Detecting (emergent) continuous symmetry of criticality via subsystem's entanglement spectrum

TL;DR

The paper addresses the challenge of identifying the underlying continuous symmetry at quantum critical points without assuming a specific low-energy EFT. It proposes a general diagnostic based on the entanglement spectrum (ES) of a small subsystem, mapped to an entanglement Hamiltonian at $T_E=1$, where a tower-of-states (TOS) structure reveals the symmetry via $E_S(L)-E_0(L) \propto S(S+N-2)$. The authors demonstrate the method on several (2+1)$\,$D systems: a dimerized Heisenberg model with $(2+1)$D $O(3)$ criticality, the $J$-$Q_3$ model with emergent $SO(5)$ symmetry at a deconfined quantum critical point, and the checkerboard $J$-$Q$ model with emergent $O(4)$ symmetry; they show the ES TOS is robust to finite size and compatible with QMC and DMRG using cornerless cuts. The approach enables unbiased determination of universality classes and symmetry content for unknown quantum critical systems, guiding the construction of low-energy effective field theories and facilitating the study of emergent symmetries across tensor-network and QMC frameworks.

Abstract

The (emergent) symmetry of a critical point is one of the most important information to identify the universality class and effective field theory, which is fundamental for various critical theories. However, the underlying symmetry so far can only be conjectured indirectly from the dimension of the order parameters in symmetry-breaking phases, and its correctness requires further verifications to avoid overlooking hidden order parameters, which by itself is also a difficult task. In this work, we propose an unbiased way to numerically identify the underlying (emergent) symmetry of a critical point in quantum many-body systems, without prior knowledge about its low-energy effective field theory. Through calculating the reduced density matrix in a very small subsystem of the total system numerically, the Anderson tower of states in the entanglement spectrum clearly reflects the underlying (emergent) symmetry of the criticality. It is attributed to the fact that the entanglement spectrum can observe the broken symmetry of the entanglement ground-state after cooling from the critical point along an extra temperature axis.

Figures (9)

• Figure 1: A general phase diagram of the EH. $J$ is a tunable parameter of the original Hamiltonian (OH). $T_E$ is the temperature of the EH. The reduced density matrix $\rho_A=e^{-H_E}$ can be considered a Gibbs mixed state of EH at $T_E=1$, and the dashed line $T_E=1$ is the ground-state phase of the OH. The yellow point is not only the quantum phase transition point of the OH but also the thermal phase transition point of the EH. In this way, the EH naturally introduces an extra temperature space, and SSB will occur through reducing $T_E$ (cooling) from the upper critical curves or points. Because the ES reflects SSB in the ground state at $T_E=0$, different symmetries ($S_1$, $S_2$, $S_3$) in finite $T_E$ phase transitions can be detected.
• Figure 2: Model and phase diagram. (a) Spin-1/2 dimerized AFM Heisenberg model: the strong bonds $J_2>0$ are indicated by thick lines, and the weak bonds $J_1> 0$ are indicated by thin lines. The dashed lines are used to illustrate the bipartition into two subsystems. The red dashed lines illustrate the cutting method, in which $A$ is a ring and the other part is denoted as $B$. (b) $J-Q$ model: $J>0$ is the AFM Heisenberg interaction, and the six-spin $Q$ interaction covers the entire lattice. (c) Diagram of the dimerized AFM Heisenberg model with $J_1=1$, in which the quantum critical point is $J_c=J_{2}/J_{1}=1.90951(1)$NSMa2018anomalousMatsumoto2001. (d) The phase diagram of the $J-Q_3$ model with $J=1$ and for which $Q_c=Q/J=1.49153$.
• Figure 3: Entanglement spectrum of the columnar Heisenberg model with $L_x=L_y=8$. The subsystem is chosen as a chain $L^{sub}_x=8$ with a PBC and various $J=J_2/J_1$. The QCP is $J_c=J_2/J_1=1.90951$. In (a) and (b), the tower is proportional to $S(S+1)$, which reflects the $O(3)$ SSB here. All the error bars are smaller than the data symbols.
• Figure 4: Entanglement spectrum of the $J-Q_3$ square lattice model with $L_x=L_y=8$. The subsystem is chosen as a chain with size $L_x^{sub}=8$ and various $Q$ values. We choose $J=1$ as the energy unit, and the results are obtained by RDM-QMC method. The different linear relationships $S(S+N-2)$ reflect different degrees of symmetry breaking. All the error bars are smaller than the data symbols.
• Figure 5: Entanglement spectrum of the checkerboard $J-Q$ model with $L_x=L_y=8$ at the first-order phase transition point. The subsystem is chosen as a chain of size $L_x^{sub}=8$. The linear relation $S(S+2)$ is associated with the $O(4)$ SSB here. All the error bars are smaller than the data symbols.