum entanglement L L 1 L 1 Hˆ = ˆ + k k + ˆk k k entropies in quantum matter featuring topological phase tran- hi mˆ i nˆi+1 A xˆi yˆi+1 sitions. By resorting to a GS description for the photon i=1 k=1 i=1 k=1 i=1 subsystem, as supported by DMRG calculations, we found a γ L + ˆk k + ˆ, simple but powerful relation between the photon Fano factor, B zˆi C (A1) single-mode quadrature amplitudes and the light-matter Rényi k=1 i=1 entropy. Singularities in the latter can then be of help for char- acterizing topological phase transitions and their connection where L is the size of the chain. Here Aˆk, Bˆk, and Cˆ are to nonmonotonic nonlocal correlations in a fermionic chain. operators that act on the global site (cavity). On the other ˆ k k k k k We also provide evidence of how the topological phase can be hand, hi,ˆmi ,ˆni ,ˆxi ,ˆyi , andz ˆi are single-site chain operators. modified with both on-site as well as hopping terms of photon- Additionally, α, β, and γ denote, respectively, the minimum fermion interactions, yielding in some cases to a more robust number of operators needed to conform terms corresponding topological phase. The possibility of extracting nonlocal or to nearest-neighbor interactions in the chain (), nearest- topological information of the Kitaev chain from the photonic neighbor interactions in the chain with the global site (λ1), field itself should be highly timely given the continuous chal- and on-site chain terms coupled with the cavity (λ0). It is lenges to assess in a clean way Majorana features in transport important to note that in this way we resort to a generalized experiments. Moreover, our results also open novel questions 1D system which consists of L + 1 sites, where the global site which motivate further studies of the role of decoherence on is located at the left edge of the 1D arrangement. Denoting the this quantum light-matter system. site 0 as the global site, we designed an MPO for this type of Hamiltonian with matrices W i defined as follows: L = L (a) For site L, Wa Wa,1, the matrix will be the first ACKNOWLEDGMENTS column vector of the corresponding matrix for the bulk. (b) For i ∈ [1, L − 1], W i ∈ T (d × d) with d = 2 + 2β + The authors acknowledge the use of the Universidad de γ + α and T (m × n) the set of tensors with size m × n, with los Andes High Performance Computing (HPC) facility in elements carrying out this work. J.J.M.-A., F.J.R., and L.Q. are thankful ⎧ for the support of MINCIENCIAS, through the project “Pro- ⎪1ifˆ n = m = a, with ducción y Caracterización de Nuevos Materiales Cuánticos ⎪ ⎪ a ∈{1, d}∪{b|b = 2k + 1, k ∈ [1,β]} de Baja Dimensionalidad: Criticalidad Cuántica y Transi- ⎪ ⎪ ∪{c|c = k + 2β + 1, k ∈ [1,γ]}, ciones de Fase Electrónicas” (Grant No. 120480863414) and ⎪ ⎪ from Facultad de Ciencias-UniAndes, projects “Quantum ⎪ k = , = , ∈ ,β , ⎪yˆi if n 2k m 1 with k [1 ] thermalization and optimal control in many-body systems” ⎪ ⎪ (2019–2020) and “Excited State Quantum Phase Transitions ⎪ k = + , = , ∈ ,β , ⎪xˆi if n 2k 1 m 2k with k [1 ] in Driven Models—Part II: Dynamical QPT” (2019–2020). ⎨⎪ C.T acknowledges financial support from the Ministerio de W i = zˆk if n = 2β + k + 1, m = 1, with k ∈ [1,γ], Economía y Competitividad (MINECO), under Project No. n,m ⎪ i ⎪ MAT2017-83772-R. F.J.G.-R. acknowledges funding support ⎪ ⎪nˆk if n = 2β + γ + k + 1, m = 1 with k ∈ [1,α], from the Ministerio de Ciencia e Innovación, under Project ⎪ i ⎪ No. PID2019-109007GA-I00. ⎪ ⎪mˆ k if n = d, m = 2β + γ + k + 1 with k ∈ [1,α], ⎪ i ⎪ ⎪ ⎪hˆ if n = d, m = 1, APPENDIX A: MATRIX PRODUCT OPERATOR ⎪ i ⎪ REPRESENTATION ⎩ 0 otherwise. To find the ground state of the fermion-photon system with DMRG, it is necessary to find a way to write the Hamiltonian (c) For the global site, i.e., site i = 0, we have a row vector in Eq. (1) in a matrix product operator (MPO) representation. with the form: The latter can be interpreted as describing any system operator ⎧ of interest as a product of matrices that only contains opera- ⎪Cˆ if m = 1, ⎪ tors of a single site, in a 1D arrangement of the system. For ⎪ Hˆ ⎪ instance, a Hamiltonian acting over a 1D lattice with L sites ⎪Aˆk if m = 2k + 1, k ∈ [1,β], Hˆ = L i i ⎪ is required to be represented as i=1 W , where W is a ⎨ matrix that only contains operators of site i. W 0 = Bˆk if m = 2β + k + 1, k ∈ [1,β], m ⎪ The Hamiltonian we consider here describes a chain, with ⎪ nearest-neighbor interactions, coupled to a global site [a single ⎪ ⎪1ifˆ m = d, cavity field in the case of the Hamiltonian Eq. (1)], thus having ⎪ ⎩⎪ a starlike geometry. The cavity field is assumed to interact 0 otherwise. with the whole chain. The Hamiltonian can thus be written

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In this way, the Hamiltonian of Eq. (1) is represented with the and the hopping terms in the form μ → μeff ≡ μ − 2λ0x and → ≡ − λ Hˆ MF Hˆ MF following MPO under a Jordan-Wigner transformation: t teff t 1x, thus defining K . The Hamiltonian C is the sum of all bosonic terms, and Hˆ MF consists of the W i∈[2,L] Const ⎛ ⎞ remaining constant terms. 1ˆ 0 000000 The canonical partition function is given by ⎜ ⎟ σˆx 0 000000 ⎜ ⎟ Z = tr[exp[−βHˆ MF]], (B2) ⎜ λ ⎟ ⎜ √1 σ ˆ ⎟ ⎜ 0 ˆx 100000⎟ β = / ⎜ 4 L ⎟ with 1 (kBT ) and kB the Boltzmann constant, and is ⎜ σ ⎟ Z = Z ∗ Z ∗ ⎜ ˆy 0 000000⎟ the product of three different terms, namely C K = ⎜ λ ⎟, Z . Following the common procedure to diagonalize the ⎜ 00√1 σˆ 10000ˆ ⎟ Const ⎜ 4 L y ⎟ Kitaev Hamiltonian through a Bogoliubov-de Gennes quasi- ⎜ λ ⎟ ⎜ √0 σˆ 0000100ˆ ⎟ description [30,43,70], we find ⎜ 2 L z ⎟ ⎜ σ ⎟ 1 ⎝ ˆy 0 000000⎠ Hˆ MF = ω ˆ† ˆ − , K (x) 2 k (x) d dk (B3) μ σ −σ ˆ k 2 2 ˆz 00000ˆy 1 k (A2) ⎛ ⎞ where dˆ (dˆ†) is the annihilation (creation) operator of Bogoli- ˆ k k 1 ubov quasiparticles in momentum space k at the first Brillouin ⎜ σ ⎟ ⎜ ˆx ⎟ ⎜ ⎟ zone. The relation is ⎜ 0 ⎟ ⎜ σ ⎟ ⎜ ˆy ⎟ ω (x) = [t cos(k) + μ /2]2 + 2 sin2(k). (B4) L = ⎜ ⎟, k eff eff W ⎜ 0 ⎟ (A3) ⎜ λ ⎟ This results in the chain partition function Z = ⎜ √0 σˆ ⎟ K ⎜ 2 L z⎟ 2 cosh(βω ). The cavity term can be diagonalized by ⎝ σ ⎠ k k y the displacement of the bosonic field, from which it is μ σ straightforward to obtain the partition function for the cavity √ 2 z √ √ 0 † term as well, given by W = (ωaˆ aˆ + λ0LXˆ 02 LXˆ 02 LXˆ 2 LXˆ 0 1)ˆ , (A4) exp [βφ2/ω] ZC = , (B5) whereσ ˆ ,ˆσ , andσ ˆ are the Pauli matrices. (1 − exp [−βω]) x y z √ with φ = L[λ1D + λ0(1 − Sz )]/2. For the constant term, the APPENDIX B: MEAN-FIELD APPROACH effect in the partition function is trivial, namely

In Sec. III we obtained the MF interaction term. This al- ZConst = exp [−βL(λ0Sz − λ1D)x]. (B6) lows us to write the Hamiltonian, Eq. (1), as the contribution Z of two independent systems corresponding to a Kitaev chain Following the product form of , the free energy, defined as F =− Z /β F = (in terms of fermionic operators) and a forced harmonic oscil- ln [ ] , is given by the addition of three terms: F + F + F F lator (bosonic operators), plus a constant energy. Then the MF K C Const. The free energy thus reads Hˆ ≈ Hˆ MF + Hˆ MF + Hˆ MF 2 Hamiltonian can be written as MF C K Const. 1 φ F = F + ln[1 − e−βω] − + L(λ S − λ D)x. The MF Hamiltonians are defined as K βL ω 0 z 1 Hˆ MF = ω † + λ + λ − ˆ , (B7) C aˆ aˆ L[ 1D 0(1 Sz )]X (B1a) μ L Therefore, Eq. (7) can be recovered from the free energy when MF † Hˆ =− − λ x [2ˆc cˆ − 1]ˆ performing the minimization ∂F/∂Sz = 0. K 2 0 j j j=1 By direct substitution of Eq. (7)onEq.(B7), the free energy per site, f ≡ F/L, results in an expression that only L − − λ † + † depends on the cavity expected value x: [t 1x] [ˆc j cˆ j+1 cˆ j+1cˆ j] = 1 j 1 f x = f x + ωx2 + λ x + − e−βω , ( ) K ( ) 0 β ln[1 ] (B8) L L +  + † † , [ˆc jcˆ j+1 cˆ j+1cˆ j ] (B1b) where the photonic part of the ground state will be defined by j=1 the x value that minimizes the free energy. The state of√ the |x L Hˆ MF = L(λ S − λ D)x. (B1c) cavity will be represented by a single coherent state . Const 0 z 1 Note that this coherent state label does not have any imaginary Thus, the eigenstates of the Hamiltonian are just products part. The reason for this is that only the position quadrature of chain and cavity states. With the MF Hamiltonian Hˆ MF explicitly appears in the Hamiltonian (Xˆ term, see Ref. [71]). being identified, we proceed to describe the thermodynam- The momentum quadrature, which is directly related to the ics of the composed system (at finite temperature T , which imaginary part of a coherent state (see Ref. [72]), is only later on will tend to zero) by simply replacing new effective implicitly regarded in the mean number of photons, aˆ†aˆ, parameters in the Kitaev Hamiltonian. The coupling with the that appears in the Hamiltonian in Eq. (1). Since the effect cavity produces a displacement of both the chemical potential of the imaginary part of a coherent state upon which the

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FIG. 6. Density plot of the Kitaev mean-field free energy, fK ≡ fK (x = 0) with β = 100 as a function of μ and . The black diago- nal lines mark down the phase transition μ =±2.

Hamiltonian of Eq. (1) acts is to add energy to the system, it sets the momentum quadrature to zero. Consequently, the mean number of photons satisfies aˆ†aˆ≡nˆ=Lx2.Last, the free energy of the Kitaev term reads 1 f (x) =− ln [2 cosh [βω (x)]], (B9) K βL k k which is shown in Fig. 6. Then, when the system supports super-radiance in the ground state, the free energy must meet FIG. 7. Characterization of the phases of the system. Panels (a), the following condition for a given x = 0: (c), and (e) [(b), (d), and (f)] are results for the chemical potential-like (hoppinglike) coupling. [(a) and (b)] Number of photons normalized 2 fK (x) + ωx + λ0x − fK (0) < 0. (B10) by the maximum value predicted by MF in the phase diagrams. [(c) and (d)] Majorana number topological invariant [Eq. (C1)] in the A state with an expected value x that satisfies the previous phase diagrams. The white regions have M =−1 which accounts for inequality will be less energetic than a state with no radiation the topological phase. The black regions correspond to M = 1, which in the composite system. This leads to a redefinition of the represents a phase that is topologically trivial. [(e) and (f)] Two ends ground state compared to the case of the isolated Kitaev correlations Q in the phase diagrams. The Kitaev-cavity parameters chain. The latter means that the state of the cavity controls are L = 100 and  = 0.6ω. the free energy of the Kitaev chain by creating effective dis- placements, which are characterized by super-radiance in the cavity. coupled system shows a super-radiant when the phase of the chain is topological, that is, when it is located in the TP of the phase diagram. Finally, the asymptotically super-radiant APPENDIX C: PHASE CHARACTERIZATION phase is the phase where the number of photons asymptot- To understand the many-body phases that are bounded by ically reaches the maximum value allowed by the physical the critical points, we may recur to different observables of restrictions [i.e., Eq. (7)]. The asymptotically super-radiant the chain and cavity. Let us start with the state of the cavity. phase emerges only with the on-site interaction; the normal- Inspired by the Dicke model, the main observable of the ized value of the number of photons rapidly approaches 1 in cavity to characterize the phase of the system is the number this phase [cf. Fig. 7(a)]. We refer to the latter behavior of the of photons [73]. From our analysis we were able to find three radiance, together with a trivial ordering of the chain, as the different behaviors of radiation in the cavity; namely, trivial, SP phase. super-radiant, and asymptotically super-radiant. All three are On the other hand, to describe the phase of the chain displayed in Figs. 7(a) and 7(b) for the two types of light- we need to assess its topological properties. The Majorana matter coupling; the borderlines in this figure were signaled number is a topological invariant that indicates the presence by the maximum in the absolute value of the derivative of the of ZEM in one-dimensional systems [29]. From our MF number of photons with respect to μ. The trivial behavior of approach we can recover the Majorana number by simply the cavity is characterized by the absence of photons as occurs replacing the effective chemical potential and hopping in its in the NP for both couplings. We refer to the super-radiant expression for the isolated Kitaev chain, which yields to behavior as that where there are photons in the cavity, but their number is lower than the maximum allowed by Eq. (7). The M = sgn(μeff − 4teff ). (C1)

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the cavity. Finally, the SP phase is topologically trivial in the chain and asymptotically super-radiant in the cavity.

APPENDIX D: DENSITY MATRIX RENORMALIZATION GROUP VS. MEAN FIELD For the on-site coupling, we can identify the two second- order phase transitions with the upper inset of Fig. 8(a) for different sizes of the chain; the result shows a rapid con- vergence to what is obtained from the MF scheme. For an isolated chain, with the parameters used for Fig. 8, the phase transition should occur at μ/2 =±1. For a chain-cavity coupled system with λ0 = 0.49ω and λ1 = 0ω, in contrast, they now occur at μ/2 =−1.04 ± 0.02 and 0.64 ± 0.02. In Fig. 8, we find that the maximum of the FF and the first derivative of the number of photons match with the critical points predicted by the second derivative of the energy. The FF curve shares a similar shape to that of the von Neumann entropy [cf. Fig. 8(b)]. This fact, along with the FF behavior, let us conclude that the chain and the cavity increase their entanglement at the critical points, thus driving the cavity into a super-Poissonian state.

FIG. 8. Observables for the on-site coupling. (a) FF of the cavity. Insets: Other expected values of the system that show singulari- ties at the critical points of phase transition; each plot depicts the corresponding expected values for different sizes of the chain. The mean-field (MF) results are shown by the dashed black line. Upper inset: Second derivative of the energy with respect to μ/2. Lower inset: First derivative of the number of photons. Each curve is nor- malized by the respective maximum. (b) von Neumann entropy of the system with a bipartition between photons and the Kitaev chain. Insets: Scaling of the critical points with the size of the chain; the MF value is shown with the dashed line. The parameters for these plots are ω = 1,  = 0.6ω, λ0 = 0.49ω,andλ1 = 0.

The Majorana number takes the value −1 when a system has a topological ordering and 1 when the ordering is trivial. In Figs. 7(c) and 7(d) we show the Majorana number behavior for different parameters in the phase diagram; the value of μeff and teff was obtained by DMRG taking the expected value x. However, we can recover more information about the state of the chain with the two end correlations Q described in Sec. IV. Since Q is a quantification of the localization of the ZEM, with higher values of Q the emergent Majoranas are expected to be more topologically protected. The behavior of Q for the phase diagram of both couplings is reported in Figs. 7(e) and 7(f); the boundaries of the phases in this figure were built from the point where Q exceeds QTrigger. The latter was defined as 10−2 by comparison with exact diagonalization as described in Sec. IV. Nevertheless, the phase transition points obtanied by the Majorana number and the definition of FIG. 9. Gaussian parameters and number of photons: The QTrigger are consistent with each other. Moreover, in Fig. 7(e) squeezing parameter r is depicted in (a) and (e), the coherent param- we can appreciate that the on-site coupling shifts the point of eter α in (b) and (f), the thermal parameter N in (c) and (g). The mean maximum localization of ZEM in the chain. number of photons obtained with DMRG (GS) is shown by symbols In summary, comparing Figs. 7(a) and 7(b) and Figs. 7(e) (lines) in (d) and (h). The on-site coupling strength is λ0 = 0.4ω for and 7(f); we can identify the definition of the phases. The NP (a), (b), (c), and (d). The hoppinglike coupling strength is λ1 = 0.4ω is topologically trivial and does not experience radiance in the for (e), (f), (g), and (h). The other parameters for the cavity-Kitaev cavity. The TP is topological in the chain and super-radiant in system are L = 100 and  = 0.6ω.

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The shift of the critical points can be understood consider- teff in the free energy can be understood again by considering ing Eq. (B8) and the Kitaev chain free energy in Eq. (B9). This the isolated Kitaev free energy. Regarding  as the indepen- last equation, at a fixed  and with λ0,λ1 = 0, is a U-shaped dent variable, holding μ constant and using λ0,λ1 = 0, the curve with a maximum at μ = 0, a curve that decays faster as free energy is again a U-shaped curve with a maximum at  = we get farther from the critical point (cf. Fig. 6). Then, the free 0(cf.Fig.6). Then x will allow super-radiance to minimize the energy of the whole system would allow super-radiance if μeff free energy of the whole system. The most remarkable result gets farther from the maximum in Eq. (B9). Since μeff >μ, of this kind of nonlocal coupling is the behavior of the cavity the system enters slightly earlier into the topological phase as a control device since there are only photons in the cavity (from −μ to μ) and leaves it highly sooner. The U-shaped free when the chain is in the topological phase. energy will require x to be the minimum possible value as we get farther from the maximum on the right. As a consequence, in such a region the number of cavity photons will asymptoti- APPENDIX E: GAUSSIAN PARAMETERS 2 cally approach n = (λ0/ω) , generating the SP. Deep into the The fundamental Gaussian information was build from region at the left of the maximum, the free energy will require DMRG ground-state expected values as discussed in the maximum x possible; then nˆ will vanish. In the transition Secs. V, VI. As mentioned in Appendix B, the imaginary part between the nonradiant and asymptotic phases, we can find of α is equal to zero, leading to α ∈ IR for all parameters; the TP, which is super-radiant for the cavity. Therefore, the thus qˆpˆ + pˆqˆ=0. Looking at the results for N, r and α the topological phase can be recognized as the phase between approximations N, r |α| and N, r 1 are justified in the peaks in the first derivative of the number of photons, as analyzed parameter window (cf. Fig. 9). We also compared shown in the lowest inset in Fig. 8(a). In the left inset of the mean number of photons obtained with DMRG and with Fig. 8(b), we can see that the critical points accurately con- the Gaussian approximation. The latter leads to verge to the result predicted by the MF. Slight disagreements 2 are higher for μc2, since the cavity, and therefore correlations, nˆ=(N + 1/2) cosh[2r] + α − 1/2, (E1) play a more significant role for that parameter region because it is easier to generate radiation. which under the approximations defined above is nˆ≈α2. For the hoppinglike interaction, the phase transition is sym- As seen in Fig. 9, the GS approximation results agree well metrical with respect to μ since now μeff = μ. The effect of with those obtained directly from DMRG simulations.

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