New critical states induced by measurement

TL;DR

The paper shows that weak local measurements and postselection can qualitatively alter the critical ground state of a Luttinger liquid. By combining bosonization, replica techniques, and a dual-field description, it identifies a measurement-induced entanglement transition controlled by the Luttinger parameter K: logarithmic entanglement with c=1 for K>1, area-law entanglement with algebraic corrections for K<1, and a marginal line at K=1 with a continuously varying effective central charge c_eff(W). Numerical (DMRG, Gaussian-state) and proposed experimental pathways (ancilla-based postselection, variable filling, and variational quantum algorithms) support the analytic predictions and suggest feasible routes to realize and study these new critical states. The work connects boundary/CFT ideas with measurement physics and highlights potential routes to observe MIPT in solid-state analogs and quantum simulators. The findings advance understanding of how measurements shape quantum criticality and entanglement structures, with implications for quantum information and many-body physics.

Abstract

Finding new critical states of matter is an important subject in modern many-body physics. Here we study the effect of measurement and postselection on the critical ground state of a Luttinger liquid theory and show that it can lead to qualitatively new critical states. Depending on the Luttinger parameter $K$, the effect of measurement is irrelevant (relevant) at $K>1$ ($K<1$). We reveal that this causes an entanglement transition between two phases, one with logarithmic entanglement entropy for a subregion ($K>1$), and the other with algebraic entanglement entropy ($K<1$). At the critical point $K=1$, the measurement is marginal, and we find new critical states whose entanglement entropy exhibits a logarithmic behavior with a continuous effective central charge as a function of measurement strength. We also performed numerical density matrix renormalization group and fermionic Gaussian state simulations to support our results. We further discuss promising and feasible routes to experimentally realize new critical states in our work.

Figures (14)

• Figure 1: (a) A schematic plot of the spinless fermion chain. $t$ and $V$ denote the hopping and the nearest-neighbor interaction, respectively, in (\ref{['eq:fermion_Hamiltonian']}), $W$ is the measurement strength in (\ref{['eq:measurement']}). (b) Phase diagram of the Luttinger liquid after weak measurement. $K$ and $W$ denote the Luttinger parameter and the measurement strength, respectively. For $K>1$, the measurement is irrelevant, and the entanglement entropy of a subregion $A$ with length $x_A$ satisfies a log-law with central charge $c=1$. For $K<1$, the measurement is relevant, and changes the entanglement entropy to an area law with a subleading algebraic correction. At $K=1$, there is a continuous critical line, at which the measurement is marginal. The entanglement entropy satisfies a log-law with an effective central charge $c_\text{eff}$ given in \ref{['eq:effective_central_charge']}. We use the red line to indicate the non-measurement case $W=0$, where the entanglement entropy is given by $S_A = 1/3 \log x_A$.
• Figure 2: The half-chain entanglement entropy as a function of different sizes at (a) $\Delta = -0.6$, (b,c) $\Delta = 0.6$. The black dots are numerical results and the colored lines are fitting curves. (a) The blue (orange) curve is given by measurement strength $W=0$ ($W=0.6$). The same slope indicates the effective central charges are the same. (b) shows the data at measurement strength $W=0$. It shows a logarithmic function with central charge $c=1$. (c) shows the data at measurement strength $W=0.6$. It shows an algebraic function with power $0.77$. (d) The algebraic power as a function of different $K$ for $K<1$. The measurement strength is $W=0.6$. The black dots show the power fitted by numerical data. The orange curve is our prediction $2/K-2$. The numerical calculation is obtained with bond dimension $\chi = 100$.
• Figure 3: (a) The half-chain entanglement entropy at the critical point as a function of different system sizes $L$. The parameter is chosen to be $t=1$, $V = 0$, $W=1$. The black dots represent the numerical data, and the orange line is a fitting with $0.04 \log L+0.0435$. (b) The effective central charge as a function of different measurement strength $W$. The black dots represent effective central charge that is extracted from fitting the numerical data. The solid curve is our prediction $c_\text{eff}$. In the inner figure, we plot $c_{\rm eff}$-$W$ on a log-linear scale.
• Figure 4: (a) Half-chain entanglement entropy as a function of $\Delta$ for different sizes $L$. The measurement strength is $W=1.0$. (b) Mutual information $I_{\rm AB}$ as a function of $L$ at critical point for different measurement strength $W$. The dots (curves) are numerical results (analytical predictions).
• Figure S1: (a) Diagram of Integral for integral \ref{['eq:integral dI3']}. (b) Diagram of Integral for integral \ref{['eq:integral dI4']}