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arXiv:2303.07848 · Physical review. B./Physical review. B

Renormalization group for measurement and entanglement phase transitions

Adam Nahum, Kay Jörg Wiese

Published Sep 12, 2023• 15 citations
arXivHTMLPDF

Abstract

Field theoretic renormalization group is applied to measurement and entanglement phase transitions. The renormalization group flows for both random tensor networks and monitored systems show that, in high dimensions, there exist distinct weak- and strong-coupling universality classes for these phase transitions. In addition, one of the field theories exhibits a supersymmetry that leads to dimensional reduction by four.

DOI
10.1103/physrevb.108.104203

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Field theoretic renormalization group is applied to measurement and entanglement phase transitions. The renormalization group flows for both random tensor networks and monitored systems show that, in high dimensions, there exist distinct weak- and strong-coupling universality classes for these phase transitions. In addition, one of the field theories exhibits a supersymmetry that leads to dimensional reduction by four.
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We provide a resolution of one of the long-standing puzzles in the theory of disordered systems. By reformulating the functional renormalization group for the critical behavior of the random field Ising model in a superfield formalism, we are able to follow the associated supersymmetry and its spontaneous breaking along the functional renormalization group flow. Breaking is shown to occur below a critical dimension d(DR) ≃ 5.1 and leads to a breakdown of the "dimensional reduction" property. We compute the critical exponents as a function of dimension and give evidence that scaling is described by three independent exponents.
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Renormalization group theory is a powerful and intriguing technique with a wide range of applications. One of the main successes of renormalization group theory is the description of continuous phase transitions and the development of scaling theory. Most courses on phase transitions focus on scaling and critical exponents, while less attention is paid to universality, renormalization group flow, and the existence of a unique fixed point, which are the ultimate reasons why scaling theory is so effective in describing continuous phase transitions. We use a combination of Monte Carlo simulations and real space renormalization group theory to determine the renormalization group flow and to show the existence of a universal fixed point in the context of the ferromagnetic Ising model.
Entanglement phase transitions in quantum chaotic systems subject to projective measurements and in random tensor networks have emerged as a new class of critical points separating phases with different entanglement scaling. We propose a mean-field theory of such transitions by studying the entanglement properties of random tree tensor networks. As a function of bond dimension, we find a phase transition separating area-law from logarithmic scaling of the entanglement entropy. Using a mapping onto a replica statistical mechanics model defined on a Cayley tree and the cavity method, we analyze the scaling properties of such transitions. Our approach provides a tractable, mean-field-like example of an entanglement transition. We verify our predictions numerically by computing directly the entanglement of random tree tensor network states.