Quenched topological boundary modes can persist in a trivial system

Topological boundary modes can occur at the spatial interface between a topological and gapped trivial phase and exhibit a wavefunction that exponentially decays in the gap. Here we argue that this intuition fails for a temporal boundary between a prequench topological phase that possess topological boundary eigenstates and a postquench gapped trivial phase that does not possess any eigenstates in its gap. In particular, we find that characteristics of states (e.g., probability density) prepared in a topologically non-trivial system can persist long after it is quenched into a gapped trivial phase with spatial profiles that appear frozen over long times postquench. After this near-stationary window, topological boundary mode profiles decay albeit, slowly in a power-law fashion. This behavior highlights the unusual features of nonequilibrium protocols enabling quenches to extend and control localized states of both topological and non-topological origins. Floquet topological engineering describes the driving of a topologically trivial system into transitory non-trivial state and there are many open questions about the underlying mechanisms. Here, the authors theoretically investigate the time evolution of a topological mode when the bulk band structure is driven into a topologically trivial structure, and show that it can survive unexpectedly long in a "frozen” regime, before gradually disappearing in the "melting” regime.

Here we argue that vestiges of spatially localized TBMs, which can be initially prepared in a gapped topologically non-trivial system, will persist long after it is quenched into a gapped trivial and uniform phase. Unlike the gapless postquench case (Fig. 1a), this persistence is characterized by a frozen (slower than powerlaw decay regime), followed by a melting (power-law decay) regime, see Fig. 1b. In particular, we find that even though TBMs are not eigenstates of the postquench phase and exist inside the postquench bulk gap, characteristics such as a spatially peaked TBM probability density persist and appear frozen over a long and tunable time window postquench. As we explain below, this frozen regime is a consequence of the curvature of the gapped post-quenched dispersion profile, and applies to generic localized initial states.
This near-stationary nature of the TBM postquench probability density contrasts with the gapless postquench case (Figs. 1a and 2a), and is particularly surprising given the rapid evolution of the postquench wavefunction (Figs. 1b and 2b). Indeed, this contrasts with recurrent Loschmidt-echo type responses found in many quenched systems [25][26][27] wherein observables oscillate as a function of time. Further, even as translational symmetry in the postquench Hamiltonian is maintained, the postquench probability density remains spatially localized (Fig. 1b) retaining its prequench profile. It decays as a power law at very long times.
As we explain in the "Results" section, the persistence of probability density arises from an interplay of TBM spatial localization and its fast dynamical evolution postquench. This will be shown to be the underlying mechanism for its frozen and melting behavioral regimes. It is a striking example of how quantum systems pushed far out-of-equilibrium can possess very unusual properties with no analog in equilibrium 10,[28][29][30] . We anticipate TBM persistence can be readily accessed in quenched systems that are directly sensitive to the probability density distribution that include in ultracold atoms setups 19,20,22,[31][32][33] , as well as in photonic 34 or circuit realizations [35][36][37][38] .

Results and discussion
Probability density and quench protocol. The persistence of TBMs can be most easily illustrated by a time-dependent Dirac Hamiltonian (two-band) that undergoes a quench at time t = 0 ( Fig. 1 where v is the Dirac velocity, r = (x, y), p = (p x , p y ), σ i , i = 1, 2, 3 are Pauli matrices, and time-varying mass In the following, we shall refer to the pre-(post-) quench Hamiltonian asĤ i (Ĥ f ). For t < 0 before the quench, we have M (x) satisfying M(x < 0) < 0 and M(x > 0) > 0, such that it describes a mass domain wall along the y-axis (x = 0). Due to the jump in sgn M(x), the domain wall carries a unit topological charge, and thus supports gapless TBM jψ p y i that linearly disperse as vℏp y . Their spatial wavefunction is well localized at the domain wall x = 0: where N is a normalization constant, and Φ(x) decays exponentially on both sides. The decay length, α, defines the spatial extent of Φ(x).
Here we will consider a TBM state jψ p y i prepared in a prequench bulk gap with |p y | < α −1 . At t = 0, the system is quenched via Eq. (2) and the mass parameter becomes uniform in space. As a result, jψ p y i no longer exist as eigenstates of the postquench Hamiltonian and evolve as exp½ÀiĤ f t=_jψ p y i. TBM jψ p y i postquench characteristics at t > 0 are most saliently captured by its probability density with explicit p y dependence suppressed hereafter for brevity. Postquench, the initial boundary-localized TBM state is no longer an eigenstate of the new spatial-translation-invariant Hamilto-nianĤ f ; instead its temporal evolution can be understood from the Larmor precession of its projected components in the postquench eigenbasis ofĤ f . Due to the tightly localized (in x) profile of the initial TBM, many postquench p x eigenstates are accessed. As a result, their destructive interference and multifrequency Larmor precession can lead to dephasing 29 and decay of the initial TBM. Indeed, for mα/(ℏv) ≲ 1, we find through direct numerical integration of Eq. (4), the probability density generically decays exponentially in time, Fig. 2a. However, when mα/(ℏv) ≫ 1, we find that the probability density no longer exhibits fast decay. Direct numerical integration of Eq. (4) yields an unusual regime wherein the initial TBM state freezes maintaining its original localized (in x) profile (Figs. 1b and 2b) at short times. As we explain below, this frozen TBM stays locked over times 0 ≤ t ≪ τmα/ℏv controlled by the size of the postquench gap m; here the characteristic time τ = α/2v. At longer times, the TBM melts, displaying a much slower powerlaw decay. As we now explain, this arises from a near-lock-step interference process between p x components enforced in the mα/ (ℏv) ≫ 1 regime.
To more concretely understand the freezing and subsequent melting of the TBM mode, we analyze the solution of 〈ρ〉 t,x obtained through direct evaluation of Eq. (4): . The quantities C(x, t) and S(x, t) represent the even and odd temporal components of the postquench dynamical evolution of the probability density and can be written as Cðx; tÞ ¼ ½ðF 0 ðx; tÞ þ F 0 ðx; ÀtÞ=2 and S x;y;z ¼ ½F x;y;z ðx; tÞ À F x;y;z ðx; ÀtÞ=2i with where j = 0, x, y, z, and the j-dependent weights as f 0 = 1 and (f x , f y , f z ) = d(p)/|d(p)|. Here ε p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is the (postquench) Dirac eigenenergy and Φðp x Þ is the Fourier transform of TBM profile Φ(x). In obtaining Eq. (5), we expanded exp½ÀiĤ f t=_ andP r in Eq. (4) as superpositions of the postquench energy eigenstates, and applied the Pauli matrix identity σ i σ j = δ ij + iϵ ijk σ k repeatedly, see full details in the "Methods".
Initially at t = 0, S(x, t = 0) = 0 and C(x, t = 0) reduces to the spatial profile Φ(x) of the topological boundary state yielding Eq. (5) that follows the profile of the TBM mode. For t > 0, however, both the arguments of S(x, t) and C(x, t) oscillate with time capturing the Larmor precession of the TBM projected components in the postquench eigenbasis. Indeed, given the wide range of p x eigenstates that the TBM state projects to, the temporal evolution of probability density in Eq. (5) involves (momentum-space) nonlocal interference between multiple distinct p x momentum modes, and can lead to a complex spatio-temporal evolution of the probability density.
When mα/(ℏv) ≲ 1, the dynamical phase factor exp½iε p t=_ in Eq. (6) rapidly oscillates as a function of p x . As a result, it suppresses the large momentum contributions to C(x, t) (and S(x, t)). Since the tight localization of the initial TBM depends on large p x momentum contributions, the initial state is rapidly eroded. On a physical level, this can be understood as an intrinsic dephasing phenomena where the multiple frequency oscillation get out of phase destroying the coherence of the initial state. In real space, this manifests as two wavetrains moving almost uniformly in opposite directions with a Fig. 1a) It displays a frozen probability density regime as well as a melting regime exhibiting a power-law decay of the TBM probability density. The prequench mass distribution is the same as in a, with a domain wall at x = 0, but the postquench mass mα/ _v = 20 is large (red). In all panels probability density snapshots are plotted using Eq. (4) taken at logarithmic time spacings. We used the initial TBM profile Φ(x) = sech(x/α) from Eq. (3).
Frozen TBMs and Wick-rotated diffusion. However, when mα/ (ℏv) ≫ 1, the Larmor precession of the postquench state for various p x components oscillate in near lock-step, see below. Indeed, in this limit, the oscillation frequency In the latter, we focus on small p y < α −1 ≪ m. As the tight localization of the TBM state arises from large p x~α −1 components, when mα/(ℏv) ≫ 1, the differences in the frequencies are severely suppressed. This near-lock-step oscillation dramatically slows the erosion of the TBM state and as we will now discuss, at short times can even halt it. To see this explicitly, we take ε p % Δ þ ð_vp x Þ 2 =ð2ΔÞ in Eq. (6) and directly integrate: with the postquench profile where G t (x) is an imaginary Gaussian kernel that results from the gapped dispersion in the large mα/(ℏv) ≫ 1 limit, see "Methods". This kernel G t (x) is the mathematical embodiment of the random interference processes between different momentum sectors, with its imaginary Gaussian nature paralleling Brownian motion scattering in a diffusive medium; it is fundamentally distinct from the dynamical behavior expected when the system is quenched into a gapless linearly dispersing Hamiltonian, e.g., m = 0 above. We note that F x,y contributions are small (suppressed in the mα/(ℏv) ≫ 1 limit) and do not contribute substantially to hρi t;x profile. For short times 0 < t ≪ mα 2 /(2ℏv 2 ) = τmα/(ℏv), the imaginary gaussian G t (x) in Eqs. (7) and (8) exhibit spatial oscillations far more rapid than Φðx þ x 0 Þ varies in x. Here τ = α/2v is an intrinsic timescale. As a result, at these short times F 0;z % N ΦðxÞe iΔt=_ , so that Cðx; tÞ % N ΦðxÞ cosðΔt=_Þ and S z % N ΦðxÞ sinðΔt=_Þ. Substituting into Eq. (5) we obtain the probability density: We note the probability density in this regime is flat to infinite order in vt/α (Fig. 2b) (blue shaded); any temporal decay in this frozen regime is non-analytic and slower than any power law. As a result, we term this a region of frozen probability density. This is particularly surprising since the prequench TBM state is not an eigenstate of the postquench HamiltonianĤ f . Perhaps even more striking is the fact that the period over which the TBM's probability density is frozen can be controlled by the postquench gap size m; increasing m gives a wider frozen window for TBM probability density, Eq. (9). This can be understood physically by noting that the time window over which the probability density remains frozen can be recast in a more physical intuitive way as the ratio of the spatial confinement of the prequench domain wall, α, and the typical group velocity of wavepackets postquench; larger m translates to slower postquench group velocities and hence longer frozen time window. At long times t ≫ mα 2 /(2ℏv 2 ), the situation is dramatically different with the imaginary gaussian G t (x) in Eqs. (7) and (8) slowly varying over large ranges of x. As a result, the frozen states "melts" with its spatial profile spreading slowly out according to the convolution Ψ t (x) and an overall probability density amplitude decaying as t −1 : This t −1 power-law melting decay (dashed line) conforms with that found from direct numerical integration in Fig. 2b (orange shaded), and even exhibits a data collapse for probability density taken at different values of x. In this regime, the spatial extent/ width of |Ψ t (x)| 2 grows as t thereby conserving the total probability density over all space. We note that the persistence of TBM postquench is consistent with the edge currents found in quantum quenches of Chern insulators 10 .
To understand this long-time behavior more intuitively, we note that G t (x) can be interpreted as a 1D heat kernel corresponding to a Wick-rotated "diffusion" process (i.e. evolution by Schrödinger's equation involving the id/dt instead of d/dt operator) with diffusion constant − iℏ/2Δ. While scattering processes in ordinary diffusion processes lead to a "Gaussianblurred" distribution characterized by a (real) Gaussian decaying kernel, the imaginary Gaussian kernel G t (x) represents the rapid interference effects from multiple non-coherent contributions in Schrödinger evolution. A large Δ results in slow spreading of the initial state, which by Eq. (8) is asymptotically governed by power-law decay, instead of Gaussian decay as in the more familiar real-time diffusion scattering processes. Physically, the slow melting TBM behavior can be likened to that of slow diffusion found in classical systems with large inertial masses; the large mα/(ℏv) ≫ 1 limit corresponds to the regime where postquench modes are energetically inaccessible.
While we have concentrated on a continuum description of the frozen and melting regimes in the main text, both frozen and melting behavior persist even on a lattice model (see detailed numerics of quench dynamics on a lattice in Supplementary Note 2) as well as at hard-wall boundaries. This agreement arises because the frozen and melting regimes are most sensitive to the long-wavelength evolution of the TBM and the mismatch between the TBM confinement, α, and the postquench Compton length scale ℏv/m. Indeed, this mismatch and the ensuing quench dynamics we describe here can manifest for other initially localized modes, topological or otherwise.
Pseudospin precession and creep current. The frozen-in-time probability density profile at short times described in Eq. (9) hides the fact that the TBM modes are not eigenstates of the postquench HamiltonianĤ f . Are all other observables similarly frozen in time as well? To further interrogate the TBM postquench, we consider its local (spatially resolved) pseudospin expectation with explicit p y dependence suppressed hereafter for notational brevity. 〈σ〉 t,x encodes the spinor-wavefunction information of the TBM. Prequench, 〈σ〉 is aligned alongŝ in theê 2 direction; here e 1,2,3 denote the x, y, z directions in a Bloch sphere. However, postquench, the eigenstates ofĤ f generically possess spinor components in all three directions. As a result, the dynamical evolution of 〈σ〉 t,x postquench involves a complex intertwining of precession and interference between wave components composing the spatially localized profile of the TBM state. Indeed, 〈σ〉 t,x possesses dynamics that generically departs from that of the Bloch equation.
To see this, we directly evaluate Eq. (11) in the same fashion as Eq. (5) above producing the closed form The first term tracks the persistence of the initial pseudospin directionŝ, while the second term represents a "pure" precession contribution; the other terms correspond to additional dynamical contributions from S, which arises physically from projection onto postquench eigenstates. Eq. (12) in fact describes the dynamical solution of any two-component state with an initial spatially inhomogeneous profile. In our case, Eq. (12) contains information on how the localization of the TBM interplays with precession effects. This generically yields a complex spatiotemporal and spin-dependent evolution, see explicit example in Supplementary Note 1. However, when mα/ℏv ≫ 1, the pseudospin expectation 〈σ〉 t,x of the TBM mode can act like a localized block spin with TBM pseudospin expectation at each x precessing in unison. Similar to the probability density described above, this also leads to a frozen regime, where the magnitude of the |〈σ〉 t,x | pseudospin is preserved for a sizeable period of time (Fig. 3a). Intuitively, in this limit, the d = (vℏp x , 0, m) precession axis points strongly towards in theê 3 direction, causing the initial pseudospinŝ to precess largely in theê 1 -ê 2 plane. Since the precession axis does not shift much across different p x and p y sectors due to large m, destructive interference is minimized and the quenched TBM states periodically return to their initial configurations: |〈σ〉 t,x | is preserved. Indeed, in this regime, various positions in the TBM have hσ x;y i x;t that oscillate in phase (Fig. 3b). At longer times t ≫ τmα/ℏv, |〈σ〉 t,x | melts, and decays as a power-law t −1 (Fig. 3a).
Since _ À1 ∂Ĥ f =∂p ¼ vðσ x ; σ y Þ, the pseudospin oscillations indicate a cyclotron-type motion of the TBM mode; in the large mα/ℏv ≫ 1, the oscillations are largely locked to a frequency 2m/h. Interestingly, when the magnitude of the pseudospin decays, the value of hσ x i or hσ y i does not come back to itself after a full revolution (Fig. 3c). This decay in pseudospin leads to a creep current: an uncompensated drift current of the TBM (see estimate in Fig. 3d) that drifts along the domain wall.
The persistence of TBM characteristics can form the basis for a strategy to extend TBM features long after the topological system that supports it is gone: namely, by quenching to a large gap wherein postquench eigenstates are energetically inaccessible. Since the persistent TBM probability density arises from the fast postquench Larmor precession enforced by large m, we expect that it is robust against inelastic scattering with energies far smaller than the gap scale as well as slowly varying disorder with typical lengths longer than the TBM width. The frozen and melting quenched probability density regimes can be readily prepared and measured in ultracold atom optical lattices 19,20,22,31,32 with its pseudospin components of the wavepacket independently extracted 31,32 , and its real-space dynamics tracked 22 or, more recently, via quantum computer digital simulations 39,40 .
While we expect that the frozen and melting quench dynamics that we discuss here to be most easily realized in ultracold-atomic optical lattices, or in photonic 34 or circuit platforms [35][36][37][38]41 wherein interactions are weak, it is interesting to consider how the quench dynamics will behave in electronic systems wherein interactions can be present. In an electronic platform, interactions can lead to dephasing that can disrupt the lock-step precession of the postquench states that underlie the Frozen and melting regime 29 . As a result, we expect the unusual frozen and melting regimes to only manifest at times shorter than the dephasing time of the electronic system.
Perhaps most striking is the contrast in behavior of TBMs across a sharp temporal boundary (as realized in the quench at t = 0) against those found at spatial boundaries. In the latter, boundary states (topological or otherwise) that exist in an energy gap typically exponentially decay over space into a gapped insulating bulk. Whereas in the former, TBM probability density persists, freezing for a long window then decaying in a power-law fashion even when there are no eigenstates in trivial gap. This underscores the stark difference between sharp spatial and temporal interfaces and displays the dichotomy between equilibrium and out-of-equilibrium responses.

Methods
Postquench dynamical evolution. In this section, we review how the TBM in Eq.
(3) of the main text and its characteristics (namely probability density and pseudospom expectation) evolve postquench. For generality, we do not make any assumption on the form of the initial state (unless otherwise stated), other than it can be spatially inhomogeneous in the x-direction. We shall also assume the most general postquench Hamiltonian that is translation-invariant in both directions. Hence our following results are applicable for a broader class of quench settings than those discussed in the main text.
Wavefunction evolution. The postquench (t > 0) system is described by the . The wavefunction of the where Φðp x Þ is the Fourier transform of Φ(x). The time-evolved wavefunction postquench can be directly evaluated as ψ p y ðr; tÞ ¼ r h je ÀiĤ f t jψ p y i.
Writing expðÀiĤ f tÞ ¼ ∑ p; ± e Çiε p t jε ± p ihε ± p j, and noting that jε ± p ihε ± p j ¼ 1 2 ðI ±dðpÞ Á σÞ p p , where rjp ¼ e ipÁr is the wavefunction of the momentum eigenstate, anddðpÞ ¼ dðpÞ=jdðpÞj, we obtain ψ p y ðr; tÞ ¼ ∑ p x ;r 0 e ipÁðrÀr 0 Þ ðcos ε p t À idðpÞ Á σ sin ε p tÞhr 0 jψ p y i ð13Þ In obtaining Eq. (13) we summed across the ± bands as well as inserted the resolution of the identity ∑ r 0 r 0 j i r 0 h j. Summing across r 0 ; p 0 x by writing out hrjψ p y i ¼ we obtain the compact form ψ p y ðr; tÞ ¼ e ip y y ½Cðx; tÞ À iSðx; tÞ Á σ Ψ ð0Þ ð14Þ where C and S are the same as the main text. We reproduce these here for the convenience of the reader In Eq. (14), we see that ψ(t, r) comprises two pseudospinor contributions, one proportional to cos ε p t in the direction of the original spinor Ψ ð0Þ , and another proportional to i sin ε p t for which Ψ ð0Þ has undergone adðpÞ Á σ spinor rotation.
Probability density dynamics. The evolution of the probability density postquench can be obtained from direct evaluation of Eq. (4) of the main text. This amounts to finding the square amplitude of the wavefunction above, jψ p y ðr; tÞj 2 . This can be evaluated as where we have suppressed the explicit x, t dependence of C, S for brevity. This expression can be readily simplified by recalling the vector identity: This yields the compact expression for probability density as whereŝ ¼ Ψ ð0Þ σ Ψ ð0Þ . Writing S Ã C À C Ã S ¼ 2iIm ðS Ã CÞ we obtain Eq. (5) in the main text.
We note, parenthetically, when d(p) describes a Dirac model, further simplifications can be made to Eq. (19). For example, for Dirac models ε p is even in p x . This forces C to be real, and Re[S] (Im[S]) to arise solely from the even(odd) components ofdðp x ; p y Þ. Hence Im ½S kê 1 and Re ½S ?ê 1 , and the time-evolved probability density reads as with the last term nonzero only for p y ≠ 0. In obtaining the above, we have used s ¼ê 2 for the initial TBM.
Pseudospin dynamics. The evolution of the pseudospin expectation (density) postquench can be evaluated in the same fashion as above. Using the C, S notation above, the pseudospin expectation density in Eq. (11) of the main text can be rewritten compactly as This expression can be readily simplified by repeated application of the identity σ a σ b ¼ δ ab I þ iϵ abc σ c and recalling the identity ϵ abc ϵ ade = δ bd δ ce − δ be δ cd . Here ϵ abc is the Levita-Cevita symbol, and a, b, c, d, e indices run over x, y, z. Focussing on the ath component of the pseudospin expectation density, we obtain Re-writing in terms of real and imaginary parts of S and C, we obtain Eq. (12) in the main text. In obtaining Eq. (12) we have noted the identity (u + iv) × (u − iv) = −2iu × v, where u and v are vectors with real components. Similar to that discussed above, simplifications to Eq. (22) arise when ε p is even in p x . This forces C to be real. In that case, Re[S] (Im[S]) arises from the integral of the even(odd) components ofdðp x ; p y Þ. IfdðpÞ is furthermore purely even/odd, Eq. (22) simplifies to and σ h i odd t;x ¼ŝðjCj 2 À jSj 2 Þ À 2 Im ½S Ã C þ 2 Re ½ðS Ã ÁŝÞS: ð24Þ In either case, the last term of Eq. (22) always disappears, since it requires S to have both real and imaginary parts.
Another useful case to study is that of the TBM modes discussed in the main text. Recalling that the TBM modes haveŝ ¼ê 2 , one can obtain the squared expected pseudospin magnitude (not to be confused with We note that Eq. (22) can also be used to deduce the expressions for general initial states withŝ ≠ê 2 .
Frozen and melting regimes. In this section, we provide a detailed description of how both frozen and melting regimes arise in the limit of large mα/ℏv ≫ 1. Before we proceed, we note C and S can be expressed as Cðx; tÞ where j = 0, x, y, z, and the j-dependent weights as f 0 = 1 and (f x , f y , f z ) = d(p)/|d (p)|. Since the dynamics of C(x, t) and S(x, t) are controlled by F j (x, t), we will analyze the temporal dynamics of F j (x, t) in the limit of mα/ℏv ≫ 1.
In so doing, we first note that in the large postquench mass limit we can write where Δ ¼ ðm 2 þ _ 2 v 2 p 2 y Þ 1=2 . We note parenthetically, when the postquench mass is large and by focussing on small p y < α −1 ≪ m/ℏv, we have Δ ≈ m.
The tight localization of the TBM state arises from large p x~α −1 components. When mα/(ℏv) ≫ 1, the differences in their precession frequency are very much suppressed and can be well approximated by Eq. (27). Applying Eq. (27) into the precession frequency of Eq. (26) we obtain where (f x , f y , f z ) ≈ (ℏvp x /Δ, ℏvp y /Δ, m/Δ), and we have re-written and integrating out p x by completing the square in the argument of the exponential function, we obtain where where we have changed dummy variables x À x 0 ! x 0 . We note that in obtaining the probability density profile in both frozen and melting regimes, F x,y contributions are small (suppressed in the large mα/ℏv ≫ 1 limit) and are negligible in hρi x;t . This is because while the integrand in F z is proportional to m/Δ → 1, the integrand in F x , F y are proportional to p x , p y respectively, which are much smaller than the gap size m.
Frozen regime. The imaginary Gaussian kernel in Eq. (31) plays a crucial role in the dynamical evolution of the probability density. At short times, G t (x) exhibit spatial oscillations far faster than the spatial variations of Φ(x). Indeed, Φ(x) varies significantly for x varying on order of the TBM width, α. In contrast, G t (x) oscillates rapidly on length scales of order ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2t_v 2 =Δ p , and for short times, this length scale can be far shorter than α. As a result, for short time windows (controllable by postquench m) we have Δα 2 /(2tℏv 2 ) ≫ 1. In this short time window this behavior picks out Φ(x) and we can approximate As a result, at these short times F 0;z ðx; tÞ % N ΦðxÞe iΔt=_ , so that Cðx; tÞ % N ΦðxÞ cosðΔt=_Þ and S z % N ΦðxÞ sinðΔt=_Þ, where we have noted that F x,y are suppressed for large mα/ℏv. Substituting into Eq. (5) in the main text we obtain the Frozen profile This frozen (i.e. near-stationary in time) profile closely conforms to that found from a direct numerical evaluation of the probability density found in Fig. 2b of the main text.
Interestingly, in this regime the imaginary Gaussian kernel in Eq. (31) varies slowly in x, and in particular, far more slowly than Φðx þ x 0 Þ varies in position. As a result, for x ≲ α close to the domain wall Ψ t % R Φðx þ x 0 Þdx 0 goes approximately as a constant. In this limit, we find that the probability density at different x ≲ α collapse onto each other and decay as displaying a universal x-independent power-law decay. Indeed, this collapse of probability density at different x positions in the long-time limit is seen in a direct numerical evaluation in Fig. 2b of the main text. Indeed, we expect that at sufficiently long times, when Ψ t (x) has spread out, larger x > α within the width of Ψ t (x) will also similarly exhibit a universal x-independent power-law decay.
Illustration: exactly solvable example. While we have obtained approximate expressions for Ψ t (x) [and hence, F j (x, t)] above, it is instructive to illustrate the "frozen" and "melting" regime behavior through an exactly solvable model. To do so we consider the case where the initial TBM state Φ(x) takes the form of a Gaussian: ΦðxÞ ¼ e Àx 2 =2α 2 , where α is the standard deviation. We can then obtain by direct integrating Eq. (31) Evidently, when t ( α 2 Δ _v 2 then we have Substituting in Eq. (30), we obtain F 0;z ðx; tÞ % e iΔt N expðÀx 2 =2α 2 Þ / N ΦðxÞe iΔt that leads to the frozen (near-stationary in time) probability density profile as discussed above.
In the opposite limit t ) α 2 Δ _v 2 , Substituting into Eq. (30), we similarly obtain the power-law decay in the probability density described in Eq. (35).

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.