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arXiv:2409.02107

Highly complex novel critical behavior from the intrinsic randomness of quantum mechanical measurements on critical ground states -- a controlled renormalization group analysis

Rushikesh A. Patil, Andreas W. W. Ludwig

Published Sep 3, 2024• 0 citations
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Abstract

We consider the effects of weak measurements on the quantum critical ground state of the one-dimensional (a) tricritical and (b) critical quantum Ising model, by measuring in (a) the local energy and in (b) the local spin operator in a lattice formulation. By employing a controlled renormalization group (RG) analysis we find that each problem exhibits highly complex novel scaling behavior, arising from the intrinsically indeterministic ('random') nature of quantum mechanical measurements, which is governed by a measurement-dominated RG fixed point that we study within an $ε$ expansion. In the tricritical Ising case (a) we find (i): multifractal scaling behavior of energy and spin correlations in the measured groundstate, corresponding to an infinite hierarchy of independent critical exponents and, equivalently, to a continuum of universal scaling exponents for each of these correlations; (ii): the presence of logarithmic factors multiplying powerlaws in correlation functions, a hallmark of 'logarithmic conformal field theories' (CFT); (iii): universal 'effective central charges' $c^{({\rm eff})}_n$ for the prefactors of the logarithm of subsystem size of the $n$th Rényi entropies, which are independent of each other for different $n$, in contrast to the unmeasured critical ground state, and (iv): a universal ("Affleck-Ludwig") 'effective boundary entropy' $S_{\rm{eff}}$ which we show, quite generally, to be related to the system-size independent part of the Shannon entropy of the measurement record, computed explicitly here to 1-loop order. - A subset of these results have so-far also been obtained within the $ε$ expansion for the measurement-dominated critical point in the critical Ising case (b).

DOI
10.48550/arxiv.2409.02107

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The impact of projective measurements on a many-body quantum state is tightly linked to its underlying entanglement structure. Critical states are particularly sensitive, as long-range entanglement allows local measurements to have global consequences. This has been extensively studied in the context of critical states described by a conformal field theory (CFT), where measurements can alter critical properties in post-measurement quantities ('measurement-altered criticality') and induce long-range entanglement with universal features ('measurement-induced entanglement'). By contrast, little is known about the role of measurements on critical states with quenched disorder, which admit no CFT description. Here, we develop a theory of measurements on one-dimensional critical states governed by infinite-randomness fixed points (IRFPs), focusing on two closely related examples: the random XXZ chain and the random transverse-field Ising model. We show that the measurement-induced entanglement decays as a power-law with universal exponent (3−5)/2(3-\sqrt{5})/2(3−5​)/2 in both models, and that the minimum number of measurements required to establish this entanglement obeys a universal scaling form. When only a finite density of measurements is made, we show that the critical properties of the state are remarkably robust: `measurement-altered criticality' is absent, with measurements either preserving the critical exponents, or destroying the criticality entirely. These results establish IRFPs as an analytically tractable arena for measurements on critical states, complementary to the CFT case. The outcome randomness that necessitates replica methods in clean systems becomes trivial at infinite randomness, with the universal response governed by the statistics induced by the quenched disorder.
The entanglement entropy (EE) of the ground state of a one-dimensional Hamiltonian at criticality has a universal logarithmic scaling with a prefactor given by the central charge ccc of the underlying (1+1)(1+1)(1+1)-dimensional conformal field theory. When the system is probed by measurements, the entanglement in the critical ground state is inevitably affected due to wave-function collapse. In this paper, we study the effect of weak measurements on the entanglement scaling in the ground state of the one-dimensional critical transverse-field Ising model. For the measurements of the spins along their transverse spin axis, we identify interesting postmeasurement states associated with spatially uniform measurement outcomes. The EE in these states still satisfies the logarithmic scaling but with an alternative prefactor given by the effective central charge ceff{c}_{\mathrm{eff}}ceff​. We derive the analytical expression of ceff{c}_{\mathrm{eff}}ceff​ as a function of the measurement strength. With both numerical simulations and analytical studies, we show that for the EE averaged over all postmeasurement states based on their Born-rule probabilities, the effective central charge is independent of the measurement strength, contrary to the usual expectation that local and nonoverlapping measurements reduce the entanglement in the system. We also examine the behavior of the average EE under (biased) forced measurements where the measurement outcomes are sampled with a predetermined probability distribution without intersite correlations. In particular, we find an optimal probability distribution that can serve as a mean-field approximation to the Born-rule probabilities and lead to the same ceff{c}_{\mathrm{eff}}ceff​ behavior. The effects of the measurements along the longitudinal spin axis and the postmeasurement correlation functions are also discussed.
We study the effects of measurements, performed with a finite density in space, on the ground state of the one-dimensional transverse-field Ising model at criticality. Local degrees of freedom in critical states exhibit long-range entanglement, and as a result, local measurements can have highly nonlocal effects. Our analytical investigation of correlations and entanglement in the ensemble of measured states is based on properties of the Ising conformal field theory (CFT), where measurements appear as (1+0)(1+0)(1+0)-dimensional defects in the (1+1)(1+1)(1+1)-dimensional Euclidean spacetime. So that we can verify our predictions using large-scale free-fermion numerics, we restrict ourselves to parity-symmetric measurements. To describe their averaged effects analytically we use a replica approach, and we show that the defect arising in the replica theory is an irrelevant perturbation to the Ising CFT. Strikingly, the asymptotic scalings of averaged correlations and entanglement entropy are therefore unchanged relative to the ground state. In contrast, the defect generated by postselecting on the most likely measurement outcomes is exactly marginal. We then find that the exponent governing postmeasurement order parameter correlations, as well as the ``effective central charge'' governing the scaling of entanglement entropy, vary continuously with the density of measurements in space. Our work establishes connections between the effects of measurements on many-body quantum states and of physical defects on low-energy equilibrium properties.