Continuum field theory of matchgate tensor network ensembles

Abstract

Tensor networks provide discrete representations of quantum many-body systems, yet their precise connection to continuum field theories remains relatively poorly understood. Invoking a notion of typicality, we develop a continuum description for random ensembles of two-dimensional fermionic matchgate tensor networks with spatially fluctuating parameters. As a diagnostic of the resulting universal physics, we analyze disorder-averaged moments of fermionic two-point functions, both in flat geometry and on a hyperbolic disk, where curvature reshapes their long-distance structure. We show that disorder drives universal long-distance behavior governed by a nonlinear sigma-model of symmetry class D with a topological term, placing random matchgate networks in direct correspondence with the thermal quantum Hall problem. The resulting phase structure includes localized phases, quantum Hall criticality, and a robust thermal metal with diffusive correlations and spontaneous replica-symmetry breaking. Weak non-Gaussian deformations reduce the symmetry to discrete permutations, generate a mass for the Goldstone modes, and suppress long-range correlations. In this way, typicality offers a route from ensembles of discrete tensor networks to continuum quantum field theories.

Figures (8)

• Figure 1: Top: Phase diagram (cf. Ref. Wang2021) of the disordered class D superconductor, spanned by disorder strength, $W$, and the parameter $a$ controlling the system's band structure. The numbers in parentheses are the Chern numbers of the different topological phases, Anderson insulator (AI), thermal quantum Hall (TQH). (The thermal metal (TM) phase is fundamentally non-topological.) The shaded region marks the parameter range for which our effective theory Eq. \ref{['eq:Scont']} is applicable. Bottom: schematic flow diagram of the coupling constants, $g$ and $\vartheta$ of the field theory. Different parameter values $(a,W)$ 'initialize' the microscopic values of these constants. Under renormalization, i.e., successive integration over field fluctuations, they flow to qualitatively different fixed points which correspond to localization, $(g,\vartheta)=(0,2\pi n)$$(1,2,5)$, quantum Hall criticality $(g,\vartheta)=(g^\ast,\pi)$, with a critical conductance $g^\ast$, or metallicity $(\infty,0)$$(4)$.
• Figure 2: Left: Fermionic tensor network on a square lattice showing the ordering of fermionic modes (clockwise red arrow) per tensor and the bond orientations (red arrows). Right: Representation of the pseudo Hamiltonian corresponding to the tensor network on the left in case the tensors are Gaussian, cf. Eq. \ref{['eq:HCA']}. The labeling of modes in the unit-cell (red) is inferred from the ordering of the fermionic modes per tensor.
• Figure 3: Band structure of the (four-band) pseudo-Hamiltonian $H= -\i (C +\bigoplus A)$ for $A_{\alpha \beta}=a \operatorname{sgn}(\beta-\alpha)$ at $a=a_-$ (left), $a=1$ (center) and $a=a_+$ (right).
• Figure 4: Graphical illustration of the contraction rule in Eq. \ref{['eq:R_moments']} (up to prefactors).
• Figure 5: Setup. We consider $c=1,\ldots,N$layers of the clean system ($N=3$ shown above) and couple the layers with local random tensors ${\tilde{A}}_x$ (red). The random tensors couple all modes at a given site to all other modes at that site and do not discriminate between layers or unit-cell positions. The $N$-layer system is then replicated by $r=1,\ldots,R$ uncoupled replicas.