Day 3 of the conference.
Guifre Vidal, Criticality, impurities and real-space renormalization, joint work with Glen Evenbly.
First an overview of the multi-scale entanglement renormalization ansatz (MERA). Basic notions: isometries, disentanglers, causal cone. Bounded causal cone width means that it is possible to efficiently evaluate the expectation value of local observables. One can also course grain a local operator by applying one level of the MERA. (For ternary MERA, a two-body term gets mapped to a two-body term.) We will focus on critical systems, and then put in impurities. To enforce scale and translation invariance, we characterize the MERA by just a single pair of tensors and repeated across the whole network. MERA seems to be a good ansatz to describe critical systems where such invariance is manifest. MERA also can exhibit polynomial decay of correlations. Define the scaling superoperator. The eigenvalues and eigenvectors of this superoperator are the scaling operators and (the exponential of) the scaling dimensions of the underlying CFT. How does one choose the optimal tensors given a local Hamiltonian? He dodges the question… doesn’t want to get into such details in this talk. Showed an example for the 1D critical Ising model with transverse field. Now let’s consider an impurity at the origin. Now we lose translation invariance, but we still have scale invariance (conformal defect). The bulk scaling operators in the absence of an impurity has zero expectation. However, in the presence of an impurity the one-point functions decay like a power of the distance to the impurity. We need to add new scaling operators and new scaling dimensions. Translation invariance is broken, we can’t possibly optimize over an extensive number of tensors (because we could never reach the thermodynamic limit). Well, desperate times call for desperate measures. Introduce something called “Principle of minimal influence” in RG flow. PMI says: Given a MERA for the initial (pure) problem, we can obtain a MERA for the problem with impurity by then varying the tensors in the causal cone for the region containing the impurity. We don’t expect the principle to work in all cases, such as in systems with spontaneous symmetry breaking where the impurity restores the symmetry. We are still exploring the limits of applicability of this principle. If you split the isometries into two “half isometries”, you get that the total number of tensors you need to optimize is still . Why is this ansatz plausible? For one thing, we can show that the scaling superoperator now exhibits power low decay with respect to the distance from the impurity. The optimization goes as follows. First optimize with respect to the bulk Hamiltonian (no impurity). Next, use the bulk tensors to map the original impurity problem to an effective impurity problem. This gives a new Hamiltonian problem with exponentially decaying interaction strengths. Finally, we can reduce the optimization via “folding in half” to that of an MPS for an open boundary problem. This same form appears in Wilson’s solution of the Kondo problem. However, the impurity MERA can be used beyond the setting of free fermions, unlike Wilson’s original approach. Showed example of transverse-field Ising model with impurity. This model allows an exact solution via CFT, and the numerics agree with the exact solution for various impurity strengths. We can extend this to more than one impurity by treating each impurity individually in their non-intersecting causal cones, and then once the causal cones intersect we do one more variation. We can also have an impurity that “cuts” the chain or a boundary between two half-chains coupled by the impurity, and even boundary conditions.
André-Marie Tremblay, Success and limitations of dynamical mean-field theory, joint work with Sordi, Sénéchal, Haule, Okamoto, Kyong and Civelli.
First recall how one “builds” a metal. We take a lattice and put orbitals on the sites. If the orbitals overlap, then it takes no energy to excite an electron. However, this doesn’t always work, e.g. in NiO. Recall the Hubbard model with on-site and hopping strength (and for nearest diagonal neighbors, etc.) When , we can Fourier transform and solve the model. When , everything is localized, and there are degenerate ground states at half filling corresponding to the different electron configurations with one electron per site. We can observe antiferromagnetism in the Hubbard model. We see a Mott transition in the region where and are of the same order, and at this point in the phase diagram neither the plane wave nor the localized wavefunctions gives a good description. One can write an effective model with Heisenberg coupling. We would like to measure Green’s functions for a single particle. Why do we care about Green’s functions? We can use them to compute the value of other observables and can measure it directly via photo emission. Fermi liquid theory: means that it is not too far from a free electron problem. We want to be able to solve in the presence of an impurity. Let’s review dynamical mean-field theory (DMFT). We begin working in infinite dimensions. Then the self-energy is a function only of the frequency. We want to compute the self-energy of the impurity problem. Then use that self-energy (which depends only on frequency) for the whole lattice. Now project the lattice onto a single site and adjust the bath so that the single site density of states obtained both ways are equal. There are some self-consistency equations. We can use a number of methods: cavity method, local nature of perturbation theory in infinite dimensions, expand around the atomic limit, effective medium theory, Potthoff self-energy functional. Use the universal self-energy functional method (see the talk by Sénéchal). Vary with respect to the parameters of the cluster (including Weiss fields). In this context, we can view DMFT as a stationary point in the variational space. What about the impurity solvers? The most trivial one is just exact diagonalization. Could also do Monte Carlo using an ergodic Markov chain. Showed a bunch of plots of these numerical methods for various models. The strengths of DMFT are: can obtain one-particle properties and phase diagrams; weaknesses include, order parameters are not coupled to observables that are quadratic in creation and annihilation operators, transport (four-point correlation functions), analytic continuation for QMC solvers. Future challenge is to find the optimum environment.
Matthias Troyer, Galois conjugates of topological phases, arXiv:1106.3267.
The question is: should we explore the computational power of non-unitary topological phases? Such phases appear through Galois conjugation of unitary topological phases. The arising models are non-Hermitian, but still have a purely real spectrum. Do they make sense? First, a review of topological phases and anyons (both Abelian and non-Abelian). Review of fusion rules. These states might appear in the Pfaffian state (Moore & Read) which has Ising anyons (level ) or in the parafermion state (Read & Rezayi) which has Fibonacci anyons (level ). How can we make more exotic states? We start with an anyonic liquid on a high-genus surface. Picture two sheets which are connected by fusing a bunch of perforations through both sheets. These create doubled (non-chiral) models. We can consider two types of quantities: flux through a hole and flux through a tube. We can build a family of basis states to describe the anyon wavefunctions by building a “skeleton” through each of the tubes. The basis states are labeled by string nets of particle labels on the skeleton lattice. Now we add the Hamiltonian for the model. We can either add flux through a tube or a hole, and each of these costs some different amount of energy. Each of these terms acts on either plaquettes or edges of the skeleton lattice. The plaquette term gives a ground state without flux through the plaquettes. This is like closing each of the holes: we get vacuum states of anyons on each of the two sheets. This is just the toric code or the Levin-Wen model. It’s a topological phase which is robust at zero-temperature to weak perturbations. This can be solved exactly, even for the excited states. Violations of the vertex constraint are chiral anyonic excitations living on the surface. Consider a finite density of anyons, and let’s pin the anyons into fixed positions. We would like to add a Heisenberg-like interaction Hamiltonian to this. We just introduce an energy difference between the various labels of the particle types. Review of the fusion basis for anyons, specialized to the Fibonacci case. Review of the F-moves ( symbols, Clebsch-Gordan), the pentagon and hexagon equations. So what does a Heisenberg chain for anyons look like? where is the F-move that exchanges two nearest-neighbors in a chain and is a projector onto a maximally entangled state. Ran numerical simulations to compute the gap… but it was gapless! The gap decreased like 1/L, so we guess it is a critical model. We can compute the central charge using the connection with entanglement entropy. It turns out that the model is exactly solvable by using the Temperley-Lieb algebra. They studied these interacting chains for various values of the level . There are lots of equivalences known to CFTs depending on the central charge and the sign of the coupling (antiferromagnatic or ferromagnatic). Plot of the energy spectrum, with very good agreement between exact solution and numerics. When there are finite density interactions, there are gapless edge modes and a nucleated liquid. This foreshadows the situation in 2D. You get a nucleated liquid with a finite bulk gap and edge states. A novel quantum Hall liquid is nucleated inside the parent liquid. Now we move to the Galois conjugation. Review of Galois theory. Different roots of a polynomial are Galois conjugates of each other. The algebraic properties are the same, but the geometric properties are different. As an example, consider again the Fibonacci case. The elements of the F tensor element can be expressed in terms of the golden ratio , or in terms of . However, this second one (Yang-Lee theory) is non-unitary. But if you look at the spectrum of the antiferromagnetic Yang-Lee model, you get a completely real spectrum! The central charge is negative (-3/5). The vacuum state is not the lowest energy state in this model. Similar conclusions for the ferromagnetic case. How does the entanglement entropy scale? It can’t still obey since the central charge is negative, right? Let’s look at the doubled Yang-Lee model. We can redefine the inner product to get a self-adjoint Hamiltonian. (Bender et al., PT-symmetric quantum mechanics). We can solve this doubled YL model exactly, just like with the Levin-Wen models before. Now the code property that defines topological order has a “left-right” eigenvector version, since these are now distinct notions. There are several ideas one might have to turn it into a Hermitian model: Square the plaquette terms, or use just the projector onto the left or right eigenspaces. But these approaches don’t work at giving a topological phase which is robust… they are sensitive to local perturbations. “Hermitianizing” destroys the left-right code property. Might there by some local transformation to a Hermitian model? If not for Fibonacci, then maybe for some other non-unitary TQFT? Norbert: can you use a tensor network description of the ground state to make a Hermitian parent Hamiltonian? Which ground state do you choose? The left or the right? Main result: Galois conjugates of TQFTs where the braid group acts densely in cannot be realized as ground states of quasilocal Hermitian Hamiltonians. Non-Abelian topological phases offer universal quantum computation; the minimal models of CFT are Heisenberg models of non-Abelian anyons; Galois conjugation of topological phases gives intriguing non-unitary theories and non-Hermitian models; but non-unitary phases will not appear in physical systems. Tobias: can’t we add a single extra spin which couples to all the spins and make the model Hermitian?
Renato Renner, Aisenstadt Lecture, An information-theoretic view of thermalization.
Consider a very trivial system of a classical binary random variable (for classical) which takes the values 0 or 1 with equal probability. Now let’s introduce an observer who has some knowledge about the . Suppose that with probability and with probability . Now when we condition the probability of on the state of we get a new distribution. A Bayesian would just call this an update. Suppose now that a new observer has quantum information about the system. So now the conditional state of is a quantum state. If we try to condition the state of on the quantum information , this becomes more interesting. For example, in quantum cryptography there is no reason to assume that the adversary is classical so it makes sense to consider this. Now let’s move on to thermodynamics. Consider a Szilard engine, i.e. a box with one particle in it and a piston that can compress to the middle of the box. If we let the “gas” expand adiabatically or compress it, which gives or takes work, then we find that the work satisfies to transition between the two states. From information theory, we know that a two-state system stores information. To erase information costs units of energy. Landauer showed that in fact this is optimal (Landauer’s principle), and this is independent of the physical representation of the bit. The argument is, if you could erase with a lower energy cost, then you could use the Szilard engine to extract work for free from heat (which violates the second law). If we have conditional knowledge about the state of the Szilard engine, then we can extract a different amount of work, namely where is the binary entropy and describes our (potentially quantum) knowledge about the state of the system. (Technical note: this is the von Neumann entropy in the asymptotic setting and the max entropy in the single-shot setting.) Consider a computer with some registers and . I would like to erase them. If I erase them in sequence, then the total work required to do this would just sum the individual works using the previous formula. If instead, we did some joint operation we would only pay , which is in general less than the sum of the individual entropies. That is, in general we have , so it makes sense to take advantage of the joint information. We could use instead a formula in terms of conditional entropies by just changing the formula in the obvious way. It turns out that this formula is the correct formula, , where is the system and represents the observer. If we take advantage of this conditional information, then we can serially achieve the same work as the optimal joint protocol. This is a manifestation of the so-called chain rule for entropy: . Now let’s discuss thermalization. There is some large system (including an environment). If the total system is in a pure quantum state, how can it thermalize? What does that mean? We can look only at the subsystem, and there is a reduced density operator. Recall Alioscia’s talk for an example of some conditions for when the subsystem thermalizes. However, if two separate subsystems are separately thermal, that does not imply that the joint system is thermal. If instead we condition on the other subsystem instead, we could indeed reach this conclusion. As before, we cannot simply condition on quantum information, but the conditional entropies are well-defined. We can introduce the decoupling lemma, which guarantees that the joint state is nearly separable. It is a sufficient condition, phrased in terms of entropies conditioned on the observer, for decoupling. Namely, decoupling holds if we typically have where the total system has degrees of freedom and the system has . This is tight. If we apply this condition to thermalization. If the subsystem is sufficiently small, then the system will thermalize.
Tobias Osborne, The continuum limit of a quantum circuit: variational classes for quantum fields, arXiv:1102.5524 and arXiv:1005.1268, joint work with Cirac, Eisert, Haegeman, Verschelde, Verstraete.
Review of quantum fields. The most several ways one might “solve” a QFT: perturbation theory, Monte Carlo, and the variational methods. There were problems using the variational method in QFT, namely sensitivity to UV and poor reproduction of correlations. Review of MPS. Recall that MPS can be build from a staircase circuit. Review of area law result due to Hastings. As Tobias himself has proved, log-time evolution can be simulated efficiently, too. Also, a review of MERA. We can view it as a quantum circuit which takes ancillas and prepares a state. How can we pass to the continuum for these variational classes? We can use the quantum circuit preparation prescriptions of MPS and MERA and change the time step size for each gate to an infinitesimal quantity. That’s the main idea. Some nice features of cMPS: it is possible to analytically compute all the correlation functions. There is a built-in UV cutoff. There is generic clustering of correlations. Can build cMERA by alternating scaling transformations and local interactions for infinitesimal time steps, and take the limit. There is one main difference between MERA and cMERA: flexibility in the UV cutoff. The cMERA allow a smooth cutoff, but regular MERA only offer a lattice cutoff. These classes also obey entropy/area laws, rigorously in the case of cMPS, and there is a heuristic argument for cMERA. Gave an example for a simple bosonic system where the expressions are analytic for cMERA.