Generalized Landau Paradigm for quantum phases and phase transitions

Abstract

The Landau paradigm is a central dogma for understanding phase and phase transitions in condensed matter systems, yet for decades it has been known that a variety of quantum phases exist beyond the framework. Is there a more general framework that provides a systematic understanding of phase and phase transitions in quantum many-body systems? Recent developments of the notion of generalized symmetry and generalized gauging seem to point to a way to generalize the Landau Paradigm. In this essay, we discuss how `beyond Landau' phases and phase transitions can be captured by a generalized Landau paradigm in terms of the breaking of generalized symmetries, often induced by the generalized gauging procedure facilitated through the topological holography formalism. We also discuss what needs to be understood to make the generalized Landau paradigm useful in the study of quantum phase and phase transitions.

Figures (6)

• Figure 1: (a) $1+1$D and (b) $2+1$D sandwich structures. A $D+1$-dimensional sandwich has a $(D+1)+1$-dimensional topological order in the bulk, a finite height, and a gapped boundary at the top. The symmetry of the sandwich acts in the bulk, while the bottom boundary contains all the dynamics.
• Figure 2: The $2+1$D $Z_2$ topological phase. Fractional excitations $e$, $m$, and $f$ are created at the end of string operators. Closed loop operators, both contractible and non-contractible, generate the $1$-form symmetry of the phase.
• Figure 3: The Jordan-Wigner transformation in the SymTFT formalism. The trivial/topological superconducting chain is realized in a sandwich structure with the $2+1$D $Z_2$ topological order in the bulk, the $f$-condensed boundary at the top, and (a) the $e$-condensed and (b) $m$-condensed boundary at the bottom, respectively. The Jordan-Wigner transformation corresponds to changing the top boundary to $e$-condensed without modifying the bulk or the bottom boundary. The vertical tunneling of the $e$ excitation (red line in c) indicates spontaneous breaking of the $Z_2$ symmetry.
• Figure 4: Generalized gauging (changing the top boundary from $\mathcal{C}$-condensed to $\mathcal{A}$-condensed) maps general gapped phases (a) to SB phases (c) and the phase transition between general gapped phases (a,b) to SB transitions (c,d). Blue dots indicate transition points between gapped phases. The red line represents the tunneling of fractional excitations between matching condensates on the two boundaries, which becomes the order parameter in the spontaneous breaking of $a$-symmetry.
• Figure 5: The Toric Code model on a two-dimensional square lattice. Red edges represent the $X$ operator, green edges represent the $Z$ operator. The Hamiltonian is a sum of the plaquette terms $A_p$ and the vertex terms $B_v$. Closed string operators along nontrivial cycles $W_x$,$W_y$,$V_x$,$V_y$ act non-trivially in the ground space. Open string operators $V_{\mathcal{L}}$ and $W_{\tilde{\mathcal{L}}}$ create fractional excitations $e$ and $m$ at the two ends.