Quantum Coherence in a Compass Chain Under an Alternating Magnetic ﬁeld

Quantum coherence in a compass chain under an alternating magnetic ﬁeld

Wen-Long You,1, 2 Yimin Wang,3 Tian-Cheng Yi,1 Chengjie Zhang,1 and Andrzej M. Ole´s4, 5 1College of Physics, Optoelectronics and Energy, Soochow University, Suzhou, Jiangsu 215006, China 2Jiangsu Key Laboratory of Thin Films, Soochow University, Suzhou 215006, China 3Communications Engineering College, Army Engineering University, Nanjing, Jiangsu 210007, China 4Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany 5Marian Smoluchowski Institute of Physics, Jagiellonian University, Prof. S.Lojasiewicza11, PL-30348 Kraków,Poland (Dated: 25 April, 2018) We investigate quantum phase transitions and quantum coherence in a quantum compass chain under an alternating transverse magnetic field. The model can be analytically solved by the Jordan-Wigner transformation and this solution shows that it is equivalent to a two-component one-dimensional (1D) Fermi gas on a lattice. We explore mutual effects of the staggered magnetic interaction and multi-site interactions on the energy spectra and analyze the ground state phase diagram. We use quantum coherence measures to identify the quantum phase transitions. Our results show that l1 norm of coherence fails to detect faithfully the quantum critical points separat- ing a gapped phase from a gapless phase, which can be pinpointed exactly by relative entropy of coherence. Jensen-Shannon divergence is somewhat obscure at exception points. We also propose an experimental realization of such a 1D system using superconducting quantum circuits.

I. INTRODUCTION where ρi,j is the element of the density matrix with i

and j being the row and the column index. Cl1 (ρ) is Many physical phenomena in quantum information sci- a geometric measure that can be used as a formal dis- ence have evolved from being of purely theoretical inter- tance measure. In parallel, the relative entropy has been est to enjoying a variety of uses as resources in quan- established as a valid measure of coherence for a given tum information processing tasks. Throughout the de- basis: velopment of the resource theory of entanglement, vari- Cre(ρ) = S(ρ||ρdiag) = S(ρdiag) − S(ρ), (2) ous measures were established. However, entanglement is not a unique measure of quantum correlation because where S(ρ) = −Tr(ρ log2 ρ) stands for the von Neumann separable states can have nonclassical correlations. The entropy and ρdiag is the incoherent state obtained from ρ

concept of quantum coherence has recently seen a surge by removing all its oﬀ-diagonal entries. Cl1 and Cre are of popularity since it serves as a resource in quantum in- known to obey strong monotonicity for all states. Since formation tasks [1], similar to other well-studied quantum Cre is similar with relative entropy of entanglement, it has resource such as the entanglement [2], quantum correla- a clear operational interruption as the distillable coher-

tions [3], and the randomness [4]. Baumgratz et al. [5] ence. Meanwhile, Cl1 takes an operational interpretation introduced a rigorous framework for the quantiﬁcation of as the maximum distillable coherence from a resource

coherence based on resource theory and identified easily theoretical viewpoint [25, 26]. It was shown that Cl1 computable measures of coherence. Quantum coherence is an upper bound for Cre for all pure states and qubit resulting from quantum state superposition plays a key states [27]. Moreover, the measure of quantum coherence role in quantum physics, quantum information process- based on the square root of the JS divergence is given by ing, and quantum biology. s Ideally the coherence of a given state is measured as ρ + ρ 1 1 C (ρ) = S diag − S(ρ ) − S(ρ). (3) its distance to the closest incoherent state [6–23]. Coher- JS 2 2 diag 2 ence properties of a quantum state are usually attributed to the off-diagonal elements of its density matrix with re- The JS divergence is known to be a symmetric and spect to a selected reference basis. Among a few popular bounded distance measure between mixed quantum measures, there are three recently introduced coherence states [28] and is exploited to study shareability of co- arXiv:1806.03337v1 [cond-mat.str-el] 8 Jun 2018 measures, namely, relative entropy of coherence, l1 norm herence [24]. We remark that these coherence measures quantum coherence [5], and Jensen-Shannon (JS) diver- are all basis dependent [29]. gence [24]. The JS divergence and the l1 norm of co- Currently, approaches adopted from quantum informa- herence obey the symmetry axiom of a distance measure tion theory are being tested to explore many-body theory while the relative entropy does not obey such distance from another perspective and vice versa. Various quan- properties. The l1 norm of coherence sums up absolute tum resource measures have been exploited to character- values of all off-diagonal elements ρi,j (with i 6= j) of the ize the state of many-body systems and the associated density matrix ρ, that is, quantum phase transitions (QPTs). A QPT occurs at zero temperature and is engraved by a qualitative change X solely due to quantum fluctuations as a non-thermal pa- Cl1 (ρ) = |ρi,j|, (1) i6=j rameter is varied. Quantum entanglement and quantum 2 discord have been proven to be fruitful to investigate overview of the 1D compass model with staggered mag- QPTs. For instance, entanglement entropy changes at netic fields is presented in Sec. II. We consider the cases some (but not all) QPTs [30]. Investigation of entangle- in the presence of a uniform and an alternating transverse ment spectra is therefore very useful and helps to identify magnetic field, and discuss a possible experimental real- a possible QPT when the entanglement entropy in the ization using superconducting quantum circuits in Sec. ground-state changes by a finite value when Hamiltonian III. The model is extended by adding three-site interac- parameters are varied. One of easily accessible param- tions in Sec. IV. Next the model is exactly solved and eters is a magnetic field which might control producing QPTs are studied. We present the calculations of quan- quantum matter near a quantum critical point (QCP) in tum coherence measures in Sec. V. A final discussion and spin chains [31]. Small systems of interacting spins in summary are presented in Sec. VI. a two-dimensional (2D) compass model with perturbing interactions could also be used for quantum computation [32]. II. THE MODEL AND ITS ANALYTICAL Quantum coherence has emerged from an information SOLUTION physics perspective to address different aspects of quantum correlation in a many-body system. Comparing with We begin with a generic 1D QCM [33] on a ring of entanglement measures, quantum coherence is expected N sites, where N is even. The Hamiltonian describes to be capable of detecting QPTs even when the entan- a competition between two pseudospin τ = 1/2 compo- glement measures fail to do so. One can easily recognize x y nents, {σi , σi }, which reads, that entanglement may be a form of coherence and the converse is not necessarily valid. For instance, a product N/2 X state (|0i+|1i)⊗(|0i+|1i), for a two-qubit system carries HQCM = (J1 X2i−1,2i + J2 Y2i,2i+1) , (4) coherence but not entanglement. That is to say, quan- i=1 tum coherence incarnates a different feature of a quantum state from entanglement. On the other hand, coherence and has the highest possible frustration of interactions. x x y y α measures can be used as a resource in quantum com- Here Xi,j ≡ σi σj , Yi,j ≡ σi σj , and σi is a Pauli matrix. puting protocols, and one may claim that they are more J1 (J2) stands for the amplitude of the nearest neighbor fundamental. So a comparative study of these measures interaction on odd (even) bonds. This model owns a par- for characterizing QPTs in various spin chain models is a ticular intermediate symmetry, which allows for N/2 mu- potential research topic, which may be valuable in both tually commuting Z2 invariants Y2i−1,2i (X2i,2i+1) in the physical theory and experiment. absence of the transverse field term, and presents distinct To test the validity of this approach, the Ising model features. The ground state possesses a macroscopic de- (N/2−1) is the most transparent example of the importance of ex- generacy of at least 2 in the structure of the spin actly soluble models as guides along this difficult path. Hilbert space [33, 36]. The intermediate symmetries also All coherence measures are able to locate the Ising-type admit a dissipationless energy current [37]. The 1D QCM second-order transition. Here we consider another promi- Eq. (4) can be transformed to the fermion language and nent model dubbed as a one-dimensional (1D) compass next diagonalized, see the Appendix. In fact, the model model for a p-wave superconducting chain, which sustains can be described by a two-component 1D Fermi gas on a more complex physical phenomena than the Ising model, lattice as displayed in Eq. (A.3). such as macroscopic degeneracy [33], pure classical fea- The 1D QCM in Eq. (4) may be supplemented by a ~ tures [34], and suppressed critical revival structure [35]. possibly spatially inhomogeneous Zeeman field hi, given The purpose of this paper is to investigate QPTs and by quantum coherence in the 1D quantum compass model N (QCM) under an alternating transverse magnetic field. X H = ~h · ~σ . (5) The motivation is twofold. On the one hand, previous h i i i=1 investigations revealed that the 1D compass model could exhibit miscellaneous phases via modulation of external In a realistic structure, the crystal fields surrounding fields. An exotic spin-liquid phase can emerge through the odd-indexed sites and even-indexed sites are differ- a Berezinskii-Kosterlitz-Thouless (BKT) QPT under a ent. The presence of two crystallographically inequiva- uniform magnetic field. We would like to verify whether lent sites on each chain with a low symmetry of the crys- such a transition is robust under realistic inhomogeneity tal structure leads to staggered gyromagnetic tensors. of external fields [36]. The calculations take into account It has been recently shown that a spatially varying both uniform and staggered fields. On the other hand, magnetic field can be induced by an effective spin-orbit its exact solvability provides a suitable testing ground interaction. The alternating spin environment is repre- for calculating accurately coherence measures to detect sented by the staggered Dzyaloshinskii-Moriya interac- QPTs. We investigate the coherence of this model in the tion and Zeeman terms. The staggered magnetic field thermodynamic limit and its connection to QPTs. plays an important role in understanding the field depen- The remaining of the paper is structured as follows. An dence of the gap in Cu benzoate antiferromagnetic chain 3

[38–41] and in Yb4As3 [42]. We remark that there is in- The advantage of the result given by Eq. (11) is that e0 is creasing interest in the effects of the staggered field mo- independent of δ as well as of the signs of J1 and J2. The tivated by the experimental work on a number of mate- intersite correlators are given by the Hellmann-Feynman rials. The interplay between the staggered Zeeman fields theorem: and dimerized hopping on the topological properties of 2 X J1 + J2 cos k Su-Schrieffer-Heeger (SSH) model has received much at- hσx σx i = − , 2i−1 2i N p 2 2 2 tention only recently [43], after the rapid progress in the k J1 + J2 + 2J1J2 cos k + 4h synthesis of 1D heterostructures [44–47]. y y 2 X J2 + J1 cos k Without the loss of generality, we consider a stag- hσ2iσ2i+1i = − . N pJ 2 + J 2 + 2J J cos k + 4h2 ~ ~ k 1 2 1 2 gered magnetic field, i.e., h2i−1=h1zˆ, h2i=h2zˆ, ∀i = (12) 1, ··· ,N/2. Be aware that hi = giµBBi here is the reduced magnetic field containing the g-factor and the For δ > h, an interchange between h and δ is performed Bohr magneton µ . An effective staggered magnetic field B in Eqs. (11-12). might be attributed to the alternating g tensor in an ap- Since a QPT occurs only when the gap closes, looking plied uniform field [48]. Thereby, we define an average for gapless points in the energy spectrum may indicate magnetic field h = (h + h )/2 and a field difference 1 2 this transition. The lower mode ε reduces to a zero- δ = (h − h )/2. The 1D compass model in external k,2 2 1 energy flat band for h = ±δ, corresponding to either field, h2i−1 = 0 or h2i = 0. This undermines the limited con- H = HQCM + Hh, (6) dition for the existence of a macroscopic degeneracy in the ground-state manifolds. The zero-energy flat band is exactly soluble and we obtain its zero-temperature is fragile against an infinitesimal external uniform mag- phase diagram, see below. For the sake of clarity, we netic field for δ = 0. A uniform field will remove the briefly describe the procedure to diagonalize the Hamil- ground-state degeneracy and the bands are no longer de- tonian Eq. (6) exactly in the Appendix. generate. The result here implies that a magnetic field Thus the Hamiltonian (6) in the Bogoliubov-de Gennes applied on one sublattice still makes the zero-energy flat (BdG) form in terms of Nambu spinors is: band intact. 1 X Interestingly, the model possesses local symmetries H = Υ† Hˆ Υ , (7) 2 k k k that one can find in the absence of field terms at odd k sites. If field h2i−1 is vanishing at odd sites and at even where Υ† = (a† , b† , a , b ). In this circumstance, the sites it takes any random values, then any eigenstate has k k k −k −k (N/2−1) Hamiltonian (6) reads 2 degeneracy for a ring of length N. These degen- eracies follow from the symmetry operators, ˆ r i Hk = 2δΓzz − 2hΓz0 + Tk (Γzx − Γyy) − Tk(Γzy + Γyx), y z x (8) S2i ≡ σ2i−1 ⊗ σ2i ⊗ σ2i+1, (13) a b x,y,z x,y,z with Γab = τ ⊗ σ , ∀ a, b = x, y, z, and τ /σ be- and are activated when the field is absent at odd sites. ing the Pauli matrices acting on particle-hole space and Such symmetry operators (13) anti-commute for the 0 0 spin space, respectively, and τ = σ is a 2×2 unit ma- neighbors, i.e., {S2i,S2(i+1)} = 0, while they commute r i trix. Here Tk and Tk are the real and imaginary parts of otherwise. ik Tk = J1+J2e . The BdG Hamiltonian Eq. (8) respects a particle-hole symmetry defined as CHˆ (k)C−1 = −Hˆ (−k) with C = Γx0K, where K is the complex conjugate op- III. POSSIBLE EXPERIMENTAL erator. As a consequence the energy levels appear in REALIZATION USING SUPERCONDUCTING conjugate pairs such as ε(k) and −ε(−k). The diagonal QUANTUM CIRCUITS form of the Hamiltonian Eq. (8) is then given by,

2 The unique features of this rich model (6) motivate us X X 1 H = ε γ† γ − . (9) to consider its possible physical implementations to ad- k,j k,j k,j 2 vance our understandings. It is well known that super- k j=1 conducting circuit systems have become one of the lead- The spectra consist of two branches of energies εk,j (with ing platforms for scalable quantum computation, quan- j = 1, 2), given by the following expressions: tum simulation and demonstrating quantum optical phenomena because of its exotic properties such as con- p 2 2 p 2 2 εk,1(2) = |Tk| + 4h ± |Tk| + 4δ . (10) trollability, ﬂexibility, scalability, and compatibility with The ground-state energy per site for h > δ may be ex- micro-fabrication [49–51]. Various models of many-body pressed as systems have been proved to be able to be simulated by superconducting circuits, such as the Kitaev lattice [52], 2 X q e = − J 2 + J 2 + 2J J cos k + 4h2. (11) the Heisenberg spin model [53], the fermionic model [54], 0 N 1 2 1 2 k the 1D Ising model [55] and anisotropic quantum Rabi 4

C energy of the circuit, and secondly choosing the average (a) V g1  g1 1 e1 phase drop ϕi of each charge qubit as the canonical co- z y ordinate, the Hamiltonian of the entire system can then be obtained by a Legendre transformation. We consider the situation when the frequency of the C C m LC oscillator is much larger than the frequency of the qubit. In this case the LC oscillator is not really ex- C L cited and the corresponding terms can be removed fromg1 Vg1 e1 the total Hamiltonian. Even though, the LC oscillator’s virtual excitation still produces an effective coupling be- Cg2 Cg3 Vg2  Vg3  tween the corresponding charge qubits. For charge qubit 2 e2 3 e3 with EC EJ , at very low temperature, theC two-levelm C system is formed by the charge states |0i and |1i, which denote the zero and one extra Cooper pair on the island, L (b) 1 respectively. After projecting the total Hamiltonian into the ith charge qubit’s computational basis {|0ii, |1ii}, we obtain [52], Cg2 Cg3 V  V  2 3 g2 e2 g3 e3 y X y y z X z z X x x H = Ji,j σi σj + Ji,j σi σj + hi σi , (14) y−links z−links i FIG. 1. Scheme of a circuit QED system for the physical implementation of the 1D compass chain Hamiltonian Eq. where all the charge qubits are biased at the optimal z (6): (a) Design of the basic building block, which is composed point (i.e., ngi = CgVgi/(2e) = 1/2) such that hi = 0, of three superconducting charge qubits, labeled as 1, 2, 3. x and hi = −EJ cos(πΦei/Φ0) is the effective Josephson Qubits 1 and 2 are coupled capacitively to each other via a energy of the ith charge qubit, Φ0 ≡ h/2e is the flux mutual capacitance Cm; and the coupling between the qubits quantum. The y-type Ising coupling strength 1 and 3 are provided by a commonly-shared LC oscillator. Inset: the orange circles denote the superconducting charge J y = −4ξE2 cos (πΦ /Φ ) cos (πΦ /Φ ) ≤ 0, (15) qubits; the two types of inter-qubit couplings are denoted i,j J ei 0 ej 0 as z- and y-bond, which are indicated by the blue-solid and 2 2 2 2 the red-dashed line, respectively. (b) A 1D compass chain with ξ = Lπ (2CJ + Cg + Cm) (Cg + Cm) /(ΛΦ0) , are constructed by repeating the building block in (a). tunable via the external magnetic flux threading the SQUIDs in the ith and jth charge qubits. Simultane- ously the z-type coupling strength is fixed as model [56]. In our case, the 1D compass chain in Eq. (6) 2 z e Cm can be built from superconducting charge qubits, each of Ji,j = ≥ 0, (16) which is composed of a direct current superconducting Λ quantum interference device (dc SQUID) with two iden- 2 2 with Λ = (2CJ + Cg + Cm) − Cm. A detailed analysis tical Josephson junctions. For ith charge qubit, the gate of circuit quantization can be found in Ref. [52]. voltage V applied through the gate capacitance C can gi gi An intuitive understanding of the coupling mechanism be used to control the charge, and the magnetic flux Φ ei in the Hamiltonian Eq. (14) would be the following. Each piercing the SQUID can be used to control the effective charge qubit is coupled to its left or right nearest neighbor Josephson energy, E (Φ ) = 2E cos(πΦ /Φ ). J ei J ei 0 via a capacitor or an LC oscillator. The appearance of a It has been demonstrated that charge qubits can be capacitor modifies the electrostatic energy of the system, coupled to each other for all the individual interactions x x and thus provides the z-type Ising coupling. On the other of Ising type [49], i.e., ∝ σi σi+1 via a mutual inductance y y z z hand, the magnetic energy of the inductor is biased by [57], ∝ σi σi+1 via an LC oscillator [58], and ∝ σi σi+1 a current composed of contributions from both of the via a capacitor [59]. This provides us a promising way to two qubits, and thus the virtually excited LC oscillator implement the 1D compass chain with the superconduct- induces the y-type Ising coupling. Then implementing a z ing charge qubits. As shown in Fig. 1, a charge qubit Q iπσj /4 unitary rotation around the y axis, i.e., U ≡ j e , is placed at each node, and is then connected to its two x † z z † x nearest neighbors with two types of couplers, i.e., a ca- one can find Uσi U = σi , Uσi U = −σi , and then Eq. pacitor for the z-type bond and an LC oscillator for the (14) can be recast into the Hamiltonian Eq. (6). y-type bond. For the sake of simplicity and without the loss of generality, we assume all the charge qubits to be identical IV. EFFECT OF THREE-SITE INTERACTIONS such that Cgi ≡ Cg, EJi ≡ EJ , ECi ≡ EC . Follow- ing the standard quantization procedure of the circuit by To make the model as general as possible and still ex- firstly writing down the kinetic energy and the potential actly soluble, we introduce in addition three-site interac- 5 tions of the (XZX+YZY)-type into Eq. (6) ,

N X H3−site = J3 (Xi−1,i+1 + Yi−1,i+1) Zi, (17) i=1 where J3 characterizes their strength. Such interactions (a) between three adjacent sites emerge as an energy cur- 1.0 rent of a compass chain in the nonequilibrium steady 0.5 states [37]. Three-site interactions violate the interme- Δ diate symmetry and elicit exotic phenomena. This gen- 0.0 eralized version of the 1D QCM has been shown to host a diversity of nontrivial topological phases and an emer- 2 gent BKT QPT under the interplay of a perpendicular Zeeman field and multi-site interactions [60]. 2 We next turn to the discussion of the physical imple- 0 mentation of the 1D QCM including the three-site in- δ 0 teraction with superconducting circuits. As supercon- h ducting circuits offer advantages of easy tunability and -2 -2 scalability, in principle, many-body interactions in superconducting systems could be designed using Josephson- junction-based couplers in a graph structure [61–63]. However, the effective many-body coupling terms may emerge with a much weaker strength. An alternative and practical strategy to generate many-body interactions would be the simulation protocols employing the 1.0 (b) microwave fields with appropriate frequency conditions, as have been studied in nuclear magnetic resonance systems [64], optical lattices [65], and superconducting cir- 0.5 Δ cuits [66, 67]. Therefore, we expect that the three-site interactions of the sort of XZX+YZY-type as in Eq. (17) would be built in similar fashions. However, an in-depth 0.0 study of experimental implementation of this particular 2 model will be left for future investigation. 2 The generalized Hamiltonian of the 1D QCM which 0 δ 0 includes the three-site (XZX+YZY) interaction is h -2 -2 H = HQCM + Hh + H3−site. (18) Eq. (18) describes a 1D sp-chain with inter-band interactions and hybridization between orbitals [68]. The three- FIG. 2. Three-dimensional plot of the energy gap as a func- site interactions can be converted into fermionic form tion of h and δ for: (a) J3 = 0 and (b) J3 = 1. Parameters P † H3−site = 4J3 k cos k ckck. We note that the spectra are as follows: J1 = 1, J2 = 4. can be pinpointed at commensurate momenta k = ±π/2 regardless of the value of J3. Hence the eigenspectra (10) can be renormalized with −2h → Fk = −2h + 2J3 cos k, Figs. 3(f)-3(i). Altogether, the number of Fermi points as evidenced in Eq. (A.11). at which the linear dispersion relation is found changes The main features and the evolution of these profiles as h increases. Indeed, here the topological transition under staggered fields with increasing magnetic field h belongs to the universality of the Lifshitz transition. are depicted in Fig. 3. We observe that the ground It is also easy to see that the Weyl points collapse at state of the system is complicated under the interplay of h = ±|J3 + δ| with increasing h. In the gapless phase three-site interactions and staggered magnetic field. As h the crossings between bands exhibit a linear dispersion rises from large negative values, εk,2 closes the gap grad- relation [see Figs. 3(c)-3(g)] and thus define effective ually and finally touches ε = 0 at momentum k = π for 1D Weyl modes. One notes that the nodes appear and h = −|J3 + δ|. Further increase of h bends εk,2 down- disappear only when two nodes are combined, as a char- wards, leading to ε|k|>|kic|,2 < 0 with an incommensurate acteristic of Weyl fermions in a three-dimensional or 2D momentum kic. An additional crossing at k = π revives superconductor [69, 70] and in topological superfluidity for h = −|J3 −δ|. We can see that the number of crossing [71–74]. It is noticed with the emergence of two Weyl points at zero energy grows from 0 to 4 in Figs. 3(a)- points at its extremities (k = ±π) and their collapse at 3(e), and then decreases with further increase of h, see the center of the Brillouin zone (k = 0). In this region 6 2 2 (a) 2 (b) (c) k

0 0 0 -2 -2 -2 -1 0 1 -1 0 1 -1 0 1 2 1 0.5 (d) (e) (f) k

0 0 0 -1 -0.5 -2 -1 0 1 -1 0 1 -1 0 1 2 (g) 2 (h) 2 (i) k

0 0 0 -2 -2 -2 -1 0 1 -1 0 1 -1 0 1 k/π k/π k/π

FIG. 3. The electron spectra εk,2 and its corresponding hole spectra −εk,2 for increasing magnetic ﬁeld h in Eq. (18): (a) h = −3, (b) h = −2.5, (c) h = −2.0, (d) h = −1.5, (e) h = 0, (f) h = 1.5, (g) h = 2, (h) h = 2.5, and (i) h = 3. Parameters are as follows: J1 = 1, J2 = 4, J3 = 2, and δ = 0.5.

2 the low-energy Hamiltonian with Weyl nodes in a 1D d i m e r system can be reduced to describe the two Bogoliubov bands that cross zero energy. The resulting phase diagram of the model Eq. (18), S L I S L I obtained by the exact solution using the Jordan-Wigner transformation, is shown in Fig. 4. For clarity we have S L I I considered the entire plane of ﬁelds, although the phase diagram is obviously symmetric under reﬂection from the

δ 0

P M P M h1 = h2 and h1 = −h2 lines, which also symmetrize the spectrum and entanglement. The quantum phase boundaries are determined by the following condition: S L I S L I |h| + |δ| = J3. (19) d i m e r For large h, the system changes to a disordered param- - 2 agnetic (PM) phase, in which the z-axis sublattice mag- - 2 - 1 0 1 2 netizations are unbiased, as shown in Fig. 5(a). On the h contrary, the dimer phase is the one in which z-axis magnetization at odd and even sites has a staggered order. FIG. 4. Phase diagram of the compass chain (18) in an alter- The staggered magnetic susceptibilities are vanishing nating transverse field. The shaded regions mark the gapless in the both gapped phases, while they are finite in the spin-liquid phase, in which SLI (SLII) denotes the spin-liquid gapless phases. Besides, as shown in Fig. 5(b), the near- phase I (II) with 2 (4) Fermi nodes. The dashed line marks y y est neighbor correlation hσ2i−1σ2ii clearly shows non- the path δ = 0. Other parameters: J1 = 1, J2 = 4, J3 = 1. analytical behavior at the QCPs, and the counterpart x x hσ2i−1σ2ii is smooth. One can notice that a singular behavior can be detected by taking the first derivative of 7

x x y y hσ2i−1σ2ii and hσ2iσ2i+1i with respect to h. Since QPTs are caused by nonanalytical behavior of (a) /=1.5 ground-state energy, QCPs correspond to zeros of εk,2. /=1.0 νz 1.0 The gap vanishes as ∆ ∼ (h−hc) , where ν and z are the

i /=0.5

correlation-length and dynamic exponents, respectively.

z 2i

The gap is determined by the condition, ∂εk=k0,2/∂k = 0 <

and one ﬁnds ∆ = mink |εk,2|. This implies that the h - - minimum is suited at either k0 = 0 or k0 = π, depend- i 0.5

ing which mode has a lower energy. One ﬁnds the crit- 2i-1

ical exponents satisfy νz = 1, as revealed in Fig. 2(b). z

The expansion of the energy spectra at the criticality h around the critical mode k0, i.e., at ∆k ≡ k − k0 1, 2 p 2 2 ∂εk,2 ∼ 2J3δ(∆k) / (J1 + J2 cos k0) + 4δ . The 0.0 quadratic dispersion in Fig. 3(b) suggests a dynamical -3 -2 -1 0 1 2 3 exponent z = 2 and hence ν = 1/2, which is different h from the generic QCM in the absence of three-site interactions [75, 76]. Remarkably, the ground state develops weak singular- ities at δ = 0. For δ = 0 the phase boundaries are pinpointed at hc = ±J3 with an incommensurate critical −1 momentum k0 = cos (h/J3), as presented in Fig. 2(b). One can find that the system transforms from the gapped phase to the gapless phase passing through an unconven- tional field-induced QCP, where infinite-order QPTs oc- cur by tuning h along the path (Fig. 4, dashed line) to approach the QCPs, with no broken-symmetry order pa- rameter. We can identify that hc = ±J3 are multi-critical points, where h − δ = ±J3 and h + δ = ±J3 meet [60]. One finds the critical exponents that follow νz = 2 by ob- serving the gap scaling. The dependence of low-energy excitations on k shows that z = 2 in the gapless phase while z = 4 at QCPs. It has been shown that z can be extracted from the measurement of the low-temperature specific heat and entropy in the Tomonaga-Luttinger liq- FIG. 5. Effect of alternating magnetic field in the 1D compass uid phase [77]. model (18): (a) the difference between odd-site and even- site magnetization for δ = 0.5, 1.0 and 1.5 with J3 = 1; (b) α α the nearest-neighbor correlations hσi σi+1i (α = x, y) on odd bonds and even bonds with J3 = 2, δ = 0.5. The symbol ? in V. QUANTUM COHERENCE MEASURES x x y y (b) marks the position of hσ2i−1σ2ii = hσ2i−1σ2ii = 0. Other parameters are: J1 = 1, J2 = 4. In the representation spanned by the two-qubit product states we employ the following basis, with {|0ii ⊗ |0ij, |0ii ⊗ |1ij, |1ii ⊗ |0ij, |1ii ⊗ |1ij}, (20) 1 u = 1 ± hσzi ± hσzi + hσzσzi , (23) where |0i (|1i) denotes spin up (down) state, and the ± 4 i j i j two-site density matrix can be expressed as, 1 x x y y z± = hσi σj i ± hσi σj i , (24) 3 4 1 X D a a0 E a a0 1 ρij = σ σ σ σ , (21) z z z z 4 i j i j ω± = 1 ± hσi i ∓ hσj i − hσi σj i . (25) a,a0=0 4

a x y z Note that the formula can be simplified when the sys- where σi stands for Pauli matrices {σi , σi , σi } with a = z z 1, 2, 3, and for a 2 × 2 unit matrix with a = 0. Since tem is translation invariant, i.e., hσi i = hσj i for arbitrary i and j, such that ω+ = ω−. Under the staggered mag- the Hamiltonian has Z2 global phase-flip symmetry, the z netic field, the magnetization densities {hσ2i−1i} at odd two-qubit density matrix reduces to an X state, z sites and {hσ2ii} at even sites are inequivalent. As is  u+ 0 0 z−  disclosed in Fig. 5(a), the difference of the z-axis magne- 0 w+ z+ 0 tizations is nonvanishing in the gapless regions and dimer ρij =  ∗  , (22)  0 z+ w− 0  phase. By means of the Wick theorem, it is well known ∗ z− 0 0 u− that two-site correlation functions can be expressed as an 8

C l 1 C 0.4 JS C

0.2

0.0 FIG. 6. Quantum coherence measures on (a) odd bonds and -5 0 5 (b) even bonds for increasing magnetic field h with J1 = 1, h J2 = 4, J3 = 2, δ = 0.5. The legend shown in (b) is the same with (a). The symbol ? marks the position of h? = −3.647. The dashed lines correspond to positions of ±|J3 ± δ|. Inset FIG. 7. Quantum coherence measures of next nearest neigh- in (a) shows the second-order derivative of the ground-state bor qubits for increasing magnetic field h with J1 = 1, J2 = 4, energy e0. J3 = 2, δ = 0.5. The dashed lines mark the positions of ±|J3 ± δ|. expansion of Pfaffians [78, 79]. One easily finds that X X quantum coherence can be simplified to: S(ρij) = − ξm log ξm − ξn log ξn, (26) 2 2 x x y y m=0,1 n=0,1 Cl1 (ρ) = max hσi σj i , hσi σj i . (29) where For our purposes, the correlation and coherence measures in the ground state of the quantum compass chain under 1 n z z m h x x y y 2 ξm = 1 + hσ σ i + (−1) hσ σ i − hσ σ i an alternating transverse magnetic field for two spins are 4 i j i j i j 1/2 investigated in the following for comparison. Figures 6 z z 2i + hσi i + hσj i , (27) and 7 display the results for the relative entropy, the l1 norm quantum coherence and the JS divergence of two 1 n z z n h x x y y 2 qubits in the ground state of the compass chain. We find ξn = 1 − hσi σj i + (−1) hσi σj i + hσi σj i 4 Cl1 (ρ) ≥ Cre(ρ) holds, as was conjectured in Ref. [27]. 1/2 z z 2i The conjecture was proved only for the pure state and + hσi i − hσj i . (28) not yet proven for mixed state, and our findings indicate its validity. Recently diagonal discord was proposed to be an eco- Looking at the relative entropy, the l1 norm quantum nomical and practical measure of discord [80], which com- coherence and the JS divergence on the odd bonds pre- pares quantum mutual information with the mutual in- sented in Fig. 6(a), three measures become zero simul- formation revealed by a measurement that corresponds taneously at h? = −3.647. The relative entropy shows a to the eigenstates of the local density matrices. As long smooth local minimum at this special point, which is dif- as the local density operator is nondegenerate, diagonal ferent from nonanalytical behaviors of its counterparts. discord is easily computable without optimization over The null point of coherence measures corresponds to a all possible local measurements, which make the discord- factoring point, where the intersite-correlators on the like quantities unamiable. The reduced density matrix x x y y Q weak bonds hσ2i−1σ2ii and hσ2i−1σ2ii vanish, see Fig. for a single ith-qubit has a local eigenbasis η = |ηihη| 5(b). As a consequence, the density matrix (22) becomes z when hσi i 6= 0, and then a local measurement follows diagonal in the orthogonal product bases. This factoring P Q Q πi(ρij) = η( η ⊗Ij)ρij( η ⊗Ij). Diagonal discord point is found to be δ independent, and such a accidental D¯ i(ρij) characterizes the reduction in mutual informa- inflexion will be absent in the quantum coherence mea- tion induced by πi(ρij) and takes a similar form to the sures for the even bonds, see Fig. 6(b), and the next- relative entropy as D¯ i(ρij) = S(ρij||πi(ρij)). nearest-neighboring qubits, see Fig. 7. One finds the In terms of the two-qubit X state in Eq. (22), D¯ i(ρij) quantum coherence on even bonds is larger than that on is identical to Cre(ρij). Without the loss of generality, we odd bonds, and the next-nearest neighbor coherence is a mainly use Cre(ρ) hereafter although it has two-fold im- little smaller. plications in quantum correlations. Besides, the l1 norm For δ = 0 the system undergoes a BKT phase transi- 9 tion, and the quantum coherence exhibits a local max- multi-critical point is found. imum; see Fig. 10(a) in Ref. [60]. For δ 6= 0, the Since a quantum phase transition is driven by a purely transitions for increasing h belong to second-order phase quantum change in the many-body ground-state corre- transitions, as is verified in inset of Fig. 6(a). In this re- lations, the notion of quantum coherence appears natu- spect, we find that the coherence measures exhibit either rally and is suited to probe quantum criticality. To this a nonanalytical behavior or an extremum across QCPs, end, we provide a study of associated exhibited quantum indicating sudden changes take place. correlations in this model using a variety of quantum After a closer inspection we find the l1 norm quantum information theoretical measures, including the relative coherence displays anomalies at QCPs hc = −J3 − δ, entropy and the l1 norm quantum coherence along with ±(J3 −δ), but it misses the QCP at hc = J3 +δ. We also the Jensen-Shannon divergence. These alternative frame- observe that there are superfluous kinks of the l1 norm of works of coherence theory stem mainly from different no- coherence in regions around h = 0, as shown in Figs. 6(a) tions of incoherent (free) operations. It is thus desirable and 7. The artificial turning points can be ascribed to and interesting to find any interrelation between them. the definition of l1 norm quantum coherence in Eq. (1). We then compare the respective kinds of insights that Further, this norm does not exhibit any anomaly on the they provide. We have found that the continuous phase strong bonds, as shown in Fig. 6(b). From such a com- transitions occurring in this model can be mostly faith- parison one can find that relative entropy [Eq. (2)] can fully detected by examining quantum information theo- faithfully reproduce quantum criticality. Despite of their retical measures. We also discern some differences. The formal resemblances, the JS divergence and the l1 norm l1 norm quantum coherence (defined as the sum of ab- quantum coherence, as different perspectives of quantum solute values of off diagonals in the reduced density ma- coherence, embody infidelity of density matrix and are trix) of odd bonds develops a singular behavior at non- insufficient to readout the locations of QCPs. criticality, which is caused by the absolute operator in the definition (29). Moreover, the l1 norm quantum coherence of even bonds does not exhibit any anomaly across VI. DISCUSSION AND SUMMARY the critical points. A closer inspection reveals that the l1 norm quantum coherence of even bond shows an in- In this paper we consider a one-dimensional Hamilto- flection point and the transition point can be easily cap- nian with short-range interactions that includes three- tured looking at its derivative with respect to h, which site interactions and alternating magnetic fields. The would display an extremum. On the contrary, the rela- one-dimensional quantum compass model is a paradig- tive entropy and the Jensen-Shannon divergence show matic scenario of quantum many-body physics, which is pronounced anomalies, either a sharp local maximum more subtle than the Ising model, and hence hosts richer or a turning point. That is to say that they faithfully phase diagrams. For a second-order quantum phase tran- sense the rapid change of quantum correlation, resulting sition from a gapped Néelphase to a gapped paramag- in a clear identification of quantum phase transitions. netic phase, tools of quantum information theory can al- Also the l1 norm quantum coherence and the Jensen- ways be employed to characterize successfully the tran- Shannon divergence become nonanalytical at exception sition points. Usually achieving a complete and rigorous points. Despite formal similarity, different measures of quantum-mechanical formulation of a many-body system quantum coherence have their respective scope for de- as desired is obstructed by the complexity of quantum tecting the quantum criticality. From such comparison, correlations in many-body states. we believe that the relative entropy is more credible than others. The spin chain in the present model is efficiently solv- able using the standard Jordan-Wigner and Bogoliubov Summarizing, our results suggest that the diagonal transformation techniques. Adopting the exact solvabil- entries of the density operator are indispensable to ex- ity we describe the phase diagram of the model as a tract information across the quantum critical points. In function of its parameters. The perpendicular Zeeman other words, quantum phase transitions are cooperative field and and three-site interactions spoil the interme- phenomena where competing orders induce qualitative diate symmetry in the generic compass model, and thus changes in many-body systems. The figures of merit of they destroy the ground-state degeneracy of the quantum these measures might be crucial to the optimizing basis. compass chain. The tunability of the staggered mag- Furthermore, we proposed an experimental scheme using netic field entails the ground state can be among para- superconducting quantum circuits to realize the compass magnetic phase, dimer phase, and spin-liquid phases, in chain with alternating magnetic fields. which the number of Fermi points falls into two cate- gories. Except the multi-critical points, the phase transitions are of second order. The critical exponents can be ACKNOWLEDGMENTS extracted from low-energy spectra and gap scalings. Our investigations show a uniform magnetic field can drive We thank Wojciech Brzezicki for insightful discus- the spin-liquid phase to the paramagnetic phase through sions. W.-L. Y. acknowledges NSFC under Grant the Berezinskii-Kosterlitz-Thouless transition, where a Nos. 11474211 and 61674110. Y. Wang acknowl- 10 edges China Postdoctoral Science Foundation Grants Following the standard Jordan-Wigner transformation, Nos. 2015M580965 and 2016T90028. C. Zhang acknowl- we rewrite the Hamiltonian in the momentum space by edges NSFC under Grant Nos. 11504253 and 11734015. taking a discrete Fourier transformation for plural spin A. M. O. kindly acknowledges support by Narodowe Cen- sites with the periodic boundary condition (PBC). trum Nauki (NCN, National Science Centre, Poland) un- r r der Project No. 2016/23/B/ST3/00839. 2 X 2 X c = e−ikja , c = e−ikjb , 2j−1 N k 2j N k k k (A.4) Appendix: Diagonalization of the Hamiltonian with discrete momenta as 2nπ N N N We are considering a 1D quantum compass model with k = , n = − − 1 , − − 3 ,..., − 1 . the three-site (XZX+YZY) terms under staggered mag- N 2 2 2 netic fields in Eq. (18), which can be rewritten as (A.5) Next the discrete Fourier transformation for plural spin N/2 sites is introduced for the PBC. The Hamiltonian takes X x x y y H = J1σ2i−1σ2i + J2σ2iσ2i+1 the following form which is suitable to apply the Bogoli- i=1 ubov transformation: N/2 X h X H = T a† b − T ∗a b† + T a† b† − T ∗a b + (h1 zˆ · ~σ2i−1 + h2 zˆ · ~σ2i) k k k k k k k k −k k k −k i=1 k i N † † † † + Fk a a + b b + 2δ a a − b b + Nh , X x z x y z y k k k k k k k k + J3 σi−1σi σi+1 + σi−1σi σi+1 . (A.1) i=1 (A.6) ik Here, J1 and J2 denote the coupling strength on odd and where Tk = J1 +J2e and Fk = 2J3 cos k −2h. After the even bonds, respectively. h1 and h2 are the transverse Fourier transformation, H is then transformed into a sum ˆ external magnetic fields applied on odd and even sites, of commuting Hamiltonians Hk describing a different k respectively. Finally, J3 is the strength of (XZX+YZY)- mode each. Then we write the Hamiltonian in the BdG type three-site exchange interactions. form in terms of Nambu spinors: First, we use a Jordan-Wigner transformation which 1 X † ˆ maps explicitly a pseudo-spin model to a free-fermion H = Υ HkΥk, (A.7) 2 k system whose properties can always be computed effi- k ciently as a function of system size [78]: where z †  F + 2δ T 0 T  σj = 1 − 2c cj, k k k j ∗ T Fk − 2δ −T−k 0 † ˆ  k  σx = eiφj c + c , Hk = ∗ (A.8), j j j  0 −T−k −Fk − 2δ −Tk  ∗ ∗ T 0 −T −Fk + 2δ y iφj † k k σj = ie cj − cj , (A.2) † † † and Υk = (ak, bk, a−k, b−k). In momentum space, time with φj being the phase accumulated by all earlier sites, reversal (TR) symmetry and particle-hole (PH) symme- P † i.e., φj = π l

N/2 TR operator T is simply a complex conjugation K and Xn h † † † † i operator C = τ K as the PH transformation. The H= J c c − c c + c c − c c x 1 2i−1 2i 2i−1 2i 2i−1 2i 2i−1 2i system Eq. (A.8) belongs to topological class D with i=1 h i topological invariant Z2 in one dimension, which satisﬁes † † † † ˆ ˆ x 0 x + J2 c2ic2i+1 − c2ic2i+1 − c2ic2i+1 − c2ic2i+1 H(−k)C = −CH(k). Here C = τ ⊗ σ K, where τ and σ0 are the Pauli matrices acting on PH space and spin † † + h1 1 − 2c2i−1c2i−1 + h2 1 − 2c2ic2i space, respectively. ˆ † † o The diagonalized form of Hk can be achieved by a four- + 2J3 c2j−1c2j+1 + c2jc2j+2 + H.c. . (A.3) dimensional Bogoliubov transformation which connects † † the original operators {ak, bk, a−k, b−k}, with two kind The fermion version of this model corresponds to a † † dimerised p-wave superconductor, in which the electrons of quasiparticles, {γk,1, γk,2, γ−k,1, γ−k,2}, as follows, also generate next-nearest neighbor hopping. Such a  γ†   †  two-component 1D Fermi gas on a lattice is realizable k,1 ak † † with current technology, for example on an optical lat-  γk,2  ˆ  b    = Uk  k  . (A.9) tice by using a Fermi-Bose mixture in the strong-coupling  γ−k,1   a−k  limit [81]. γ−k,2 b−k 11

Hˆk is diagonalized by a unitary transformation (A.9), Two-point correlation functions for the real Hamilto- nian Eq. (A.1) can be expressed as an expansion of Pfaf- X † ˆ ˆ † ˆ ˆ ˆ † X 0† 0 H = ΥkUkUk HkUkUk Υk = Υk DkΥk. (A.10) ﬁans using the Wick theorem, k k

The obtained four eigenenergies {εk,j} (j = 1, ··· , 4) q 2 2 p 2 2 εk,1(2) = |Tk| + Fk ± |Tk| + 4δ , G−1 G−2 · Gi−j q ε = − |T |2 + F 2 ∓ p|T |2 + 4δ2, (A.11) G0 G−1 · Gi−j+1 k,4(3) k k k hσxσxi = , (A.13) i j ...... ˆ † ˆ ˆ in the diagonalized Hamiltonian matrix Dk = Uk HkUk Gj−i−2 Gj−i−3 · G−1 are the excitations in the artiﬁcially enlarged PH space G1 G0 · Gi−j+2 where the positive (negative) ones denote the electron G2 G1 · Gi−j+3 (hole) excitations. The ground state corresponds to the hσyσyi = , (A.14) i j . . .. . state in which all hole modes are occupied while the elec- . . . . tron modes are vacant. The PH symmetry indicates here Gj−i Gj−i−1 · G1 † † that γk,4 = γ−k,1 and γk,3 = γ−k,2. So the spectra consist z z z z hσi σj i = hσi ihσj i − Gj−iGi−j, (A.15) of two branches of energies εk,j (with j = 1, 2), and 1 Hˆ = ε γ† γ − γ γ† k 2 k,1 k,1 k,1 −k,1 −k,1 1 † † + εk,2 γk,2γk,2 − γ−k,2γ−k,2 † † 2 where Gr = h(c0 − c0)(cr + cr)i and r = j − i represents 2 the distance between the two sites in units of the lattice X 1 = ε γ† γ − . (A.12) constant. k,j k,j k,j 2 j=1

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