Measurement Induced Asymmetric Entanglement in Deconfined Quantum Critical Ground State

Abstract

In this work, we numerically study the effect of weak measurement on deconfined quantum critical point(DQCP). Particularly, we consider the ground state of an one-dimensional spin $1/2$ system with long range exchange interactions($K$), which shows analogues phase transition to DQCP in the thermodynamic limit. This system is in the ferromagnetic phase below the critical exchange interaction $K_c$ and in the valance bond solid phase above $K_c$. The weak measurement is carried out by coupling a secondary ancilla system to the critical system via unitary interactions and later measuring the ancilla spins projectively. We numerically calculate entanglement entropy,correlation length, and order parameters of leading post-measurement states using uniform matrix product state representation of the quantum many-body state in the thermodynamic limit. We report asymmetric restructuring of entanglement of the post measurement states across the phase boundary under weak measurements. Especially, the trajectory $\left(\downarrow \downarrow\right)$ describing a uniform measurement outcome given the all ancilla spins initiated in the same $\left(\downarrow \right)$ state, shows anomalous entanglement when increasing the strength of weak measurement. The bipartite entanglement entropy strongly increases when $K<K_c$ whereas it weakly decreases when $K>K_c$. We argue with numerical evidences that observed asymmetry in entanglement would lead to a weak first order phase boundary in the thermodynamic limit. We also discuss important aspects in experimental observation of measurement induced effects linked to the strength of weak measurement and probability of post-measurement states.

Figures (6)

• Figure 1: The lower panels depict $\Delta S$, the shift of bipartite entanglement entropy of post-measurement states from that of critical ground state for 3 leading measurement outcomes, against the strength of weak measurement. The measurement outcomes $\left(\downarrow \downarrow \right)$, $\left(\uparrow \uparrow \right)$, and $\left(\uparrow \downarrow \right)$ denote the periodic extension of the spin states($z$-basis) in side the parenthesis in an infinitely long chain. The blue solid lines with squares represent the $\Delta S$ of ferromagnetic(zFM) phase when $K<K_c$. The red solid lines with circles represent the $\Delta S$ of valance bond solid (VBS) phase when $K>K_c$. Note the asymmetric development of $\Delta S$ between two phases under the weak measurement. The top left panel shows a schematic of the infinitely long critical system coupled to ancilla system and the ancilla spins are measured projective with a measurement apparatus.
• Figure 2: a.) Numerically calculated order parameters as a function of the long range exchange interaction $K$. The results are calculated at $\chi=192$ using VUMPS algorithm. The dotted and the dashed lines represent the $\langle O_{zFM}\rangle$ and $\langle O_{VBS}\rangle$ of the ground state of the critical system. The post-measurement states are calculated at $\alpha=0$ and $u=1/10$. The order parameters of the measurement outcome $\left(\downarrow \downarrow\right)$ with $X$ and $Z$ type measurements share the same dotted and dashed lines. The $\langle O_{zFM}\rangle$ and $\langle O_{zAFM}\rangle$ for the measurement outcome $\left(\uparrow \downarrow \right)$ also share the same dotted line for $Z$ and $X$ type measurements respectively. The $\langle O_{VBS}\rangle$ for $X$ type measurement is represented by the solid brown line and the that of $Z$ type measurement is represented by the solid red line. b.) The numerically calculated correlation length $\xi(\chi)$ at different values of MPS bond dimension $\chi$. The ground state and the post-measurement states at $\alpha=0$ shares the same correlation length $\xi(\chi)$. The peak value of $\xi(\chi)$ at the vicinity of the critical point diverges in the limit $\chi \rightarrow \infty$. The inset shows the entanglement entropy $S(\chi)$ versus the $\ln(\xi(\chi))$ yielding the slope $\approx 1/6$ with effective central charge $c\approx 1$.
• Figure 3: a. Correlation length $\xi(\chi)$ as a function of $K$ at $\alpha=0.001$ following $Z$ type weak measurements. The gray vertical dashed line is drawn at the pseudo critical coupling $K_c(\chi)\approx 0.549726$ at $\chi=192$. The black dashed curve is the $\xi(\chi)$ of the critical ground state. The blue dashed curve with open squares represent the $\xi(\chi)$ of the measurement outcome $\left(\downarrow \downarrow\right)$. The $\xi(\chi)$ increases significantly when $K<K_c(\chi)$ and decreases weakly when $K>K_c(\chi)$. The orange dashed curve with open squares represent the $\xi(\chi)$ of the measurement outcome $\left(\uparrow \downarrow\right)$ and shows overall reduction of $\xi(\chi)$. The gap at the $K_c(\chi)$ is denoted by $\Delta \xi$. The inset shows the variation of $\Delta \xi$ against the bond dimension for the measurement outcome $\left(\uparrow \downarrow\right)$-orange curve and $\left(\downarrow \downarrow\right)$-blue curve. The plots b. and c. shows the bipartite entanglement entropy $S(\chi)$ as a function of $K$ at $\alpha=0.001, 0.002, 0.003, 0.004$ for measurement outcome $\left(\uparrow \downarrow\right)$ and $\left(\downarrow \downarrow\right)$ respectively. In plot b. $S$ decreases when increasing the $\alpha$ and the rate of decrement is higher when $K<K_c(\chi)$. Note the direction and the length of red arrows drawn proportional to the rate of decrement. In plot c. $S$ increases when increasing $\alpha$ when $K<K_c(\chi)$(note the red up arrow) and $S$ decreases weakly when increasing $\alpha$ when $K>K_c(\chi)$ (note the red down arrow).
• Figure 4: Shift of entanglement entropy $\Delta S$ versus the unitary parameter $u$. a.) at $K=0.54$ ($<K_c$) and b.) at $K=0.56$ ($>K_c$), for measurement outcomes $\left(\downarrow \downarrow \right)$, $\left(\uparrow \downarrow \right)$ and $\left(\uparrow \uparrow \right)$ when $\alpha=0.004$.
• Figure 5: Numerically calculated order parameters as a function of $|K-K_c(\chi)|$. In the left panel the 3 sets of curves in three different colors are for post measurement states $\ket{\psi_{(\uparrow \downarrow)}^X}$-(orange symbols), $\ket{\psi_{(\downarrow \downarrow)}^{X/Z}}$-(red symbols) and $\ket{\psi_{(\uparrow \downarrow)}^Z}$-(brown symbols) from top to bottom. In the right panel, the $\langle O_{zFM} \rangle$ and $\langle O_{zAFM} \rangle$ of different post-measurement states coincide. Note in the regions $K<K_c$ and $K>K_c$, the order parameters follows approximately the same power-law exponent $\beta$ for different $\chi$.