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Effective and exact holographies from symmetries and dualities

Zohar Nussinov, Gerardo Ortiz, Emilio Cobanera

TL;DR

This work unifies two broad routes to holographic-like dimensional reduction: an effective mechanism (EQDR) that bounds $D$-dimensional boundary observables by $d$-dimensional theories when ${\bf d}$-GLS symmetries are present, and exact dualities that map $D$-dimensional systems to $d$-dimensional ones. The EQDR theorem provides rigorous inequalities $\langle f \rangle^d_l \le \langle f \rangle^D \le \langle f \rangle^d_u$ for boundary observables, enabling systematic identification of effective reduction; symmetry considerations further sharpen these results and constrain symmetry breaking. The paper then develops two exact reduction frameworks: (i) a density-of-states (DOS) approach showing that identical local DOS $g_{0}(\epsilon)$ yields identical free energies across dimensions in large-$n$ vector theories (and under high temperature/weak coupling limits), and (ii) a bond-algebra duality approach that yields explicit dimension-changing dualities for models with topological order, such as extensions of the toric code, color codes, and the XXYYZZ model, often reducing to one-dimensional systems. These results have implications for quantum-information storage and robustness, map a broad class of higher-dimensional theories to lower-dimensional descriptions while preserving locality, and offer a unified lens for AdS-CFT-type ideas in condensed-matter and lattice-field contexts. Overall, the work provides a versatile toolkit—EQDR bounds, DOS dualities, and bond-algebra dualities—for understanding when and how higher-dimensional systems effectively or exactly behave as lower-dimensional ones, with concrete examples in TQO and quantum memory physics.

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.

Effective and exact holographies from symmetries and dualities

TL;DR

This work unifies two broad routes to holographic-like dimensional reduction: an effective mechanism (EQDR) that bounds -dimensional boundary observables by -dimensional theories when -GLS symmetries are present, and exact dualities that map -dimensional systems to -dimensional ones. The EQDR theorem provides rigorous inequalities for boundary observables, enabling systematic identification of effective reduction; symmetry considerations further sharpen these results and constrain symmetry breaking. The paper then develops two exact reduction frameworks: (i) a density-of-states (DOS) approach showing that identical local DOS yields identical free energies across dimensions in large- vector theories (and under high temperature/weak coupling limits), and (ii) a bond-algebra duality approach that yields explicit dimension-changing dualities for models with topological order, such as extensions of the toric code, color codes, and the XXYYZZ model, often reducing to one-dimensional systems. These results have implications for quantum-information storage and robustness, map a broad class of higher-dimensional theories to lower-dimensional descriptions while preserving locality, and offer a unified lens for AdS-CFT-type ideas in condensed-matter and lattice-field contexts. Overall, the work provides a versatile toolkit—EQDR bounds, DOS dualities, and bond-algebra dualities—for understanding when and how higher-dimensional systems effectively or exactly behave as lower-dimensional ones, with concrete examples in TQO and quantum memory physics.

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- 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.

Paper Structure

This paper contains 23 sections, 97 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Systems of study
  3. Effective dimensional reduction via bounds
  4. A theorem on effective quantum dimensional reduction (EQDR theorem)
  5. The EQDR theorem in quantum field theory and many-body physics
  6. A basis-dependent version of the EQDR theorem
  7. The role of symmetries
  8. Exact dimensional reduction: general scope
  9. Exact dimensional reduction from the density of states
  10. Preliminaries and statement of the DOS theorem
  11. Free energy of large-$n$ vector systems
  12. Examples of dimensional reduction in large-$n$ theories
  13. Weak coupling or high temperature
  14. Dimensional reductions and density of states functional theories
  15. Exact dimensional reduction from bond algebras
  16. ...and 8 more sections

Figures (5)

  • Figure 1: A honeycomb lattice described as a triangular Bravais lattice with a basis.
  • Figure 2: Honeycomb lattice with $L_1=3,\ L_2=2$, and periodic boundary conditions (twelve distinct sites). The top panel shows the plaquette bonds $W_{(1,{\bm{m}})}$ introduced in Eq. \ref{['w1']}, numbered from one to six, and the four transverse fields $\sigma^z$ as green crosses (the spin labelled as three is not part of the Hamiltonian, but it appears in one of its ${\bf d}$-GLSs). The bottom panel shows the plaquette bonds $W_{(2,{\bm{m}})}$. The green (blue, red) dots stand for spins $\sigma^z$ ($\sigma^x,\ \sigma^y$). The middle panel represents the dual, one-dimensional ladder model defined in Eq. \ref{['dual_ely']}. The duality mapping $\Phi_{{\sf d}}$ defined in Eq. \ref{['d_iso_ly']} is indicated by corresponding labels in the illustrations of both models. For example, the plaquette bond in the bottom panel labelled as $\tilde{1}$ maps to the link $\sigma^z_{2,1}\sigma^z_{2,2}$ indicated by a green horizontal bar, and labelled by $\tilde{1}$ as well. The dual model satisfies periodic boundary conditions.
  • Figure 3: Numbering function $\phi$ used to describe the duality between $H_{\sf HETC}$ and a one-dimensional model, illustrated for a lattice with $L_1=4$ and $L_2> 4$. $\phi$ numbers the hexagons, or equivalently, the sites of the underlying triangular Bravais lattice (large numbers). Since there is one point of the basis per hexagon, $\phi$ numbers these as well (small numbers). The green crosses represent the transverse-field bonds $\sigma^z$ present in $H_{\sf HETC}$. Periodic boundary conditions are considered.
  • Figure 4: The basis vectors ${\bm{a_1}},{\bm{a_2}},{\bm{a_3}}$ for the fcc Bravais lattice. For any site ${\bm{m}}$ in an fcc lattice, there is one and only one octahedron with ${\bm{m}}$ as its lowest vertex.
  • Figure 5: The fcc Bravais lattice can be built as an $ABAB$ stack of two square lattices ($A$ and $B$), displaced relative to each other (we show a lattice elongated in the stacking direction to make the figure more clear). Since the planar lattices $A$ and $B$ are both bipartite, the sites of an fcc lattice can be grouped into four disjoint classes. Any two octahedra belonging to sites in the same class share at most one vertex.