Nonanalyticity of circuit complexity across topological phase transitions
Zijian Xiong Dao-Xin Yao Zhongbo Yan




















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    Phys. Rev. BPhys. Rev. APhys. Rev. Lett.∥Τ𝒯σΓ q Q k r x l p E B D H K e d A nρ̂ jℱ𝒜⟨⟩↔←→
        
        yaodaox@mail.sysu.edu.cnyanzhb5@mail.sysu.edu.cnState Key Laboratory of Optoelectronic Materials and Technologies, School of Physics, Sun Yat-Sen University, Guangzhou 510275, China
        The presence of nonanalyticity in observables is a manifestation of phase transitions. Through
        the study of two paradigmatic topological models in one and two dimensions, in this work we show that
        the circuit complexity based on our specific quantification
        can reveal the occurrence of topological phase transitions, both in and out
        of equilibrium, by the presence of nonanalyticity.
        By quenching the system out of equilibrium, we find that the circuit complexity
        grows linearly or quadratically in the short-time regime if
        the quench is finished instantaneously or in a finite time, respectively.
        Notably, we find that for both
        the sudden quench and the finite-time quench,
        a topological phase transition in the pre-quench Hamiltonian will be manifested by the presence
        of nonanalyticity in the first-order or second-order derivative of circuit complexity with respect to
        time in the short-time regime, and a topological phase transition in the post-quench Hamiltonian will be manifested by the presence of nonanalyticity in the steady value of circuit complexity in the long-time regime.
        We also show that the increase of dimension does not remove, but only weakens the nonanalyticity
        of circuit complexity.  Our findings can be tested in quantum simulators and cold-atom systems.
        
        Nonanalyticity of circuit complexity across topological phase transitions
            Zhongbo Yan
            December 30, 2023
        =========================================================================
        
        
        
        § I. INTRODUCTION
        
        
        In the development of physics, concepts from one field sometimes are
        able to revolutionize the understanding of other fields. Recently, the
        complexity, which was originally developed in quantum information science
        to characterize how difficult to prepare one target state from certain reference state<cit.>, has
        been brought into the fields of holography and black hole physics<cit.>. Among various progresses,
        the two conjectures, namely “complexity equals volume”<cit.>  and “complexity equals action”<cit.>,
        have attracted particular attention and triggered active investigation of complexity in these fields<cit.>,
        hopefully producing new insights in understanding quantum gravity.
        
        In the study of complexity, its quantification is a central topic<cit.>.
        According to the original quantum-circuit definition,
        the complexity (in this context, it is usually dubbed
        circuit complexity) corresponds
        to the minimum number of elementary gates required to realize a
        unitary operator U which transfers the reference state |ψ_R⟩
        to the target state |ψ_T⟩, i.e., |ψ_T⟩=U|ψ_R⟩.
        As the choice of elementary gates itself has a lot of freedom, the quantification based on
        this principle is apparently  not an easy task. A breakthrough was made by Nielsen and collaborators
        who provided a  geometric interpretation to the circuit complexity<cit.>.
        Concretely, as a desired unitary operator can be generated by some time-dependent Hamiltonian
        H(t), they impose a cost function F[H(t)] which defines a Riemannian geometry
        on the space of unitary operations, then the circuit complexity is shown to
        correspond to the minimal geodesic length of the Riemannian
        geometry.
        
        The minimal-geodesic-length quantification makes the circuit complexity
        be a geometric quantity.  In contemporary physics, another geometric quantity
        of great interest is  the so-called topological invariant which mathematically  characterizes the global
        geometric property of a closed manifold. Over the past decades, this concept
        has been demonstrated to play a fundamental role in characterizing new phases of matter
        in condensed matter physics<cit.>. As the topological invariant of a phase
        is defined in terms of the wave function of ground state or the underlying Hamiltonian, a change of topological
        invariant (or say topological phase transition) thus indicates a dramatic change of the geometry of
        the manifold defined by the wave function of ground state or the Hamiltonian. Therefore, it is quite natural
        to expect that a topological phase transition can also be manifested through the circuit complexity if the reference and target states
        correspond to two distinct ground states. Very recently, Liu  et al.<cit.> did find that
        both equilibrium and dynamical topological phase transitions in the one-dimensional Kitaev model<cit.> can be revealed through
        the circuit complexity. Concretely, they found that by choosing the ground states of the Kitaev model as the reference
        and target states, the circuit complexity exhibits nonanalytic behavior at the critical points where the target ground state undergoes a
        dramatic change in topology.
        In addition, when the Kitaev model undergoes a sudden quench, they found that
        by choosing the pre-quench ground state as the reference state and
        the post-quench unitary evolution state as the target state, the circuit complexity first increases and then saturates,
        with the steady value displaying nonanalyticity when the Kiatev model is quenched across a critical point.
        
        As the cost function itself has some arbitrariness, the quantification of circuit complexity is
        not unique. It is therefore worthy to find out that whether the nonanalyticity of
        circuit complexity is preserved or not when a different quantification is adopted. Given this,
        in this work we consider a cost function distinct from Ref.<cit.> to
        quantify the circuit complexity. For generality, we further consider two
        topological models with  richer phase diagrams. As dimension is known to have strong impact on
        both classical and quantum phase transitions, here we consider that the two topological models take different dimensions
        to reveal its impact on the circuit complexity.
        To reveal whether the presence of
        nonanalyticity in the circuit complexity is generic or not when the system is quenched
        out of equilibrium, both sudden quench and  finite-time quench are investigated.
        Our main findings can be summarized as follows: (i) while
        we adopt a distinct quantification, the presence of nonanalyticity in circuit complexity across
        topological phase transitions holds.
        (ii) We find that the long-time behavior of the post-quench circuit complexity is not sensitive
        to the way of quench. Similarly to Ref.<cit.>, we find that for
        both the sudden quench and the finite-time quench,
        the steady value of circuit complexity based on our quantification in the long-time regime
        can reveal the occurrence of topological phase transitions in the post-quench Hamiltonian
        by the presence of nonanalyticity.
        (iii) The short-time growth behavior of the post-quench circuit complexity, however,
        is found to depend on the way of quench. It grows linearly/quadratically with time for
        the sudden/finite-time quench in the short-time regime. Notably, we find that
        the first-order (for sudden quench) or second-order (for finite-time quench)
        derivative of circuit complexity with respect to time in this regime
        can reveal the occurrence of topological phase transitions in the pre-quench Hamiltonian
        by the presence of nonanalyticity. It is noteworthy that unlike
        the steady value, the determination of the short-time growth behavior can be done
        very quickly since it requires
        very little information in the short-time regime,
        therefore, we believe that this finding
        is particularly useful for the experimental study of topological
        phase transitions. (iv) We find that the increase of dimension does not remove,
        but only weakens the nonanalyticity of circuit complexity, demonstrating the generality
        of the underlying physics. As in recent years equilibrium and dynamical
        topological phase transitions are of great interest both in theory<cit.> and in experiment<cit.>, our findings may shed new light on
        this active field.
        
        
        
        The paper is organized as follows. In Sec.II, we introduce our quantification of circuit complexity and
        show that across various topological phase transitions in one dimension, the circuit complexity always displays nonanalyticity
        at the critical points. In Sec.III, we consider that the one-dimensional generalized Kitaev model
        is suddenly quenched. We study the evolution of circuit complexity after the sudden quench and show
        that the short-time and long-time behavior of the post-quench circuit complexity
        can respectively reveal the occurrence of topological phase transitions in the pre-quench
        and post-quench Hamiltonian. In Sec.IV, we consider that the quench process is finished in
        a finite time and show that the conclusions for the sudden quench are preserved.
        In Sec.V, we generalize the study to two dimensions and discuss the impact of dimension on the
        circuit complexity. We conclude in Sec.VI. Some calculation details are relegated to appendices.
        
        
        
        
        
        
        § II. CIRCUIT COMPLEXITY AND TOPOLOGICAL PHASE TRANSITIONS IN ONE DIMENSION
        
        
        We start with a generalized Kitaev model which takes the form
        H=1/2∑_kψ_k^†H(k)ψ_k with
        ψ_k=(c_k,c_-k^†)^T and
        
            H(k)   =   (-t_1cos k-t_2cos 2k-μ)τ_z+(Δ_1sin k+Δ_2sin 2k)τ_y
               ≡    d_z(k)τ_z+d_y(k)τ_y,
        
        where τ_x,y,z are Pauli matrices in particle-hole space,  t_1(Δ_1) and t_2(Δ_2) represents the nearest-neighbor and next-nearest-neighbor hopping (pairing) amplitude, respectively, and
        μ is the chemical potential. For convenience, the lattice constant is set to unit throughout this work.
        
        Owing to the presence of chiral symmetry, i.e., {τ_x,H(k)}=0, this model belongs to
        the class BDI<cit.> and is known to be characterized by the winding number defined as<cit.>
            ν=i/4π∫_-π^πdk Tr[τ_xH^-1(k)∂_kH(k)].
        
        According to the above formula, we present the phase diagram corresponding to
        Δ_1=Δ_2≠0 in Fig.<ref>(a). Noteworthily, as a
        topological phase transition is associated with the closure of bulk energy gap,
        the phase boundaries in the phase diagram can be easily determined, which
        are found to be the lines satisfying μ=t_1-t_2, μ=-t_1-t_2 and μ=(t_1+t_2)/2.
        In comparison to the standard Kitaev model which only involves
        the nearest-neighbor hopping and pairing <cit.>, it is readily seen that
        the presence of additional next-nearest-neighbor hopping and pairing leads
        to a richer phase diagram, and therefore more topological phase transitions
        can be investigated to demonstrate the generality of the presence of nonanalyticity
        in circuit complexity.
        
        
        The generalized Kitaev model above describes a spinless superconductor.
        By following the  standard Bogoliubov transformation, the model can be diagonalized
        and accordingly the ground-state wave function can be obtained, which reads<cit.>
            |Ω⟩=∏_k>0|ψ_k⟩
            =∏_k>0(cos(θ_k/2)+isin(θ_k/2)c_k^†c_-k^†)|0⟩,
        
        where θ_k=arctan (d_y(k)/d_z(k)). As here the momentum is a good quantum number,
        we can treat each k independently<cit.>. For each k, one can see that the state is a superposition
        of |0⟩ and c_k^†c_-k^†|0⟩, and the superposition
        is characterized by a single parameter θ_k/2 (noteworthily, θ_k/2 and
        θ_k/2+π are equivalent as the π difference only results in a global phase
        difference to the wave function).
        Taking the ground state with θ_k^R/2 and θ_k^T/2  as the reference
        state |ψ_k^R⟩ and target state |ψ_k^T⟩, respectively,  then the effect of
        unitary operator is equivalent to doing a transport from the point θ_k^R/2 to the point θ_k^T/2
        on the unit circle. Apparently,  on the circle the length of the arc connecting θ_k^R/2 and θ_k^T/2
        provides a natural measure of the distance between the two states. Owing to the equivalence
        between θ_k/2 and θ_k/2+π, the minimal length of the arc is in fact given by
        arccos|⟨ψ_k^T|ψ_k^R⟩|, where |⟨ψ_k^T|ψ_k^R⟩|
        is known as fidelity. In fact, such a quantification  corresponds to
        the well-known quantification of circuit complexity in terms of
        inner-product metric<cit.>.
        Throughout this work, we adopt this quantification
        which has a simple and clear geometric interpretation.
        
        
        
        
        Accordingly, if we start with a ground state |Ω^R⟩ and adiabatically transfer it to
        another ground state |Ω^T⟩,
        then the corresponding circuit complexity is
        
            C=∑_k>0arccos|⟨ψ_k^T|ψ_k^R⟩|
            =∑_k>0arccos|cos(Δθ_k/2)|,
        
        where Δθ_k=θ_k^T-θ_k^R. For each k, one can see that the maximum
        value is π/2, which corresponds to that |ψ_k^R⟩ and | ψ_k^T⟩ are orthogonal.
        For comparison, the formula for circuit complexity in Ref.<cit.> takes the form C=∑_k(Δθ_k/2)^2.
        While in essence the two formulas only differ by a square-root operation, in the following one will see
        that remarkable differences will arise in the behaviors of circuit complexity.  To simplify the analysis of circuit complexity,
        below we will restrict ourselves to ground state evolutions
        corresponding to the variation of only one parameter of the model. Concretely, we
        will fix (t_1, Δ_1,Δ_2) and only vary either μ or t_2.
        
        
        
        
        
        In Fig.<ref>(b), we present the circuit complexity
        associated with the variation of μ. The result clearly demonstrates the presence of nonanalyticity
        in the C-μ^T curve. The nonanalytic behavior becomes even more apparent by
        performing a first-order derivative, i.e., dC/dμ^T.
        As shown in  Fig.<ref>(c), divergence appears exactly at the critical
        points of topological phase transitions. Furthermore, it is readily found that in Fig.<ref>(c)
        dC/dμ^T does not depend on μ^R, that is, the first-order derivative of
        circuit complexity does not depend on the choice of reference state. Noteworthily, such an independence
        of reference state is absent in Ref.<cit.>, indicating that
        different quantifications  may lead to remarkable different behaviors in the circuit complexity.
        
        Let us now give a simple explanation of the presence of nonanalyticity. For this generalized
        Kitaev model, its topological invariant, the winding number, characterizes the number of times that the
        vector ()=(d_y(),d_z()) winds around the origin when k goes from -π to π.
        When a topological phase transition occurs, the number of times that the vector ()
        winds around the origin has a discrete change. Accordingly, the angle θ_k, which characterizes
        the orientation of the vector (),
        must jump at some k. Apparently, Δθ_k for these k also jump at the critical points.
        Then according to Eq.(<ref>), it is readily seen
        that the circuit complexity will be nonanalytic at the critical points.
        
        
        
        
        Before proceeding, here we define a  quantity,
        
            γ(λ)≡∑_k>0lim_δλ0^+arccos|⟨ψ_k(λ+δλ)|ψ_k(λ)⟩|/δλ,
        
        where λ denotes some  parameter of the Hamiltonian. The physical meaning
        of this quantity is quite obvious. It  characterizes the growth rate of circuit complexity
        when a state is adiabatically transferred to its nearby state.  For the sake of discussion
        convenience, we name
        it  adiabatical growth rate. According to Eq.(<ref>), a short calculation reveals<cit.>
            γ(λ)=∑_k>01/2|dθ_k/dλ|.
        
        As mentioned above, when a topological phase transition occurs, Δθ_k
        will dramatically change for some k. Therefore, it is quite obvious that γ(λ)
        is also nonanalytic at the critical points. Interestingly,  we find that |dC/dμ^T| and γ(μ)
        coincide with each other, as shown in Fig.<ref>(c). The divergence of γ(μ) at
        the critical points indicates
        that the adiabatical growth rate goes
        divergent when the state gets close to a critical point. In other words,
        when the reference state gets more close to a critical point, it becomes
        more difficult to prepare the target state in an adiabatical way.
        Noteworthily, the overlap of the adiabatical growth rate and the first-order derivative  is not accidental. As shown in
        Fig.<ref>(d), it also appears when we fix μ and vary t_2. This overlap is apparently related
        to the result that the first-order derivative of circuit complexity is independent of the reference state, indicating that
        this property is tied to our quantification.
        
        
        
        § III. SUDDEN QUENCH AND CIRCUIT COMPLEXITY EVOLUTION
        
        
        In equilibrium, as the Hamiltonian
        and the ground state are tied to each other, the topological invariants defined in terms of
        them are equivalent. When out of equilibrium,
        however, as the underlying instantaneous wave function in general does not correspond to the ground state of
        the instantaneous Hamiltonian, the topological invariants defined in terms of instantaneous Hamiltonian (labeled as ν_H) and
        instantaneous wave function (labeled as ν_w) are not guaranteed to be equivalent. In particular, when a system is isolated
        from the environment, the wave function will follow unitary evolution, and so ν_w will keep its value
        no matter how  ν_H changes<cit.>.
        Therefore, for an isolated system out of equilibrium, a topological phase transition
        can only be defined as a change of ν_H. As out of equilibrium, transitions associated
        with a change of ν_H are usually dubbed dynamical topological phase transitions.
        In the following, we focus on such isolated systems and first
        investigate the evolution of
        circuit complexity after a sudden quench<cit.>.
        
        For concreteness, we consider that for t<0, the system is described by H_i and stays at its ground state |Ω^i⟩.
        At t=0, the  Hamiltonian is suddenly quenched to H_f, and afterwards it keeps as H_f. Accordingly, the wave function
        at t>0 is given by |Ω(t)⟩=e^-iH_ft|Ω^i⟩. For each k, we have
        |ψ_k(t)⟩=e^-iH_f(k)t|ψ_k^i⟩. Thus, the post-quench circuit complexity
        is given by C(t)=∑_k>0arccos|⟨ψ_k(t)|ψ_k^i⟩|.
        A short calculation reveals<cit.>
            C(t)=∑_k>0arccos√(1-sin^2Δθ̃_ksin^2 (E_f(k)t)),
        
        where Δθ̃=θ_k^f-θ_k^i with θ_k^f(i)=arctan (d_y^f(i)(k)/d_z^f(i)(k)),
        and E_f(k)=√((d_y^f(k))^2+(d_z^f(k))^2).
        
        
        
        
        By fixing μ_i, we show the evolution of post-quench circuit complexity for a series of μ_f in Fig.<ref>(a). One can see
         that after the quench, the circuit complexity will first grow linearly with time (see the inset)
        and then saturate  with some degree of oscillations. While adopting  a different quantification,
        we find that similarly to Ref.<cit.>, here the steady value in the long-time regime also
        exhibits nonanalyticity at the critical points where the topology of the post-quench Hamiltonian
        H_f undergoes a change, as shown in Fig.<ref>(b).
        Noteworthily, as oscillations always appear, we take the average of C(t) over a sufficiently long time as
        the steady value. Throughout this work we take C_s=(∫_t_i^t_fC(t)dt)/(t_f-t_i), with
        t_i=50 and t_f=130.
        
        
        From above it is readily seen that obtaining the steady value requires us to know the evolution of C(t) for quite a long time,
        therefore, using the steady value to detect the occurrence of
        topological phase transitions is not very efficient.
        Surprisingly,  we find that the growth rate of post-quench circuit complexity
        within the linear growth regime in fact can also reveal the occurrence of topological phase transitions.
        In order to distinguish from the equilibrium case,
        here we name it  dynamical growth rate.  As the circuit complexity grows linearly right after the quench,
        the dynamical growth rate is thus simply given by
        
            χ≡lim_δ t 0^+C(δ t)/δ t=∑_k>0|sinΔθ̃_k|E_f(k),
        
        or equivalently, χ≡∂_tC|_t0^+. One can see that for each k, the dynamical growth rate is proportional to the energy. If starting from different pre-quench Hamiltonian H_i
        and quenching to the same H_f (e.g.,  H_f can be chosen to describe a trivial phase
        for which hopping and pairing are turned off, i.e., H_f=-μτ_z),
        then as Δθ̃_k  displays
        distinct winding behavior for H_i with distinct v_H, χ will exhibit
        nonanalyticity at the critical points where
        the topology of the pre-quench H_i undergoes a change.
        In Fig.<ref>(c), we present the result for the case with a fixed μ^f.
        It is readily seen that nonanalyticity does appear at every critical points.
        
        According to Eq.(<ref>), it is quite obvious that the determination of  dynamical growth rate requires very little information
        right after the quench, therefore using it to reveal the occurrence of
        topological phase transitions is
        much more convenient and accurate than
        using the steady value suggested in Ref.<cit.>.
        As a final remark of this section, we point out that the linear growth behavior right after
        the quench is also specific to our quantification. For comparison, in Ref.<cit.>
        the circuit complexity does not grow with time in a linear way
        right after the quench. For their quantification, the slope of the circuit complexity
        in the extremely short-time regime is time-dependent, and therefore it is hard to give a sharp definition
        of dynamical growth rate as here, suggesting that a good choice of
        the quantification may benefit its application.
        
        
        
        
        § IV. FINITE-TIME QUENCH AND CIRCUIT COMPLEXITY EVOLUTION
        
        
        For the sudden quench, the
        change from the initial Hamiltonian to the final
        Hamiltonian is completed instantaneously. In this section
        we consider finite-time quench for which one of the parameters
        in the Hamiltonian changes from
        its initial value at t≤0 to the final value
        with a finite speed.
        To be specific, we focus on the chemical potential and take
        μ(t)=μ^i+vtΘ(t_q-t)
        +(μ^f-μ^i)Θ(t-t_q), where
        t_q=|μ^f-μ^i|/|v| is the quench duration time 
        and Θ(x) is
        the step function with Θ(x)=1/0 for x>/<0.
        This expression means that the chemical potential
        is quenched from μ^i to μ^f with a
        finite speed v. When v goes to infinity,
        it returns to the sudden quench addressed in
        the previous section.
        
        
        As the Hamiltonian changes with time in the domain
        0<t<t_q, the circuit complexity in this domain
        is given by
        
            C(t)=∑_k>0arccos |⟨ψ_k^i |Te^-i∫_0^tH(k,t')dt'|ψ_k^i⟩|,
        
        where T denotes the time ordering operator. When t>t_q,
        
            C(t)=∑_k>0arccos |⟨ψ_k^i |e^-iH_f(k)(t-t_f)Te^-i∫_0^t_fH(k,t')dt'|ψ_k^i⟩|.
        
        
        By fixing μ^i and t_q, we show the evolution of circuit complexity
        for a series of μ^f in Fig.<ref>(a). Similarly to the sudden-quench
        case, we find that the circuit complexity will saturate with some degree of oscillations
        in the long-time regime (not shown explicitly) and its steady value C_s
        also displays nonanalyticity at the critical points where the topology of the post-quench Hamiltonian
        undergoes a change,  as shown in Fig.<ref>(b).
        The similarity in the long-time regime between the sudden quench and the finite-time quench
        is naturally expected since after t_q the dynamics of the whole system
        is governed by the same time-independent Hamiltonian H_f for both cases.
        When looking at the short-time regime in Fig.<ref>(a),
        we find that the circuit complexity does not grow linearly
        as in the sudden-quench case. Instead, it grows quadratically with time
        right after the start of the quench. To better show the quadratical growth,
        we present the evolution of ∂_t C in Fig.<ref>(c).
        It is readily seen that right after the start of the quench, ∂_t C
        grows linearly with time like the circuit complexity after a sudden quench.
        From Fig.<ref>(c), one can also find that while ∂_t C
        depends on the quench speed, its linear growth behavior
        is robust against the variation of the quench speed. The linear growth behavior
        of ∂_t C leads us to expect that ∂_t^2 C,
        which can be understood as the acceleration of the circuit complexity,
        will display nonanalyticity at the critical points where the topology of the pre-quench
        Hamiltonian undergoes a change.  The numerical results confirm
        the above expectation, as shown in Fig.<ref>(d).  Therefore,
        for the finite-time quench, the short-time behavior
        and the long-time behavior of the circuit complexity can also respectively
        reveal the occurrence of topological phase transitions in the pre-quench Hamiltonian
        and in the post-quench Hamiltonian.
        
        
        
        
        
        
        
        
        
        § V. CIRCUIT COMPLEXITY AND TOPOLOGICAL PHASE TRANSITIONS IN TWO DIMENSIONS
        
        
        Since a topological phase transition is always associated with a dramatic change
        of the global geometric property of the underlying ground state wave function or
        Hamiltonian, it is natural to expect that the circuit complexity will also exhibit
        nonanalyticity in higher dimensions. To demonstrate this explicitly, here we consider
        a paradigmatic “·τ” model in two dimensions for concreteness.
        The model reads H=1/2∑_ψ_^†H()ψ_, with H()=()·τ
        and <cit.>
            d_x()   =   sin k_x,    d_y=-sin k_y,
            
            d_z()   =   t_1(cos k_x + cos k_y) + t_2cos k_xcos k_y-μ,
        
        where the d_x and d_y terms together describe a chiral p-wave pairing,
        and the d_z term is the kinetic energy of normal state. As for this Hamiltonian only particle-hole
        symmetry is preserved, it belongs to the class D that in two dimensions is characterized by
        the first-class Chern number,
        
            n=1/4π∫_BZ d^2k ()·(∂_k_x()×∂_k_y()),
        
        where =(d_x,d_y,d_z)/√(d_x^2+d_y^2+d_z^2).
        The geometric meaning of this formula also has a winding interpretation:
        it counts the number of times that the three-component vector
        () winds around the origin when the momentum transverses the whole first Brillouin
        zone. It indicates that when a topological phase transition takes place, the orientation of the vector
        () will also change dramatically at some momenta.
        By using the above formula, we present the phase diagram in Fig.<ref>(a).
        
        
        
        
        Following the steps in one dimension,
        we find that for an adiabatical evolution of ground states, the circuit complexity is
        
            C=∑_k_x>0,k_yarccos|cos(Δθ_k/2)|,
        
        where Δθ_k takes the same definition as in Eq.(<ref>), but here
        θ_k^T(R)=arctan (√((d_x^T(R)())^2+(d_y^T(R)())^2)/d_z^T(R)()).
         As for this model all topological
        phase transitions occur at the time-reversal invariant momenta,
        including (k_x,k_y)=(0,0), (0,π), (π,0), and (π,π), at which
        d_x() and d_y() vanish identically, it is readily seen
        that across a critical point, the orientation of the vector ()  will
        reverse its direction at the respective momenta. Then according to Eq.(<ref>),
        it is clear that nonanalyticity will also be present in the circuit complexity.
        
        Similarly to the one-dimensional model, in the following we also consider that
        only μ and t_2 are variables. In Fig.<ref>(b), we fix t_2 and present several C-μ^T curves corresponding to different μ^R. In comparison to Fig.<ref>(b), it is readily seen that the increase of dimension enhances the smoothness of the C-μ^T curve.
        Nevertheless, it is as expected that the circuit complexity also exhibits  nonanalyticity at all critical points, and
        this feature becomes quite obvious  after performing a first-order derivative, as
        shown in Fig.<ref>(c), where one can see that the overlap of |dC/dμ^T| and γ(μ) also holds.
        If fixing μ and varying t_2, the results also demonstrate the presence of nonanalyticity at the critical points and the overlap of
        |dC/dt_2^T| and γ(t_2), as shown in Fig.<ref>(d).
        
        
        The evolution of  circuit complexity after a sudden quench is presented in Fig.<ref>(a).
        It is readily seen that the long-time and short-time behaviors are quite
        similar to those in one dimension. Due to this similarity, here we will not discuss
        the more complicated finite-time quench.
        From Fig.<ref>(b)(c),  one can find that owing to the increase of dimension,
        the C_s-μ^f curve, as well as
        the χ-μ^i curve both become somewhat smoother. Nevertheless, the nonanalyticity
        at the critical points holds and can be revealed by performing a first-order derivative (see the inset in Fig.<ref>(b)).
        
        Through Fig.<ref> and Fig.<ref>, we demonstrate that the circuit complexity can also reveal
         the occurrence of  topological phase transitions in higher dimensions. However, as the increase of dimension is shown to weaken the nonanalyticity,
        it indicates that the application of this quantity to detect higher dimensional topological phase transitions
        requires a higher precision of measurements.
        
        
        
        
        § VI. CONCLUSIONS
        
        
        In this work, we have adopted a simple quantification of the circuit complexity distinct from Ref.<cit.> and
        demonstrated that it can reveal the occurrence of topological phase transitions by
        the presence of nonanalyticity. By studying the evolution of circuit complexity after a
        quench, either sudden or finite-time, we find that the
        growth behavior in the short-time regime and
        the steady value in the long-time regime can respectively
        reveal the occurrence of
        topological phase transitions in the pre-quench Hamiltonian
        and in the post-quench Hamiltonian by the presence of nonanalyticity.
        Since the growth behavior in the short-time regime can be determined by very
         few measurements, its nonanalytic behavior is expected to be
        very useful for the experimental
        study of topological phase transitions.
        Furthermore, we have investigated the effect of dimension and shown that the increase of dimension does not remove, but only weakens the nonanalyticity.
        As the circuit complexity based on our quantification can be determined by measuring the fidelity, the test of our predictions
        is well with the current experimental accessibility. For instance,
        the two-band models studied can be simulated in a superconducting qubit, and
        the evolution of states can be measured by performing  quantum state tomography<cit.>.
        In light of the active study of equilibrium and
        dynamical topological phase transitions
        in quantum simulators and cold-atom systems<cit.>, our predictions can also be tested in these platforms,
        hopefully providing some new insights into understanding  various phase transitions.
        
        
        
        
        § ACKNOWLEGEMENTS
        
        
         We would like to acknowledge helpful discussions with Chengfeng Cai.
        Z.X. and D.X.Y. are supported by NKRDPC Grants No. 2017YFA0206203, No. 2018YFA0306001, NSFC-11574404, NSFG-2015A030313176, National Supercomputer Center in Guangzhou, and Leading Talent Program of Guangdong Special Projects. Z.Y. would
        like to acknowledge the support
        by Startup Grant (No. 74130-18841219) and NSFC-11904417.
        
        
        
        
        
         §.§ Appendix A: Derivation of
         the circuit complexity under the inner-product metric
        
        
        As mentioned in the main text, |ψ_k⟩ is a superposition of
        |0⟩ and c_k^†c_-k^†|0⟩. Therefore, under
        the basis (c_k^†c_-k^†|0⟩,|0⟩), |ψ_k⟩
        can be written as a two-component spinor,
        
            |ψ_k⟩=(
                               [ isin(θ_k/2);  cos(θ_k/2);             ]).
        
        A two-component spinor can be mapped to a three-component vector in terms
        of the Pauli matrices,
        
            ψ_k=⟨ψ_k|σ|ψ_k⟩=-(0,sinθ_k,cosθ_k)^T.
        
        Let us consider that the target state is infinitely close to the reference state, i.e.,
        |ψ^R⟩=|ψ_k⟩, and |ψ^T⟩=|ψ_k+dψ_k⟩,
        or equivalently, ψ^R=ψ_k, and ψ^T=ψ_k+dψ_k.
        The normalization requires ψ_k· dψ_k=0.
        As the two states are infinitely close to each other, to leading order, the unitary operator which transforms the reference state to
        the target state can be written as
        
            U=𝐈+i𝐅(ψ_k)·σ+...
        
        where 𝐅(ψ_k) is an infinitely small parameter to be determined.
        As |ψ^T⟩=U|ψ^R⟩ and ψ^T=ψ_k+dψ_k=⟨ψ^T|σ|ψ^T⟩, it is readily found that to leading order of 𝐅(ψ_k).
        
            dψ_k   =   i⟨ψ_k|(σ𝐅(ψ_k)·σ
            -𝐅(ψ_k)·σσ)|ψ_k⟩
               =   -2⟨ψ_k|𝐅(ψ_k)×σ|ψ_k⟩.
        
        A solution of the above equation is
        
            𝐅(ψ_k)=1/2(dψ_k×ψ_k+αψ_k),
        
        where α denotes  an infinitely small constant. Therefore,
        to leading order, the unitary operator for an infinitely small change
        of the state is given by
        
            U=𝐈+i/2(dψ_k×ψ_k+αψ_k)·σ.
        
        
        Ignoring global phase, U belongs to the SU(2) group. The standard inner-product metric on SU(2) is<cit.>
            ds^2   =   1/2Tr[dU^†dU]
               =   (dψ_k×ψ_k+αψ_k)
            ·(dψ_k×ψ_k+αψ_k)/4
               =   dψ_k· dψ_k+α^2/4,
        
        which is minimized when α=0. Accordingly, the inner-product metric is
        ds^2=dψ_k· dψ_k/4=dθ_k^2/4.
        Therefore, the circuit complexity, which corresponds to the geodesic length, is given by
        
            ds=|dθ_k|/2=arccos|⟨ψ_k+d ψ_k|ψ_k⟩ |.
        
        Above we have used that
        
            |ψ_k+dψ_k⟩=(
                               [ isin((θ_k+dθ_k)/2);  cos((θ_k+dθ_k)/2);                    ]).
        
        Performing a summation over k, we reach the expression of circuit complexity in Eq.(<ref>) of
        the main text.
        
        
        
         §.§ Appendix B: The wave function of ground state
        
        
        We start with the Hamiltonian
        H=1/2∑_kψ_k^†H(k)ψ_k with
        ψ_k=(c_k,c_-k^†)^T and
        
            H(k)   =   (-t_1cos k-t_2cos 2k-μ)τ_z+(Δ_1sin k+Δ_2sin 2k)τ_y
               ≡    d_z(k)τ_z+d_y(k)τ_y.
        
        By performing a standard Bogoliubov transformation, i.e.,
        
            c_k   =   cosθ_k/2α_k+isinθ_k/2α_-k^†,
            
            c_-k^†   =   isinθ_k/2α_k+cosθ_k/2α_-k^†,
        
        where θ_k=arctan (d_y(k)/d_z(k)), the Hamiltonian
        will be diagonalized as
        
            H   =   1/2∑_kE_k(α_k^†α_k-α_-kα_-k^†)
               =   ∑_k>0E_k(α_k^†α_k+α_-k^†α_-k)+...,
        
        where “...” stands for some unimportant constant, and E_k=√(d_y^2(k)+d_z^2(k)).
        The ground state satisfies α_k|Ω⟩=0 for arbitrary k. One can find that
        the solution is
        
            |Ω⟩=∏_k>0|ψ_k⟩=∏_k>0
            (cosθ_k/2+isinθ_k/2c_k^†c_-k^†)|0⟩,
        
        where |0⟩ represents the vacuum, i.e., c_k|0⟩=0. As k is a good quantum number,
        we can treat each k independently. For each k, |ψ_k⟩ is a superposition of
        |0⟩ and c_k^†c_-k^†|0⟩. Let us focus on a specific k and consider
        that the reference state is given by |ψ_k^R⟩=(cosθ_k^R/2
        +isinθ_k^R/2c_k^†c_-k^†)|0⟩, and the target state
        is given by |ψ_k^T⟩=(cosθ_k^T/2
        +isinθ_k^T/2c_k^†c_-k^†)|0⟩. The inner product of
        the two states, which is also known as fidelity, is given by
        
            |⟨ψ_k^T | ψ_k^R⟩|   =   |cos(θ_k^T/2)cos(θ_k^R/2)+sin(θ_k^T/2)sin(θ_k^R/2)|
               =   |cos((θ_k^T-θ_k^R)/2)|
               ≡   |cos(Δθ_k/2)|.
        
        
         §.§ Appendix C: The growth rate of adiabatic circuit complexity
        
        
        Here we give a derivation of Eq.(<ref>) in the main text. Consider a ground state
        |Ω^R⟩=∏_k>0|ψ_k(λ)⟩ as the reference state,
        where λ refers to the parameter to vary, and consider its near neighbour
        |Ω^T⟩=∏_k>0|ψ_k(λ+δλ)⟩ as the
        target state, then according to Eq.(<ref>) in the main text, we have
        
            δ C(λ)=∑_k>0arccos|⟨ψ_k(λ+δλ)|ψ_k(λ)⟩ |.
        
        To characterize the growth rate of circuit complexity when  λ is adiabatically varied away
        from its reference state's value, a natural definition of the growth rate is
        
            γ(λ)≡lim_δλ→ 0^+δ C(λ)/δλ=∑_k>0lim_δλ→ 0^+arccos|⟨ψ_k(λ+δλ)|ψ_k(λ)⟩ |/δλ.
        
        As δλ is infinitely small, it's justified to make a Taylor expansion,
        
            ⟨ψ_k(λ+δλ)|ψ_k(λ)⟩   =   ⟨ψ_k(λ)|ψ_k(λ)⟩+⟨dψ_k(λ)/dλ|ψ_k(λ)⟩δλ
                  +1/2⟨d^2ψ_k(λ)/dλ^2|ψ_k(λ)⟩(δλ)^2+....
        
        According to the expression of |ψ_k(λ)⟩, one can easily find that the linear-order term vanishes.
        Therefore, up to second-order of δλ,
        we have
        
            |⟨ψ_k(λ+δλ)|ψ_k(λ)⟩|=1-1/8(dθ_k/dλ)^2(δλ)^2.
        
        Accordingly, we have
        
            δ C(λ)   =   ∑_k>0arccos |⟨ψ_k(λ+δλ)|ψ_k(λ)⟩|
               =   ∑_k>0arccos(1-1/8(dθ_k(λ)/dλ)^2(δλ)^2)
               =   ∑_k>0√(2)√(1/8(dθ_k(λ)/dλ)^2(δλ)^2)+...
               =   ∑_k>01/2|dθ_k(λ)/dλ||δλ|,
        
        and
        
            γ(λ)   =   lim_δλ→ 0^+δ C(λ)/δλ
               =   ∑_k>01/2|dθ_k(λ)/dλ|.
        
        To see that γ(λ) is nonanalytic at critical points, we take a special case of the model in Eq.(<ref>) for illustration.
        Concretely, we take t_2=Δ_2=0 and t_1=Δ_1=1. Accordingly, we have
        
            H(k)   =   (-cos k-μ)τ_z+sin kτ_y.
        
        For this Hamiltonian,  topological phase transitions  take place at
        μ=±1. As  only μ is a variable, we let λ=μ, then we have
        
            γ(μ)   =   ∑_k>01/2|dθ_k(μ)/dμ|
               =   1/2∑_k>0sin k/(cos k+μ)^2+sin^2k
               =   L/2∫_0^πdk/2πsin k/(cos k+μ)^2+sin^2k
               =   L/4π∫_-1^1dx/1+μ^2+2μ x
               =   L/4πμln|1+μ/1-μ|,
        
        where L denotes the system size. One can see that γ(μ) exhibits
        logarithmic divergence exactly at the two critical points μ=±1.
        
        According to Eq.(<ref>), one can understand that the presence of divergence in γ(μ) at a critical point
        is because θ_k will undergo a dramatic change across a topological phase transition. To see this intuitively, we go back to
        the more general model in Eq.(<ref>)  and  plot θ_k-k
        curves near the critical points. As shown in Fig.<ref>, θ_k will undergo a sudden jump at some momenta
        when the system goes across a topological phase transition. Such sudden jumps lead to the presence of divergence in γ(μ).
        
        
        
         §.§ Appendix D: Post-quench circuit complexity
        
        
        In the main text, we have considered a global quench of the Hamiltonian.
        Concretely,  we consider that before quench, the system is described by H_i and stays at its ground state |Ω^i⟩.
        At t=0, the underlying Hamiltonian is suddenly quenched to H_f, and afterwards it keeps as H_f. For an isolated system,
        the wave function follows a unitary evolution, i.e., |Ω(t)⟩=e^-iH_ft|Ω^i⟩ for t>0.
        For each k, we have |ψ_k(t)⟩=e^-iH_f(k)t|ψ_k^i⟩. Thus, the post-quench circuit complexity
        is given by C(t)=∑_k>0arccos|⟨ψ_k(t)|ψ_k^i⟩|.
        
        To obtain the concrete expression of C(t), we need to know |ψ_k(t). According
        to the unitary evolution,
        
            |ψ_k(t)⟩   =   e^-iH_f(k)t|ψ_k^i⟩
               =   e^iE_f(k)t⟨ψ_k,-^f|ψ_k^i⟩|ψ_k,-^f⟩
            +e^-iE_f(k)t⟨ψ_k,+^f|ψ_k^i⟩|ψ_k,+^f⟩,
        
        where |ψ_k,±^f⟩ are the eigenstates of H_f(k),
        with |ψ_k,+^f⟩ and |ψ_k,-^f⟩ corresponding to
        eigenenergy E_f(k) and -E_f(k), respectively. Here
        E_f(k)=√((d_y^f(k))^2+(d_z^f(k))^2).
        According to Eq.(<ref>), one has
        
            |ψ_k,-^f⟩   =   (cosθ_k^f/2+isinθ_k^f/2c_k^†c_-k^†)|0⟩,
            
            |ψ_k,+^f⟩   =   (isinθ_k^f/2+cosθ_k^f/2c_k^†c_-k^†)|0⟩,
        
        then a short calculation reveals
        
            ⟨ψ_k(t)|ψ_k^i⟩   =   e^-iE_f(k)t⟨ψ_k,-^f|ψ_k^i⟩⟨ψ_k^i| ψ_k,-^f⟩
                  +e^iE_f(k)t⟨ψ_k,+^f|ψ_k^i⟩⟨ψ_k^i| ψ_k,+^f⟩
               =   e^-iE_f(k)tcos^2Δθ_k/2+
            e^iE_f(k)tsin^2Δθ_k/2,
        
        where Δθ_k=θ_k^f-θ_k^i. Therefore,
        
            |⟨ψ_k(t)|ψ_k^i⟩|   =   √(cos^2(E_f(k)t)+cos^2Δθ_ksin^2(E_f(k)t))
               =   √(1-sin^2Δθ_ksin^2(E_f(k)t)).
        
        Accordingly, we obtain the expression for post-quench circuit complexity,
        
            C(t)=∑_k>0arccos√(1-sin^2Δθ_ksin^2 (E_f(k)t)).
        
        
         §.§ Appendix E: Growth rate of post-quench circuit complexity
        
        
        In the main text, we have shown that right after the quench, the post-quench circuit complexity grows
        linearly with time. As the slope of a line can be determined by any two points on the line, thus the growth rate
        of post-quench circuit complexity can be defined as
        
            χ≡lim_δ t→ 0^+C(δ t)-C(0)/δ t=lim_δ t→ 0^+C(δ t)/δ t.
        
        Above we have used  C(0)=0. According to Eq.(<ref>), we have
        
            χ   =   ∑_k>0lim_δ t→ 0^+arccos√(1-sin^2Δθ_ksin^2 (E_f(k)δ t))/δ t
               =   ∑_k>0lim_δ t→ 0^+arccos(1-1/2sin^2Δθ_k(E_f(k)δ t)^2
            +𝒪((δ t)^3))/δ t
               =   ∑_k>0lim_δ t→ 0^+√(2)√(1/2sin^2Δθ_k(E_f(k)δ t)^2+𝒪((δ t)^3))/δ t
               =   ∑_k>0|sinΔθ_k|E_f(k).
        
        To see that χ is nonanalytic at the critical points, we again consider the special model in Eq.(<ref>) for illustration.
        Concretely, we consider
        
            H_i(k)   =   (-cos k-μ^i)τ_z+sin kτ_y,
            
            H_f(k)   =   -μ^fτ_z.
        
        Accordingly, we have
        
            χ   =   ∑_k>0|sinΔθ_k|E_f(k)
               =   ∑_k>0sin k/√((cos k+μ^i)^2+sin^2k)|μ^f|
               =   L|μ^f|/2π∫^π_0dksin k /√((cos k+μ^i)^2+sin^2k)
               =   L|μ^f|/2π∫_-1^1dx/√(1+(μ^i) ^2+2μ^i x)
               =   L|μ^f|/2π|1+μ^i|-|1-μ^i|/2μ^i,
               =   {[      L|μ^f|/2π,        |μ^i|≤1,; L|μ^f|/2π|μ^i|,        |μ^i|>1.;                 ].
        
        The nonanalyticity of χ at the two critical points μ^i=±1 is
        obvious.
