Dynamical Quantum Phase Transitions

Author: Xu-Dong Liu | Date: April 20, 2026

1. Total Phase

1.1 Loschmidt Amplitude

The Loschmidt amplitude is defined as the overlap between the initial state and the time-evolved state:

$g_k(t) = \langle\Psi(0)|e^{-iH_k t}|\Psi(0)\rangle$

In polar form, $g_k(t) = r_k e^{i\varphi_R}$ , where $r_k = |g_k(t)|$ . The total phase $\varphi_R$ is extracted as:

$\varphi_R = -i \ln \left( \frac{g_k(t)}{|g_k(t)|} \right)$

1.2 Density Matrix and Statistical Analogy

Using the analogy $P_n = e^{-\beta E_n} / Z$ , we construct the density matrix in momentum space. For $H_k = \omega_k \hat{n}_k \cdot \vec{\sigma}$ , we have:

$\begin{aligned} \rho_k(0) &= \frac{e^{-\beta H_k}}{\text{Tr } (e^{-\beta H_k})} \\ &= \frac{\cosh(\beta \omega_k) \sigma_0 - (\hat{n}_k \cdot \vec{\sigma}) \sinh(\beta \omega_k)}{\text{Tr} [ \cosh(\beta \omega_k) \sigma_0 - (\hat{n}_k \cdot \vec{\sigma}) \sinh(\beta \omega_k) ]} \\ &= \frac{\cosh(\beta \omega_k) \sigma_0 - (\hat{n}_k \cdot \vec{\sigma}) \sinh(\beta \omega_k)}{2 \cosh(\beta \omega_k)} \\ &= \frac{1}{2} \left( \sigma_0 - \tanh(\beta \omega_k) (\hat{n}_k \cdot \vec{\sigma}) \right) = \frac{1}{2} \left( \sigma_0 - m (\hat{n}_k \cdot \vec{\sigma}) \right) \end{aligned}$

Here, $m = \tanh(\beta \omega_k)$ acts as the effective magnetization.

2. DYNAMICAL PHASE

In non-Hermitian systems ($[H, H^\dagger] \neq 0$), we use biorthogonal quantum mechanics. The right and left eigenvectors satisfy $H|s^r\rangle = s\omega|s^r\rangle$ and $H^\dagger|s^l\rangle = s\omega^*|s^l\rangle$ with $\langle s^l | s'^r \rangle = \delta_{s,s'}$ .

[Image of biorthogonal basis vectors in non-Hermitian systems]

The time-evolution operators are expanded as:

$\begin{aligned} U^r &= \sum_{s=\pm} e^{-is\text{Re}[\omega]t + s\text{Im}[\omega]t} |s^r\rangle\langle s^l| \\ U^{l\dagger} &= \sum_{s=\pm} e^{is\text{Re}[\omega]t + s\text{Im}[\omega]t} |s^r\rangle\langle s^l| \end{aligned}$

Evaluating the traces for $U^{l\dagger}U^r$ using the biorthogonality condition:

$\begin{aligned} \text{Tr}(U^{l\dagger}U^r) &= \sum_{s=\pm} \langle s^l| U^{l\dagger}U^r |s^r\rangle = e^{2\text{Im}[\omega]t} + e^{-2\text{Im}[\omega]t} = 2 \cosh[2\text{Im}(\omega)t] \\ \text{Tr}(U^{l\dagger}HU^r) &= \sum_{s=\pm} \langle s^l| H U^{l\dagger}U^r |s^r\rangle = \omega e^{2\text{Im}[\omega]t} - \omega e^{-2\text{Im}[\omega]t} = 2\omega \sinh[2\text{Im}(\omega)t] \\ \text{Tr}(U^{l\dagger}HU^r H) &= \sum_{s=\pm} \langle s^l| H^2 U^{l\dagger}U^r |s^r\rangle = \omega^2 e^{2\text{Im}[\omega]t} + \omega^2 e^{-2\text{Im}[\omega]t} = 2\omega^2 \cosh[2\text{Im}(\omega)t] \end{aligned}$

For the initial state $\rho(0) = \frac{1}{2}(\sigma_0 - \frac{m}{\omega}H)$ , the integrated dynamical phase $\varphi_D$ is:

$\begin{aligned} \varphi_D &= -\frac{1}{\hbar} \int_{0}^{\tau} \frac{\text{Tr}[U^{l\dagger} \frac{1}{2}(\sigma_0 - \frac{m}{\omega}H) U^r H]}{\text{Tr}[U^{l\dagger} \frac{1}{2}(\sigma_0 - \frac{m}{\omega}H) U^r]} dt \\ &= -\frac{1}{\hbar} \int_{0}^{\tau} \omega \frac{\sinh[2\text{Im}(\omega)t] - m \cosh[2\text{Im}(\omega)t]}{\cosh[2\text{Im}(\omega)t] - m \sinh[2\text{Im}(\omega)t]} dt \\ &= -\frac{1}{\hbar} \int_{0}^{\tau} \omega \frac{\tanh[2\text{Im}(\omega)t] - m}{1 - m \tanh[2\text{Im}(\omega)t]} dt \end{aligned}$

3. HERMITIAN SYSTEMS

In the Hermitian limit ( $H = H^\dagger$ , $\text{Im}[\omega] = 0$ ), the left and right eigenvectors become identical ($|s^l\rangle = |s^r\rangle = |s\rangle$). As $\tanh[2\text{Im}(\omega)t] \to 0$ , the phase accumulation becomes strictly linear:

$\varphi_D = -\frac{1}{\hbar} \int_{0}^{\tau} \omega (-m) dt = \frac{m\omega\tau}{\hbar}$

4. SPECIAL CASE: EXACT REAL SPECTRUM IN QUANTUM WALKS

For certain non-Hermitian quantum walks, the energy spectrum is $E_\pm = \pm \omega(k,\theta) + i\ln \eta(k,\theta)$ , where $\omega$ is real. The non-Hermiticity acts as a global scalar $e^{2\ln \eta \cdot t}$ , which cancels out in the density matrix:

$\rho(t) = \frac{e^{2\ln \eta \cdot t} \rho(0)}{e^{2\ln \eta \cdot t} \text{Tr}[\rho(0)]} = \rho(0)$

The observable dynamical phase (real part) accumulates linearly, identical to Hermitian systems:

$\varphi_d(k,\tau) = \text{Re}\{\varphi_D\} = \frac{m \omega(k,\theta) \tau}{\hbar}$

5. GEOMETRIC PHASE AND WINDING NUMBER

The geometric phase is $\varphi_G(k,t) = \varphi_R(k,t) - \varphi_d(k,t)$ . The winding number $\nu(t)$ acts as a dynamical topological order parameter:

$\nu(t) = \oint_{\text{BZ}} dk \frac{\partial \varphi_G(k,t)}{\partial k}$

[Image of winding number jumps at critical times]

Note: Quantized jumps in $\nu(t)$ occur exactly at critical times $t_c$ where the Loschmidt amplitude vanishes ($g_k(t_c) = 0$), signaling a Dynamical Quantum Phase Transition.