Plan de recursos humanos

This chapter will describe in considerable detail the quantum critical properties of the quantum N = 1 model in one spatial dimension. All of the results in this section are believed to be exact, but the physically oriented reader should not be turned off by this: we will keep technical details to a minimum, and show how the exact results can be obtained by physical arguments which do much to illustrate the main underlying principles. Most of the important concepts of this book will appear in the simple model under consideration: the remainder of the book is largely a description of similar phenomena in more complicated settings. This is thus one of the central chapters of this book, and a careful reading is urged.

We will study the D = 1 case of (1.5) which is HI =−JX

i

gˆσ^x_i + ˆσ_i^zσˆ_i+1^z

. (4.1)

As we have discussed in Part 1 and will establish in this chapter, HI

exhibits a phase transition at T = 0 between an ordered state with the Z2 symmetry broken, and a quantum paramagnetic state where the symmetry remains unbroken. The quantum-classical mapping QC ensures that this transition will be in the universality class of the D = 2 classical Ising model.

There has been a great deal of theoretical work on the ground state correlations of HI [306, 382, 336, 35]. However, properties of the or-der parameter ˆσz at T > 0, which are our primary interest here, have been studied much less: methods relying upon knowledge of all the ex-act eigenstates and eigenfunctions of HI do yield explicit results for equal-time correlators [306, 333, 38, 425], but results for unequal-time correlators have been restricted to T =∞ [71, 380, 381] or to precisely

51

52 The Ising chain in a transverse field

the critical coupling [337, 245] (seen below to be g = 1). There is also an approach which relies upon deriving non-linear partial differential equa-tions satisfied by the T > 0 unequal time correlators [338, 281, 297]

but these have not so far been solved to yield the physical correla-tors. Our discussion of the low T dynamics here will follow the intuitive phenomenological approach developed recently in Ref [442]: despite its seeming inexactness, its results are believed to be asymptotically exact, and this will be supported by evidence from numerical computations.

We will use our discussion of the quantum critical point of HI and its vicinity to introduce some basic concepts and tools. These include the central idea of a scaling transformation to characterize the scaling limit theory of the quantum critical point, the scaling dimension of opera-tors and coupling constants about the critical theory, and the dynamic critical exponent z. Another very useful, but much less familiar con-cept, is that of the reduced scaling function, and it will be introduced as an essential tool towards understanding the mechanism of emergence of classical behavior in limiting regimes of the phase diagram.

We will describe the properties of HI by focusing on an especially important observable: the dynamic two-point correlations of the order parameter ˆσ^z (as discussed in Section 3.1, correlators of the field φ in (3.11) will be similar to those of ˆσ^z)

C(xi, t) ≡ hˆσ^z(xi, t)ˆσ^z(0, 0)i

= Tr

e^−HI/Te^iHItσˆ^z_ie^−iHItσˆ₀^z

/Z (4.2)

where Z = Tr(e^−HI/T) is the partition function, and xi = ia is the x-coordinate of the i’th spins with a the lattice spacing. Here, and in the remainder of this book, we will always use the symbol t to represent real physical time. Occasionally we will also find it convenient to consider the correlation at an imaginary time τ ; this is defined by the analytic continuation it→ τ from (4.2) with τ > 0

C(xi, τ ) = Tr

e^−HI/Te^HIτσˆ_i^ze^−HIτσˆ^z₀

/Z. (4.3)

Compare this definition with (2.30); from the discussion in Chapter 2 it should be clear that C(x, τ ) is the correlator of the classical D = 2 Ising model (3.2) on an infinite strip of width 1/T and periodic boundary conditions along the ‘imaginary time’ direction. In all our subsequent discussion on the correlators like C, we will consistently use the argument t when referring to real time correlators as in (4.2), and the argument τ for imaginary time correlators as in (4.3). We will also deal with the

The Ising chain in a transverse field 53 dynamic structure factor, S(k, ω), which is simply the Fourier transform of C(x, t) to wavevectors and frequencies

S(k, ω) = Z

dx Z

dt C(x, t)e^{−i(kx−ωt)}. (4.4) This is a useful quantity because it is directly proportional to the cross section in scattering experiments in which the probe (usually neutrons) couples to σ^z. If energy of the scattered neutron is integrated over, then the cross section in proportional to the equal-time structure factor, S(k), defined by

S(k)≡ Z dω

2πS(k, ω), (4.5)

which is clearly also the spatial Fourier transform of C(x, 0). The num-ber of arguments of S will specify whether we are referring to the dy-namic or equal-time structure factor. The identity (ˆσ_i^z)² = 1 implies that C(0, 0) = 1, and leads to the following sum rule for the dynamic structure factor

Z dkdω

(2π)²S(k, ω) = 1. (4.6)

Finally, also useful is the corresponding dynamic susceptibility χ(k, ωn) which is most conveniently defined by a Fourier transform in imaginary time

χ(k, ωn)≡ Z 1/T

0

dτ Z

dx C(x, τ )e^{−i(kx−ωnτ )} (4.7) where ωn = 2πnT , n integer, is the usual Matsubara imaginary fre-quency arising from the restriction to periodic functions along the imag-inary time direction. We may also perform the analytic continuation to real frequencies by iωn→ ω + iδ (where δ is a positive infinitesimal) and obtain the dynamic susceptibility, χ(k, ω). As was the case for C, it will be symbol for the frequency (ωn or ω) which will distinguish whether we are referring to the imaginary or real frequency susceptibility. The dynamic susceptibility measures the response of the magnetization σ^z to an external field which couples linearly to σ^z and is oscillating at a wavevector k and frequency ω. In the limit that the external field becomes time-independent, the response is given by the static suscepti-bility, χ(k) defined by

χ(k)≡ χ(k, ω = 0). (4.8)

54 The Ising chain in a transverse field

Again, the number of arguments of χ will specify whether we are refer-ring to the dynamic or static susceptibility.

From (4.2), (4.3), (4.4) and (4.7) it is clear that there should be a rela-tionship between the two real frequency correlators S(k, ω) and χ(k, ω).

This is the so-called fluctuation-dissipation theorem, and is established by expressing all the above correlators in terms of the (possibly un-known) exact eigenstates of HI and their matrix elements; such an anal-ysis may be found in many text books, and we will not reproduce it here (see, e.g., Ref. [146]). It is conventional to decompose χ(k, ω) into its real and imaginary parts by χ(k, ω) = Reχ(k, ω) + iImχ(k, ω), and the required relationship is then

S(k, ω) = 2

1− e^−ω/TImχ(k, ω). (4.9) A Kramers-Kronig transform also connects the real and imaginary parts of χ(k, ω):

Reχ(k, ω) =P Z ∞

−∞

dΩ π

Imχ(k, Ω)

Ω− ω , (4.10)

whereP labels the principal part. The spectral analysis also shows that Imχ(k, ω) is an odd function of ω, while Reχ(k, ω) is an even function of ω. From (4.9), the dynamic structure factor satisfies S(k,−ω) = e^−ω/TS(k, ω).

We will begin this chapter by developing a simple physical picture of the possible ground and excited states of HI by examining the large and small g limits in Section 4.1. The exact spectrum will be determined in Section 4.2 and this will show the existence of a quantum critical point at g = 1. The universal continuum quantum theory of the vicinity of g = 1 will be obtained in Section 4.3. Equal time correlators for T > 0 will be discussed in Section 4.4, and the dynamical properties of the different T > 0 regimes will be examined in Section 4.5.

We note that the reader may also wish to examine the recent book by Chakrabarti, Dutta and Sen [82] which discusses aspects of quantum Ising models in one and higher dimensions.

4.1 Limiting cases at T = 0

We begin by examining the spectrum of HI under strong (g ≫ 1) and weak (g ≪ 1) coupling limits, which were discussed briefly in Sec-tion 1.4.1. The analysis is relatively straightforward in these limits, and two very different physical pictures emerge. The exact solution, to be

4.1 Limiting cases at T = 0 55 discussed later, shows that there is a critical point exactly at g = 1, but that the qualitative properties of the ground states for g > 1 (g < 1) are very similar to those for g ≫ 1 (g ≪ 1). One of the two limiting descriptions is therefore always appropriate, and only the critical point g = 1 has genuinely different properties at T = 0.

4.1.1 Strong coupling g≫ 1

The g =∞ ground state was presented in (1.7), where we also discussed the nature of the 1/g corrections. We found a quantum paramagnetic ground state, invariant under the Z2 symmetry (1.11), with exponen-tially decaying ˆσ^z correlations as in (1.9).

What about the excited states ? For g =∞ these can also be listed exactly. The lowest excited states are

|ii = | ←iⁱY

j6=i

| →i^j, (4.11)

obtained by flipping the state on site i to the other eigenstate of ˆσ^x(the eigenstates of ˆσ^xwere defined in (1.8)). All such states are degenerate, and we will refer to them as the “single-particle” states. Similarly, the next degenerate manifold of states are the two-particle states|i, ji, ob-tained by flipping the states at sites i and j, and so on to the general n-particle states. To first order in 1/g, we can neglect the mixing be-tween states bebe-tween different particle number, and just study how the degeneracy within each manifold is lifted. For the one-particle states, the exchange term ˆσ^z_iˆσ_i+1^z in HI is not diagonal in the basis of the| →i,

| ←i states, and leads only to the off-diagonal matrix element

hi|H^I|i + 1i = −J (4.12)

which hops the ‘particle’ between nearest neighbor sites. As in the tight-binding models of solid state physics [27], the Hamiltonian is therefore diagonalized by going to the momentum space basis

|ki = 1

√N X

j

e^ikxj|ji, (4.13)

where N is the number of sites. This eigenstate has energy (we have added an overall constant to HI to make the energy of the ground state zero)

εk = Jg

2− (2/g) cos(ka) + O(1/g²),

(4.14)

56 The Ising chain in a transverse field

where a is the lattice spacing. The lowest energy one-particle state is therefore at ε0= 2g− 2J

Now consider the two particle states. At g =∞, the subspace of two particle states is spanned by the states (generalizing (4.11))

|i, ji = | ←iⁱ| ←i^j Y

h6=i,j

| →i^h, (4.15)

where i 6= j. Also notice that |i, ji = |j, ii, and so we may restrict our attention to i > j. Alternatively, we can say that the states are symmetric under interchange of the particle positions i, j and so we treat the particles as bosons. At first order in 1/g, these states will be mixed by the matrix element (4.12); this will couple|i, ji to |i±1, ji and i, j± 1i for all i > j + 1, while |i, i − 1i will couple only to |i + 1, i − 1i and |i, i − 2i. For i and j well separated, we can ignore this last case, and the two particles will be independent of each other, with the matrix elements for each particle identical to those considered above for single particles. So the particles will acquire momenta k1, k2 (say), and the total energy of the two particle state will be Ek = εk + ε^′_k, and its total momentum k = k1+ k2. However, when i and j approach each other, we will have to consider mixing between these momentum states arising from the restrictions in the matrix elements noted above. This is a problem in ordinary scattering theory, treated in many elementary quantum mechanics texts. (In this discussion, we are assuming that there are no two-particle bound states, a fact that can be verified by a full solution of the two-particle Schr¨odinger equation at order 1/g.) The scattering of the two incoming particles with momenta k1, k2 will conserve total energy, and total momentum up to a reciprocal lattice vector. For small k1, k2 (which is our primary interest here), these conservation laws allow only one solution in d = 1: the momenta of the particles in the final state are also k1 and k2. The existence of a single final state is a special feature of d = 1, while a sum over an infinite number of momenta in the final state is required for the problems in d > 1 we will consider later. By this reasoning, we can conclude that the wavefunction of the two particle state will have the following wavefunction for i≫ j

e^i(k1xi+k2xj)+ Sk1k2e^i(k2xi+k1xj)

|i, ji. (4.16) The quantity Sk1k2 is of central importance, and is the S matrix for two particle scattering. Upon interpreting the stationary scattering state in (4.16) from the perspective of a time-dependent scattering problem, in

4.1 Limiting cases at T = 0 57 which particles scatter from an incoming wave corresponding to the first term in (4.16), to an outgoing wave corresponding to the second term, the S matrix can be related (just as in familiar scattering theory) to the time-evolution operator of HIfrom the infinite past to the infinite future, and must therefore be a unitary matrix. In the present situation with a single final state, the S matrix is a complex number of unit modulus.

The reader is urged to go through the simple exercise of computing the S matrix from the Schr¨odinger equation at order 1/g. The result turns out to be remarkably simple; we find

Sk1k2=−1, (4.17)

for all momenta k1, k2. We will not give an explicit derivation of this result here (a detailed discussion of the computation of such S matrices in general spin models may be found in Ref [117]). Instead, will present a simple argument in the next paragraph which shows that a result like (4.17) holds in the limit of small k1, k2 for a generic Ising chain with additional further neighbor exchange couplings; the validity of (4.17) at all momenta is a special feature of the nearest neighbor exchange model (4.1). Our argument will also show that (4.17) continues to hold at higher orders in 1/g for small k1, k2.

Transform to the center of mass frame of the two particles, and con-sider the Schr¨odinger equation for their relative co-ordinates x = xi−x^j. Taking for simplicity, a repulsive delta function potential uδ(x) between them (the result does not require this special form), we can write down the schematic Schr¨odinger equation



− d²

dx² + uδ(x)



ψ(x) = Eψ(x), (4.18)

where x is their relative co-ordinate and ψ(x) is the wavefunction in the center of mass frame. We make a simple argument based upon dimensional analysis. Notice from (4.18) that u has the dimensions of an inverse length. The S matrix is a dimensionless quantity, and can only be a function of u and the relative momentum k = k1− k². Dimensionally, this can only be of the form S = f (u/k) where f is some unknown function. We are interested in the limit k → 0, which is given by the value of f (∞). However, conceptually, it is much simpler to obtain f(∞) by taking u→ ∞ at fixed k. So to slowly moving particles, the potential appears effectively impenetrable. This means that ψ(x) should vanish at x = 0, and its bosonic symmetry under particle exchange implies that it has the form ψ∼ sin(k|x|/2) for small x. Comparing with (4.16), we

58 The Ising chain in a transverse field

conclude that f (∞) = −1, and so (4.17) holds universally in the limit of small momenta.

We have now described the manner in which 1/g perturbations lift the degeneracy of the g =∞ two particle eigenstates (4.15). The energy of a two-particle state with total momentum k is given by Ek = εk1+ εk2

where k = k1+k2. Notice that for a fixed k, there is still an arbitrariness in the single particle momenta k1,2 and so the total energy Ek can take a range of values. There is thus no definite energy momentum relation, and we have instead a ‘two-particle continuum’. It should be clear, however, that the lowest energy two-particle state in the infinite system (its “threshold”) is at 2ε0. Similar considerations apply to the n-particle continua, which have thresholds at nε0.

At higher orders in 1/g we have to account for the mixing between states with differing numbers of particles. Non-zero matrix elements like h0|H^I|i, i + 1i = −J (4.19) lead to a coupling between n and n + 2 particle states. It is clear that these will renormalize the one-particle energies εk. However qualitative features of the spectrum will not change, and we will still have renormal-ized one-particle states with a definite energy-momentum relationship, and renormalized n≥ 2 particle continua with thresholds at nε⁰. Note especially that the integrity and stability of the one particle states is not modified at any order in 1/g: the one particle state with energy εk is the lowest energy state with a momentum k, and this protects it from decay.

Upon explicitly carrying out these higher order computations for the particular nearest neighbor model HI, some rather ‘miraculous’ features emerge for this special Hamiltonian: as already noted, the result (4.17) holds not only at small k1, k2, but at all momenta and at all orders in 1/g (there are also no processes in which the number of outgoing particles does not equal the number of in-going ones). This remarkable fact appears quite mysterious at this stage, but will be explained rather simply in Section 4.2 using a mapping of HI to fermionic variables.

The spectrum described above has important consequences for the dynamic structure factor S(k, ω). Inserting a complete sets of states between the operators in the definition (4.4) we see that T = 0

S(k, ω) = 2πX

s

| h0|ˆσ^z(k)|si |²δ(ω− E^s), (4.20) where the sum over s extends over all the eigenstates of HI with

en-4.1 Limiting cases at T = 0 59

ε

ω

Fig. 4.1. Schematic of the dynamic structure factor S(k, ω) of HIas a function of ω at T = 0 and a small k. There is a quasiparticle delta function at ω = εk, and a three-particle continuum at higher frequencies. There are additional n-particle continua (n ≥ 5 and odd) at higher energies which are not shown.

ergy Es > 0, and so at T = 0, S(k, ω) is non-zero only for ω > 0 (recall that we have chosen the ground state energy to be zero). The dy-namic susceptibility can be obtained from (4.9), and equals Imχ(k, ω) = (sgn(ω)/2)S(k,|ω|). The eigenstates and energies described above allow us to simply deduce the qualitative form of S(k, ω) which is sketched in Fig 4.1. The operator ˆσ^z flips the state at a single site, and so the matrix element in (4.20) is non-zero for the single particle states: only the state with momentum k will contribute, and so there is an infinitely sharp delta function contribution to S(k, ω) ∼ δ(ω − ε^k). This delta function is the “quasiparticle peak” and its co-efficient is the quasipar-ticle amplitude. At g =∞ this quasiparticle peak is the entire spectral density which saturates the sum rule in (4.6), but for smaller g the quasi-particle amplitude decreases and the multiquasi-particle states also contribute to the spectral density. The mixing between the one and three parti-cle states discussed above, means that the next contribution to S(k, ω) occurs above the 3 particle threshold ω > 3ε0; because there are a con-tinuum of such states, their contribution is no longer a delta function, but a smooth function of omega (apart from a threshold singularity), as shown in Fig 4.1. Similarly there are continua above higher odd number particle thresholds; only states with odd numbers of particles contribute because the matrix element in (4.20) vanishes for even numbers of par-ticles.

60 The Ising chain in a transverse field 4.1.2 Weak coupling g≪ 1

The g = 0 ground states were given in (1.10). They are two-fold degener-ate, and posses long range correlations in the magnetic order parameter ˆ

σ^z. In the present notation, the result (1.12) implies

|x|→∞lim C(x, 0) = N₀²6= 0, (4.21) where C(x, t) was defined in (4.2). The spontaneous magnetization N0

equals ±hˆσ^zi in the two ground states, corresponding to spontaneous breaking of the Z2 symmetry (1.11). All of the statements made in this paragraph clearly hold for g = 0, and will hold for some g > 0 provided the perturbation theory in g has a non-zero radius of convergence. The exact solution of the model to be discussed later will verify that this is indeed the case.

The excited states can be described in terms of an elementary domain wall (or kink) excitation. For instance the state

· · · |↑ii|↑ii+1|↓ii+2|↓ii+3|↓ii+4|↑ii+5|↑ii+6· · ·

has domain walls, or nearest neighbor pairs of antiparallel spins, between sites i + 1, i + 2 and sites i + 4, i + 5. At g = 0 the energy of such a state is clearly 2J×number of domain walls. The consequences of a small non-zero g are very similar to those due to 1/g corrections in the complementary large g limit: the domain walls become “particles” which can hop and form momentum eigenstates with excitation energy

εk= J 2− 2g cos(ka) + O(g²)

. (4.22)

The spectrum can be interpreted in terms n-particle scattering states, although it must be emphasized that the interpretation of the particle is very different from that in the large g limit. Again, the perturbation theory in g only mixes states which differ by even numbers of particles, although the matrix element in (4.20) is non-zero only for states s with an even number of particles; these assertions can easily be checked to hold in a perturbation theory in g. The structure factor S(k, ω) will have a delta function at k = 0, ω = 0, from the term in (4.20) where s = one of the ground states, indicating the presence of long-range order.

Further, there is no single particle contribution, and the first finite ω

Further, there is no single particle contribution, and the first finite ω