Effective and exact holographies from symmetries and dualities

Abstract

The theoretical basis of the phenomenon of effective and exact dimensional reduction, or holographic correspondence, is investigated in a wide variety of physical systems. We first derive general inequalities linking quantum systems of different spatial (or spatio-temporal) dimensionality, thus establishing bounds on arbitrary correlation functions. These bounds enforce an {\em effective} dimensional reduction and become most potent in the presence of certain symmetries. {\em Exact} dimensional reduction can stem from a duality that (i) follows from properties of the local density of states, and/or (ii) from properties of Hamiltonian-dependent algebras of interactions. Dualities of the first type (i) are illustrated with large-$n$ vector theories whose local density of states may remain invariant under transformations that change the dimension. We argue that a broad class of examples of dimensional reduction may be understood in terms of the functional dependence of observables on the local density of states. Dualities of the second type (ii) are obtained via {\em bond algebras}, a recently developed algebraic tool. We apply this technique to systems displaying topological quantum order, and also discuss the implications of dimensional reduction for the storage of quantum information.Comment: 53 pgs, 6 figs. Submitted to Nuclear Physics

Quantum memories at finite temperature

“…A similar model whose ground-state degeneracy depends on microscopic details (Bravyi, Leemhuis, and Terhal, 2011) is introduced by Chamon (2005). We remark, however, that unlike the cubic code, the model due to Chamon has a constant energy barrier and is well understood not to give rise to self-correcting properties (Chamon, 2005;Castelnovo and Chamon, 2012;Nussinov, Ortiz, and Cobanera, 2012;Temme, 2014). We finally remark that the study of exotic partially self-correcting systems has led to new classifications of systems under the context of fractal topological quantum field theories (Yoshida, 2013;Haah, 2014).…”

“…While we have more sophisticated methods of extracting the phase diagram of the two-dimensional Ising model due to its exact solution by Onsager (Onsager, 1944;Yeomans, 1992), the intuition developed from Peierls original argument is a very useful tool for understanding the stability of models where no exact solution is known; see, for instance, Lebowitz and Mazel (1998), Campari and Cassi (2010), and Bonati (2014. Indeed, Peierls argument is used to demonstrate the stability of the high-dimensional quantum systems.…”

Compass models: Theory and physical motivations

“…To clarify the distinction between these different symmetries, we can rephrase it in a formal way as it applies to general systems (Batista and Nussinov, 2005;Nussinov et al, 2012b). Consider a theory with fields {φ i } that is characterized by a Hamiltonian H (or action S).…”

Robust topological degeneracy of classical theories

“…In the more commensurate Type II lattice realizations of the classical Toric Code model as well as in a host of other systems, the ground state degeneracy is "holographic"-i.e., exponential in the linear size of the lattice [18,44]. This classical holographic effect is different from more subtle deeper quantum relations, for entanglement entropies, e.g., [72][73][74].…”

“…In particular, building on a generalization of Elitzur's theorem [43,44] it was shown how to construct and classify theories for which no local order parameter exists both at zero and at positive temperatures; this extension of Elitzur's theorem unifies the treatment of classical systems, such as gauge and Berezinskii-Kosterlitz-Thouless type theories in arbitrary number of space (or spacetime) dimensions, to topologically ordered systems. Moreover, it was demonstrated that a sufficient condition for the existence of topological quantum order is the explicit presence, or emergence, of symmetries of dimension d lower than the system's dimension D, dubbed d-dimensional gauge-like symmetries, and which lead to the phenomenon of dimensional reduction.…”