Universal properties of dissipative Tomonaga-Luttinger liquids: Case study of a non-Hermitian XXZ spin chain

We demonstrate the universal properties of dissipative Tomonaga-Luttinger (TL) liquids by calculating correlation functions and performing finite-size scaling analysis of a non-Hermitian XXZ spin chain as a prototypical model in one-dimensional open quantum many-body systems. Our analytic calculation is based on effective field theory with bosonization, finite-size scaling approach in conformal field theory, and the Bethe-ansatz solution. Our numerical analysis is based on the density-matrix renormalization group generalized to non-Hermitian systems (NH-DMRG). We uncover that the model in the massless regime with weak dissipation belongs to the universality class characterized by the complex-valued TL parameter, which is related to the complex generalization of the $c=1$ conformal field theory. As the dissipation strength increases, the values of the TL parameter obtained by the NH-DMRG begin to deviate from those obtained by the Bethe-ansatz analysis, indicating that the model becomes massive for strong dissipation. Our results can be tested with the two-component Bose-Hubbard system of ultracold atoms subject to two-body loss.

Figure 1

Numerical solution of the Bethe equations [Eq. (45)] for the ground state of the NH XXZ model with $L=2M=250$ and $\mathrm{\Phi }=0$. The model parameter is set to ${\mathrm{\Delta }}_{\gamma }=0.8-0.3i$. Green dots show spin rapidities ${\lambda }_j(j=1,\cdots ,M)$.

Figure 2

Ratio between the critical exponents $K_{\varphi }$ and $K_{\theta }$ calculated from the exact result (64) on the NH XXZ model.

Figure 3

Numerical results (solid lines) obtained by the NH-DMRG and exact results (dotted lines) by the Bethe-ansatz solution. (a), (b) Critical exponents $K_{\varphi }$ and $K_{\theta }$. (c), (d) Velocity of excitations as a function of dissipation $-\mathrm{Im}{\mathrm{\Delta }}_{\gamma }$. Color plots show the data for $\mathrm{Re}{\mathrm{\Delta }}_{\gamma }=0.1,0.2,0.3,\cdots ,0.9$ from top to bottom in (a) and (b), and from bottom to top in (c) and (d). The system size is set to $L=130$ and up to 200 states are kept during the NH-DMRG sweep.

Figure 4

Numerical results of correlation functions versus $|i-j|$ for (a) $\mathrm{Re}{\mathrm{\Delta }}_{\gamma }=0.3$ and (b) $\mathrm{Re}{\mathrm{\Delta }}_{\gamma }=0.9$ obtained by NH-DMRG. The correlation functions are calculated with $i=(L-r+1)/2$ and $j=(L+r+1)/2$ for odd $r$ and $i=(L-r+2)/2$ and $j=(L+r+2)/2$ for even $r$. The system size is set to $L=100$ and up to 170 states are kept during the NH-DMRG sweep.

Figure 5

Numerical results (solid lines) obtained by the fitting of correlation functions with NH-DMRG and exact results (dotted lines) obtained by the Bethe-ansatz solution of the critical exponent $K_{\varphi }$ as a function of dissipation $-\mathrm{Im}{\mathrm{\Delta }}_{\gamma }$. Color plots show the data for $\mathrm{Re}{\mathrm{\Delta }}_{\gamma }=0.1,0.2,0.3,\cdots ,0.9$ from top to bottom. The system size is set to $L=100$ and up to 170 states are kept during the NH-DMRG sweep.

Figure 6

NH-DMRG results of energy gaps in a finite system as a function of dissipation $-\mathrm{Im}{\mathrm{\Delta }}_{\gamma }$. Color plots show the data for $\mathrm{Re}{\mathrm{\Delta }}_{\gamma }=0.1,0.2,0.3,\cdots ,0.9$ from bottom to top in (a), (c), and (d), and from top to bottom in (b). The system size is set to $L=130$, and up to 200 states are kept during the NH-DMRG sweep.

Figure 7

Comparison of the energy gaps obtained by NH-DMRG (solid lines and broken lines) and the exact diagonalization (dotted lines). As the ground states for $\mathrm{Re}{\mathrm{\Delta }}_{\gamma }>1$ are doubly degenerate in an infinite system, spin gaps for the two lowest states are plotted. The system size is set to $L=14, \mathrm{Re}{\mathrm{\Delta }}_{\gamma }=1.5$, and up to 40 states are kept during the NH-DMRG sweep. We use the double-precision data for both NH-DMRG and the exact diagonalization (see text).