Dynamics of an inhomogeneous quantum phase transition

We argue that in a second order quantum phase transition driven by an inhomogeneous quench density of quasiparticle excitations is suppressed when velocity at which a critical point propagates across a system falls below a threshold velocity equal to the Kibble-Zurek correlation length times the energy gap at freeze-out divided by $\hbar$. This general prediction is supported by an analytic solution in the quantum Ising chain. Our results suggest, in particular, that adiabatic quantum computers can be made more adiabatic when operated in an"inhomogeneous"way.


INTRODUCTION
A quantum phase transition is a qualitative change in the ground state of a quantum system when one of the parameters in its Hamiltonian passes through a critical point. In a second order transition a continuous change is accompanied by a diverging correlation length and vanishing energy gap. The vanishing gap implies that no matter how slowly a system is driven through the transition its evolution cannot remain adiabatic near the critical point. As a result, after the transition the system is excited to a state with a finite correlation lengthξ whose size shrinks with increasing rate of the transition. This scenario, known as Kibble-Zurek (KZ) mechanism (KZM), was first described in the context of finite temperature transitions [1,2]. Although originally motivated by cosmology [1], KZM at finite temperature was confirmed by numerical simulations of the time-dependent Ginzburg-Landau model [3] and successfully tested by experiments in liquid crystals [4], superfluid helium 3 [5], both high-T c [6] and low-T c [7] superconductors, and convection cells [8]. Most recently, spontaneous appearance of vorticity during Bose-Einstein condensation driven by evaporative cooling was observed in Ref. [9]. However, the quantum zero temperature limit, which is in many respects qualitatively different, remained unexplored until recently, see e.g. Refs. [10][11][12][13][14][15][16][17][18][19][20]. The recent interest is motivated in part by adiabatic quantum computation or adiabatic quantum state preparation, where one would like to cross a quantum critical point as adiabatically as possible, and in part by condensed matter physics of ultracold atoms, where it is easy to manipulate parameters of a Hamiltonian in time and which, unlike their solid state physics counterparts, are fairly well isolated from their environment. In fact, an instantaneous quench to the ferromagnetic phase in a spinor BEC resulted in finite-size ferromagnetic domains whose origin was attributed to KZM [21]. However, since the transition rate was effectively infinite in that experiment, the KZ scaling relation between the average domain sizeξ and the quench rate has not been verified.
The KZM argument is briefly as follows [2,12]. When a transition is driven by varing a parameter g in the Hamil-tonian across an isolated critical point at g c , then we can define a dimensionless distance from the critical point as When ǫ → 0 the correlation length ξ in the ground state diverges as ξ ∼ |ǫ| −ν , and the energy gap ∆ between the ground state and the first excited state vanishes as ∆ ∼ |ǫ| zν . Setting = 1 from now on, a diverging ∆ −1 ∼ |ǫ| −zν is the shortest time scale on which the ground state can adjust adiabatically to varying ǫ(t). A generic ǫ(t) can be linearized near the critical point ǫ = 0 as where the coefficient τ Q is called a quench time. Assuming that the system was initially prepared in its ground state, its adiabatic evolution fails at aǫ when the timet left to crossing the critical point equals the shortest time scale ∆ −1 on which the ground state can adjust. Solving this equality, we obtain Fromǫ the evolution becomes impulse, i.e. the state does not evolve but remains frozen in the ground state atǫ, until −ǫ when the evolution becomes adiabatic again. In this way, the ground state atǫ with a KZ correlation lengthξ becomes the initial excited state for the adiabatic evolution after −ǫ. In particular,ξ −1 determines density of quasiparticles excited during the phase transition in D dimensions. Note that when τ Q is large, thenǫ is small and the linearization in Eq. (2) is self-consistent because the KZM physics happens very close to the critical point between −ǫ and +ǫ.

II. INHOMOGENEOUS TRANSITION
As pointed out in the finite temperature context [22], in a realistic experiment it is difficult to make ǫ exactly homogeneous throughout a system. For instance, in the superfluid 3 He experiments [5] the transition was caused by neutron irradiation of helium 3. Heat released in each fusion event, n + 3 He → 4 He, created a bubble of normal fluid above the superfluid critical temperature T c . As a result of quasiparticle diffusion, the bubble was expanding and cooling with a local temperature T (t, r) = exp(−r 2 /2Dt)/(2πDt) 3/2 , where r is a distance from the center of the bubble and D is a diffusion coefficient. Since this T (t, r) is hottest in the center, the transition back to the superfluid phase, driven by an inhomogeneous parameter proceeded from the outer to the central part of the bubble with a critical front r c (t), where ǫ = 0, shrinking with a finite velocity v = dr c /dt < 0. A similar scenario is likely in the ultracold atom gases in magnetic/optical traps. The trapping potential results in an inhomogenous density of atoms ρ( r) and, in general, a critical point g c depends on atomic density ρ. Thus even a transition driven by a perfectly uniform g(t) will be effectively inhomogeneous, with the surface of critical front, where ǫ = 0, moving with a finite velocity. According to KZM, in a homogeneous symmetry breaking transition, a state after the transition is a mosaic of finite ordered domains of average sizeξ. Within each finite domain orientation of the order parameter is constant but uncorrelated to orientations in other domains. In contrast, in an inhomogeneous symmetry breaking transition [22], the parts of the system that cross the critical point earlier may be able to communicate their choice of orientation of the order parameter to the parts that cross the transition later and bias them to make the same choice. Consequently, the final state may be correlated at a range longer thanξ, or even end up being a ground state. In other words, the final density of excited quasiparticles may be lower than the KZ estimate in Eq. (6) or even zero.
From the point of view of testing KZM, this inhomogeneous scenario, when relevant, may sound like a negative result: an imperfect inhomogeneous transition supresses KZM. However, from the point of view of adiabatic computation or adiabatic state preparation it is the KZM itself that is a negative result: no matter how slow the homogeneous transition is there is a finite density of excitations (6) which decays only as a small power of transition time τ Q . From this perspective, the inhomogeneous transition may be a way to suppress KZ excitations and prepare the desired final ground state adiabatically.
To estimate when the inhomogenuity may actually be relevant, in a similar way as in Eq. (2), we linearize the parameter ǫ(t, n) in both n and t near the critical front where ǫ(t, n) = 0: Here n is position in space, e.g. lattice site number, α is a slope of the quench and v is velocity of the critical front.
When observed locally at a fixed n, the inhomogeneous quench in Eq. (9) looks like the homogeneous quench in Eq. (2) with The part of the system where n < vt, or equivalently ǫ(t, n) < 0, is already in the broken symmetry phase. The orientation of the order parameter chosen in this part can be communicated across the critical point not faster than a threshold velocitŷ v ≃ξ t .
When v ≫v the communication is too slow for the inhomogenuity to be relevant, but when v ≪v we can expect the final state to be less excited than predicted by KZM. Given the relation (10), the condition (11) can be solved either asv or as a relation between the threshold transition time and the slope,τ This relation means that, for a given inhomogenuity α, the transition is effectively homogeneous when τ Q ≪τ Q , but the inhomogenuity becomes relevant when the transition is slow enough, τ Q ≫τ Q . In the homogeneous limit α → 0, the threshold transition timeτ Q → ∞. The threshold velocity in Eq. (11) appeared for the first time in the context of finite temperature classical phase transitions [22] where it looks formally the same, but the underlying physics is qualitatively different: the scalesξ andt are determined not by the gap of a quantum Hamiltonian, but the relaxation time of an open classical system. Nevertheless, the key mechanism that non-zero order parameter penetrates from the symmetry broken phase into the symmetric phase ahead of the critical front seems to be the same.
In the next Section we rederive results (12,13,14) from a different perspective.

III. KZM IN SPACE
References [23,24] considered a "phase transition is space" where ǫ(n) is inhomogeneous but timeindependent. In the same way as in Eq. (9), this parameter can be linearized in n − n c , near the static critical front at n = n c where ǫ = 0. The system is in the broken symmetry phase where n < n c and in the symmetric phase where n > n c . In the first "local approximation", we would expect that the order parameter behaves as if the system were locally uniform: it is nonzero for n < n c only, and tends to zero as (n c − n) β when n → n − c with the critical exponent β. However, this first approximation is in contradiction with the basic fact that the correlation (or healing) length ξ diverges as ξ ∼ |ǫ| −ν near the critical point and the diverging ξ is the shortest length scale on which the order paramater can adjust to (or heal with) the changing ǫ(n). Consequently, when approaching n − c the local approximation (n c − n) β must break down when a local correlation length ξ ∼ [α(n c −n)] −ν equals the distance remaining to the critical point (n c − n). Solving this equality with respect to ξ, we obtainξ From n−n c ≃ −ξ the evolution of the order parameter in n becomes "impulse", i.e, the order parameter does not change until n−n c ≃ +ξ in the symmetric phase where it begins to follow the local ǫ(n) again and quickly decays to zero on the same length scale ofξ. This "KZM in space" predicts that a nonzero order parameter penetrates into the symmetric phase to a depth The critical point is effectively "rounded off" on the length scale ofξ. As a consequence, we expect a nonzero gap scaling as∆ as opposed to the local approximation, where we would expect gapless quasiparticles near the critical point. We expect the finite gap in Eq. (18) to prevent excitation of the system even when the critical point n c in Eq. (24) moves with a finite velocity, n c (t) = vt, up to a threshold velocitŷ v ∼ξ which is identical with thev in Eq. (13).
In the following Sections we test these predictions in the quantum Ising chain.

IV. QUANTUM ISING CHAIN
The model is For N → ∞, a uniform system with g n = g has two critical points at g = ±1 separating a ferromagnetic phase, when |g| < 1, from two paramagnetic phases, when |g| > 1. We focus on the critical point at g = 1 when ǫ = g − 1. Given z = 1 and ν = 1, we expect v ≃ 1 independent of either τ Q or α. The quench is exactly solvable for homogeneous g [13,15], but even in an inhomogenous case some useful analytic insights can be obtained as follows. After Jordan-Wigner transformation to spinless fermionic operators c n , σ x n = 1 − 2c † n c n and σ z n = − c n + c † n m<n (1 − 2c † m c m ), Eq. (20) becomes with ω m ≥ 0. Here u ± nm ≡ u nm ± v nm .

V. ISING CHAIN: KZM IN SPACE
To begin with, we consider the ground state of the quantum Ising chain in a static inhomogeneous tranverse field g n which can be linearized near the critical point g = 1 as compare with Eq. (15). The chain is in the (broken symmetry) ferromagnetic phase where n < n c and in the (symmetric) paramagnetic phase where n > n c . We want to know if the nonzero ferromagnetic magnetization Z n = σ z n in the ferromagnetic phase penetrates across the critical point into the paramagnetic phase and what is the depth δn of this penetration.
Since in a homogeneous system quasiparticle spectrum is gapless at the critical point only, we expect low energy quasiparticle modes u ± n,m to be localized near the critical point at n c where we can use the linearization in Eq. (24). We also expect that these low energy modes are smooth enough to treat n as continuous and make a long wavelength approximation in Eq. (23). Under these assumptions, we obtain a longwavelength equation After some algebra, its eigenmodes can be found as where is a rescaled position, ψ m≥0 (x) are eigenmodes of a harmonic oscillator satisfying and ψ −1 (x) = 0. As expected, the modes in Eq. (27) are localized near n = n c where x = 0. A typical width of the lowest energy eigenmodes is δx ≃ 1, or equivalently When α ≪ 1 then δn ≫ 1 and the long wavelength approximation in Eqs. (25,26) is self-consistent. Thus δn in Eq. (30) is the relevant scale of length near n c and we expect that this δn determines the penetration depth of the spontaneous ferromagnetic magnetization into the paramagnetic phase. We test this prediction by a numerical solution for an inhomogeneous transverse magnetic field which is shown in Fig. (1) with a variable slope α. This field can be self-consistenly linearized near n = n c as in Eq. (24) because, when the slope α ≪ 1, the predicted δn ≃ α −1/2 is much shorter than the width α −1 of the tanh. Figures 2A and B show how the spontaneous ferromagnetic magnetization Z n = σ z n from the ferromagnetic phase, where n < n c , penetrates into the paramagnetic phase, where n > n c . In particular, the collapse of the rescaled plots in Fig. 2B demonstrates that the penetration depth is δx ≃ 1 equivalent to δn ≃ α −1/2 , as predicted in Eqs. (17,30) and Ref. [23]. Paramagnetic spins near the critical point are biased towards the direction of spontaneous magnetization chosen in the ferromagnetic phase.

Moreover, the analytic solution (27) implies a finite (relevant) gap∆
in accordance with the scaling ∼ α 1/2 predicted by the general Eq. (18) and the numerics in Ref. [23]. This gap is the energy of the lowest relevant (even parity) excitation of two quasiparticles.

VI. ISING CHAIN: INHOMOGENEOUS TRANSITION
Let the critical front in Eq. (24) and Fig. 1 move with a velocity v > 0: A t-dependent version of the long-wavelength Eq. (26), can be solved exactly for both v < 2 and v > 2 with qualitatively different solutions in the two regimes. Not incidentally, v = 2 is the maximal velocity of quasiparticles at the critical point whose dispersion is ω = 2|k| for small |k| ≪ π.
A. Case of v < 2 When v < 2 equation (34) has solutions where m = 0, 1, 2, ..., the phase ϕ = arcsin(v/2)/2, and is a rescaled position. When v → 0 we recover the static solutions (27). In the reference frame of x v , which is co-moving with the critical point, the solutions (35) are stationary modes with ω m ≥ 0 so there are no quasiparticles in the system, and, in particular, no kinks where g n = 0. As shown in Figs. 2C and D, ferromagnetic correlations penetrate across the critical point into the paramagnetic phase to a depth δx v ≃ 1 equivalent to The penetration depth δn v shrinks to 0 when v → 2 − suggesting communication problems across the critical point when v > 2. The same δn v is a typical width of the lowest eigenmodes in the spectrum (35). As it shrinks to 0 when v → 2 − , the eigenmodes become inconsistent with the long-wavelength approximation in Eq. (34).

B.
Case of v > 2 When v > 2 then equation (34) can be mapped to a homogeneous transition. Indeed, we replacẽ introducing local timet measured from the moment the critical point passes through n, and simultaneously make a transformation bringing Eq. (34) to a new form Here Up to an unimportant rotation of a Pauli matrix σ y → σ v and the momentumdependent energy shift 4 iv ∂ñ, the new Eq. (41) is a homogeneous version of the old Eq. (34), but with a longer effective quench timeτ Consequently, a quasimomentum representation (ũ + ,ũ − ) = (a k , b k ) exp(ikñ − 4ikt/v)/ √ 2π brings the homogeneous Eq. (41) to the Landau-Zener form: where s = kt is a new time variable and δ k = 1/4k 2τ Q is a new transition rate. The Landau-Zener formula p k = exp(−π/2δ k ) gives excitation probability for a quasiparticle k and density of excited quasiparticles is where Λ ≃ 1 is an ultraviolet cut-off. The integral is accurate forτ Q ≫ 1. When v ≫ 2 the density is the same as the density after a homogeneous quench with the same τ Q , see Ref. [13], but when v → 2 + then d is suppressed below the "homogeneous" density d KZM by the factor (1 − 4/v 2 ) 3/4 .

C. Numerical results
Since the long-wavelength Eq. (25) does not give self-consistent long-wavelength solutions when v → 2, we simulated the exact time-dependent version of the Bogoliubov-de Gennes equations  Figure 4 demonstrates good quantitative agreement between Eq. (43) and numerical results, despite the breakdown of the long-wavelength approximation near v = 2.

VII. CONCLUSION
We made the general estimate Eqs. (11,12,13,14) when an inhomogeneous quench cannot be considered homogeneous with respect to KZM. Then we solved the problem in detail in the particular case of the quantum Ising chain where z = 1 and the threshold velocityv = 2 is equal to velocity of quasiparticles at the critical point. Excitation of kinks is dramatically suppressed when a critical front propagates slower thanv and the ferromagnetic phase is able to communicate its choice of ferromagnetic polarization to the paramagnetic phase ahead of the front. In contrast, when the front is much faster thanv the communication across the front is not efficient enough and kinks are excited as in the homogeneous KZM. However, even abovev density of excited kinks is suppressed below which is significantly less than 1 when v is close tov + . Thus the general estimates (11,12,13,14) are confirmed by our solution of the quantum Ising chain, but we leave their interesting implications when z = 1 or in more than one dimension for future exploration.
The estimates and the solution suggest that "inhomogeneous" adiabatic quantum computers can be more adiabatic than their "homogeneous" counterparts.