Phase transition in space: how far does a symmetry bend before it breaks?

Wojciech H Zurek, Uwe Dorner

Abstract

We extend the theory of symmetry-breaking dynamics in non-equilibrium second-order phase transitions known as the Kibble–Zurek mechanism (KZM) to transitions where the change of phase occurs not in time but in space. This can be due to a time-independent spatial variation of a field that imposes a phase with one symmetry to the left of where it attains critical value, while allowing spontaneous symmetry breaking to the right of that critical borderline. Topological defects need not form in such a situation. We show, however, that the size, in space, of the ‘scar’ over which the order parameter adjusts as it ‘bends’ interpolating between the phases with different symmetries follows from a KZM-like approach. As we illustrate on the example of a transverse quantum Ising model, in quantum phase transitions this spatial scale—the size of the scar—is directly reflected in the energy spectrum of the system: in particular, it determines the size of the energy gap.

Keywords:

1. Introduction

Near the critical point of a second-order phase transition both the relaxation time (which determines ‘reflexes’ of the system) and the healing length (which sets the scale on which its order parameter ‘heals’ in space, i.e. returns to its equilibrium value) diverge:Embedded Image(1.1)Embedded Image(1.2)where ϵ is a dimensionless parameter, which measures the distance from the critical point. For instance, when the transition is caused by the change of the temperature T or by varying the parameter g of a Hamiltonian, ϵ is, respectively, given byEmbedded Image(1.3)where TC and gC are the critical values.

Divergences in τ and ξ that appear near the critical point as a consequence of these two equations are often referred to as critical slowing down (for obvious reasons) and as critical opalescence (as the fluctuation on a scale ξ that becomes large in the vicinity of a critical point cause, in some systems, variations of the optical properties on scales ∼ξ, which lead to opalescence). These divergences will play a crucial role in our discussion.

Phase transitions are usually investigated as equilibrium phenomena in homogeneous systems. But many intriguing questions arise when a system is driven at a finite pace from one phase to another. That this will inevitably happen in the cosmological setting was first pointed out by Kibble (1976, 1980) who noted that, owing to relativistic causality, cosmological phase transitions in a variety of field theoretic models necessarily lead to the formation of topological defects (such as monopoles or cosmic strings) and may have dramatic astrophysical consequences.

It was later pointed out by one of us (Zurek 1985, 1993) that analogues of cosmological phase transitions can be studied in the laboratory, and that the equilibrium critical scalings—equations (1.1) and (1.2)—can be used to predict various aspects of non-equilibrium dynamics of symmetry breaking, including the density of topological defects left behind by a non-equilibrium second-order phase transition (Zurek 1985, 1993, 1996; Kibble 2003).

The resulting theory (known as the ‘Kibble–Zurek mechanism’ or KZM) uses critical scalings of the relaxation time and of the healing length to estimate the size Embedded Image of the domains that choose the same broken symmetry (Zurek 1985, 1993, 1996). Owing to the universality of phase transitions, KZM can be applied on many energy scales, from the (cosmologically relevant) high-energy settings (including experiments such as RHIC or LHC), all the way to Bose–Einstein condensates at ultra-low temperatures.

In all of these situations, a frequent prediction is that the process of symmetry breaking will lead to the formation of topological defects. This is essentially inevitable when the broken symmetry phase (characterized by the homotopy group) permits their existence (Kibble 1976, 1980, 2003; Zurek 1985, 1993, 1996). Following the transition, topological defects should appear with the density of about one defect unit (e.g. one monopole or one Embedded Image-sized section of a string) per Embedded Image-sized domain. The value of Embedded Image, which results from this non-equilibrium process, can be deduced from the equilibrium near-critical behaviour. Estimating this size is therefore essential.

Here, we first summarize the key ideas and equations that lead to such estimates. We shall then investigate the behaviour of the order parameter in an inhomogeneous system, where the change between phases is imposed by a slowly varying externally controlled parameter, such as the field in the quantum Ising model. We shall calculate how far does the transition region between the two phases persist; how much does a symmetry that prevails on one side of the critical line ‘bend’ before it breaks, spontaneously, on the other side.

2. Dynamics of symmetry breaking

We consider a second-order phase transition that is traversed at a finite rate set by the quench time scale τQ,Embedded Image(2.1)The system will be able to adjust its state adiabatically as long as the rate of change imposed from the outside is slow compared with its reaction time given by τ, equation (1.1). This change from nearly adiabatic to approximately impulse behaviour will happen at an instant Embedded Image whenEmbedded Image(2.2)Thus, the state of the system's order parameter will in effect ‘freeze’ atEmbedded Image(2.3)Its evolution will restart only Embedded Image after the critical point is passed. The instant Embedded Image plays a key role in the establishment of the fluctuations that seed structures (such as topological defects) in the broken symmetry phase.1

The key instant Embedded Image corresponds toEmbedded Image(2.4)which in turn sets the characteristic spatial scale given by the corresponding healing lengthEmbedded Image(2.5)This is the estimate of the size of regions that break symmetry in a more or less coordinated manner (Zurek 1985, 1993, 1996). Our derivation subverts equilibrium properties of the system, its scaling in the vicinity of the critical point, to predict non-equilibrium consequences of the quench. The density of topological defects left behind in the wake of a phase transition is the best known (but not the only) example of such predictions of KZM.

These predictions were tested, extended and refined with the help of numerical simulations (Laguna & Zurek 1997, 1998; Yates & Zurek 1998; Antunes et al. 1999; Stephens et al. 1999, 2002; Hindmarsh & Rajantie 2000), and verified in a variety of increasingly sophisticated and reliable experiments in liquid crystals (Chuang et al. 1991; Bowick et al. 1994), superfluids (Hendry et al. 1994; Bäuerle 1996; Ruutu et al. 1996; Dodd et al. 1998), superconductors (Carmi et al. 2000; Monaco et al. 2002; Maniv et al. 2003; Monaco 2003, 2006, 2008) as well as other systems (Ducci et al. 1999; Casado et al. 2001, 2007). The majority of the experiments to date are consistent with KZM. The case of superfluid 4He may be an exception. There the initial reports of detection of KZM vortices (Hendry et al. 1994) were retracted (Dodd et al. 1998) after it turned out that the observed copious vorticity was inadvertently induced by stirring. It is still not clear if 4He behaviour is really at odds with the KZM predictions re-evaluated in view of the refined numerical estimates (Antunes et al. 1999). One problem is that the decay of vortex tangle generated by KZM may be faster than that generated by stirring: KZM leads to anticorrelations between neighbouring vortices, so phase-ordering kinetics is expected to proceed faster than when vortices are correlated, as when they originate from turbulent flows. An accessible recent summary of the experimental situation due to Kibble (2007) is a highly recommended and up to date complement to this brief discussion. Further experiments that may allow for a better control of various parameters would be obviously welcome. Gaseous Bose–Einstein condensates are one recent and very promising proving ground (Anglin & Zurek 1999; Dziarmaga et al. 2002, 2003) that has already yielded some exciting results (Sadler et al. 2006; Lahaye 2007; Scherer et al. 2007).

Recently, KZM theory was applied to quantum phase transitions (Dziarmaga 2005; Polkovnikov 2005; Zurek et al. 2005; Cherng & Levitov 2006; Damski & Zurek 2006, 2007, 2008; Dziarmaga 2006; Caneva et al. 2007; Cincio et al. 2007; Cucchietti et al. 2007; Fubini et al. 2007; Lamacraft 2007; Mostame et al. 2007; Saito et al. 2007). There, the system is at T=0 and the nature of its ground state changes discontinuously as a result of a continuous change of some parameter of its Hamiltonian. At the critical value of that parameter, the gap—energetic price of the lowest excitations, given by the difference between the energy of the ground and first excited states—is at its minimum, and it disappears in the limit of an infinite system. In a finite system, the quantum phase transition is then described by an avoided level crossing. The basic observation due to Damski (2005) is that KZM can be used to analyse the avoided level-crossing process, and that it yields results in excellent agreement with Landau–Zener theory (Zener 1932; Landau & Lifshitz 1958). As we shall see below, the analogy is based on the observation that, far away from the avoided level crossing, the quantum system usually starts its enforced evolution adiabatically, as was the case for second-order phase transitions. However, it can only react on the relaxation time scale given by the inverse of the gap. Therefore, near the avoided crossing, where the gap closes, its reflexes deteriorate, and the impulse approximation in the immediate vicinity of a quantum critical point is appropriate. Below, we give an illustration of KZM in action in the exactly solvable and paradigmatic quantum Ising model.

3. Quench in a quantum Ising model

According to Sachdev (1999), the quantum Ising model is one of the two prototypical examples of quantum phase transitions. It represents a chain of spins with the HamiltonianEmbedded Image(3.1)where Embedded Image are Pauli operators; W is the Ising coupling; and J(t) is due to the external (e.g. magnetic) field that attempts to align all spins with the x-axis. The phase transition from the paramagnetic state where all the spins are aligned with x by the strong external field (e.g. Embedded Image) to the low-field ferromagnetic and degenerate (in the large N limit) ground state that ‘lives’ in the Hilbert subspace spanned by the broken-symmetry basis Embedded Image and Embedded Image takes place when Embedded Image. Therefore, as suggested by equations (1.3) and (2.1), a key role is played by the relative coupling given byEmbedded Image(3.2)and we assume that it depends linearly on t.

All the relevant properties depend on the size of the gap Δ between the ground state and the first excited state. In an infinite system, the gap is given byEmbedded Image(3.3)The spectrum (and the gap) of a finite quantum Ising system is illustrated in figure 1. The gap sets the energy scale and is reflected in the relaxation time and the healing lengthEmbedded Image(3.4)Embedded Image(3.5)the latter given by the product of τ and the speed of sound Embedded Image (see Sachdev 1999), where a is the distance between spins. The divergence of τ and ξ is the critical slowing down and the analogue of critical opalescence, respectively.

Figure 1

Energies of the lowest excitations of the Ising chain for N=20 with open (rather than periodic) boundary conditions. The ground and first accessible excited states that define the gap are marked with a thicker line.

The instants Embedded Image when the behaviour switches from adiabatic to impulse happen when the reaction time of the system, equation (3.4), is the same as the time scale on which its Hamiltonian is altered (given by Embedded Image), i.e.Embedded Image(3.6)and thusEmbedded Image(3.7)As before, for Embedded Image, the state of the system will continue to adjust adiabatically to changes imposed by the decreasing J(t). However, at Embedded Image, before the critical point the evolution will cease, and it will restart only at Embedded Image after the transition, with the initial state similar to the one ‘frozen out’ at Embedded Image.

Using the relative coupling Embedded Image associated with Embedded Image, we getEmbedded Image(3.8)Embedded Image(3.9)Following KZM, we now predict the appearance of O(1) defects per Embedded Image. Their density should be approximatelyEmbedded Image(3.10)per spin. This is only an estimate: simulations of classical second-order phase transitions yield defect densities that scale with τQ in accordance with this reasoning, but are lower than the relevant power of Embedded Image of equation (3.9) so that a ‘unit of defect’ is often separated by approximately 10–15 Embedded Image (e.g. Laguna & Zurek 1997, 1998; Yates & Zurek 1998; Antunes et al. 1999; Stephens et al. 1999, 2002; Hindmarsh & Rajantie 2000).

There is a fundamental difference between quantum and thermodynamic phase transitions. In the thermodynamic case, ‘real’ fluctuations exist above the critical point. They can initiate the symmetry-breaking process; in effect, choose how symmetry breaks in domains that appear after the transition. In the case of quantum phase transitions, there are only ‘quantum fluctuations’, but they are virtual, so one cannot be certain that they will have an analogous effect on the post-transition state. Therefore, an explicitly quantum approach to the quantum Ising model is needed.

As we have already noted, a key feature of a quantum phase transition is the gap Δ. Actually, as can be seen in figure 1, the relevant gap (i.e. the gap between the ground and first accessible states) is not the symmetric Δ but rather 2Δ on the approaching side. Nevertheless, in the quantum Ising model, the gap disappears at the critical point when the system is infinite. When N<∞, this critical gap becomes small, but does not disappear (figure 1). This is important, and allows us to propose a purely quantum approach. Instead of density of defects in an infinite system, we can compute (as a function of quench time scale τQ) the length of the spin chain that, with probability p of about a half, can remain defect-free (in a ground state) after the quench. An excited state would (most likely) contain just a single excitation (e.g. a kink). Therefore, the inverse of the length that limits the excitation probability to approximately 0.5 should correspond to about half of the kink density.

The two lowest accessible levels of H in the vicinity of the critical point (figure 1) exhibit an avoided level crossing and we can calculate the excitation probability of the system driven through this avoided crossing using the Landau–Zener formula (LZF; Zener 1932; Landau & Lifshitz 1958),Embedded Image(3.11)Here, Embedded Image is the minimum gap between the two levels and v is the velocity with which the transition is imposed on the system given by Embedded Image far away from the avoided crossing. The KZM approach was shown by Damski to provide an excellent approximation to LZF (Damski 2005; see also Damski & Zurek (2006, 2008) for extensions). Using LZF, we compute the size Embedded Image of the spin chain that will probably remain in the ground state in the course of the quench with probability p=0.5. Equation (3.11) translates into a condition for the rate of quench that produces a kink with probability p,Embedded Image(3.12)Using Embedded Image (see equation (3.2)) and Embedded Image for the gap upon ‘closest approach’, we getEmbedded Image(3.13)This relates the size Embedded Image of a chain that will remain defect-free with the probability 1−p to the quench rateEmbedded Image(3.14)This LZF estimate is (surprisingly) accurate for p<0.5 even though there are many levels in the spectrum of the quantum Ising model. Such accuracy was surprising when this analysis was first presented (Zurek et al. 2005). Dziarmaga (2005, 2006) soon after demonstrated that this is no accident: in effect, the phase transition in the quantum Ising model can be decomposed into a collection of independent avoided level crossings.

We can now directly compare KZM, equation (3.10), and LZF predictions for defect densityEmbedded Image(3.15)The two estimates exhibit the same scaling with the quench rate and the parameters of H, equation (3.1). LZF predicts fewer defects than ‘raw KZM’ (Embedded Image when p is set, somewhat arbitrarily, to 0.5). This is not a big surprise: as seen in the numerical simulations, confirmed by the experiments and verified analytically in specific models, equations (3.9) and (3.10) provide correct scalings, but tend to overestimate densities (e.g. Laguna & Zurek 1997, 1998; Yates & Zurek 1998; Antunes et al. 1999; Stephens et al. 1999, 2002; Hindmarsh & Rajantie 2000). Figure 2 indicates that this conclusion holds also for the quantum Ising model.

Figure 2

(a) Number of kinks per spin in the quantum Ising model after a quench starting in the ground state at J=5W and ending at J=0, plotted as a function of the quench rate Embedded Image for N=50, 60, 70, 80, 90, 100 (solid lines; bottom to top). The scaling Embedded Image predicted by KZM is consistent with the simulations (see Dorner et al. (2003) for details of the numerical method). Agreement improves with the size of the system: for 100 spins, a fit gives Embedded Image (dashed line). As in the classical case (Laguna & Zurek 1997, 1998; Yates & Zurek 1998; Stephens et al. 1999, 2002; Hindmarsh & Rajantie 2000) equation (3.10) is an overestimate; the best fit is Embedded Image. (b) The total number of kinks for Embedded Image spins (dashed lines; bottom to top) after a quench as a function of the quench rate Embedded Image. Both the scaling Embedded Image predicted by KZM (solid straight lines; equation (3.10)), and the LZF estimate p (solid bend lines) when less than one kink is expected are valid. The straight lines are linear fits in the range (0.025, 0.25) yielding slopes between 0.66 and 0.58. Numerical data include these used in (a) but we now go beyond the expected range of validity of KZM. For sufficiently slow quenches, LZF provides reliable predictions. Very fast quenches are ‘all impulse’, levelling off of the expected number of kinks, as is indeed seen.

Figure 2 also shows that the kink density scales approximately as Embedded Image, equation (3.10), in the region of the validity of KZM, i.e. for Embedded Image less than 1 (so that quench is quasi-adiabatic at the beginning and at the end, but impulse near the critical point, i.e. when at least one defect is expected). The prefactor which is approximately 0.16 (0.12 if the steeper slope on the approach in figure 1 is taken) is not far from the previous experience (Laguna & Zurek 1997, 1998; Yates & Zurek 1998; Stephens et al. 1999, 2002; Hindmarsh & Rajantie 2000). For very slow quenches (Embedded Image, or Embedded Image), i.e. if the system is nowhere convincingly ‘impulse’, LZF is surprisingly accurate. We conclude that the two approaches work well in complementary regimes of quench rates, and predict the same scaling of the size of the broken symmetry domains with quench time.

The importance of the behaviour of the gap (and, in particular, of the smallest gap Embedded Image) for the quantum estimate of defect density is, in addition to the broader theme of connections between KZM and LZF, our principal general conclusion at this point. We shall return to discuss the gap in the quantum Ising model and the usefulness of the extensions of KZM and LZF to more complicated behaviours of the near-critical gap below.

4. Statics of symmetry breaking

Consider now a situation where the all-important dimensionless ϵ is time independent, but depends on the location in space instead. We suppose, in analogy with equations (2.1) and (3.2), that in the vicinity of the critical point ϵ(x) is approximately linear:Embedded Image(4.1)Clearly, sufficiently far from x=0, the order parameter will settle into an equilibrium state corresponding to the local value of ϵ. We are, however, left with an interesting question: how far, and how, does the influence of the critical point propagate?

There is an intriguing analogy between the freeze-out in time we have discussed in the preceding section and the freeze-out in space we are led to consider here. When ϵ changes in space slowly compared with the local healing length given by Embedded Image, equation (1.2), the order parameter will be able to adjust in space to the changes of ϵ, i.e. there should be a well-defined local value of the order parameter, local healing length ξ and local relaxation time τ, etc. However, very close to x=0, the system cannot heal fast enough: the healing length becomes large, much larger than x. Consequently, the critical opalescence—the divergence of Embedded Image—will open up a ‘scar’ in the order parameter that does not properly heal. Our aim now is to describe the consequences. To find out the size of the scar, we write down the spatial analogue of equation (2.2),Embedded Image(4.2)for the ‘adiabatic-impulse’ borderline point, Embedded Image. Thus,Embedded Image(4.3)Consequently, and in accord with previous discussion, we conclude that the order parameter will in effect freeze (or, to put it differently, that the scar opened up by the transition through the critical region will heal) at a distanceEmbedded Image(4.4)from the critical point. The distance Embedded Image corresponds toEmbedded Image(4.5)We are really done—there is just a spatial x, rather than both x and t, as before. But let us (as a consistency check) take a few more steps, and use this to estimate Embedded Image to calculate the characteristic spatial scale given by the corresponding healing length,Embedded Image(4.6)We know this already (see equation (4.4)). This is now the estimate of the size of the scar, the size of the region that is ‘still thinking’ about how to break symmetry. Earlier, our derivation of the freeze-out time and of the resulting analysis of the dynamics of symmetry breaking subverted equilibrium properties of the system—its scaling in the vicinity of the critical point—to predict non-equilibrium consequences of the quench. We have now repeated this subversive strategy in a new setting, by using the healing length (i.e. strictly speaking, defined in a homogeneous system) to find out the consequences of inhomogeneity that must be there in the vicinity of the critical point if the phase transition takes place in space. The structure of the scar in the order parameter that connects the two phases is now of interest per se.

There are two specific values of ν that are often encountered: as we have seen in §3, for the quantum Ising model ν=1. Therefore,Embedded Image(4.7)and the size of the scar is simply a geometric average of the two relevant lengths in the problem. We shall see below that, as a consequence, in the quantum Ising model the presence of such a scar widens a critical gap. In a mean field theory, we have ν=1/2. Consequently,Embedded Image(4.8)The thickness of the boundary between the two layers is a compromise that involves the characteristic healing length of the order parameter on the one hand, and the typical scale of the externally imposed inhomogeneity of the Hamiltonian on the other hand. The extent to which each of them has its say depends on the scaling exponent ν. The most obvious test of these predictions would then explore variations of the order parameter in the vicinity of an externally imposed inhomogeneity.

5. Opening a gap with a wedge of a field

To illustrate our considerations with a concrete example we return to the quantum Ising model we have already employed in the previous section. The Hamiltonian we shall use now has the same form as before, equation (3.1), but the field J, which before was the same for every spin, now has a form illustrated in figure 3. As a result, sufficiently far to the left of Embedded Image, the system will be in a paramagnetic phase, while to the right a ferromagnetic phase should be dominant.

Figure 3

Illustration of the spatially dependent field J(x) in the transverse quantum Ising model we consider. The parameter J(x) corresponds to J(t) from Hamiltonian (3.1) but now depends on position, Embedded Image. The spins are located at positions Embedded Image with Embedded Image.

Our task is to characterize the state of such a system. In a homogeneous quantum Ising spin system, the correlation function given byEmbedded Image(5.1)is a well-defined object, whose behaviour is thoroughly explored (Sachdev 1999). Indeed, the decay of such correlation functions is used to define the coherence length (which is in effect often equal to the healing length we have employed earlier). In a homogeneous system, Embedded Image obviously does not depend on the ‘reference spin’ l: each spin lives in an identical neighbourhood, so ζ(k) is translationally invariant with respect to l. But when a system is inhomogeneous—for example, when a phase transition occurs at some point within the system—the correlation function will obviously depend on where the reference spin l is. We therefore consider a conditional correlation function,Embedded Image(5.2)It depends explicitly on the location of the reference spin l. Formally, ζ(k) is given by Embedded Image averaged over all l. In a homogeneous case Embedded Image is independent of l, so this averaging is trivial. On the other hand, in an inhomogeneous system we are investigating here, such averaging would be counterproductive: the averaging would obscure precisely the signature we are looking for, that is, the imprint made by the inhomogeneity of the Hamiltonian on the state of the system.

In figure 4a, we show Embedded Image for the fixed reference spin l located in the ferromagnetic region. The behaviour of Embedded Image seen there is a good illustration of our previous discussion: for spins located in the ferromagnetic region, Embedded Image. However, when k assumes values such that Embedded Image falls within the paramagnetic region, the correlation function vanishes, Embedded Image. The transition between these two extremes is our focus of interest. We can characterize the size of the scar by quantifying the width of the transition regime. The most obvious way to do this is through the inverse of the slope of Embedded Image, where Embedded Image. The resulting width of the transition region is plotted in figure 4b. Evidently, the behaviour predicted by the KZM-like discussion for the quantum Ising model, equation (4.7), is reflected in the correlations between spins. Such correlation functions should be experimentally accessible, and may be a good way to test our predictions.

Figure 4

(a) Conditional correlation function, Embedded Image, for N=200, Embedded Image, Embedded Image and l=160. The reference spin located at Embedded Image is in the ferromagnetic region. (b) Width of the transition region of the conditional correlation function Embedded Image versus Embedded Image for N=200, Embedded Image and l=160. The dashed line is a fit to the data leading to Embedded Image.

We note that some of the effects of the inability of the order parameter to adjust to the inhomogeneous variations of the Hamiltonian may become apparent even if the system does not cross ‘all the way’ into the other phase. For instance, just coming close to the critical point may lead to the structures with sizes that can be estimated using the above approach. Similarly, when the same phase is separated by a narrow strip when the parameter that controls its phase crosses the critical point (i.e. a strip in space within which ϵ has a different sign than outside it), an extension of the above discussion should be applicable. There should be, therefore, lots of opportunities to experimentally test the above equations and plenty of variations of the homogeneity–inhomogeneity theme. One such obvious variation involves adopting a nonlinear spatial dependence of ϵ on x, i.e. a dependence that is different from equation (4.1). But there are clearly many more. We shall not attempt to enumerate them here.

Let us instead point out a less obvious consequence of a spatial inhomogeneity that will be especially important in quantum phase transitions: its effect on the gap. The spectrum of an inhomogeneous Ising system as a function of the location of the point Embedded Image, the place where J(x) becomes 0 (figure 3), is shown in figure 5. When this figure is compared with figure 1, the spectrum of a homogeneous system, two striking differences in the behaviour of the gap emerge: (i) instead of a sharp ‘corner’ where J/W=1, there is now a plateau which extends over a range of values of Embedded Image and (ii) moreover, this plateau is lifted above where it was (at Embedded Image) for a chain of N spins. In short, the sharp gap minimum we had before becomes now a wider, extended ‘bottleneck’. The spatial structure induced by the inhomogeneity in the state of the system should have an effect on the energetics. Indeed, one could venture a guess (based on ‘general principles’) that the size of the gap, which is the property of the whole inhomogeneous spin chain, will change with the size of the scar, Embedded Image. One could moreover speculate (and here the general principles are becoming too general for comfort) that the size of the gap should be inversely proportional to (some power of) Embedded Image. We shall now pose this very same question more precisely, and arrive at the estimate of the basic parameters of the bottleneck more directly.

Figure 5

Energies of the lowest excitations for N=50, Embedded Image versus Embedded Image. The ground and first accessible excited states that define the gap are marked with a thicker line.

The inhomogeneous Ising system we are dealing with can be typically divided into three regions: a ferromagnetic domain (where the Ising coupling W dominates), a paramagnetic domain (where J(x) is large) and what is in-between these two, the near-critical domain (where Embedded Image is small). The energetic price of excitations in either ferromagnetic or paramagnetic domains is large compared with the size of the gap, which (we remind the reader) in a near-critical chain of L spins is given by Embedded Image.

Our aim is to compute the size of the bottleneck gap in the inhomogeneous case of figure 3. We shall do that by a self-consistent estimate suggested by the above division of the whole inhomogeneous system into three domains. We first note that the energetic price of the excitations outside the near-critical regime is prohibitively large whenever these regimes are well defined, so such excitations will not be relevant for the calculation of the size of the gap (which is defined as the ‘price’ of the least energetically expensive excitation). On the other hand, the near-critical domain should be able to support collective excitations that, when the size of that domain is L, should be able to support collective eigenstates with energies given by Embedded Image. Thus, all we have to do is to find that size L. One might be at this stage tempted to venture a guess that Embedded Image, but let us proceed cautiously, and follow a line of reasoning based on energetics we have outlined above. To this end, we propose a self-consistency condition,Embedded Image(5.3)which in effect defines the near-critical domain. A plot that illustrates this inequality is shown in figure 6.

Figure 6

Illustration of inequality (5.3), which defines a length Embedded Image of the critical region.

At this point, the relevant Δ is not known. But (as Embedded Image is small in the near-critical domain) it is natural to employ Embedded Image, where now L stands for the number of spins that contribute to the collective state that defines the gap, i.e. the spins that fit inside the near-critical domain. Using Embedded Image of figure 3, we solve the above equation to obtain the extent, in space, of the near-critical domain that defines the size of the gap,Embedded Image(5.4)or, in number of spins (rather than in distance),Embedded Image(5.5)This immediately yields an estimate of the gap size,Embedded Image(5.6)With the help of these results, we now understand the basic structure of the eigenenergies, and, in particular, of the gap seen in figure 5. Figure 7 shows the size of the gap as a function of the slant Embedded Image. The square root dependence predicted by equation (5.6) is evident. Indeed, even the prefactor obtained through our simple estimate above is close to the one that is obtained from the numerical experiment. There are obvious limits to the validity of our derivation that are also in evidence with figure 7: when α is so small that the system as a whole is approximately homogeneous, the near-critical domain extends over the whole system, Embedded Image is comparable with, or larger than, N, and the asymptotic size of the gap Embedded Image for the homogeneous system is attained. On the other hand, when α is so large that the size of the scar becomes comparable with a, the spacing between the spins, the near-critical region disappears altogether and the size of the gap is set by 2W.

Figure 7

The size of the smallest gap between the ground and first accessible excited states, the level of the plateau in figure 5, as a function of the slope Embedded Image for N=50 spins. The dotted line is a fit to the data between Embedded Image and Embedded Image leading to Embedded Image. The horizontal dashed line corresponds to the minimum gap of a homogeneous spin chain.

6. Discussion

We have presented an approach to symmetry-breaking phase transitions that occur in space. As in the original KZM (which employs equilibrium near-critical scalings of the relaxation time τ and of the healing length ξ), it rests on the near-critical behaviour of ξ. While the healing length is defined in a homogeneous system, we have employed its variation with ϵ (dimensionless parameter that measures the distance from the critical point) to predict what happens near a critical point that occurs in space in an obviously inhomogeneous system. The resulting theory yields the size of the scar, of the transition region between the two phases.

We have applied these results to the quantum Ising model. There the gradient of the external field leads to a gradual realignment of the spins. The border between the paramagnetic and ferromagnetic phases has a structure that can be characterized by the correlations between spins, which change between the two patterns of alignment. We have seen that this change is gradual, and happens (as was predicted by our theory) on the scale inversely proportional to the square root of the gradient of the field. The simple theory leading to equation (4.6) results in correct scaling behaviour. As expected, a more specific calculation based on the quantum Ising model leads to a different prefactor (equations (5.4) and (5.5)). In effect, we now have two different measures of the size of the scar, one based on the correlations (figure 4) and the other on the size of the gap. We do not know which one of them (if any) is ‘correct’, but we point out that they can differ.

Existence of the ‘transition scar’ has a dramatic effect on the eigenspectrum of the system. In particular, instead of the simple avoided level crossing where the minimum gap helps set the probability that the system will remain in its ground state, we are now dealing with an extended ‘bottleneck gap’. The parameters of this bottleneck gap are (including a prefactor, somewhat in contrast with the case of the scar size estimates) in a surprisingly good accord with the theory.

One is now tempted to consider quantum phase transitions that occur both in space and in time. For thermodynamic phase transitions, this situation can be treated by the KZM-based approach of Kibble & Volovik (1997). Their conclusion was, in effect, that when the critical line moves through space with velocities in excess of Embedded Image, the quench that leads to spontaneous symmetry breaking will proceed as if it were homogeneous, and KZM estimates of defect densities relevant for homogeneous quenches will apply. On the other hand, in the limit of slowly propagating phase fronts defect production will be suppressed, as the symmetry-breaking choices made by the order parameter will be simply propagated with the phase front. This conclusion seems to be only weakly influenced by the spatial gradient of ϵ, and is now supported by numerical studies (e.g. Dziarmaga et al. 1999).

Our preliminary numerical simulations of the same problem in the quantum case indicate that there the situation is more complicated. Part of the problem can be traced to the fact that the dynamics of a quantum phase transition is strictly reversible. Thus, the energy deposited in the order parameter stays in the order parameter, and for example, in the quantum Ising model, has ‘nowhere to go’ except into the kinks. As a result, the rate of the creation of kinks now depends on the spatial gradient of ϵ, and does not become suppressed as dramatically as before when the front velocity falls below Embedded Image. Moreover, in quantum phase transitions, the limit when less than a single kink is expected in the whole system is explicitly quantum, and (as we have seen earlier) different from (although compatible with) the KZM scaling. However, in an inhomogeneous case, the simple Landau–Zener formula (suitable for gaps of figure 1) is obviously inapplicable. We have tried out its natural generalization (which approximates the spectrum of figure 5) with two obvious avoided level crossings connected with a plateau (figure 8a). The fit is imperfect. The resulting prediction of the probability of a kink (figure 8b) is also not very satisfying. But this calculation (while it predicts spurious oscillations, and overestimates the rate by about an order of magnitude) does capture one essential feature: the dependence of the probability p on the quench rate Embedded Image is now no longer exponential, as it was in the original LZF, but it becomes quadratic. Such quadratic dependence appears automatically, if rather surprisingly—given the exponential nature of LZF, equation (3.11)—in the avoided level-crossing problem when the transition starts at the place of the nearest approach of the two levels (Damski & Zurek 2006; Cucchietti et al. 2007), rather than far away from the avoided crossing, as is usually assumed (Chuang et al. 1991). The behaviour we observe may be due to the asymmetry of the gap. In effect, as the slanted potential of figure 3 traverses the spin chain, the system will generally approach the plateau of the bottleneck gap with a slope that is quite different from the slope on the other end of the gap, where it exits. Therefore, recent study (Damski & Zurek 2006, 2008)—showing that when avoided level crossing is traversed starting at the minimum of the gap, the probability of excitation p on the quench rate Embedded Image is no longer exponential but that it becomes quadratic—becomes relevant. For instance, in the case of the asymmetric gap of figure 8a, one can imagine that the system is delivered essentially in its ground state to the bottleneck of the gap. While Embedded Image moves along the flat part of the gap, the Hamiltonian of the system is unchanged, so no transitions happen. However, as the gap opens up, now with a steep slope, one is really starting an avoided level-crossing transition from a system that is still in its ground state near the end of the plateau. Clearly, conclusions of Damski & Zurek (2008) and Cucchietti et al. (2007) apply.

Figure 8

(a) Energy of the first accessible state for N=80, Embedded Image (solid line). The slant is moving with a constant velocity Embedded Image across the spin chain, i.e. Embedded Image. Embedded Image is chosen such that the flat part of the spectrum begins at t=0. The dashed lines are linear fits to the solid line in order to get parameters for a Landau–Zener type two-level Hamiltonian the excited state of which is given by the dotted line. (b) Upper line: solution of a Landau–Zener type problem with an energy spectrum given by the dotted line in (a). The function oscillates rapidly around the value Embedded Image (dotted line). The lower line shows the excitation probability for N=80, Embedded Image. The dashed line is a fit leading to Embedded Image.

While these preliminary insights are encouraging, more work is needed to gain a complete picture. In addition to the obvious intrinsic interest of this problem, there are applications that may benefit from its thorough understanding. Let us mention one, slightly speculative: adiabatic quantum computing rests on the idea that a known ground state of a simple Hamiltonian can be adiabatically transformed into an initially unknown ground state that solves a problem encoded into the structure of another Hamiltonian. The rate at which this transition can be accomplished is limited by the size of the gap: faster transitions will result in errors appearing at the rate given by LZF, equation (3.11). This will severely limit the speed of the adiabatic computation; its duration will increase exponentially with the inverse of the size of the gap. But we have seen above that a gap can be widened, and the rate of transitions can be brought down from exponential to quadratic in Embedded Image. It remains to be seen whether (and how) these changes can be used to suggest improvements in adiabatic quantum computing, but the preliminary results we have reached here are certainly suggestive.

Acknowledgments

This research was supported by the LDRD programme funded by DoE at the Los Alamos National Laboratory. The research of U.D. was supported by a Marie Curie Intra-European Fellowship within the Sixth European Community Framework Programme and by the EPSRC (UK) through the QIP IRC (GR/S82176/01) and EuroQUAM project EP/E041612/1.

Footnotes

  • One contribution of 16 to a Discussion Meeting Issue ‘Cosmology meets condensed matter’.

  • It is not yet clear, however, whether these seeds are sown before or after the transition. There is numerical evidence that in thermodynamic transitions the instant Embedded Image after the quench plays a dominant role (Antunes et al. 2006). On the other hand, in quantum phase transitions both instants play a similar role (Damski & Zurek 2006), so that the effective power laws that follow are a compromise (usually ‘won’ by the steeper dependence). Fortunately, when critical exponents are the same on both sides of the transition—as is the case in this paper—this distinction is of no importance.

References

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