User:Sylvain Ribault/Bootstat2021/Gruzberg

Lectures on random critical points by Ilya Gruzberg.

Cover different apparently unrelated systems. Try to show that they have common features.

We have a lattice with spacing $$a$$ that provides a UV cutoff or regularization. We do not care if it is renormalizable, as we always have a lattice. Moreover, the system is always of finite size $$L$$, very large with respect to lattice size. This provides an IR regularization, and also serves as a scaling variable. We have expectation values of the type
 * $$ \langle O \rangle \sim \left(\frac{a}{L}\right)^{x_0}

$$ We are talking about critical points that may not be conformally invariant. We do not know. However, the bootstrap program is not restricted to conformal field theory.

General remarks about disorder
Imperfections of the lattice: impurities, dislocations, etc. These have to be included in the description of the solid. Dynamics of impurities typically have large time scale. So we consider them as constant: this is called quenched disorder.

The impurities break translation invariance. We cannot use momentum space. We should think about an ensemble of disordered systems (samples). Nominally, different samples have the same properties. (Density of impurities.) But different samples have different arrangements of impurities, and lead to different measurements. Average quantities or distributions of quantities can be translationally invariant.

Phase transitions exhibit universal scaling behaviour. Evolution of observables with system size? Need functional renormalization group to follow distribution functions, not just a few coupling constants. But we are not going to do functional RG. We will focus on extracting low moments of observables, such as expectation values.

Some quantities are self-averaging, ex. the free energy. Become non-random (narrow distribution) in thermo. limit. Some other quantitites have broader distributions in scaling limit, lead to multifractality. Free energy is self-averaging because it is extensive. Local magnetization is not of this type.

Techniques for extracting low moments: replica, supersymmetry.

Consider the Hamiltonian
 * $$ H = -\sum_{ij} J_{ij}S_iS_j - \sum_i h_i S_i

$$ where $$ S_i \in S^{n-1}$$ or $$S_i\in\{\pm 1\}$$. Or in some group coset. Random bonds $$J_{ij}$$, random fields $$h_i$$. Subject to some probability distribution, with very little correlation between different fields and/or bonds. So we assume them to be IID (independent identically distributed) with distributions $$P_J,P_h$$ that do not depend on position. This makes the ensemble of Hamiltonians translationally invariant.

Correlation functions: connected correlators $$ \langle \prod_j S_{i_j} \rangle_c$$.
 * $$ \langle O\rangle = \frac{1}{Z} \sum_{\{S\}} O e^{-H} \, \ Z = \sum_{\{S\}}  e^{-H}

$$ The free energy $$F= -\log Z$$ is a function of $$X=\{J,h\}$$. We have
 * $$ m_i =\langle s_i\rangle = -\frac{\partial F}{\partial S_i}$$. Disordered average: average over $$X$$.

\overline{\langle S_{i_1}\cdots S_{i_n}\rangle_c} = \overline{ \frac{1}{Z[X]} \cdots } $$ difficult to compute as we average a ratio.

Spin glass example: $$ \bar m = \overline{\langle s_i \rangle} = 0 $$ but $$ \overline{m^2}= \overline{\langle s_i^2 \rangle} = q_{EA}\neq 0 $$.

Want to know if disorder is relevant or irrelevant at the critical point.

Replica method
Method for averaging over disorder. Write
 * $$ F[X] = \lim_{n\to 0} \frac{1-Z[X]^n}{n} $$

and for $$n\in \mathbb{N}^*$$ we write $$ Z[X]^n$$ as the partition function of a system made of $$n$$ copies of the original system. Each spin gets an index $$a$$ with $$n$$ values. The average $$Z_n = \overline{Z[X]^n}$$ is not so hard to perform. Then, analytically continue to non-integer $$n$$, and take the limit $$n\to 0$$. This is the essence of the replica method. It can be used to compute connected correlators:

\overline{\left\langle O_1(r_1)O_2(r_2)\right\rangle} = \lim_{n\to 0} \left\langle O_1^a(r_1)O_2^a(r_2)\right\rangle_{Z_n} $$ Take same replica index for connected correlator, different indices for disconnected correlators.

No guarantee that analytic continuation in $$n$$ exists. In perturbation theory quantities are typically polynomial, but there is a debate on whether the trick is sound non-perturbatively. Sometimes, complicated results are obtained by the replica trick.

Replica symmetry i.e. permutation group $$S_n$$ is broken in spin glasses. It is remarkable that this breaking of replica symmetry leads to correct results, in some models where alternative approaches (such as SUSY) are feasible.

Non-unitarity
$$ F_n = -\log Z_n$$ is the free energy of the replicated system. In the limit $$n\to 0$$, we have $$Z_n=1$$ and $$F_0=0$$. It is known that systems with $$Z=0$$ generically exhibit logarithmic correlation functions, so the corresponding CFTs must be log-CFTs.

Mechanism: for finite $$n$$ we have discrete spectrum, which collapses as $$n\to 0$$ and form a Jordan block, so that dilation operator is no longer diagonalizable. This is very similar to phenomenon in solving differential equations.

This argument about logs relies on some internal symmetry. In the replica approach this is provided by $$S_n$$, in the SUSY approach there is also an internal symmetry.

So we should face the issue of non-unitarity. In 2d, $$F=0$$ is closely connected to $$ c=0$$.

Multifractality
Consider a local observable $$O$$ and compute $$\overline{\langle O\rangle^q}\sim L^{-x_q}$$. The dimensions $$x_q$$ are convex as fct of $$q$$ and become negative if $$q$$ is large enough. This is not unphysical: it means the distribution of observables becomes broader as system grows. Operators with negative dimensions do not appear in Hamiltonian, so they do not make the system unstable. But they show up when we measure the right quantities.

Harris criterion
To know if disorder is relevant or irrelevant.

Product of two spins leads to operator $$E(r)$$ with scaling exponent $$x_E$$ such that $$ x_E+\frac{1}{y_+}=d$$. The correlation length exponent is $$ \nu = \frac{1}{y_+}$$. Consider

S[J] = S^* +\int d^dr J(r)E(r) $$ Do replicas, average over disorder, obtain effective action that no longer depends on disorder.
 * $$ Z_n = \int \prod_a dS_a \overline{e^{-S_n[J]}} = \int \prod_a DS_a e^{-S_{eff}}

$$ where
 * $$ S_{eff} = \sum_a S_a^* + \kappa_1 \int d^dr \sum_a E_a(r) + \kappa_2 \cdots

$$ OPEs are very simple:
 * $$ E_a\cdots E_b = \delta_{ab} 1 + \delta_{ab}E_a + (E_aE_b)$$

Linear term in effective action is inocuous as it just shifts critical temperature. The interesting term is quadratic in $$E$$. We need to find dimension, to know if this is relevant or not. Two-point function of quadratic operator is quartic in $$E$$, and easy to compute, as replicas are decoupled. The dimension of $$EE$$ is $$x_{EE}=2x_E$$.

We infer that the disorder is irrelevant if $$d\nu >2$$.This is the Harris criterion. It applies to systems where disordered is well-behaved, but fails in other systems such as polymers.

The same argument can be generalized to random fields rather than random bonds. It turns out random fields are always relevant.

Let us stick with bond disorder, and consider the case when the criterion is violated i.e. disorder is relevant. We believe the singularity survives but belongs to a different universality class. Theorem: we flow to different fixed point that obeys the Harris criterion. Case of 3d Ising: $$d\nu = \frac{3}{3-x_E}$$ where $$ x_E= 1.412625$$, criterion violated not by much. For $$ d\in \{2,4\}$$ we have $$ d\nu = 2$$ i.e. strict marginality.

Marginal disorder: could be marginally relevant or marginally irrelevant. 2d Ising has marginally irrelevant disorder, giving logarithmic corrections to scaling. Clean fixed point of 2d Ising is given by Majorana fermions.

Strong disorder: probability distribution

P[J_{ij}] = (1-p) \delta(J_{ij}-J) + p \delta(J_{ij}+J) $$ Transition ferromagnet-paramagnet (phase diagram), with also a spin glass phase, and a multicritical fixed point where the three phases meet. Nishimori line in phase diagram goes through fixed point, is an RG trajectory, on this line we have gauge symmetry. In 2d the spin glass is supposed to exist only a zero temperature. ($$T= \frac{1}{J}$$)

We may hope to study these fixed points using bootstrap. But how could we apply the bootstrap to fixed points we know little about? If there is a symmetry such as SUSY this can give hints.

Quantum spin models
We wrote classical spin models. To define quantum spin models we allow spins to be generators of a Lie algebra:
 * $$ H = -\sum_{ij} J_{ij}\hat{S}_i\hat{S}_j - \sum_i h_i \hat{S}_i

$$ Now the different terms in the Hamiltonian do not commute with each other. Now the competition is no longer between energy and entropy, but between different terms in the Hamiltonian (?), leading to quantum phase transitions.

Time now plays a special role. There is a dynamical critical exponent $$z$$. If it is not one, space-time is anisotropic, and we lose Lorenz invariance and therefore conformal invariance. But $$z=1$$ in many cases.

Example:

H= -Jg \sum \hat{\sigma}^z \hat{\sigma}^z - J \sum \hat{\sigma}^x $$ For $$g >>1$$ there is order, for $$g<<1 $$ disorder. There is a critical coupling with a fixed point in the same universality class as classical Ising?

Trotterize partition function: write it in terms of matrix elements for small time intervals.

Harris criterion for quantum systems involves $$d$$ not $$d+z$$. The disorder is perfectly correlated along the time direction. This is called columnar disorder. Not good for the bootstrap.

Anderson transitions
Adressed using the bootstrap by Hikami using Gliozzi's methods. Not sure the bootstrap addresses the right critical point.

Anderson localization
Single particle problem:
 * $$ H=H_0 + U(r) \, \ H_0 = \frac{p^2}{2m} $$

where $$U$$ is a random potential with
 * $$ \langle U(r)U(r') \rangle = u \delta(r-r') $$

There are localized states in the system. For a clean system, the free wavefunction is periodic. Adding a weak disorder, wave is less periodic. For a large potential, the wavefunction becomes localized, exponentially decaying away from a point $$ r_0$$.

The existence of localized states is a rigorous, mathematically-proven statement. Proving the existence of extended states is harder, although obvious physically.

Fermions: filling states in the system one by one, what matters for conductance is what happens at Fermi surface. Density of states in clean system is $$\nu(E)\propto \sqrt{E}$$, while in disordered systems density is nonzero for $$E<0$$, with lowest states localized. There is a metal/insulator transition when energy reaches $$E_c$$. This is called the Anderson transition. This is a continuous transition, correlation length $$\xi(E)\sim |E-E_c|^{-\nu}$$. Density of states is smooth across transition. To see it we need the Green's function
 * $$ G^\alpha(r,r';E) = \left\langle r|\frac{1}{E+i\alpha O^+-H}| r'\right\rangle

$$ Need to average over disorder. Disordered average of single-particle Green function would be short-range because disorder randomizes phase of wavefunction. Need a square to get rid of random phases.

Disorder average
Let's do the disorder average using the supersymmetry method, introduced by Efetov. We write $$G^\alpha$$ as a path integral Since the partition function is a Gaussian integral, it is just a determinant. Its inverse is a bosonic path integral:
 * $$ G^\alpha(r,r';E) = -i\alpha \frac{1}{Z} \int D\chi \chi(r)\chi^*(r') e^{i\alpha\int d^dr \chi^*(r)(E+i\alpha 0^+-H)\chi(r)}

$$ leading to

\overline{G^\alpha(r,r';E)} = -i\alpha \int D\phi D\chi \chi(r)\chi^*(r') e^{-S_{eff}} $$ where

S_{eff} = \int d^dr \left[ \phi^*(E-H_0)\phi +\chi^*(E-H_0)\chi +\frac{u}{2}(\phi^*\phi +\chi*\chi)^2\right] $$ where there is no denominator because the partition function is one. In particular, at the critical point if it exists, the central charge is zero. We introduced a boson $$\phi$$, and we get a Lie algebra symmetry $$u(1|1)$$. This is a global supersymmetry which has nothing to do with the spacetime on which fields live.

Adjoint representation of Lie superalgebra has indecomposable structure of dimension $$4$$. The action is invariant, can be thought of as part of an indecomposable representation. At the critical point, the stress-energy tensor is also part of such a representation, has a logarithmic partner $$t$$. The indecomposable structure holds before and after disorder average, on and off criticality.

Non-linear sigma model
Manipulate action to make it quadratic in the original fields. If disorder is weak, can do a saddle point approximation and get a non-linear sigma model

S[Q] = \frac{1}{t} \int d^dr \text{Str}\left[-(\nabla Q)^2 + i\omega \Lambda Q \right] $$ where $$Q \in \frac{U(1|1)}{U(1)\times U(1)} $$ or some larger supercoset. It is $$\Lambda$$ which breaks the group symmetry into a coset.

If replicas are used, the rank of the coset depends on the number of replicas. With supersymmetry, we get a supercoset. We have to impose a constraint $$Q^2=1$$ coming from the saddle point.

Properties of the system are controlled by the coupling $$t$$.

Altland-Zirnbauer classification of these systems, includes three classical matrix ensembles. Each one of the ten classes corresponds to one supercoset. Some supercosets have nontrivial topology, allowing us to add theta terms to the action.

Why use the non-linear sigma model rather than the original quartic action? Argument is analogy to QCD, where pions are effective fields (analogous to $$Q$$) and easier to study than original quarks (analogous to $$\phi,\chi$$). But at the critical point the original fields may be better, and the non-linear sigma model might be useless, see Zirnbauer's talk.

Symmetry classification
3 Wigner-Dyson classes for random matrices, depending on symmetries such as time reversal. If $$T$$ is broken we have Hermitian matrices $$H=H^\dagger$$ (unitary class). If $$T$$ is preserved we can have $$H=H^T$$ or $$H=\sigma_3 H^T\sigma_3$$ (orthogonal, symplectic classes). If we view $$ X=iH$$ as an element of a Lie algebra, which is the tangent space of some coset space $$U(N)=U(N)\times U(N)/U(N)$$, $$U(N)/O(N)$$, $$U(N)/Sp(N)$$.

Altland and Zirnbauer considered 3 more symmetry classes called chiral classes. There are also 4 Bogoliubov-de Gennes classes, originating from mean field superconductors.

These 10 classes all have potential for a critical point, depending on dimension.

Multifractality
Transition between localization and extended states at $$E=E_c$$. At the critical energy, wavefunctions are neither periodic nor really localized, but multifractal: very non-uniform.

Break system of size $$L$$ into boxes of size $$a$$, with probabilities $$p_i$$. Define

P_q = \sum_{i=1}^N p_i^q \, \ \overline{P}_q = N \overline{p_i^q} \sim \left(\frac{a}{L}\right)^{\tau(q)} $$ This defines critical exponents $$\tau(q)$$ (infinitely many). Maybe not the best observable to bootstrap, as it depends on system size. Extended states: $$p_i\sim \frac{1}{N}$$ and $$\tau(q)=(d-1)q$$. With localized states $$\tau(q) = 0$$. At the critical point,
 * $$\tau(q) =(d-1)q +\Delta(q)$$

Let $$p(r)\sim r^\alpha$$ where $$\alpha$$ is a local exponent, subject to some distribution $$f(\alpha)$$, then $$\tau(q)$$ is its Legendre transform. The function $$\tau(q)$$ has nice properties: non-decreasing, $$\tau(0)=-d$$, $$\tau(1)=0$$.

Consider the local density of states
 * $$ \nu(r, E) = \sum_n \delta(E-E_n) |\psi_n(r)|^2

$$ Can smear it in energies with parameter $$\eta$$. Focus on critical energy. In a finite system, many states can be called critical. The sum reduces to the global density of states

\nu_\eta(r, E) \sim |\psi_E(r)|^2 \nu_\eta(E) $$ Define
 * $$ O_q(r) \sim \left[L^d\nu(r, E_c)\right]^q

$$ then $$ \langle O_q(r)\rangle \sim L^{-x_q} $$. Then
 * $$ \tau(q) = d(q-1) +x_q -q x_1 $$

We have $$x_0=0$$ of course, but also $$x_{q_*}=0$$. For Wigner-Dyson we have $$q_*=1$$. So we have an operator with zero dimension, which is not the identity. We have abelian fusion (conservation of momentum) for these observables.

The field theory may not be exhaustive: multifractal correlation functions may not all be described by the field theory, they do not necessarily behave like field theory correlators, only some special combinations do. For example
 * $$ \langle O_{q_1}(r_1) O_{q_2}(r_2)\rangle \sim r_{12}^{x_{q_1+q_2}-x_{q_1}-x_{q_2}} L^{-x_{q_1+q_2}}

$$ This depends on system size unless $$ q_1+q_2=0$$, which is however still not conformally invariant as in general $$x_{-q}\neq x_q$$. We can also kill the dependence on system size by $$q_1+q_2=q_*$$, in which case we have a CFT two-point function thanks to $$ x_{q_*-q} = x_q$$, a symmetry that can be justified from the sigma model.

In 2d we consider a correlation function with $$\sum q_i =q_*$$. Only one conformal block by Abelian fusion. We can show that $$x_q \propto q(q_*-q)$$. Conformal symmetry constrains the theory to be a Gaussian free field theory. In higher dimensions, could global conformal symmetry be so restricting? i.e. abelian fusion plus conformal invariance fix $$x_q$$?

Argument for abelian fusion i.e. momentum conservation: it follows from a target space symmetry, which survives quantization. Since it is a target space symmetry, it commutes with conformal symmetry, so it does no constrain descendants, only primaries.