Lieb-Schultz-Mattis Theorem in Open Quantum Systems

Abstract

The Lieb-Schultz-Mattis (LSM) theorem provides a general constraint on quantum many-body systems and plays a significant role in the Haldane gap phenomena and topological phases of matter. Here, we extend the LSM theorem to open quantum systems and establish a general theorem that restricts the steady state and spectral gap of Liouvillians based solely on symmetry. Specifically, we demonstrate that the unique gapped steady state is prohibited when translation invariance and U (1) symmetry are simultaneously present for noninteger filling numbers. As an illustrative example, we find that no dissipative gap is open in the spin-1/2 dissipative Heisenberg model while a dissipative gap can be open in the spin-1 counterpart -- an analog of the Haldane gap phenomena in open quantum systems. Furthermore, we show that the LSM constraint manifests itself in a quantum anomaly of the dissipative form factor of Liouvillians. We also find the LSM constraints due to symmetry intrinsic to open quantum systems, such as Kubo-Martin-Schwinger symmetry. Our work leads to a unified understanding of phases and phenomena in open quantum systems.

Figures (5)

• Figure 1: Dissipative Heisenberg XXZ model ($J=J_z=1.0$, $\gamma = 1.0$) for (a, b) spin half $S=1/2$ ($L=8$) and (c, d) spin one $S=1$ ($L=5$). The $\mathrm{U} \left( 1 \right)$ charge is $S^{z}_{\pm} = 0$, i.e., half filling (a, b) $\nu = 1/2$ for $S=1/2$ and (c, d) $\nu = 1$ for $S=1$. The $\mathrm{U} \left( 1 \right)$ flux $\phi$ in the ket space is inserted. (a, c) Lindbladian spectrum with $\phi = 0$ for (a) $S=1/2$ and (c) $S=1$. (b, d) Real part of the Lindbladian spectrum around the steady state $\lambda = 0$ for (b) $S=1/2$ and (d) $S=1$ as a function of the flux $\phi$.
• Figure S1: Complex-spectral flow of the dissipative Heisenberg XXZ model with dissipators $L_n = \sqrt{\gamma} S_{n}^{z}$ ($J = J_z = 1.0$, $\gamma = 1.0$) for (a, b) spin half $S=1/2$ ($L=8$) and (c, d) spin one $S=1$ ($L=5$). The $\mathrm{U} \left( 1 \right)$ flux $\phi$ in the ket space is inserted. The $\mathrm{U} \left( 1 \right)$ charge is (a) $S_{\pm}^{z} = -2$ ($\nu = 1/4$), (b) $S_{\pm}^{z} = -1$ ($\nu = 3/8$), (c) $S_{\pm}^{z} = -2$ ($\nu = 3/5$), and (d) $S_{\pm}^{z} = -1$ ($\nu = 4/5$).
• Figure S2: Dissipative Heisenberg XXZ model with dissipators $L_n = \sqrt{\gamma}\,(S_{n}^{z})^2$ and spin one $S=1$ ($L=5$, $J=J_z=1.0$, $\gamma = 1.0$). The $\mathrm{U} \left( 1 \right)$ charge is $S^{z}_{\pm} = 0$, i.e., half filling $\nu = 0$. The $\mathrm{U} \left( 1 \right)$ flux $\phi$ in the ket space is inserted. (a) Lindbladian spectrum with $\phi = 0$. (b) Real part of the Lindbladian spectrum around the steady state $\lambda = 0$ as a function of the flux $\phi$.
• Figure S3: Dissipative fermionic model ($L=8$, $J=1.0$, $\gamma = 1.0$) with the half filling $\nu = 1/2$. The $\mathrm{U} \left( 1 \right)$ flux $\phi$ in the ket space is inserted. (a) Lindbladian spectrum under the periodic boundary conditions with $\phi = 0$ (red dots) and open boundary conditions (blue dots). (b) Real part of the Lindbladian spectrum around the steady state $\lambda = 0$ as a function of the flux $\phi$.
• Figure S4: Complex-spectral flow of Lindbladians in the half filling [(a, d) $\nu = 1/2$ and (b, c) $\nu = 1$]. The $\mathrm{U} \left( 1 \right)$ fluxes $\phi_{+}$ and $\phi_{-}$ are inserted in both ket and bra spaces so that the Lindbladians will be invariant under modular conjugation (i.e., $\phi_{+} + \phi_{-} = 0$). (a, b, c) Dissipative Heisenberg XXZ model with (a) dissipators $L_{n} = \sqrt{\gamma}\,S_{n}^{z}$ and spin $S=1/2$ ($L=8$, $J=J_z=1.0$, $\gamma = 1.0$), (b) dissipators $L_{n} = \sqrt{\gamma}\,S_{n}^{z}$ and spin $S=1$ ($L=5$, $J=J_z=1.0$, $\gamma = 1.0$), and (c) $L_{n} = \sqrt{\gamma}\,(S_{n}^{z})^2$ and spin $S=1$ ($L=5$, $J=J_z=1.0$, $\gamma = 1.0$). (d) Dissipative fermionic model with dissipators $L_{n} = \sqrt{\gamma}\,c_{n+1}^{\dag} c_{n}$ ($L=8$, $J=1.0$, $\gamma = 1.0$).