5.1.1 Disordered systems and replicas
Perturbation theory and Harris criterion.
Let us consider a ferromagnet with quenched disorder, or to be more concrete, a spin model with Hamiltonian 𝐻 = −∑
𝑖,𝑗𝐽𝑖,𝑗𝑆𝑖𝑆𝑗 in 𝑑 dimensions, where 𝐽𝑖,𝑗 are quenched random variables, with short-range correlations only. Near the critical point of the pure system, the continuum limit of the system can be represented as
𝒮 =𝒮∗+
∫
d𝑑𝑟𝐽(𝑟)𝜖(𝑟) +. . . (5.1)
where𝒮∗ the action of the pure system at the critical point,𝜖(𝑟) is the energy operator, and 𝐽(𝑟) is a random field with 𝐽(𝑟) = 0 and 𝐽(𝑟)𝐽(𝑟′) = 𝜆𝛿(𝑟−𝑟′). Obviously, the disordered theory is not conformally invariant, since it is not even invariant under translations; however, once averaged over, the disorder may be a relevant perturbation that will drive the system to a new RG fixed point, with conformal invariance restored (in the averaged system). To see this, we need a clear field theory description of averaged observables. The typical problem when dealing with disordered systems is to compute averaged correlation functions, of the spin operator for example
⟨𝜎(𝑟1)𝜎(𝑟2)⟩= 1
𝑍Tr𝜎(𝑟) e−𝒮[𝜎(𝑟)]𝜎(𝑟1)𝜎(𝑟2). (5.2) The main issue here is the partition function in the denominator. A very convenient (and standard) way around this is to introduce replicas 𝑎 = 1, . . . , 𝑛, and to take the formal limit 𝑛 → 0 in the end. This amounts to computing the free energy as log𝑍 = lim𝑛→0𝑍𝑛𝑛−1. Doing so, we find
𝑍𝑛 = Tr𝜎𝑎(𝑟)e−∑𝑎𝒮𝑎+𝜆2∫d𝑑𝑟∑𝑎∕=𝑏𝜖𝑎(𝑟)𝜖𝑏(𝑟)+..., (5.3) where we have dropped less relevant terms. The dimension of the perturbation is 2(𝑑−𝜈−1) where𝜈is the thermal exponent at the pure critical point, hence the disorder is relevant if (Harris criterion)
𝑑𝜈 <2. (5.4)
When the disorder is relevant, the system will typically flow to a new random fixed point, for which we expect conformal invariance to be restored. This new fixed point is what we will study in the following, and we will argue that we expect it to be a good candidate for a LCFT. Random fixed points may be accessed perturbatively in the dimension of the perturbation, this can be done for example for the Ising model in 𝑑= 2 +𝜀dimensions, or for the 2𝐷 Potts model with𝑄= 2 +𝜀 states (seee.g.[166]), or the disordered 𝑂(𝑛 = 1−𝜀) model [167, 168]. For a discussion of the effects of quench disorder on first-order phase transitions, see [169].
𝑛 →0 limit and logarithmic correlations.
Following Cardy [18, 53], we now argue that random fixed points in disordered systems must contain logarithmic observables. To do so, let us consider an operator𝜙𝑎 (𝑎= 1, . . . , 𝑛) in the replicated theory that transforms as a ‘vector’ under permutations of the replicas. Typically, one can think of 𝜙 as the energy operator in a spin model.
More formally, {𝜙𝑎}𝑎=1,...,𝑛 is a representation of the permutation group 𝑆𝑛, where 𝑛 is the number of replica. This representation is reducible as the linear combination Φ = ∑
𝑎𝜙𝑎 is clearly invariant under permutations. The 𝑛−1 remaining fields ˜𝜙𝑎 = 𝜙𝑎− 1𝑛Φ (satisfying ∑𝜙˜𝑎 = 0) then transform irreducibly under 𝑆𝑛. Because 𝑆𝑛 is a global symmetry of the replicated theory, we expect the scaling fields of the theory to transform irreducibly under the symmetric group 𝑆𝑛. We thus expect Φ and ˜𝜙𝑎 to have different scaling dimensions ΔΦ(𝑛) and Δ𝜙˜(𝑛). More precisely, because of the 𝑆𝑛
symmetry, their two-point functions must take the form
⟨Φ(𝑟)Φ(0)⟩= 𝑛𝐴(𝑛)
𝑟2ΔΦ(𝑛), (5.5)
⟨𝜙˜𝑎(𝑟) ˜𝜙𝑏(0)⟩= 𝐴(𝑛)˜ (
𝛿𝑎𝑏− 𝑛1)
𝑟2Δ𝜙˜(𝑛) , (5.6)
where 𝐴(𝑛) and ˜𝐴(𝑛) are regular functions of 𝑛, with 𝐴(0) ∕= 0 and ˜𝐴(0) ∕= 0. Since we ultimately want to take the limit 𝑛 → 0 to obtain the physical properties of the disordered model, we see that ˜𝜙𝑎 is ill-defined in that limit. The 1/𝑛 pole in the correlation function of ˜𝜙 is actually the reason why logarithms appear when 𝑛 = 0, this is obviously reminiscent of the𝑐→0 catastrophe described in chapter 2. As argued by Cardy [18], averaged physical quantities are well-defined if and only if𝐴(0) = ˜𝐴(0) and Δ = ΔΦ(0) = Δ𝜙˜(0). Getting back to the original system, one finds that
⟨𝜙(𝑟)⟩⟨𝜙(0)⟩= lim
𝑛→0⟨𝜙1(𝑟)𝜙2(0)⟩=−2𝐴(0) 𝜇(0)
log𝑟+ Cst
𝑟2Δ , (5.7)
⟨𝜙(𝑟)𝜙(0)⟩ − ⟨𝜙(𝑟)⟩⟨𝜙(0)⟩= lim
𝑛→0(⟨𝜙1(𝑟)𝜙1(0)⟩ − ⟨𝜙1(𝑟)𝜙2(0)⟩) = 𝐴(0)
𝑟2Δ , (5.8) where 𝜇= −lim𝑛→0 𝑛
Δ𝜙˜(𝑛)−ΔΦ(𝑛). To reformulate these results in a more familiar lan-guage, let Ψ = 𝜇(𝑛)𝜙1, with 𝜇(𝑛) = −Δ𝜙˜(𝑛)−𝑛ΔΦ(𝑛). Under a scale transformation
𝑟 −→Λ𝑟, one can readily check that it transforms as
Ψ(Λ𝑟) = Λ−Δ(Ψ(𝑟)−log ΛΦ(𝑟)), (5.9) which means that the dilatation operator𝒟acts on Ψ as𝒟Ψ = ΔΨ+Φ. The two-point functions of these fields then read
⟨Φ(𝑧)Φ(0)⟩ = 0, (5.10a) Some remarks: other operators and Replica-Symmetry-Breaking.
It is important to notice that the argument above works only if the operator 𝜙 transforms trivially under some eventual extended symmetry. For example [18], because of the ℤ2 spin-field symmetry of the Ising model, the spin operators{𝜎𝑎}𝑎=1,...,𝑛 in the replicated theory transform irreducibly under the symmetry groupℤ2×𝑆𝑛, so that the argument above does not apply in that case. However, because of the𝑐→0 argument, we still expect logarithms to occur in higher-rank correlation functions. For instance, one can easily argue that the expected Jordan cell for the stress-energy tensor in the disordered theory implies that Another important point is that the above argument holds only if the 𝑆𝑛symmetry is not broken. This seems to be the case for example for the disordered two-dimensional Potts model for𝑄 >2 [170]. However, there is no reason for this to be true in general, and it would be very interesting to generalize the argument above to the case of replica symmetry breaking. The representation theory is more involved in that case, but it remains manageable at least for the one-step replica symmetry breaking (1RSB). In a nutshell, instead of𝑛 replica, one has 𝑙 groups of 𝑘 elements each, with 𝑛=𝑙×𝑘. The symmetry group then becomes a so-called wreath product of permutation groups [171].
Labeling the fields 𝜙(𝑖)𝑎 , with 𝑖 = 1, . . . , 𝑙 and 𝑎 = 1, . . . , 𝑘, Φ = ∑
𝑎,𝑖𝜙(𝑖)𝑎 is still an invariant, but the (𝑛 − 1)-dimensional representation ˜𝜙(𝑖)𝑎 ≡ 𝜙(𝑖)𝑎 − 1𝑛Φ becomes reducible and is broken into two representations with dimension 𝑙−1 and 𝑛−𝑙. These representations are spanned by the fields𝜃𝑖 =∑
𝑎𝜙˜(𝑖)𝑎 and ˆ𝜙(𝑖)𝑎 = ˜𝜙(𝑖)𝑎 −𝑘1𝜃𝑖. Constructing correlation functions as before and taking the limit 𝑘 → 0 while keeping 𝑙 fixed, one
obtains logarithmic correlation functions as in the replica-symmetric case.
5.1.2 Disordered systems and SUSY
In this last chapter, we shall mostly focus on replica-like approaches similar to the 𝑛 → 0 argument introduced in the previous sections. However, as mentioned in the introduction, there is another convenient way to deal with disordered systems in the case where there are no interactions. Let us imagine that one wants to compute the average of
⟨𝒪⟩= 1
𝑍[{ℎ(⃗𝑟)}]Tr𝜙
(𝒪e−𝐻[𝜙,{ℎ(𝑟)}])
, (5.13)
where ℎ(𝑟) are quenched disordered variables, 𝜙 are bosonic degrees of freedom and 𝐻[𝜙] is Gaussian –i.e. non-interacting. Using the properties of Grassmanian Gaussian integrals, one can write the denominator as 𝑍−1 = Tr𝜓e−𝐻[𝜓] when 𝜓 are fermionic variables, and then average over disorder configurations. This is the so-called su-persymmetry approach [28]. As a concrete example, suppose we start with a 𝑑 + 1 dimensional random quantum problem with Hamiltonian 𝐻 =−∇2+𝑉, describing a single particle in a disordered potential 𝑉 with𝑉 = 0 and𝑉(𝑟)𝑉(𝑟′) =𝜆𝛿(𝑟−𝑟′). The Using supersymmetry and averaging over the disorder, one ends up with
±𝑖𝐺±(𝐸)(𝑟, 𝑟′) = We see that the averaged Green function can be computed from a pure field theory where bosonic and fermionic degrees of freedom are effectively interacting because of the disorder. If the SUSY problem is conformal, it must have central charge 𝑐 = 0 since 𝑍SUSY = 1, and so this means that averaged quantities in the original disordered system will behave algebraically with critical exponents given by a 𝑐= 0 theory.
We therefore see that disorder systems must correspond to 𝑐= 0 CFTs1, and thus must be described by Logarithmic CFTs [18, 19]. Examples of problems described by 𝑐= 0 LCFT after mapping onto a SUSY model include the transition between plateaus
1. In the replica approach, this corresponds to the fact that the central charge of the replicated theory𝑐(𝑛) goes to𝑐= 0 as𝑛→0.
𝒟𝑖 𝒟𝑗
Figure 5.1: 2𝑁-leg watermelon operators in the Potts model correspond to the insertion of 𝑁 FK clusters that will persist until they are taken out by another watermelon operator.
in the IQHE, or for instance, the so-called Nishimori point in the two-dimensional random-bond Ising model [30]. In the following, we will not use this supersymmetry approach which corresponds more to the algebraic framework developed in the previous chapters, but instead we shall argue that the replica approach can be used to study geometrical problems as well.