Table 3 A comparison of different pure and mixed state geometric phases of long-range SSH chain for infinite neighbors (\(r\rightarrow \infty\)). The Berry phase and Uhlmann phase are calculated through Eqs. (4) and (10) respectively. Here the geometric phases are quantized in the units of \(\pi\) with \(t^{\prime }=1\). The top and bottom tables represents the Hermitian and non-Hermitian SSH chains respectively.
From: Mixed state behavior of Hermitian and non-Hermitian topological models with extended couplings
Region | Berry phase | Uhlmann phase |  |
|---|---|---|---|
\(\alpha <1\) \(t>-t^{\prime }(2^{1-\alpha }-1)\zeta [\alpha ]\) | \(-\pi /2\) | Undefined | Â |
\(\alpha <1\) \(t<-t^{\prime }(2^{1-\alpha }-1)\zeta [\alpha ]\) | \(\pi /2\) | Undefined | Â |
\(\alpha >1\) \(-t^{\prime } Li_{\alpha }(1)<\mu\) \(<-2\lambda (2^{1-\alpha }-1)\zeta [\alpha ]\) | 1 | 1 | Â |
\(\alpha >1\) \(t<-t^{\prime } Li_{\alpha }(1)\) \(t>-t^{\prime }(2^{1-\alpha }-1)\zeta [\alpha ]\) | 0 | 0 | Â |
Region | \(\alpha\) | Berry phase \(\left( W_B=\frac{W1+W2}{2}\right)\) | Uhlmann phase \(\left( W_U=\frac{W1+W2}{2}\right)\) |
|---|---|---|---|
\(t<-\delta -\text {Li}_{\alpha }(-1)\) | \(\alpha <1\) | \(\frac{0+1}{2}=\frac{1}{2}\) | Undefined |
\(\delta -\text {Li}_{\alpha }(-1)\) \(<t<\) \(-\delta -\text {Li}_{\alpha }(-1)\) | \(\alpha <1\) | \(\frac{0+1/2}{2}=\frac{1}{4}\) | Undefined |
\(t>\delta -\text {Li}_{\alpha }(-1)\) | \(\alpha <1\) | \(\frac{-1/2+0}{2}=-\frac{1}{4}\) | Undefined |
\(-\delta -\text {Li}_{\alpha }(1)\) \(<t<\) \(\delta -\text {Li}_{\alpha }(1)\) | \(\alpha >1\) | \(\frac{0+1}{2}=\frac{1}{2}\) | \(\frac{0+1}{2}=\frac{1}{2}\) |
\(\delta -\text {Li}_{\alpha }(-1)\) \(<t<\) \(-\delta -\text {Li}_{\alpha }(-1)\) | \(\alpha >1\) | \(\frac{1+0}{2}=\frac{1}{2}\) | \(\frac{1+0}{2}=\frac{1}{2}\) |
\(\delta -\text {Li}_{\alpha }(1)\) \(<t<\) \(-\delta -\text {Li}_{\alpha }(-1)\) | \(\alpha >1\) | \(\frac{1+1}{2}=1\) | \(\frac{1+1}{2}=1\) |
