Topologically driven nonequilibrium phase transitions in diagonal ensembles

We identify a new class of topologically driven phase transitions when calculating the Hall conductance of two-band Chern insulators in the long-time limit after a global quench of the Hamiltonian. The Hall conductance is expressed as the integral of the Berry curvature in the diagonal ensemble. Even if the topological invariant of the wave function is conserved under unitary evolution, the Hall conductance as a function of the energy gap in the post-quench Hamiltonian displays a continuous but nonanalytic behavior, that is it has a logarithmically divergent derivative as the gap closes. The coefficient of this logarithmic function is the ratio of the change of Chern number in the ground state of the post-quench Hamiltonian to the energy gap in the initial state. This nonanalytic behavior is universal in two-band Chern insulators.

Introduction.-Thediscovery of quantum Hall effects [1,2], i.e., a quantized Hall conductance in the ground state which jumps from one plateau to another inspired the study of topological order [3,4] to characterize different topological phases outside the conventional framework of spontaneous symmetry breaking.Since then considerable effort has been devoted to understanding topological order or symmetry protected topological order (SPT) in the ground state.More recently, the nature of topological order and SPT order for a state driven out of equilibrium has attracted attention, in particular for quantum quenches of the Hamiltonian [5][6][7][8][9][10][11][12][13].
Imagine an isolated system originally in the ground state of Ĥi and suddenly change the Hamiltonian to Ĥf .The wave function follows a unitary time evolution, while the local observables in the long time limit settle to the prediction of the diagonal ensemble [14], which in some cases can be reduced to a thermal ensemble or a generalized Gibbs ensemble [15,16].Since topological order and SPT order cannot be expressed as a local observable, its identification in a nonequilibrium state is far from trivial.In the toric code model, the topological entropy in the long time limit is found to be the same as its initial value under several families of Ĥf [5,6].The topological order is insensitive to Ĥf whether the ground states of Ĥi or Ĥf are in the topologically trivial or nontrivial phases [5][6][7], coinciding with a universal argument for gapped spin liquids [17].Similarly, in the Fermi gas on a honeycomb lattice which essentially simulates the Haldane model, the Chern number is proved to be conserved under unitary evolution [18].However, in the two-dimensional topological superfluid, the winding number of the retarded Green's function depends on Ĥf in the phase diagram [8,9], even if the winding of the Anderson pseudo spin texture is conserved [10].Also in the one-dimensional case, an analysis of tunneling spectroscopy by coupling the system to an auxiliary thermal bath shows that the SPT order is mostly determined by Ĥf [11].But in topological superconductors with proximity-induced superconductivity, the Majorana order parameter [12] or the entanglement spectrum [13] indicate that the quenched state is topologically trivial whenever Ĥi and Ĥf are in different topological phases.
To clarify the issue of SPT order far from equilibrium, we appeal to a measurable physical quantity, namely the Hall conductance in a paradigmatic model with nontrivial Chern numbers, i.e., the Dirac model [19].In the Dirac model SPT order explains the phase transition signaled by a jump of the Hall conductance.We find that (i) the Chern number of the unitarily evolving wave function is conserved and uniquely decided by Ĥi , and (ii) the Hall conductance of the quenched state in the diagonal ensemble is continuously changing as Ĥf changes with a logarithmically divergent derivative whenever Ĥf changes the Chern number.We thus identify a new class of topologically driven phase transitions with an exotic critical behavior, which is quite different from the orthodox one with a discontinuous Hall conductance but a zero derivative everywhere.The discrepancy in the SPT order obtained from the Chern number (based on unitary time evolution) and the Hall conductance is attributed to the fact that the latter must be calculated according to the diagonal ensemble, in which the coherence between different eigenstates of Ĥf in the wave function is lost in the long-time limit.In this experimentally relevant sense the SPT order of quenched states depends on Ĥf .
Real-time dynamics of the Chern number.-TheDirac model is a paradgim for two-dimensional topological insulators breaking time-reversal symmetry [19].Its Hamiltonian is where ĉ k = ĉ k1 , ĉ k2 T is the fermionic operator and k sums over the whole momentum plane.The singleparticle Hamiltonian H k is written as which is quantized and changes only at the phase boundary M = 0 or B = 0.The Hall conductance of the ground state is simply the Chern number in units of e 2 /h.At the time t = 0, we suddenly change the Hamiltonian from Ĥi = Ĥ(M i , B i ) to Ĥf = Ĥ(M f , B f ).Then the wave function evolves according to where |u k (t) is the single-particle wave function obeying Because momentum is a good quantum number both in Ĥi and Ĥf , it is natural to generalize the definition of the Chern number for the time-dependent wave function in the following way: This real-time Chern number characterizes the topological property of the wave function |Ψ(t) , and can be reexpressed as where S denotes the k x -k y plane oriented in the k zdirection and A(t) = u k (t)|▽ k |u k (t) is the Berry phase.C(t) is determined by the poles of A(t) and must remain quantized at all times since locally deforming A(t) cannot change it.In fact, the two poles of A(t) at k = 0 and k = ∞ have conserved residues under a unitary evolution [20], so that for arbitrary Ĥi and Ĥf we have The Chern number of the wave function never changes however the system is driven out of equilibrium, coinciding with the no-go theorem proved by D'Alessio and Rigol [18].This result suggests that the SPT order of a wave function is generally conserved after a quench if the Hamiltonian in real space contains only local operators [17].
Hall conductance in the diagonal ensemble.-Theobservation that C(t) is insensitive to Ĥf does not imply the absence of nonequilibrium phase transitions because C(t) is not a measurable physical quantity.In this paper, a nonequilibrium phase transition unambiguously indicates the nonanalytic behavior of observables as the postquench Hamiltonian Ĥf changes.We choose the Hall conductance as the indicator of nonequilibrium phase transitions.Notice that in the ground state the Hall conductance is directly related to the Chern number.
It is well known that the Hall conductance cannot be expressed as the expectation value of a local operator with respect to the wave function, but must be written as the long-time response to an external electric field in linear response theory.This fact reflects the topological nature of the Hall conductance and is related to the observation that in order to measure the Hall conductance, one must couple the system to auxiliary reservoirs.However, coupling to reservoirs unavoidably introduces decoherence and therefore in the long-time limit the far-fromequilibrium system will be described by the diagonal ensemble and not the unitarily evolved wave function of the isolated system.This motivates us to pursue a definition of SPT order and topologically driven nonequilibrium phase transitions by studying the Hall conductance in the diagonal ensemble, which is the experimentally relevant setting.In the long-time limit, the off-diagonal terms of the density matrix in the eigenbasis of Ĥf are averaged out [14].The time-averaged expectation value of an operator Ô can be expressed as lim where |E are the eigenstates of Ĥf and ρ is diagonal in the basis |E with the probabilities | E|Ψ(0) | 2 .If the system relaxes its long time properties must obey lim t→∞ Ψ(t)| Ô|Ψ(t) = Tr[ Ô ρ].We can then describe the system by ρ, the so-called diagonal ensemble [14].While this argument is based on non-degenerate eigenenergies, the applicability of the diagonal ensemble has also been shown in many integrable quantum many-body models [21,22].
We then build our formalism on the diagonal ensemble with the density matrix written as where |u f kα is the eigenvector of H f k and α = ± denotes the upper and lower bands with the positive and negative eigenvalues ±| d f k | respectively.n kα is the occupation number of the band α and can be expressed as the overlap where |u i k− is the lower-band eigenvector of the initial Hamiltonian H i k , and is in fact the initial wave function.The total occupation at each k is conserved to be n k+ + n k− ≡ 1. Eq. ( 8) is obtained by averaging out the offdiagonal elements in |u k (t) u k (t)| .Now we calculate the Hall conductance of the diagonal ensemble in linear response theory [23], i.e., we replace the equilibrium density matrix in linear response theory by the diagonal ensemble ρ.This replacement does not cause any problem in the formalism because ρ is timeinvariant satisfying [ρ, Ĥf ] = 0. We can then express the Hall conductance as the current-current correlation in the diagonal ensemble: where S denotes the area of the system and is conven- operator along the β-direction with e denoting the charge of the particle.Following the process for obtaining the celebrated TKNN number [3], we reexpress the dimensionless Hall conductance C neq := σ H /(e 2 /h) as [20] The dimensionless Hall conductance is the integral of the weighted mixture of Berry curvatures in different bands of the final Hamiltonian.In the case of Ĥi = Ĥf (no quench), the occupation is n k− = 1 and n k+ = 0 everywhere in the momentum plane, and C neq is just the Chern number of the initial state as we expect.But generally as Ĥi = Ĥf , n kα ∈ [0, 1] is a continuous function of the momentum k, so that C neq is not quantized any more but can take an arbitrary value in the range [−1 , 1].
In this two-band model, it is well known that the Berry curvatures in different bands are opposite to each other.By using this and the conservation law n k+ + n k− ≡ 1, we reexpress Eq. ( 11) as C neq = d k 2 cos θ • C, where C denotes the lower-band Berry curvature of Ĥf , and cos θ := (2n k− − 1) is the occupation factor evaluated as ) with θ the angle between the initial and final d k vectors.Noticing that both C and cos θ are rotationally invariant in the k x -k y plane, we carry out the azimuthal integration and make the substitution k = k 2 , and finally obtain where the Berry curvature is and the occupation factor is with It is worth comparing the real time Chern number C(t) in Eq. ( 4) with the dimensionless Hall conductance C neq in Eq. (11).The former reflects the topology of the wave function, being quantized but not measurable, while the latter is a true observable but nonquantized.They are both integrals of the Berry curvature, but C(t) is derived from the wave function while C neq follows from the diagonal ensemble where the coherences in the wave function are lost.Decoherence plays a crucial role in understanding the SPT order of a quenched state in the long-time limit, i.e., a nonequilibrium steady state.
Nonanalytic behavior of the Hall conductance.-Weare now prepared to discuss the nonanalytic behavior in the Hall conductance C neq , which is a function of (M i , B i , M f , B f i.e., the parameters of Ĥi and Ĥf .This function satisfies the properties: We first show the function C neq (M f , B f ) at some special points of (M i , B i ).Due to the above symmetries, we only consider the two cases M i , B i > 0 and M i > 0, B i < 0. As shown in Fig. 1, C neq is a continuous function of (M f , B f ) in the whole parameter space (we will strictly prove this for arbitrary (M i , B i ) in the next section).This result is astonishing if we consider the fact that the Chern number of the ground state has a jump whenever M or B changes its sign (see Eq. ( 2)).By driving the system out of equilibrium, we smoothen the Hall conductance.Whatever Ĥi is, C neq has a similar shape, reminiscent of the function (sgn(M f ) + sgn(B f ))/2, i.e., the Chern number of the post-quench Hamiltonian Ĥf .In the quadrant M f , B f > 0 (M f , B f < 0), C neq approximately takes a positive (negative) value, while it is close to zero when M f and B f have different signs.Even if the initial state is topologically trivial (see the right panel), we can still measure a finite Hall conductance as Ĥf is nontrivial, but it cannot reach the quantized values ±e 2 /h.When the initial state is nontrivial (see the left panel), the Hall conductance is suppressed as Ĥf deviates from Ĥi , and can even change the sign as M f and B f both change their signs.While C neq remains continuous, the key finding of this paper is that whenever the post-quench Hamiltonian crosses the boundary M f = 0 (B f = 0), the derivative of the Hall conductance We plot ∂Cneq ∂M f as a function of ln |M f | in Fig. 2. As M f → 0, the curves asymptote to straight lines with the slope −1/(2|M i |), independent to Ĥf or from which side M f goes to zero.As B f → 0, the same divergence happens to ∂Cneq ∂B f because C neq is symmetric in M f and B f according to Eq. ( 15).We identify a nonequilibrium phase transition when the Chern number of Ĥf changes.This phase transition displays a universal [24] critical behavior, a logarithmically-divergent derivative of the Hall conductance with a prefactor depending only upon the pre-quench Hamiltonian Ĥi .This critical behavior is exotic compared to that of ground-state phase transitions, in which the Hall conductance has a zero derivative everywhere within one topological phase, but displays a discontinuity at the phase boundary.
This phase transition reveals different nonequilibrium phases which share the common symmetries of the Dirac model.Apparently, the broken symmetry picture does not account for this transition, which must be ically driven.Interestingly, the topological invariant of the wave function C(t) fails to reveal the nonequilibrium phase transition displayed in an observable, namely the Hall conductance.Roughly speaking, one can assign the Chern number of Ĥf to each nonequilibrium phase to distinguish them, and this assignment coincides with the character of C neq plotted in Fig. 1.
Finally, we briefly mention how to prove the continuity of C neq and the logarithmic divergence of its derivative (Eq.( 16)) at the phase boundary.We only consider the boundary M f = 0, since M f and B f are symmetric to each other.Recalling C neq = ∞ 0 d k cos θ • C, we can express its derivative as at M f = 0.A straightforward observation is that both cos θ( k) and C( k) are smooth functions in the range (0, ∞).However, they do not uniformly converge to The second integral is smooth in M f , derived from the asymptotic behavior of (cos θ • C) at k → ∞, or to be strict, by making a substitution k → 1/ k in the integral.In fact, k = ∞ is a true singularity at the other boundary B f = 0, where k = 0 is a regular point, because cos θ and C are invariant under the simultaneous substitutions k ↔ 1/ k and M i/f ↔ B i/f .If there is any nonanalytic behavior in C neq , it must come from the first integral denoted by C η neq .Interestingly, we can take an arbitrary small η so that the term d i in (cos θ) converges to a constant d i = |M i |, and then obtain The calculation of this integral is straightforward since the integration function is rational.We write the result as C η neq = F (η)−F (0) with F denoting the original function.F (η) is smooth in M f , while F (0) for small M f is expanded as where the higher order term O goes to zero faster than M f ln |M f | and does not contribute to the discontinuity of C η neq or ∂C η neq /∂M f [20].The function , which directly leads to Eq. ( 16) and the continuity of C neq [25].
It is worth explaining why C neq is continuous but the Chern number C = ∞ 0 d k C is not.We notice that the energy gap closes at the phase boundary, while the occupation factor cos θ in the vicinity of the singularity is close to zero corresponding to the occupation n k+ = n k− = 1/2.The equally-weighted mixture of different bands cancels the discontinuity caused by the singularity, and therefore smoothens the Hall conductance.
Conclusions.-Insummary, we find a new class of topologically driven phase transitions in quenched states of the Dirac model, characterized by a continuous Hall conductance with a logarithmically-divergent derivative (Eq.( 16)).We obtain the Hall conductance by applying linear response theory in the diagonal ensemble of the system, which is the physically correct description of the long-time limit in a far-from-equilibrium quench setup.The topological invariant of the real-time wave function fails to predict this phase transition, which can only be correctly addressed by the diagonal ensemble where decoherence effects are taken into account.Our finding indicates the possibility of exotic topological phase transitions in systems far from equilibrium.

Calculation of the real-time Chern number C(t)
We express the real-time Chern number as where the Berry phase is T denoting the two-component wave function.It is straight forward to calculate the wave function and obtain and where k + = k x +ik y , and d i/f and d z i/f are the length and the z-component of the vector d i/f k , respectively.We divide the Berry phase into two parts: x + k 2 y , we immediately know that ▽ k × A 1 must be zero, so that A 1 does not contribute to C(t).We again divide A 2 into the irrelevant term A 2a with a zero curl and the relevant term A 2b with its imaginary part written as where x and y denote the two unit vectors in the momentum plane.Now we reexpress the Chern number by the relevant field Im[ A 2b ] as Im[ A 2b ] is a vortex field with two poles at zero and infinity, respectively.Applying the Kelvin-Stokes theorem in an annulus with inner radius r and outer radius R, and then taking the limit r → 0 and R → ∞, we have where |Im[ A 2b ]| k=R denotes the length of the vector Im[ A 2b ] on the circle of radius (k = R).The first limit evaluates π(1 − sgn(B i )) , while the second limit evaluates π(1 + sgn(M i )) , being both time-independent.In other words, the residues of the two poles at zero and infinity are both constants, leading to a constant Chern number: Calculation of the Hall conductance Cneq We briefly mention how to express the nonequilibrium Hall conductance as the integral of the Berry curvature.Our derivation is a straightforward extension of the work by Thouless et al. [1].
In linear response theory, the Hall conductance is written as where η is an infinitesimal number, accounting for the adiabatic switch-on of an external electric field, and ω is the frequency of this field with the limit ω → 0 corresponding to the dc-conductance.
The nonequilibrium density matrix is known to be ρ Notice that ρ is a product state, because the conversation law α n kα = 1 constrains the system within a subspace of the Fock space where the empty or doubly-occupied states at each momentum are excluded.We can then reexpress σ H in the first-quantization language as σ H = lim  The expression of F is quite lengthy, even being just a combination of fundamental functions.For simplicity, we only show its value at zero: What we are interested in is the behavior of F (0) in the neighborhood of the phase boundary M f = 0. Noticing that at M f = 0, 1 − 4B f M f can be expanded into we substitute this expression into Eq.( 31) and obtain The first term is a constant independent to M f , while the second term continuously goes to zero as M f → 0, however, its derivative with respect to M f is divergent.
All the other terms go to zero faster than the second term and their derivatives are finite at M f = 0, so that the nonanalytic behavior of ∂C neq /∂M f is decided by the second term.
where σ denotes the Pauli matrices and d k = (k x , k y , M − Bk 2 ) with two parameters M and B. The ground state is well known to be classified by the Chern number

2 = −ie 2 S= α ǫ kα |u f
are interested in the dc Hall conductance which should be a real number, we take the real part of σ H , which is found to beReσ H = σ H + σ * H k,α,βn kα (ǫ kα − ǫ kβ ) 2 ǫ kα denotes the eigenvalue of H f k .We make use of the relation H f k FIG.2.∂Cneq/∂M f as a function of (ln |M f |) at different (Mi, Bi, B f ).Note that in the two curves at Mi = 1 we simultaneously plot the data of M f → 0 + and M f → 0 − , which are in fact undistinguishable at small |M f |.