Zeros of Loschmidt echo in the presence of Anderson localization

TL;DR

This paper investigates the behavior of the Loschmidt echo and dynamical free energy in the Anderson model after a disorder quench, revealing periodic zeros linked to localization phenomena and connecting dynamical phase transitions with localization-delocalization transitions.

Contribution

It introduces a novel analysis of Loschmidt echo zeros in Anderson localization, showing their periodic nature and relation to spectral width, supported by numerical evidence.

Findings

Loschmidt echo exhibits periodic zeros in localized regimes

Dynamical free energy diverges logarithmically at zeros

Zeros are connected to spectral width and phase transitions

Abstract

We study the Loschmidt echo and the dynamical free energy of the Anderson model after a quench of the disorder strength. If the initial state is extended and the eigenstates of the post-quench Hamiltonian are strongly localized, we argue that the Loschmidt echo exhibits zeros periodically with the period $2\pi /D$ where $D$ is the width of spectra. At these zeros, the dynamical free energy diverges in a logarithmic way. We present numerical evidence of our argument in one- and three-dimensional Anderson models. Our findings connect the dynamical quantum phase transitions to the localization-delocalization phase transitions.

Equations

$$\mathcal{L}(t)=⟨\Psi (0)|e^{-i\hat{H}t}|\Psi (0)⟩,$$

$$f(t)=-\underset{N\to \infty }{\lim }\frac{1}{N}\ln {\left|\mathcal{L}(t)\right|}^2,$$

$$\hat{H}=-g\sum _{⟨i,j⟩}({\hat{a}}_i^{†}{\hat{a}}_j+H.c.)+\sum _i{u}_i{\hat{a}}_i^{†}{\hat{a}}_i,$$

$$|\varphi (t)⟩=\sum _{n=1}^L⟨{\alpha }_n⟩\varphi (0)e^{-i{ϵ}_{n}t}|{\alpha }_n⟩.$$

$$\begin{array}{rl}\mathcal{L}(t)=& ⟨\varphi (0)⟩{\varphi (t)}^N\\ =& {\left(\sum _{n=1}^L{\left|⟨\varphi (0)⟩{\alpha }_n\right|}^2{e}^{-i{ϵ}_{n}t}\right)}^N.\end{array}$$

$$l(t)=\frac{1}{L}\sum _{n=1}^L{e}^{-i{ϵ}_{n}t}.$$

$$\begin{array}{r}P({ϵ}_1,{ϵ}_2,\cdots ,{ϵ}_L)=1/D^L,\end{array}$$

$$\begin{array}{rl}l(t)=& {\int }_{-D/2}^{D/2}d{ϵ}_1\cdots d{ϵ}_{L}P({ϵ}_1,{ϵ}_2,\cdots ,{ϵ}_L)\frac{1}{L}\sum _{n=1}^L{e}^{-i{ϵ}_{n}t}\\ =& \frac{e^{-itD/2}\left(e^{itD}-1\right)}{iDt}.\end{array}$$

$$f(t)=-\ln \frac{{\sin }^2\left(Dt/2\right)}{{\left(Dt/2\right)}^2}.$$

$$\Delta =\frac{1}{L^2}\sum _{n=1}^L{\left({\left|⟨\varphi (0)⟩{\alpha }_n\right|}^2-\frac{1}{L}\right)}^2.$$

Full text

I Introduction

Since Anderson’s seminal paper in 1958 anderson , Anderson localization has been extensively studied. In recent years, great progress was made in simulating the nonequilibrium dynamics of closed quantum systems by using ultracold atoms bloch . A question then arises as to what is the influence of Anderson localization on the nonequilibrium dynamics. An especially interesting protocol of driving a system out of equilibrium is by a quantum quench, i.e. by suddenly changing the Hamiltonian of the system. For a homogeneous integrable system such as a noninteracting Fermi gas, the local observable relaxes to a steady value after a quench rigol06 ; rigol07 . And this steady value can be predicted by the generalized Gibbs ensemble (GGE) rigol06 . But if the post-quench Hamiltonian has localized eigenstates, the observable exhibits an everlasting oscillation with its average deviating significantly from the prediction of GGE caneva11 ; gramsch ; ziraldo ; ziraldo13 ; he . This everlasting oscillation comes from the pure-point spectrum associated to the localized eigenstates ziraldo .

An important quantity characterizing the nonequilibrium dynamics is the Loschmidt echo

[TABLE]

where $\ket{\Psi(0)}$ denotes the pre-quench quantum state and $\hat{H}$ the post-quench Hamiltonian. $\mathcal{L}(t)$ might become zero at some critical times $t^{*}$, at which the dynamical free energy in thermodynamic limit is nonanalytic. The dynamical free energy is usually defined as

[TABLE]

where $N$ denotes the number of particles. The nonanalyticity of $f(t)$ at $t=t^{*}$ is dubbed a dynamical quantum phase transition (DQPT) heyl .

Ever since the original paper of Heyl et al. heyl , the DQPTs have been theoretically addressed in various models CK ; fagotti ; Eck ; FA ; Jpg ; MSc ; HeylT ; dora ; heyl14 with the experimental observations also realized recently flaschner ; jurcevic . It was argued that the Loschmidt echo exhibits zeros if the pre-quench and post-quench Hamiltonians have topologically-different ground states dora , or if the pre-quench state breaks some symmetry which is recovered after the quench heyl14 . The relation between the DQPTs and the topological phase transitions (or the symmetry-breaking phase transitions) is then established.

While most theoretical works on DQPTs focus on the quantum systems with phase transitions caused by broken symmetries or accompanied by the close of energy gap, DQPTs driven by disorders are rarely studied except of a recent work on the one-dimensional Aubry-André model with quasi-periodic potentials yang . In this paper we make a further step towards connecting the DQPTs to the localization-delocalization transitions. This is done by proving that zeros of Loschmidt echo and nonanalyticity of the dynamical free energy occur if an extended initial state is quenched into a strongly-localized regime. Besides a proof based on the properties of the wave functions and the spectrum in a localized phase, we also provide numerical evidence in one- and three-dimensional Anderson models. Different from the quasi-periodic system with deterministic quasi-random potentials, we will discuss the more general disordered systems with random potentials of white-noise type and focus our study on the quench dynamics from an initial extended state to the strongly-disordered regime. We find that the critical times in the case of white-noise potentials are $t^{*}_{n}=\frac{2\pi n}{D}$ with $n$ an integer and $D$ the spectrum width of $\hat{H}$, which are different from the critical times in the case of quasi-periodic potentials - the zeros of the Bessel function yang . Our results will serve as a benchmark for understanding the characteristics of the Loschmidt echo and the dynamical free energy after a more general quench with the initial and the post-quench Hamiltonians at arbitrary disorder strength.

The contents of the paper are arranged as follows. In Sec. II, we present a general argument about the zeros of the Loschmidt echo and the nonanalyticity of the dynamical free energy. The numerical evidence is given in Sec. III. Sec. IV is a short summary.

II DQPT in the strong-disorder limit

The Hamiltonian of the Anderson model is in general expressed as

[TABLE]

where $\langle i,j\rangle$ denotes a pair of neighbor sites and $u_{i}$ is the on-site potential which is an independent random number distributed uniformly in the interval $[-W/2,W/2]$ with $W$ denoting the disorder strength. $g$ is set to the unit of energy throughout the paper. We focus on bosons in this paper. $\hat{a}^{{\dagger}}_{i}$ and $\hat{a}_{i}$ denote the bosonic creation and annihilation operators, respectively.

Let us suppose that the system is initially prepared in an extended state, e.g., in the ground state of the Hamiltonian (3) at $W=0$. In the initial state, all the bosons occupy the lowest energy level, the single-particle wave function of which is denoted as $\ket{\phi(0)}$. The system is then driven out of equilibrium by suddenly changing the Hamiltonian to $\hat{H}$ at finite $W$. Since we do not consider the interaction between bosons, the many-body wave function keeps a product state during the time evolution. Furthermore, all the bosons have the same wave function at arbitrary time, which is denoted as $\ket{\phi(t)}$. We use $\ket{\alpha_{n}}$ to denote the single-particle eigenstate of the post-quench Hamiltonian with the corresponding eigenenergy $\epsilon_{n}$, where $n=1,2,\cdots,L$ and $L$ is the total number of sites. We then obtain

[TABLE]

If the system contains $N$ bosons, the Loschmidt echo of the many-body wave function becomes

[TABLE]

Now we discuss the case in which the system is quenched into the strongly-disordered regime. In this regime, all the single-particle eigenstates $\ket{\alpha_{n}}$ are strongly-localized like a $\delta$-function. Recall that the initial single-particle state $\ket{\phi(0)}$ is extended over the whole system like a plane wave. Therefore, the overlap $\braket{\phi(0)}{\alpha_{n}}$ is approximately $1/\sqrt{L}$. With this in mind, we find that $\left|\braket{\phi(0)}{\alpha_{n}}\right|^{2}$ is $1/L$ which is independent of $n$. And the Loschmidt echo becomes $\mathcal{L}(t)=l(t)^{N}$, where the single-particle Loschmidt echo is

[TABLE]

The sum of $e^{-i\epsilon_{n}t}$ is determined by the single-particle levels $\epsilon_{n}$, which depend on the configuration of disorder and are in fact random numbers. Let us study the joint probability density $P(\epsilon_{1},\epsilon_{2},\cdots,\epsilon_{L})$ of these random levels. In the Hamiltonian (3), the disordered potential $u_{i}$ is a random number uniformly distributed in the range from $-W/2$ to $W/2$. In the localized phase, the probability density of the nearest-neighbor level spacing is the Poisson distribution ($\sim e^{-s}$) which results from $\epsilon_{1},\cdots,\epsilon_{L}$ being independent and uniformly-distributed random numbers altshuler ; izrailev ; shklovskii . Therefore, in the localized phase we have

[TABLE]

where $D$ is the width of the single-particle spectra. In the strong disorder limit (large $W$ limit), the disordered potentials govern the Hamiltonian (3), and $\epsilon_{1},\cdots,\epsilon_{L}$ are then no more than the disordered potentials which are of course independent and uniformly-distributed random numbers according to the definition. In this limit the width of spectra $D$ is equal to $W$.

Equipped with the knowledge of $P(\epsilon_{1},\epsilon_{2},\cdots,\epsilon_{L})$, we can now calculate the Loschmidt echo, which after averaged over the level distribution becomes

[TABLE]

It is clear that both the single-particle and many-body Loschmidt echo vanish periodically at the critical times $t^{*}_{n}=2\pi n/D$ with $n=1,2,\cdots$ a positive integer. The vanish of Loschmidt echo causes the nonanalyticity of the dynamical free energy which according to Eq. (2) evaluates

[TABLE]

The dynamical free energy is divergent at the critical times $t^{*}_{n}$. Both the Loschmidt echo and the dynamical free energy signal periodically-occurred DQPTs at $t^{*}_{n}=2\pi n/D$. And the dynamical free energy is not continuous at these DQPTs.

III DQPTs in one- and three-dimensional Anderson models

We numerically study the Loschmidt echo and the dynamical free energy for the one- and three-dimensional Anderson Hamiltonians. In our calculation we choose the periodic boundary condition. At $W=0$, the single-particle eigenstates of the Anderson model are plane waves. In one dimension, the eigenstates are all localized in the presence of infinitesimal $W$. While in three dimensions, the eigenstates are localized only if the disorder strength $W$ is beyond a critical value $W_{c}$. The initial state is set to the ground state of $\hat{H}$ at $W=0$. A finite $W$ is switched on at $t=0$ and the Loschmidt echo and the dynamical free energy are then calculated. In the calculation, we perform an average over different disorder configurations by sampling the disorder potentials for many times until the results converge.

Let us first see the single-particle Loschmidt echo $l(t)=\braket{\phi(0)}{\phi(t)}$. The single-particle eigenenergy at $W=0$ is $\epsilon=-2g\cos k$ in one dimension or $\epsilon=-2g\left(\cos k_{x}+\cos k_{y}+\cos k_{z}\right)$ in three dimensions, thereafter, the ground state $\ket{\phi(0)}$ is a plane wave of zero wave vector. The amplitude of $\ket{\phi(0)}$ at arbitrary site is $1/\sqrt{L}$. Figs. 1(a) and (b) display the absolute value $\left|l(t)\right|$ for the one-dimensional Anderson model at different $W$ and $L$. The numerical results fit well with Eq. (8) if the disorder strength is strong ($W>50$). For fixed $W$, the fit between numerics and Eq. (8) becomes even better as the system’s size increases. As $L$ increases, the minimum of $l(t)$ goes towards zero. We then expect that in the limit $L\to\infty$ the Loschmidt echo does become zero at some critical times. And the value of the critical times also fits well with our prediction, i.e., $t^{*}_{n}D=2\pi n$.

Fig. 1(c) and (d) show the absolute value of the Loschmidt echo for the three-dimensional Anderson model at different $W$ and $L$. In both one-dimensional and three-dimensional cases, the Loschmidt echo always fits Eq. (8) at large $W$, e.g. at $W=1000$. By analyzing the change of $l(t)$ with increasing $L$, we also find in three dimensions that in the limit $L\to\infty$ the single-particle Loschmidt echo becomes zero periodically at the critical times $t^{*}_{n}=2\pi n/D$.

Next we discuss the dynamical free energy whose nonanalyticity unambiguously defines the DQPTs and the critical times. The dynamical free energy is related to the single-particle Loschmidt echo by $f(t)=-\left\langle\ln\left(\left|l(t)\right|^{2}\right)\right\rangle$ where $\langle\rangle$ denotes the average over different disorder configurations.

Fig. 2 shows the dynamical free energy at different $L$ and $W$. Eq. (9) is plotted at the same time for comparison, in which $D$ is also averaged over different disorder configurations. The numerical results fit well with Eq. (9) at large $W$, e.g. at $W=1000$ (see Fig. 2(a) and (c)). As the disorder strength increases, the fit becomes even better. It is clear that the dynamical free energy displays a peak periodically at the critical times $t^{*}_{n}D=2\pi n$. We also compare the dynamical free energy at different $L$. As the system’s size increases, the peak of $f(t)$ becomes higher, and the shape of $f(t)$ is closer to that of Eq. (9) (see Fig. 2(b) and (d)).

The similar nonanalytic behavior of $f(t)$ has been observed in many other models, such as the XXZ model heyl14 . Here we would like to emphasize that the nonanalyticity of $f(t)$ in the Anderson models is caused by the localization-delocalization transition, different from the DQPTs caused by broken symmetries in the XXZ model heyl14 or by the close of the energy gap in the topological insulatorsdora .

At small $W$ (e.g., at $W=20$ for one dimension or at $W=50$ for three dimensions), we clearly see the difference between numerics and Eq. (8) or Eq. (9) (see Fig. 1(a),(c) and Fig. 2(a),(c)). This difference is caused by the failure of the two assumptions in the above derivation of Eq. (8) and (9). First, the single-particle eigenstates are not $\delta$-like at small $W$ so that their overlap with the initial wave function ($\left|\braket{\phi(0)}{\alpha_{n}}\right|^{2}$) deviates from $1/L$ but depends on $\ket{\alpha_{n}}$. Fig. 3(a) and (b) plot the eigen wave functions of the one-dimensional Anderson Hamiltonian at different $W$. At $W=1$ and $W=5$, the eigen wave function deviates obviously from the $\delta$-function, even if it is approximately the $\delta$-function at $W=100$. We quantify the deviation of $\left|\braket{\phi(0)}{\alpha_{n}}\right|^{2}$ from $1/L$ by defining the overlap variance

[TABLE]

$\Delta$ is zero if $\left|\braket{\phi(0)}{\alpha_{n}}\right|^{2}$ equals $1/L$ independent of $\ket{\alpha_{n}}$, but $\Delta$ is finite if $\left|\braket{\phi(0)}{\alpha_{n}}\right|^{2}$ deviates from $1/L$. In Fig. 3(c), we see $\Delta=1/L^{2}$ at $W=0$ and decays towards zero as $W$ increases. In one dimension, the overlap variance already decays to $0.015$ at $W=50$. But the decay of $\Delta$ is much slower in three dimensions, and it is at $W=180$ when $\Delta$ decays to $0.015$. This explains why the deviation of the numerics from the analytical results at small $W$ is much larger in three dimensions than that in one dimension.

Second, $P(\epsilon_{1},\epsilon_{2},\cdots,\epsilon_{L})$ in Eq. (7) is not a constant at small $W$. Since $P(\epsilon_{1},\epsilon_{2},\cdots,\epsilon_{L})$ is a multivariate function which is difficult to plot, we alternatively plot $P(\epsilon)=\int d\epsilon_{2}\cdots d\epsilon_{L}P(\epsilon,\epsilon_{2},\cdots,\epsilon_{L})$ in Fig. 4 which is no more than the probability density of single-particle levels (or the normalized density of states). If $P(\epsilon_{1},\epsilon_{2},\cdots,\epsilon_{L})$ is a constant, $P(\epsilon)$ must also be a constant. We see that $P(\epsilon)$ does be approximately a constant as $W$ is as large as $50$, but apparently varies in the spectrum with two peaks at the edge and a valley at the center for small $W$ (e.g. $W=2$). This is not difficult to understand. At small $W$, the hopping term in the Hamiltonian (3) dominates. It is well known that the density of states has two peaks at the edge of the spectrum for the hopping Hamiltonian $\hat{H}_{hop}=-g\sum_{\langle i,j\rangle}\left(\hat{a}^{\dagger}_{i}\hat{a}_{j}+H.c.\right)$. But at large $W$, the random potentials in the Hamiltonian (3) dominate, leading to a constant density of states, as we argued below Eq. (7).

IV Conclusion and outlook

In this paper, we study the Loschmidt echo and the dynamical free energy after a quench of white-noise potentials. We find the periodically-occurred dynamical quantum phase transitions with a period $2\pi/D$ where $D$ is the width of the spectrum. By using the properties of the wave functions and the spectra in the strongly-localized region, we argue that the dynamical free energy is divergent at DQPTs. We present numerical results in one- and three-dimensional Anderson models to support our arguments.

Finally, we would like to mention that the measurement of the absolute value of the Loschmidt echo $\left|\mathcal{L}(t)\right|$ has been experimentally realized flaschner ; jurcevic . In a string of ions simulating interacting transverse-field Ising models by using $\text{Ca}^{+}$ ions, the non-equilibrium dynamics at DQPTs induced by a quantum quench is detected directly by measuring $\left|\mathcal{L}(t)\right|$ while projecting the many-body wave function at arbitrary time onto a chosen initial statejurcevic . With time-resolved state tomography, the topological DQPTs of ultracold atoms in optical lattices is measured by a full access to the evolution of the wave function flaschner . In view of the experimental realization of the Anderson model in the optical lattice BILLY ; ROATI , we then expect that a new type of the nonanalyticity of the dynamical free energy caused by the localization-delocalization transition predicted in this paper can be observed in cold atoms.

acknowledgments

This work is supported by NSF of China under Grant Nos. 11304280, 11774315 and 11374266, and the Program for New Century Excellent Talents in University. S. C. is supported by the National Key Research and Development Program of China (2016YFA0300600) and NSFC under Grants No. 11425419, No. 11374354 and No. 11174360.