Topological Blocking in Quantum Quench Dynamics
We study the non-equilibrium dynamics of quenching through a quantum critical point in topological systems, focusing on one of their defining features— ground state degeneracies and associated topological sectors. We present the notion of “topological blocking”, experienced by the dynamics due to a mismatch in degeneracies between two phases and we argue that the dynamic evolution of the quench depends strongly on the topological sector being probed. We demonstrate this interplay between quench and topology in models stemming from two extensively studied systems, the transverse Ising chain and the Kitaev honeycomb model. Through non-local maps of each of these systems, we effectively study spinless fermionic -wave paired superconductors. Confining the systems to ring and toroidal geometries, respectively, enables us to cleanly address degeneracies, subtle issues of fermion occupation and parity, and mismatches between topological sectors. We show that various features of the quench, which are related to Kibble-Zurek physics, are sensitive to the topological sector being probed, in particular, the overlap between the time-evolved initial ground state and an appropriate low-energy state of the final Hamiltonian. While most of our study is confined to translationally invariant systems, where momentum is a convenient quantum number, we briefly consider the effect of disorder and illustrate how this can influence the quench in a qualitatively different way depending on the topological sector considered.
pacs:71.10.Pm, 75.10.Jm, 03.65.Vf
Over the past years, there has been a revival of interest in the topics of topological systems and non-equilibrium critical dynamics stemming from the latest advances exhibited in a variety of condensed matter and cold atomic systems Dziarmaga10 (); Kibble76 (); Zurek96 (); Damski05 (); Anatoly05 (); Calabrese05 (); Levitov06 (); Mukherjee07 (); Sengupta08 (); Mondal08 (); Patane08 (); Degrandi08 (); Bermudez09 (); Perk09 (); Polkov08 (); Vishveshwara10 (); Pollmann10 (); Singh10 (); Sondhi12 (); Sondhi13 (); Patel13 (); Mostame13 (); Foster13 (). The synergy of the two topics, namely quench dynamics in topological systems, is still in its infancy Bermudez09 (); Vishveshwara10 (); Sondhi12 (); Sondhi13 (); Patel13 (), but promises to form a rich and complex avenue of study. While previous works have targeted the formation of edge states and bulk defects that are characteristic of topological systems, in this work we focus in particular on the role of ground-state degeneracy, another key characteristic of topological order.
Our work highlights special features of quenches that involve initializing a system in the ground state of a phase with a particular topological order and dynamically evolving this state through a topological phase transition, i.e. the Hamiltonian is time dependent and the ground-states of the initial and final Hamiltonians have differing topological order. We consider topological aspects of systems having periodic boundary conditions, i.e., rings or tori, where the effect of degeneracies is clear cut. This is different from open bounded systems, where the dynamics can be complicated by edge effects and from infinite systems where topological aspects can often be completely hidden. Most dramatically, we find a phenomenon which we call topological blocking: due to mismatch in degeneracies, some of the ground states of a topological system have no overlap with any of the ground states on the other side of the transition, regardless of how slowly the quench is performed.
We expect that our central observations apply to a wide range of topological systems. Our general setting involves two gapped phases having different degeneracies separated by a gapless critical point (or more generally, a gapless region). Topological blocking is best seen by initializing the system in the phase having higher degeneracy. Over the evolution of the quench, as shown in Figure 1 some of the topological sectors of this phase are forced to be lifted in energy as they pass through the gapless point so that no states in those sectors appear as ground states in the new phase. Nevertheless the states in the original topological sectors may remain topologically distinct from each other, so they cannot be connected by the action of local operators. Hence, in a quantum quench between the phases, an initial state in a sector that has its energy lifted evolves within that sector. The time evolved state after the quench thus has zero overlap with any of the ground states in the final phase.
The role of the topological sector, while directly obvious for topological blocking, is also apparent when considering state evolution within the sector. We find that an effective indicator of sectoral-dependence is the overlap of the time-evolved state with the lowest energy state of the instantaneous quenched Hamiltonian within the same sector (sectoral ground state). Figure 2 shows an example illustrating such time-dependent wave function overlaps for a quench from a doubly degenerate phase to a non-degenerate phase; the overlaps within the two sectors, labeled by parity, show a clear difference in their evolution during the quench, exhibiting the most pronounced features in the vicinity of the critical point. While the quantitative difference is obvious, under certain easily accessible circumstances, there can also be a qualitative difference if, unlike the absolute ground state, some of the sectoral ground states in the post-quench phase are not separated by a gap from the spectrum of excited states. It is worth mentioning here that these systems still respect the well-studied Kibble-Zurek mechanism Kibble76 (); Zurek96 (); Damski05 (); Anatoly05 (); Levitov06 (); Mukherjee07 (), which applies to systems having local as well as topological order and predicts power-law scaling as a function of quench rate in various quantities related to post-quench excitations. The dependence on topological sectors rides above such scaling and among the typical Kibble-Zurek quantities, such as residual energy or defect density, is most strongly manifest in wave function overlaps.
The interplay between topology and quench dynamics provides new insights into each of these respective aspects. Our treatment shows that the quench dynamics between phases that have different ground state degeneracies acts as a fine probe of topological order and examines some of its more subtle issues. For example, the notion of topological blocking highlights the fact that the number of topologically distinct subspaces (sectors) of the Hilbert space of a system may exceed the ground state degeneracy; there may be topological sectors which are “hidden” at low energy, but which nevertheless play a role in quantum quenches. In terms of quench physics, we bring attention to the concept that there typically exist multiple sectors in a system having topological order, which could show distinctly different dynamics. Understanding these quenches is also essential for the implementation of topologically fault tolerant quantum computation schemes Nayak08 (); Alicea11 (); Vishveshwara11 () where collective transitions between topological and non-topological phases (see for example Ref. Kells2013, ) represent a potential source of decoherence. The topological blocking mechanism and the fact that the “hidden” topological sectors need not be gapped (as we show below), presents a further complication for such schemes.
In what follows, we perform an analysis of the features we discussed above within the context of two topological systems that can effectively be described as spinless fermionic -wave paired superconductors. We first study the quantum Ising chain in a transverse magnetic field, perhaps one of the most celebrated systems in condensed matter for offering a tractable solution and rich physics, one with plentiful studies even in the context of quenching Levitov06 (); Mukherjee07 (); Calabrese2012 (); Essler2012 (); Fagotti13 (); Barouch1970 (). The second system, the Kitaev honeycomb model, too is special in its analytically soluble structure Kitaev2006 (); Chen2007 (); Baskaran07 (); Kells2008 (); Kells2009 (); Schmidt2007 (); Kells2010 (); Knolle13 () and has also received significant attention in the context of quenching Sengupta08 (). The transverse Ising model maps to a -wave superconducting chain Kitaev2001 () while the honeycomb lattice model maps to a superconductor coupled to a gauge field Kitaev2006 (); Chen2007 (); Kells2008 (); Kells2009 (), the latter thus a natural two-dimensional extension of the former.
In these superconducting systems, topological sectors are identified in terms of fermion parity, which is naturally accounted for in the ring and torus topologies for the one- and two-dimensional cases, respectively. In the transverse-Ising systems, the quench involves going from a topological phase having double degeneracy associated with even and odd parity to a non-topological phase having a unique ground state characterized by one of the two parities. In the spin language, the phase with the two degenerate ground states corresponds to a ferromagnet that spontaneously breaks local Ising symmetry while that with the unique ground state corresponds to a phase with spin-polarization along the magnetic field. In the honeycomb lattice model the relevant phases are an Abelian phase having the topological order of the toric code, which is four-fold degenerate, and a non-Abelian phase with Ising type topological order and three-fold degeneracy. In this model the transition is topological both in the spin language and in the fermionic language. In both models, we carefully pinpoint how topological blocking comes about, using the structure of the Bogoliubov-deGennes (BdG) Hamiltonians and perform a detailed analysis of the difference in post-quench behavior for quenches within different topological sectors.
The mapping in the transverse-Ising system between a model having local symmetry and one with topological order begs for a comment on the relevance of our analyses to systems having spontaneous symmetry breaking and local order. As with topological systems, in quenching through a spontaneous symmetry breaking transition, the symmetry broken phase would typically have larger ground state degeneracy than the unbroken phase and, if the quench dynamics preserves the symmetry, a similar blocking phenomenon can occur; some symmetry breaking states would be lifted away from the ground state energy in the unbroken phase. In fact, much of our analysis would apply for these systems and it would be worth studying sectoral dependences in the context of local order as well. However, an important distinction of topological blocking is the non-local nature of topological symmetries. Thus, unlike in spontaneous symmetry broken systems, the key features of topological blocking discussed in this work should be robust against local perturbations of the Hamiltonian of the system.
An overview of the paper is as follows. Section II discusses the transverse-Ising case in depth, starting with a brief introduction, followed by its superconductor description, a discussion of degeneracies and the quench protocol, an explanation of topological blocking in terms of parity arguments, and finally detailed studies of quench behavior for different topological sectors. Section III gives a similar treatment of the Kitaev honeycomb model. In Sec. IV, we perform initial studies of quenches in these systems in the presence of disorder as a means of demonstrating robustness against local perturbations as well as the marked difference in topological sectors in situations where the blocked sector can access a slew of low-lying excitations. We conclude with a short summary and outlook in Sec. V.
Ii The transverse Ising model
The transverse Ising model in one dimension is one of the best studied exactly solvable models, (see Ref. Sachdev_book, for a thorough treatment). As is commonly done to solve almost any aspect of the model, the non-local Jordan-Wigner transformation is used to map it to a beautiful prototype of a topological system - a spinless fermionic, one-dimensional -wave superconductor. Here, after introducing the model, we reiterate the fermionization procedure, taking into account the subtleties associated with periodic boundary conditions and fermion parity. We carefully describe the link between fermion parity, topological degeneracy, the topological sectors on either side of the transitions and their associated sectoral ground states. With these considerations in place, we show how topological blocking naturally comes about. We then study the dynamics of the quench in each topological sector, focusing on the overlap between the time-evolved initial ground state and instantaneous sectoral ground states. Our analytic treatment uses the Landau-Zener formalism typically applied of late to related quenches in homogeneous systems Damski05 (); Anatoly05 (); Levitov06 (); Mukherjee07 () and we corroborate it with numerical studies.
The most frequently encountered form of the Hamiltonian for the transverse Ising model is given by
Here, denote spin Pauli matrices, an Ising ferromagnetic coupling, a Zeeman magnetic field in the -direction, and nearest neighbors and . (We set Planck’s constant throughout this paper). If we take and , the system has two phases, ferromagnetic and paramagnetic. The ordered Ising ferromagnet along the -direction occurs for while the paramagnetic phase occurs for . The two phases are separated by a quantum critical point at .
The ground state degeneracies of the two phases can be discerned by looking at the Hamiltonian in some simple limits. In the paramagnetic limit, , we see that the ground state is simply the non-degenerate state fully polarized along the direction of the Zeeman magnetic term,
where, for the spin state on a single site, and in the eigenvalue basis of . The overbar denotes the quantum state for the entire collection of sites. In the opposite ferromagnetic limit, , there are two degenerate ground states given by superpositions of
where and are the eigenstates of . The system is symmetric under a global rotation around the -axis, given (up to a global phase) by the string operator
This non-local operator maps the and states into each other, while is left invariant. After fermionization, is associated with fermion parity and topological degeneracy; note that is conserved even if the couplings in Eq. (1) are allowed to be functions of space.
ii.1 Fermionized topological superconductor and solution
The original fermionic solution for the transverse Ising chain can be traced to Pfeuty Pfeuty1970 () who used a transformation similar to Lieb, Schultz and Mattis Lieb1961 (). Indeed, the fermionic dispersion relation for the transverse Ising can be seen to be identical to that of the model solved by Lieb, Schultz and Mattis. Here too we employ their extensively used Jordan-Wigner transformations to define the position space fermionic excitations (see, for example, Refs. Levitov06, ; Mukherjee07, ; Calabrese2012, )
The state given in Eq. (2) is therefore the fermionic vacuum state. At any site , we have . Hence gives the parity of the total fermion number,
In terms of fermion operators the Hamiltonian takes the superconducting form
where is the number of sites on the ring. This superconducting Hamiltonian for spinless fermions has an on-site chemical potential , nearest-neighbor hopping of strength , and anomalous -wave pairing terms also of strength . A generalization of this model having can be obtained by considering an spin chain instead of an Ising spin chain Mukherjee07 (); the main results of this section also hold for this case.
The boundary conditions of the system are encoded in the operator . To select the periodic sector we replace the operator with its eigenvalue corresponding to an odd number of fermions. To select the antiperiodic sector we replace the operator with the eigenvalue , corresponding to even parity.
The Hamiltonian can be written in momentum space as a sum of BdG Hamiltonians
The BdG Hamiltonians can be diagonalized by a Bogoliubov transformation. Namely, we may write
in terms of the Bogoliubov-Valatin operators
(We will see below that the modes with and require a special analysis since they satisfy . Further, for these modes; hence, .) We see that in both phases of the model, the excitation energy is gapped for all ; the minimum energy lies at with . At the critical point , the system is gapless and for .
With regard to the topological aspects of the superconductor, the ferromagnetic phase, having a double ground state degeneracy, maps to a topological phase and the non-degenerate paramagnetic phase to a topologically trivial phase. This can be seen from standard Berry’s phase analyses of the momentum eigenstate spinor structure Niu2012 (). Alternatively, it is common to consider the Kitaev chain, a finite open chain version of the Hamiltonian in Eq. (7), which naturally lacks the term associated with the (anti)periodic boundary conditions of the ring geometry. The topological phase then has free Majorana modes at each end which lie at zero energy if the chain length is much larger than the decay length of these end modes. The Majorana end modes together form a Dirac fermion state which can either be occupied or unoccupied, thus accounting for the double degeneracy and fermion parity. As alluded to above and detailed in what follows, for the ring geometry, which we confine ourselves to, the connection between topological degeneracy and fermion parity is more subtle.
ii.2 Topological degeneracy
We now describe the ground states of the model in terms of the occupation numbers of the fermionic modes and explain in detail how the topological sectors of the Ising chain are connected to fermion parity. In particular, we show that there is always a ground state of the system with even fermion number, while a ground state with odd fermion number exists only in the ferromagnetic phase. In the paramagnetic phase, the lowest energy state with odd fermion number is part of a band which is gapped away from the true (even fermion number) ground state. A schematic of the spectrum of the model highlighting these features is shown in Fig. 3.
We focus first on the case where the number of sites is even. In the even-fermion antiperiodic sector, the allowed momenta are then given by with integer . Crucially, note that the values of do not include and . The ground state is given by
where spans the restricted set of momenta described above. The energy of this state is given by, where the sum respects the quantization condition on .
In the odd-fermion periodic sector the allowed momenta are given by with integer . These include the momenta , which need to be treated carefully. In the ferromagnetic phase occurring for , we have ; hence . From Eq. (II.1) we see that the contribution of this mode to the Hamiltonian is then just , and thus the fermionic state with the mode occupied has the lower energy compared to that with the mode unoccupied. We also have , so that , and similar arguments show that the energetically favorable state has the mode unoccupied. Hence the ground state is given by
As this state is annihilated by all the it has an energy given by . In this phase, the values of become arbitrarily close to those of the even-fermion sector and for we get a two-fold degenerate ground state.
To get an intuitive picture of how the degeneracy arguments derived from parity considerations connect with the spin picture described earlier, we can analyze the limit . For any value of , we then have two degenerate ground states given by all or all as shown in Eq. (3). In terms of states with fermionic occupation numbers and at site , the two ground states are given by and . We then see that the sum and difference of these states respectively give states which have an even and odd number of fermions, recalling that is even.
The situation is quite different in the paramagnetic phase which occurs for . The odd-fermion parity sector has a state with with , so that . In principle, having the fermionic and modes unoccupied would be the lower energy state. However, this would violate the odd parity of the sector. Given that as a function of , has the smallest value for , the state defined in Eq. (13) still does the best in terms of minimizing the energy within the odd sector. In this case, , so we are looking at the state in Eq. (12) with an extra excitation. This state is the lowest state of a band which can be obtained by exciting the system at nonzero momentum using instead of . Thus, Eq. (13) corresponds to the sectoral ground state in the paramagnetic phase. However, the state now possesses energy . In the limit , we see that the ground state in the odd-fermion sector lies at an energy which is higher than the ground state in the even-fermion parity sector by a finite amount equal to .
Now let us briefly discuss what happens if is odd. Then in the even-fermion antiperiodic sector, the allowed momenta are given by with integer , which includes the term but not . In both the ferromagnetic and paramagnetic phases, the even sectoral ground state is still given by Eq. (12) (with the appropriate momentum quantization) and this state continues to be the absolute ground state. In the odd-fermion periodic sector, the allowed momenta are given by with integer , which includes the term but not . Here too, Eq. (13) remains the odd sectoral ground state and is another absolute ground state in the ferromagnetic phase but has higher energy in the paramagnetic phase. The situation is therefore similar in many ways to the case where is even.
To summarize, in the thermodynamic limit , the ground state of the system in the ferromagnetic phase has a double degeneracy, with one ground state lying in each of the sectors (even- and odd-fermion). In the paramagnetic phase, there is a unique ground state which lies in the even-fermion sector. The sectoral ground state in the odd-fermion sector lies in a band which is separated by a finite gap from the ground state in the even-fermion sector.
ii.3 Quenching Dynamics
We now turn to the quench dynamics caused by slowly varying the transverse field in time, starting at at in the ground state of the ferromagnetic phase and ending at at in the paramagnetic phase. Note that the time evolution does not mix the even- and odd-fermion sectors; hence we will consider the time evolution in the two sectors separately.
Quench protocol:- We consider a linear time dependence of the form
By a slow variation, we mean that the dimensionless quantity . Our analysis of quench dynamics partially follows those extensively performed in the context of Kibble-Zurek physics Damski05 (); Anatoly05 (); Levitov06 (); Mukherjee07 () with the crucial difference that we explicitly consider fermion parity and momentum quantization associated with the topological sectors.
For any given set of modes (except 0 and ), the quench couples the two states in the occupation number basis and . In this basis, the relevant dynamics is governed by the Hamiltonian
where Eqs. (9) imply that
The instantaneous eigenvalues of the Hamiltonian in Eq. (15) have a minimum difference gap of at . In our problem, the value of depends on . Further, the initial and final values of are given by
which also depend on .
For each value of , we study the quenching dynamics numerically as follows. We first calculate the quantities and in Eqs. (II.1-II.1) at the initial time with the initial value . We then compute the time ordered evolution operator
by dividing the time into steps of size each (with ) and calculating
where . We then calculate
Finally we compute the ground state overlap by using the Onishi formula Ring04 () which, for our matrices, amounts to
where the time independent quantities and are those given in Eq. (II.1) and encode the instantaneous ground state. Here, the subscript indicates the fermion parity, and consequently, the boundary conditions. The product over runs over the entire Brillouin zone from to and, as discussed in previous sections, is restricted to certain values that depend on fermion parity. For a given momentum pair, the probability of being in the excited state of the Hamiltonian is
This excitation probability governs much of the post-quench behavior. A plot of for a number of -values can be seen in Fig. 5
Analysis:- Because the fermion number parity is conserved throughout the quench, we observe the topological blocking behavior described in the introduction. Initializing the system in the ground state of the ferromagnetic/topological phase in the odd parity sector, we observe that, even at adiabatically slow quench rates, this state does not evolve to the overall ground state (which has even fermion number), but rather to the sectoral ground state in the odd fermion number band.
At non-adiabatic quench rates, we therefore consider the overlap of the time-evolved state with the sectoral ground state of the final Hamiltonian. Figure 2 shows a representative case for the overlap as a function of time for the odd- and even-fermion sectors; the two curves are clearly different. We now analyze the detailed behavior of the time-evolved states, focusing on the contributions of each of the momentum modes and on the differences between sectors.
To begin with, we consider a simple problem in which the time in Eq. (15) goes from to , so that the value of is irrelevant. If we start in the ground state of at , the probability of ending in the excited state of at is given by the Landau-Zener expression Landau32 (); Majorana32 (); Vitanov96 ()
This expression gives the correct limits and 1 in the adiabatic () and sudden () limits respectively. Note that the momenta and are special; for these modes and therefore for any quenching time . Namely, these states do not change at all under quenching, and they change abruptly from the ground state to the excited state when crosses zero.
In the limit , Eq. (23) shows that the excitation probability is equal to 1 for and , and becomes negligible when deviates from those points by an amount which is much larger than . However, for our quench protocol, we see from Eq. (17) that the initial and final times, , are functions of ; the time at which the two eigenvalues of the Hamiltonian are separated by the smallest amount () is crossed only if and , i.e., if . Hence, the excitation probability is dominated only by the region near ; for exactly , the two-level system undergoes a level crossing and . The modes near never reach the minimum gap region, and for exactly , the two-level system remains in the ground state with . The behavior of the Landau-Zener transition exhibited by sets of -modes and the evolution of the special mode is shown in Figure 4.
In the adiabatic limit, we see that in the even-fermion sector, if we start in the ground state given in Eq. (12) at , we reach the ground state in Eq. (12) at . However, in the odd-fermion sector, if we start in the ground state in Eq. (13) at , we reach the state in Eq. (13) at which is the ground state in that sector but which, as discussed above, is separated from the ground state of the final Hamiltonian by a finite gap. (Note that in the odd-fermion sector, the state with momentum does not change with time since the off-diagonal matrix element makes it impossible to have a transition between the two eigenstates of the Hamiltonian). Hence, an adiabatic time evolution takes a system from the initial ground state to the ground state of the final Hamiltonian in certain sectors but not in others, with the different sectors being distinguished from each other by a topological quantity, namely, the fermion parity in our model. This explicitly demonstrates topological blocking in this system.
Overlap at the final time:- At , the overlap between the final state reached and the actual ground state in a particular sector is given by
In the limit , we know that is significant only for a range of of the order of near . Let us consider the thermodynamic limit and define a dimensionless scaling variable
Using the fact that the momenta in the even- and odd-fermion sectors are given by and , where , we can express the overlaps in the even- and odd-fermion sectors as
where we have made the approximation in Eq. (23) since only the low lying modes contribute a significant excitation probability. For the same reason we have changed the upper limit from to since the overlap rapidly approaches 1 once becomes a number of order, say, , under the assumption .
A factor-by-factor comparison of the two expressions in Eqs. (26) shows that is larger than for any value of . We therefore have the interesting result that the overlap between the final state and the sectoral ground state is higher in the odd-fermion sector than in the even-fermion sector, even though the final state in the odd-fermion sector has zero overlap with the ground state of the final Hamiltonian.
We can write the logarithms of the overlaps in Eqs. (26) as sums over . In the limit , i.e., for , the sums can be approximated by integrals. Ignoring the difference between and in Eqs. (26), which amounts to ignoring some subleading terms, we find that in both even- and odd-fermion sectors,
Overlap at intermediate times:- We now look at the overlap between the state reached at a finite time and the ground state at that time. This is given by the expression
As has been analyzed in the context of Landau-Zener transitions Landau32 (); Majorana32 (); Vitanov96 (), the analytic form of can be expressed in terms of Weber functions. Numerically we find that for a certain range of values of , the overlap of the system shows pronounced oscillations around (i.e., when is going through the critical value of ) before settling down at at a value which is around , i.e., not very close to either 0 or 1. We can estimate this range of values of by looking at the overlap as a function of time for some individual values of the momentum . Assuming that , we find the following. For (but not equal to 0), we have an almost sudden process. Hence the overlap stays close to 1 till we get close to , and then it rapidly changes to a very small value. Clearly, this would make the overlap of the system (which is a product of the overlaps for all values of ) very small. On the other hand, for , we have an almost adiabatic process and the overlap stays close to 1 at all times; such values of therefore make very little difference to the overlap of the system. Only if do we get a final overlap which is around . (This is consistent with Eq. (23) since ). These different kinds of behavior are shown in Fig. 5 for and , , and . Thus, the behavior of the overlap of system that we are looking for, namely, oscillations near the critical point before settling down to a value around only occurs if the smallest non-zero value of satisfies . Then this value of makes the dominant contribution to the overlap of the system at all times since all the higher values of contribute factors close to 1 to the overlap. Since the smallest non-zero value of , where and in the even- and odd-fermion sectors respectively, the value of where the final overlap of the system is around is about and for even- and odd-fermion sectors respectively.
Figure 5 shows oscillations in the overlap near the critical region which is equal to 50 for our choice of parameters. We can understand this by mapping the time evolution with the Hamiltonian in Eq. (15) to the Schrödinger equation of a particle moving in an inverted harmonic potential Landau32 (); Vitanov96 (). If we define the upper and lower components of the two-component wave function associated with the state by and , we can eliminate, say, to obtain the equation
Since we are interested in the behavior of the solution of Eq. (29) when is small (and ), we will ignore the last term, , in comparison with the other terms like . The dominant behavior of the solutions of Eq. (29) is then given by . This explains the oscillations around . Further, as moves away from zero, oscillates more and more rapidly; this is qualitatively confirmed by the plots in Fig. 5.
To summarize the discussion in the last two paragraphs, the overlap in Eq. (28), in general, either stays close to 1 at all times or drops rapidly from 1 to zero when the system crosses the quantum critical point at . The intermediate behavior in which the overlap drops to a value which is about halfway between zero and 1 when crosses occurs only when Eq. (28) is dominated by the smallest non-zero value of , and that value of happens to satisfy . For a system of size , the smallest non-zero value of is given by and in the even- and odd-fermion sectors, respectively; from this we can deduce the value of at which the intermediate behavior occurs in the two sectors. When considered together, the highly sensitive nature of this quench behavior on the actual value of momentum, the dominance of a single mode in the net overlap, and the slightly different momentum quantization conditions for the two sectors, together explain the markedly different quantitative behavior shown by the overlap in the two sectors in Figure 2.
ii.4 Other Quantities
We have found the wave function overlap plotted in Fig. 2 to be the most sensitive yet direct measure of the dependence of quench dynamics on topological sectors. In this context, we briefly discuss here other quantities that are commonly studied in quench dynamics and related Kibble-Zurek physics Kibble76 (); Zurek96 (); Damski05 (); Anatoly05 (); Levitov06 (); Mukherjee07 (); Sengupta08 (); Mondal08 (); Degrandi08 (); Polkov08 (). In fact, the behavior of several quantities can be traced back to that of the probability of excitation within each set of momentum modes, namely, that of the which was first introduced in Eq. (22).
Defect density:- The well-studied Kibble-Zurek defect density is the cumulative sum of the excitation probabilities for all the modes, i.e. . In terms of Ising spins, the defect density is a measure of how many spins are pointing in the energetically unfavorable direction in the final phase. In the final state reached at , the total defect density is given by
In the adiabatic limit , only the regions near contribute to the integral, and we get the Kibble-Zurek scaling law . This scaling is exactly mirrored by the behavior of the logarithm of the overlap in Eq. (27).
As with the overlap, in distinguishing the even and odd sectors, the summation on in Eq. (30) is restricted to the allowed momenta. The defect density is less sensitive than the overlap in distinguishing between the different topological sectors for the following reason. If the excitation probability is close to 1 for any particular value of , this affects the overlap in Eq. (24) strongly since it is given by a product over all and therefore approaches zero if is close to zero for any . On the other hand, the defect density in Eq. (30) is given by a sum over all and is not dominated by any one value of ; in addition, the sum is divided by which further reduces the contribution from any single value of .
For a system of finite size , in the topologically blocked odd-fermion sector,the special mode has a level crossing and, across the phase transition, completely evolves into the excited state. Compared to the even sector, this mode thus contributes a term of order independent of the quench rate. In the thermodynamic limit, this contribution obviously vanishes while away from this limit, the degeneracy in the ferromagnet/topological phase is split due to finite size effects. However, in this degenerate phase, the splitting is exponentially small as a function of Lieb1961 (), and is always present in numerical simulations and physical systems due to their finite size. Thus, observation of the quench-independent jump and its scaling behavior of systems size would provide some indication of the difference between topological sectors.
Residual energy:- Another characteristic quantity discussed in quench dynamics is the residual energy; this measures the excess energy contained in a post-quench state compared to the ground state of the final Hamiltonian. In the transverse Ising system, the net residual energy at the end of the quench at time is given by the sum of the contributions of each momentum mode,
where the expectation value of defined in Eq. (15) is with respect to the time-evolved quench state, and is the energy of the ground state of .
The arguments made above for the defect density also hold for the residual energy. It respects the same scaling behavior and in considering the odd- and even-fermion sectors, involves restricted momentum summations. As with the defect density, in the odd-fermion sector the makes a special contribution, taking the time-evolved state completely into the excited branch. Thus, in this sector, the residual energy shows a jump of order . This too is an effect of order in that there are contributions from a total of momentum sets to the entire residual energy. Nevertheless, the jump reflects topological blocking and the difference in behavior of sectors illustrated in Fig. 3.
Entropies:- Various forms of entropy, such as the entanglement entropy, have been actively studied in the context of quenches. These measures provide an alternative picture for the manner in which the wave function evolves. In the context of topological sectors, based on the special behavior of the mode, i.e., , we find that a variant of the Renyi entropy Calabrese10 (), , would provide an effective way of distinguishing odd and even sectors:
Given the Kibble-Zurek scaling form discussed above, would behave as , where is a constant and for the even-fermion sector while, in the odd-fermion sector, is derived from the special mode. By picking to be large enough, we could force , resulting in being close to zero for the odd-fermion sector and large and negative for the even-fermion sector.
An obviously modified version of this discussion of other quench and sector-dependent quantities also holds for the Kitaev model of the subsequent section.
Iii Kitaev’s honeycomb model
We now explore a model that is truly topological in that while it possesses global topological order and associated degeneracies, it has no local order: the Kitaev honeycomb model Kitaev2006 (), shown in Fig. 6 (see also section III.2 for the full Hamiltonian). The model is very rich in and of itself and has the elegant analytic solution pioneered by Kitaev as well as various alternate analytic approaches.
Before embarking on the relevant details necessary to analyze the Kitaev model in the context of our present work, we first outline how our analysis of the Kitaev model can be understood as a direct two-dimensional extension of the analysis of the previous section. Regardless of whether the reader is familiar with the Kitaev honeycomb, this discussion should make our main results for it clear.
iii.1 A two-dimensional extension of the transverse Ising chain
In the previous section, we studied the topological description of the Ising chain in terms of a BdG description of a one-dimensional fermionic spinless -wave superconductor in a ring geometry. The Hilbert space was divided into two sectors consisting of momenta that were quantized either according to periodic or anti-periodic boundary conditions and were associated with odd- and even-fermion parity, respectively. Depending on the parameters in the Hamiltonian, the energetics either allowed the two sectors to be degenerate in ground state energy or for the odd sector to have a higher sectoral ground state energy than that of the even sector.
With regards to quench dynamics, this mismatch in energy resulted in topological blocking in that if one started in the odd sector in the degenerate phase and quenched into the non-degenerate phase, the overlap with the final absolute ground state would be zero. As for evaluating overlaps between time-evolved quenched states and the final sectoral ground state, this was done by studying the simple dynamics of decoupled pairs of momentum states . The momenta were special since they respect and they dictated the fermion parity. The overlaps clearly showed different behavior that depended on the topological (odd/even) sector.
While the Kitaev honeycomb model has several complex, rich aspects, much can be understood by simply generalizing the above to two dimensions. We will see that the Kitaev model can be mapped to a spinless two-dimensional -wave superconductor and the analog of a ring becomes a torus. Topological requirements now dictate periodic or antiperiodic boundary conditions along the two independent ( and ) directions, yielding a total of four topological sectors. Unlike in the transverse Ising case, the boundary conditions and fermion parity are not simply related. But in the commonly-studied situation that the honeycomb system has no vortices, one which we confine ourselves to, the fermion parity is constrained to be even. As a result, we find that as a function of parameter space, there exist three different phases in which all four sectors have degenerate ground states (Abelian phases). On the other hand, a fourth phase (non-Abelian phase) has its absolute ground state in three of the sectors while the fourth sector has higher sectoral ground state energy.
Thus, similar to the transverse Ising case, topological blocking occurs in one out of the four topological sectors. When evaluating overlaps between time-evolved quenched states and final sectoral ground states, the special momenta are . In Fig. 9 we show the typical overlap data for all 4 sectors over the course of a quench. By symmetry, two of the time-evolved overlaps and show identical behaviors. The overlap is generally different from these other two sectors but this is a finite size effect and quickly vanishes for large system sizes. The last overlap from the fully periodic sector is distinctly higher that the other three. This is a consequence of topological blocking. In what follows we will explain in more detail the mechanism behind this.
iii.2 Kitaev honeycomb Hamiltonian
The Kitaev honeycomb system consists of spins on the sites of a hexagonal lattice. The Hamiltonian can be written as
where denotes a directional spin exchange interaction occurring between the sites connected by a -link; see Fig. 6.
Consider now products of operators along loops on the lattice, , where . Any loop constructed in this way commutes with the Hamiltonian and with all other loops. The shortest such loop symmetries are the plaquette operators
where the numbers through label lattice sites on single hexagonal plaquette. We will use the convention that denotes the -dimer directly below the plaquette. The fact that the Hamiltonian commutes with all plaquette operators implies that we may choose energy eigenvectors such that . If then we say that the state carries a vortex at . When we refer to a particular vortex-sector we mean the subspace of the system with a particular configuration of vortices. The vortex-free sector for example is the subspace spanned by all eigenvectors such that for all .
On a torus of -spins, there are plaquette () operators. In general one has the relationship and so there are independent plaquette operators. We can find two more independent loop operators which we define as overlapping products of and or and operators which go around homologically non-trivial paths on the torus. We call two such operators, which go through the origin, and respectively, see Fig. 6. We will see that the operators and play a role similar to the operator of the one-dimensional transverse Ising model.
Counting these two operators and together with the plaquettes gives a total of independent symmetries. The different sectors are selected by choosing the respective eigenvalues , and . The remaining degrees of freedom are taken up by fermions (for example one for each -link) with the constraint on fermionic parity taken into account.
The breaking of -symmetry is essential for relating the model to a chiral -wave superconductor. Following the work of Ref. Kitaev2006, , we use the three-body term
with the second summation running over the six terms
For simplicity, in this work we will retain only the terms and .
iii.3 Fermionized solution and phase diagram
The Kitaev honeycomb Hamiltonian can be solved in several different ways. The method implicitly adopted here is the fermionization procedure used in Refs. Kells2008, ; Kells2009, . The procedure involves expressing the -dimers in terms of hard-core bosons and effective spins and then employing string operators to convert bosonic operators to fermionic ones. Importantly we can associate the presence of a fermion with an antiferromagnetic configuration of the dimer.
In the limit, the ground state manifold contains no fermions (spins connected by a link point in the same direction). The remaining degrees of freedom are specified through the eigenvalues of the plaquette operators and the loop operators and . It was shown by Kitaev [see Ref. Kitaev2006, ] that this manifold can be perturbatively mapped on the order to a toric code Hamiltonian
with where is the projector to the ferromagnetic subspace. In this limit, because the projector preserves the eigenvalues of and and because the operators and do not appear, there are four ground states (labeled by the eigenvalues and ) with no vortices. As the relative values of and become larger, the ground states acquire non-zero fermionic components. However, the overall parity of these states cannot change and it can be proved that the ground states are always vortex-free Lieb1994 (). Hence, given that the zero vortex sector in the toric code limit has no fermions, in the full Kitaev model, this sector, which contains the ground state, has even parity.
In the vortex-free sector of the Kitaev model, , and the associated translationally invariant Hamiltonian can be expressed in momentum space. In terms of fermionic momentum-space operators , the Hamiltonian takes the BdG form Kells2009 ()
Here, denotes the two-dimensional vector given by momentum components . Thus, the Kitaev honeycomb system maps to a spinless fermionic BdG Hamiltonian, which when compared to that associated with the transverse Ising chain in the previous system, can be regarded as a two-dimensional extension. All terms in carry net angular momentum and thus the superconducting gap is of -wave nature. The three-body terms in Eqs. (36-37) can be seen to open the gap in the phase of the model and provide a -symmetry breaking component that makes the system chiral.
As in Sec. II.1 for the 1D case, we diagonalize the BdG Hamiltonians by defining the Bogoliubov-Valatin operators
As with the 1D case, the modes with and require a special analysis since they satisfy . Further, for these modes; hence, . The diagonalized Hamiltonian once more takes the form
The ground state of this has the BCS form,
which is annihilated by all the , and has the energy .
The form of the dispersion in Eq. (45) enables us to derive the phase boundaries and gapped/gapless nature of the phases in the honeycomb system. We assume that . As we mentioned above, with this convention the fermions are associated with antiferromagnetic configurations of the -dimers and our vacua are toric code states on an effective square lattice Kells2009 ().
We first consider the case ; then Eqs. (42) are the same as those used in previous work involving quenches in the Kitaev honeycomb model, namely Ref. Sengupta08, with and . From the dispersion, it can be seen that the system is gapless in the range , and by symmetry, within similar constraints on and . Thus, as was originally discussed by Kitaev, the system has four phases Kitaev2006 (). The system is gapped in three of the phases, , and , having , , and respectively. These are called Abelian phases because the low-energy excitations satisfy Abelian statistics under exchanges. In the fourth phase, called , each of the is less than the sum of the other two couplings. The spectrum is gapless in this phase. (This makes it difficult to compute the statistics of the low-energy excitations since even a slow exchange of two of them inevitably produces other low-energy excitations). For instance, if and , we find that the spectrum is gapless at two points given by . The spectrum close to these points has the gapless Dirac form with the Dirac cones touching at those points.
If we now make , phase also becomes gapped, with the minimum gap occurring at the two points mentioned above if is small. The low-energy excitations in this phase are then found to satisfy non-Abelian statistics.
The four phases are separated by quantum phase transition lines on which one of the is equal to the sum of the other two couplings. As is standard, the four phases can be depicted in the triangular phase diagram shown in Fig. 8.
iii.4 Topological degeneracy
The topological nature of the four phases can be directly gleaned by putting the system on a torus. We discussed briefly above how the four-fold degeneracy of the -phases could be understood by mapping perturbatively to the toric code. Let us now see how this looks within the exact fermionic solution of the model where we can also understand the three-fold degeneracy of the non-Abelian phase and the gapless nature of the blocked sector.
We remark here that while most of the analysis for the transverse Ising system can be extended into two dimensions for the Kitaev honeycomb system, one crucial difference occurs with regard to fermion parity. In the Ising system, two sectors were allowed based on fermion parity, namely odd and even sectors, and while these were degenerate in one phase, they were not so in the other. Here, all states in the vortex-free sector have even fermion number parity, as argued after Eq. (38). As we shall see below, the degeneracies come about from different combinations of even fermion occupation.
We assume that the number of sites in the and directions are and , with the first site linked to the th site along each direction. On the torus, the diagonalized Hamiltonian has a restricted set of momentum modes in its form
where the dispersion relation is given in Eq. (45). The allowed values of in the various homology sectors on the torus are for integer , where the four topological sectors corresponding to have values of given by . The topological sectors dictate whether the wave functions are periodic or antiperiodic. The relationship between the topological sectors and the periodicity/antiperiodicity of the wave functions is simple if a little counter intuitive. For example the fully periodic sector has the quantum numbers , while the fully antiperiodic sector has quantum numbers .
We know that the ground state in the vortex-free sector has even-fermion parity. It can then be shown that in the three topological sectors corresponding to