Recomendaciones

ORDER

c CONTINUUM HIGH T

LOW T

Quantum paramagnet

Fig. 5.3. Large N phase diagram for the O(N ) rotor model with 2 < d < 3.

Qualitative features of the phase diagram apply for N > 2 and 2 < d < 3, or 1 ≤ N ≤ 2 and 2 ≤ d < 3. The dashed lines are crossovers determined by

∆±∼ T (∆±∼ |g − gc|^zν), while the full line is the locus of finite temperature phase transitions with Tc given by (5.86). There is true magnetic long-range order at all temperatures below the full line. The shaded region is where the reduced classical scaling functions apply.

by arguments similar to those used to obtain (4.56). First, we can use the definition (5.2) and the scaling dimension (5.39) to conclude dim[χ(k, ω)] = 2dim[n]− d − z = −(2 − η). Then recalling dim[T ] = z, we can obtain the scaling form

χ(k, ω) = Z T^(2−η)/zΦ_±

 ck T^1/z,ω

T,∆_± T



(5.60)

where the upper (lower) sign applies for g≥ g^c (g≤ g^c). Also it should be clear that in d = 1 only the upper sign can apply. The functions Φ_± are completely universal and complex-valued, and are chosen to have finite limits at all k and ω as ∆_± → 0 at fixed T (there is an exception to this in d = 1, where, as will shall see in Chapter 6, the function Φ+ diverges logarithmically as ∆+/T → 0; this logarithm divergence is however absent in the present N = ∞ theory). There are strong restrictions that arise from the consistency of the two functions as they approach the common point g = gc from the two sides; not only their

126 Quantum rotor models: large N limit

values must agree, but also the fact that χ(k, ω) must be analytic as a function of g at g = gc for T > 0 places many additional restrictions (the reasons for this analyticity, and its consequences will be discussed in more detail in Section 8.2.1). For the Ising chain we were able to work with a single function by defining a ∆ = ∆+ > 0 for g≥ g^c and

∆ =−∆− < 0 for g≤ g^c, but this is difficult to do in the present case as the definitions of ∆_± are quite different. Also, for the Ising chain, ∆ was a simple, analytic linear function of g, so the analyticity requirement was simply that Φ was analytic as a function of ∆ at ∆ = 0.

The prefactor Z is a non-universal constant which is non-singular at the T = 0 quantum critical point. It can be defined through (5.60) by relating it to some observable which depends upon the scale of the order parameter field. For g > gc, we can, by demanding that the form of χ(k, ω) near the quasi-particle pole at T = 0 in (5.30) (which holds even beyond N = ∞, as we saw in the Ising chain) be consistent with the scaling form (5.60), specify

Z = (constant) A

∆^η/z₊ . (5.61)

The constant can be chosen at our convenience, and merely changes the definition of the Φ_±. Alternatively, we could approach the critical point from g < gc and use (5.57) to define

Z = (constant) N₀²c²

ρs∆^η/z₋ . (5.62)

A similar scaling form can be written down for the uniform suscepti-bility from the knowledge of the scaling dimension in (5.43):

χu= T^d/z−1 c^d Φu±

∆_± T



. (5.63)

Unlike, (5.60), there is no non-universal prefactor like Z in front: this is because the unknown field scale, and the anomalous exponent η does not appear in the definition of χu: rather it is related by (5.3) to the free energy density.

The remainder of this section will present explicit results for these scaling functions at N = ∞. In this limit, the expressions in (5.22) and (5.23) specify χ and χu respectively. These are consistent with the scaling forms (5.60) and (5.63) for η = 0 and z = 1, if the Lagrange

5.4 Nonzero temperatures 127 where F_± are universal functions which will be obtained from the solu-tion of (5.21), as we shall show below. The resulting predicsolu-tions for the physical properties at T > 0 are quite simple. By Fourier transforming (5.22), we see that m/c is the correlation length. The imaginary part of (5.22) also implies that there is a gap in the spectrum equal to m. This feature is an artifact of the N =∞ limit: the response of any interacting system at T > 0 has a non-zero spectral density at all frequencies (in cer-tain cases, the response could vanish above some large ultraviolet cutoff

∼ cΛ), as there are essentially no restrictions on the set of frequencies at which all the possible thermally excited states can absorb energy. A prominent objective of the remaining chapters in Part 2 is to describe a dynamical theory for the filling in of this gap at finite temperatures.

The uniform susceptibility is obtained by evaluating the frequency summation in (5.23) by standard methods which the reader can find in text books like Refs [146] and [321]; the result is

χu= 1 with m given by (5.64).

We now determine the universal functions F_±, and will subsequently turn to a description of the physics in the various regions of Fig 5.1-5.3. The method used here introduces a number of useful tricks for the extraction of universal, cut-off independent crossover functions.

We present first the calculation on the disordered side g ≥ g^c. The first step is to subtract from (5.21) the corresponding equation (5.24) at the same coupling constants at T = 0; this gives us

Z Λ d^dk where ∆+is the gap at the current value of g. A trick we shall often use is to subtract from the summation over frequencies of any quantity, the integration over frequencies of precisely the same function; so we rewrite (5.66) as

128 Quantum rotor models: large N limit Now we use the general relation

TX valid for any positive a (again this can be established by standard fre-quency summation methods [146, 321]). Notice that the right-hand side falls off exponentially as a becomes large. This is a key property, and was the reason for considering the combination in (5.68). Applying this iden-tity to (5.67), we see that the first integration over k has an integrand which is exponentially small for large k, and hence is quite insensitive to Λ which can safely be sent to infinity. The integration over p in the second term is also ultraviolet convergent, again allowing Λ to be set to infinity. The resulting expression is then cutoff independent, and hence universal; we obtain for d > 1

Z d^dk where the number Xd was defined below (5.27). In d = 1, this equation is modified to The solution of these equations is clearly of the form (5.64); after rescal-ing momenta by c/T in (5.69), we find that the function F+(s) is deter-mined implicitly by solution of the equation

Z d^dk

for d > 1, and similarly for d = 1. We will discuss asymptotic features of the solution of these equations in the subsections below. We note here that precisely in d = 2, the equation (5.71) has a simple, explicit solution [97]

Now we turn to the ordered side, g≤ g^c, which implicitly means that we have d > 1. We assume that T is large enough that the magnetization is zero; the case of the magnetized state with T 6= 0 can be treated

5.4 Nonzero temperatures 129 similarly, and will be referred to below. Subtract from (5.21), the value of ρs/N in (5.58), and insert the value of 1/gc in (5.25). Evaluating the frequency summation as above we find

Z d^dk The solution of this is also in the form (5.64), and the function F₋(s) is given by Again, there is a simple explicit solution in d = 2 [97]

F₋(s) = 2 sinh⁻¹

e^−2πs 2



d = 2 (5.75)

With expressions for the crossover functions F_±in hand, let us discuss the physical properties of the system in different regimes of the g, T , plane for different values of d.

5.4.1 Low T on the quantum paramagnetic side, g > gc, T ≪ ∆⁺

The discussion here also applies in d = 1.

Properties of this phase are essentially identical to those of the low T quantum paramagnetic region of the Ising model in Section 4.5.2. The ground state has a gap, and non-zero T induces an exponentially small density of thermally excited triplet magnons. For the parameter m we have

m = ∆++O(e^−∆+/T). (5.76) So there is a finite correlation length c/m which has exponentially small corrections from its T = 0 value c/∆+. The N = ∞ expression (5.22) has a quasi-particle peak that remains infinitely sharp at T > 0: this is clearly incorrect for finite N , as damping must be present, and will be described in subsequent chapters. The uniform susceptibility can be computed from (5.65), and we find that it is exponentially small

χu=O(e^−∆+/T). (5.77)

130 Quantum rotor models: large N limit 5.4.2 High T , T ≫ ∆⁺, ∆₋

Again properties are the similar to those of the continuum high T region of the Ising chain as discussed in Section 4.5.3. Now we have, for d > 1 m = T F+(0) = T F₋(0) (5.78) where F+(0), F₋(0) are pure numbers. This represents a correlation length∼ c/T . In d = 1, the correlation length has an additional loga-rithmic correction [253], as can be seen from the solution of (5.70)

m = πT

ln(CT/∆⁺), (5.79)

where

C = 4πe^−γ= 7.055507955 . . . . (5.80) In a similar manner we find for the uniform susceptibility from (5.65) that in d > 1

χu=T^d−1

c^d Φu+(0) = T^d−1

c^d Φu−(0), (5.81) where Φu± are universal pure numbers which can be determined by solutions of (5.65) and (5.69); in d = 2 we have the simple result Φu±(0) = (√

5/π) ln((√

5 + 1)/2). Again, in d = 1 there are log cor-rections [253]

χu= 1

πcln(CT/∆⁺) (5.82)

which will be better understood in the following chapter.

By analogy with the Ising chain we expect that the dynamics is quan-tum relaxational with a phase coherence time∼ 1/T . However damping and relaxation are completely absent at N = ∞ and will be further discussed later.

5.4.3 Low T on the magnetically ordered side, g < gc, T ≪ ∆−

This section applies only for d > 1, as there is no such region for d = 1.

The properties in d = 1 will be analogous to the low T ordered region of the Ising chain in Section 4.5.1, but there will be important differences for 2 < d < 3.

Let us assume first that T is large enough so that hni = 0 and so (5.74) can be used to determine F₋. For d = 2, one finds that there is a

5.4 Nonzero temperatures 131 solution of (5.74) for all T , and even as T → 0 (s = ∆−/T → ∞). We find that as T → 0

m = T exp(−2π∆−/T ) = T exp(−2πρ^s/N T ). (5.83) So the correlation length∼ c/m diverges as T → 0, but remains finite for all non-zero T . This was exactly the situation as in the Ising chain, and the phase diagram for this model is therefore as shown in Fig 5.2.

We will see in subsequent chapters that, as in the case of the Ising chain, because of the very large correlation length, it is possible to develop an effective classical dynamical model of the system, and to express the result in terms of reduced scaling functions. Let us also note (from (5.65)) that the uniform susceptibility in d = 2 is given as T → 0 by

χu=2∆₋ c² = 2ρs

N c². (5.84)

This is actually an exact result even for finite N , as we will see later.

Now let us consider the case 2 < d < 3. Although there is no physical dimension in this region, the results obtained below will apply in d = 3 with cutoff-dependent logarithmic corrections we do not want to discuss here. Further, the physics of the quantum Ising model in d = 2 is expected to be similar to that of the large N solution with 2 < d < 3.

The key observation in this case is that there is no solution of (5.74) for F₋(s) above a critical value s = sc, where F₋(sc) = 0. The value of sc

is given by

s^d_c⁻¹ =

Z d^dk (2π)^d

1 k

1 e^k− 1

= 2Γ(d− 1)ζ(d − 1)

Γ(d/2)(4π)^d/2 . (5.85)

Just as was the case in the T = 0 analysis at the beginning of Section 5.3, the absence of a solution for the Lagrange multiplier m (related to F₋(s) by (5.64)) implies that there must be magnetic order for s > sc This defines a critical temperature Tc given precisely by

Tc≡ ∆−/sc (5.86)

such that the system is in the paramagnetic phase only for T > Tc: the resulting phase diagram is shown in Fig 5.3. There is a finite tempera-ture phase transition at T = Tc, and a magnetically ordered phase for T < Tc. As T approaches Tc, the conventional classical phase transition theory becomes applicable in the region |T − T^c| ≪ T^c. The classical

132 Quantum rotor models: large N limit

scaling functions of this transition emerge as reduced scaling functions of the quantum functions, in a manner very similar to the discussion on the quantum Ising chain in Section 4.5.1. One consequence of this behavior is that all the scale factors of the classical scaling functions, which are usually considered non-universal, are universally determined by the parameters ∆₋, c, and N0 of the quantum crossover functions.

We have already seen an example of this in (5.86), where Tcwas univer-sally determined by ∆₋ [427].

Let us explicitly observe the collapse of the scaling function (5.64) in this classical region. As the primary quantum crossover function has only one argument, the reduced function would have no arguments, ı.e., it is a pure power law. Indeed, solution of (5.74) for s close to but above sc gives us

m = Tc

T− T^c Tc

(d− 1)s^dc⁻¹

Xd

1/(d−2)

. (5.87)

The correlation length c/m diverges with the classical exponent νc = 1/(d− 2) with an amplitude that is universal.

The above is part of a very general lesson. Quantum critical scal-ing forms like (5.60) hold everywhere in the vicinity of the quantum critical point, including at or close to any finite temperature phase tran-sition lines that may be approaching the quantum critical point. The classical critical singularities of these finite temperature transition ap-pear as singularities of the quantum critical scaling function. Further, the amplitudes of the classical transitions, which are normally non-universal, become universal when expressed in terms of the arguments of the quantum-critical scaling function.

5.5 Applications and extensions

We have already mentioned application to double-layer antiferromagnets in Section 5.1.1.1.

We indicated in Section 5.1.1.1 that the O(3) quantum rotor model de-scribes a large class of Heisenberg antiferromagnets, and this connection will be established more generally in Chapter 13. Here we will discuss application of rotor model results to thermodynamic measurements of the uniform spin susceptibility, χu, of quantum antiferromagnets; impli-cations for other physical properties of antiferromagnets will be noted in subsequent chapters. The rotor model predictions for χu are given by

5.5 Applications and extensions 133 (5.65) in the large N limit, but computations with 1/N corrections are also available [96, 97].

The S = 1/2 square lattice antiferromagnet, found in the parent insulating compounds of the high temperature superconductors (like La2CuO4), has its low energy properties described by the O(3) quantum rotor model [83]. Very accurate results for the thermodynamic proper-ties of the former model have been obtained in precision Monte Carlo computations by Kim and Troyer [269]. In particular they obtained the T dependence of the uniform susceptibility, χu, for a wide range of temperatures; experimental measurements of χu on La2CuO4 are also available [252], but these are of lower precision than the numerical data, and as it is practically certain that La2CuO4 is a square lattice antifer-romagnet, it is appropriate to use the numerical data. For T ≪ ρ^stheir measurements are in good agreement with universal low temperature response of the continuum rotor model in (5.84) (the correction of order T /ρsto (5.84) will be derived later in (7.25) [97, 218], and this was used in the comparisons with the numerical data in Ref [269]). At larger T they observe a clear crossover which is in good agreement with the con-tinuum high T behavior in (5.81) (the leading 1/N and ρs/T computed corrections [97] to (5.81) were used in this comparison). This evidence supports the proposal, made in Ref. [96], that the S = 1/2 square lattice antiferromagnet is close enough to a quantum critical point to display the continuum high T behavior of Fig 5.2 at higher temperatures.

The low energy properties of a double-layer model of two S = 1/2 square lattice antiferromagnets coupled to each other are also described by O(3) quantum rotor model, as should be clear from the discussion in Section 5.1.1.1. Moreover, by changing the ratio of exchange couplings in this model it is possible to tune the rotor model coupling g through gcT [329, 342]. There have been a number of studies of the double-layer antiferromagnet near this critical point [443, 444, 330, 175, 136, 498, 331], and the numerical results for χuare in good agreement with the (5.81) and its 1/N corrections [97].

Normand and Rice [366, 367] have proposed an interesting recent ex-perimental realization of the quantum critical point of the d = 3 quan-tum rotor model in LaCuO2.5. This is a spin-ladder compound in which the ladders are moderately coupled in three dimensions. By varying the ratio of the intra-ladder to inter-ladder exchange it is possible to drive such an antiferromagnet across a d = 3 quantum critical point separating N´eel ordered and quantum paramagnetic phases. The uniform suscepti-bility has a T²dependence at intermediate T , which is characteristic of

134 Quantum rotor models: large N limit

the “High T” dependence in (5.81) in d = 3. The entire T dependence of χuhas been computed in Monte Carlo simulations of an S = 1/2 an-tiferromagnet on the LaCuO2.5 lattice [500] and the results are in good agreement with quantum rotor model computations like those discussed here.

6