A Supplemental Material: Remnants of Anderson localization in pre-thermalization induced by white noise

Remnants of Anderson localization in pre-thermalization induced by white noise


We study the non-equilibrium evolution of a one-dimensional quantum Ising chain with spatially disordered, time-dependent, transverse fields characterised by white noise correlation dynamics. We establish pre-thermalization in this model, showing that the quench dynamics of the on-site transverse magnetisation first approaches a metastable state unaffected by noise fluctuations, and then relaxes exponentially fast towards an infinite temperature state as a result of the noise. We also consider energy transport in the model, starting from an inhomogeneous state with two domain walls which separate regions characterised by spins with opposite transverse magnetization. We observe at intermediate time scales a phenomenology akin to Anderson localization: energy remains localised within the two domain walls, until the Markovian noise destroys coherence and accordingly disorder-induced localization, allowing the system to relax towards the late stages of its non-equilibrium dynamics. We benchmark our results with the simpler case of a noisy quantum Ising chain without disorder, and we find that the pre-thermal plateau is a generic property of weakly noisy spin chains, while the phenomenon of pre-thermal Anderson localisation is a specific feature arising from the competition of noise and disorder in the real-time transport properties of the system.

Introduction — Modern experimental advances in control of cold atoms Bloch2008 has revived the interest in non-equilibrium physics Greiner2002a and in real-time dynamics occurring in isolated quantum systems PolkovnikovRMP; Eisert2015a. Besides fundamental questions regarding eventual thermalisation of closed interacting systems, the current interest in out-of-equilibrium physics stands mainly in the possibility to engineer novel phases of matter or in realising phenomena without counterpart in traditional statistical mechanics, arising when a quantum many body system is driven far away from equilibrium for significantly long times. Noticeable examples range from Floquet topological insulators kita; Lindner11 to time crystals Monroe; Choi, encompassing pre-thermalization Berges2004a; Gring2012; Langen2016 and dynamical phase transitions Zhang.

In this work, we aim at showing how the characteristic features of an Anderson insulator – the inhibition of energy transport induced by disorder – can persist at intermediate time scales in a disordered system perturbed by Markovian noise.

The dynamics arising after a quantum quench of isolated, disordered spin models does not relax towards a steady state in the long time limit Ziraldo1; Ziraldo2, while a quatum Ising model coupled to a Markovian bath via its transverse field, thermalises efficiently with correlations spreading in a light-cone fashion Marino2012; Marino2014. Here, we merge together these two scenarios, considering the quantum quench dynamics of a disordered quantum Ising chain in one dimension (which is equivalent to a quadratic model of spinless fermions on a lattice), driven by time-dependent noisy transverse fields, and using these two previously studied cases Marino2012; Marino2014; Ziraldo1; Ziraldo2 as a benchmark for our results. Despite the fact that the localised phase of non-interacting fermions on a lattice is destroyed by the coupling to a heat bath woly; Derrico; NGADP; BanerjeeAltman; Bloch17, our findings show that on intermediate time scales transport can still be impeded by disorder, and only at longer times quantum coherence is wiped out by the noise, thermalization is established and energy is free to redistribute across the system.

This phenomenon bears analogies with prethermalization in weakly non-integrable systems Kollar2011; Bertini2015; Langen2016, where dynamics is first dominated by features of the perturbed integrable hamiltonian, and only at later times – when inelastic collisions induced by integrability breaking channels become effective, the system is capable to relax and dynamics is attracted by a thermal state. Following a similar logic, we first establish pre-thermalization in our model studying the dynamics of simple observables, as the on-site transverse magnetization, and then we show that remnants of Anderson localisation can persist at intermediate times, focusing on specific features of energy transport which become manifest when the Ising chain is prepared in a spatially inhomogeneous spin state.

A quantum Ising chain with noise and disorder — We consider the real-time dynamics of the transverse field quantum Ising chain in one dimension Sachdevbook


where are Pauli matrices acting on the site of the chain, and is a uniform transverse magnetic field. This model is characterized by two mutually dual gapped phases separated by a continuous quantum phase transition at ; it is exactly solvable by a Jordan-Wigner transformation mapping it onto a system of free fermions, which is then diagonalised by a Bogoliubov rotation Sachdevbook. This makes the Ising chain in Eq. (1) equivalent to a collection of free fermions, , with momenta where .

Figure 1: (Colors online) Expectation value of the local transverse magnetization, , as a function of time, , after a quench from to of the transverse field of the quantum Ising model Eq. (1). The green line corresponds to dynamics without noise, and sets the value of the pre-thermal plateau attained under influence of homogeneous (blue line) or inhomogeneous (red line) noise along the transverse field direction (). As discussed in the main text and shown in the figure, relaxation towards the asymptotic equilibrium value occurs faster in the latter case.
Figure 2: (Colors online) Population, , of Bogolyubov modes as a function of the momentum, , and time, , after a quantum quench of the transverse field, in the presence of spatially homogeneous (left) and inhomogeneous (right) noise. The modes close to the band edge (marked with red continuous lines) undergo a slower evolution under the effect of uniform noise, while, in the instance reported in the right panel, populations relax for any momentum swiftly towards their asymptotic infinite temperature value, . Parameters in the plots are the same as in Fig. 1: , , .

We will be interested in studying the dynamics of an Ising chain subject to an inhomogeneous time dependent noise. For this sake at we switch on a space and time-dependent Gaussian white noise, , on the top of the uniform transverse field, , on each site of the chain, as described by the operator


The Gaussian field is chosen with zero average , and is characterised by the two-point function,


delta-correlated in space and time. At a fixed time , describes an inhomogeneous configuration of transverse fields along the quantum Ising chain, from site to , drawn from a Gaussian distribution of variance ; the memoryless nature of ensures that these spatial disorder configurations are generated in an uncorrelated fashion at every time .

We are therefore considering the non-equilibrium dynamics of the model


describing a quantum Ising chain with competing time-dependent noise and spatial disorder along the direction of the transverse field. Equivalently, Eq. (4) describes disordered, non-interacting fermions on a one dimensional lattice and driven by a time-dependent Markovian noise.

The evolution of the density matrix of the system, , is ruled by the equation of motion


Following a standard procedure (see for instance Refs. Novikov1965; Budini2001; Breuerbook and the Supplemental Material), it is possible to derive a local-in-time master equation for the dynamics of the density matrix, , averaged over different realisations of the noise field, ,


which we solve numerically starting from different initial non-equilibrium conditions, in order to extract dynamics of observables of interest in this work. In the following, we will consider both quantum quenches of the transverse field – the system is prepared in the ground state of the Hamiltonian (1) with a given value of and evolved at later times under the influence of the noise and at a different value of the average transverse field – and spatially inhomogeneous spin states, as initial states for non-equilibrium dynamics.

Before discussing our results, we recall that the impact of a spatially homogeneous Markovian noise, , on the quench dynamics of the quantum Ising chain has been previously studied by two of us in Refs. Marino2012; Marino2014, using Keldysh diagrammatics methods. As in the presence of an inhomogeneous field , an analogue master equation for can be derived for the homogeneous case, , and reads


The numerical solution of Eq. (A), and the analytical results of Marino2012; Marino2014, will be used in the following to benchmark our findings with the non-equilibrium dynamics of the model (4) and its master equation (6).

Pre-thermalization induced by Markovian noise — We first show that after a quantum quench , the effect of an homogeneous, , and an inhomogeneous, , noisy transverse field have a qualitative, similar impact on the dynamics of single-site observables. In particular, we consider the local transverse magnetisation, , at a given site , and we calculate numerically the evolution of its expectation value, averaging over the density matrix . Fig. 1 shows that reaches, after a first relaxation process, a plateau with an expectation value close to the one acquired after a quantum quench of the Ising chain without noise (if , as in the homogeneous case Marino2012; Marino2014). This behaviour is akin to the phenomenon of pre-thermalization in isolated systems, since it precedes the decay of towards its actual equilibrium value, which is set by the infinite temperature state – since the Markovian noise, , can heat the system indefinitely.

The runaway of from the pre-thermal state to the asymptotic, infinite temperature one, occurs exponentially fast with a rate proportional to in the presence of the inhomogeneous field, , while in the presence of an uniform one, drops algebraically as for (Fig. 1); these latter results were found in Refs. Marino2012; Marino2014, and we have confirmed their validity from the numerical solution of the Lindblad dynamics given by Eq. (A).

The different relaxational laws in the two cases are due to the role played by the two modes , which are slow when the Ising chain is driven by an homogeneous noise field, and can significantly affect late-time dynamics.

For time-dependent perturbations proportional to the total transverse magnetisation like , the occupation number of the two Bogolyubov modes close to the band edges, and , are conserved quantities , with . This commutator vanishes continuously when the limits , or , are taken, implying that the relaxation rates, , of the modes close to the band edges vanish continuously as well, , see also Ref. Marino2014, and determining a slow, algebraic relaxation of one point functions (as the on-site transverse magnetization ), which can be expressed as bi-linears of Bogolyubov operators, and whose dynamics is accordingly determined by the expectation values – after coherences, , have been suppressed by noise-induced dephasing. In contrast, for inhomogeneous time-dependent fields as in Eq. (2), there are no soft modes slowing down quantum evolution and dissipation drives quickly the system towards the asymptotic steady state of dynamics. The two panels of Fig. 2 show a three-dimensional plot of as a function of time, , and momentum, , respectively for homogeneous (left panel) and inhomogeneous (right panel) noisy transverse fields. According to the above discussion on quasi-particles relaxation rates, the figure shows that relaxes uniformly for all momenta in the presence of competing noise and disorder, while in the presence of global noise, the modes with wave-vectors close to approach slowly their asymptotic equilibrium value.

Pre-thermal Anderson localisation — We now extend our study to the energy transport properties of the noisy chain (4). First of all, we consider as initial state , an inhomogeneous spin texture (without performing a quench of the transverse field, ), preparing a region of spins polarised along the positive direction at the centre of the chain,


the block of size in the state (8) is delimited by two domain walls, separating regions with different spin polarizations. We let evolve the system under the Lindblad dynamics (6), and study the flow of local energy,


governed by the equation


where , and , and where the average over the state has been taken. The term changes into when the noise perturbation is homogeneous, . Eq. (10) is straightforwardly derived, evolving the local energy (9) with the Lindbladian dynamics encoded in Eq. (6).

Figure 3: (Colors online) Rate of energy flow as a function of time, , in a quantum Ising chain of size , prepared with ten spins polarised in the positive direction at its centre (). The left panel corresponds to energy transport in the presence of homogeneous noise along the transverse field, while in the case of inhomogeneous noisy transverse fields (right panel), ballistic transport is inhibited by Anderson localization. Blockade of energy transport is ruled out by decoherence after a transient (see right panel again), with approaching eventually the infinite temperature state. Dynamics of has been simulated with and in the figure.

Fig. 3 shows time evolution of the expectation value of the rate of energy flow , at every site , starting from an inhomogeneous spin state of the type (8) with , in a chain of length .

The left side of Fig. 3 corresponds to evolution under the collective homogeneous noise field, , while the right side shows dynamics driven by the inhomogeneous one. The difference among the two is noticeable. In the first case, a linear light-cone propagation rules the transport of the energy, initially stored in the region of size at the center of the chain, towards the borders; this finding is consistent with the light cone structure of spin correlation functions in a quantum Ising model driven by global noise, , found in Refs. Marino2012; Marino2014.

The most striking effect is demonstrated in the second panel of Fig. 3. The fields act equally as spatial disorder and Markovian noise (see discussion after Eq. (3)), and they compete in order to determine the transport properties of the model (4). Energy transport is inhibited at short times: a disordered non-interacting model undergoes Anderson localization at any disorder strength in one dimension Anderson; Anderson2 and this is reflected in the trapping of energy within the region of size at the center of the Ising chain (cfr. with right panel of Fig. 3). However, since disorder-induced localisation originates from quantum interference among wave packets scattering against disordered lattice centres (represented by the fields on the sites ), the blockade of energy transport will persist until decoherence induced by the Markovian becomes sizeable. At that point, quantum coherence is washed out, Anderson localisation disappears, and energy is left free to spread. However, at comparable times, will approach the trivial infinite temperature state, as it occurs in the dynamics of the on-site transverse magnetization, . The effect is prominent for a disorder variance, , comparable to the transverse field, ; for smaller values of , energy transport would become sizeable again, consistently with previous studies reporting the onset of diffusion in noisy Anderson models woly; Derrico.

The phenomenon just described constitutes an instance of pre-thermal Anderson localization: despite the fact that a disordered system coupled to an infinite temperature bath cannot display a localised phase, the confining effect of disorder is active at intermediate time scales. This is reminiscent of pre-thermalization in non-integrable closed systems, where features of the weakly perturbed integrable dynamics can persist at intermediate times before eventual equilibration, ruled by integrability breaking perturbations, occurs Kollar2011; Bertini2015; Langen2016.

Perspectives — An interesting extension of our result would consist in studying the robustness of the pre-thermal Anderson phenomenon to integrability breaking perturbations of the Ising hamiltonian (1) (e.g. a next-neighbour spin-spin interaction in the transverse direction, ), in the spirit of the many-body localization problem (MBL) Nandkishore-2015; VHO; AbaninReview. Since the many body localised phase shares, at strong disorder, some features of a genuine Anderson insulator HNO; Serbyn; Imbrie; Imbriereview, we expect a qualitative similar phenomenon as the one reported in Fig. 3 to manifest (see Refs. Altman; Levi for related studies). We believe however that generalising a transient MBL behaviour to more complex spin chains (XXZ spin chain) or to different microscopic degrees of freedom (disordered Bose-Hubbard model), has the potential to highlight a richer phenomenology compared to the one established in this work. On short/intermediate time scales, where pre-thermal effects set in, this kind of extensions should be accessible with state-of-art numerical methods.

Acknowledgements — This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-17-1-0183. S.L, T.A., and G.M.P., acknowledge support from the EU Collaborative Project QuProCS (Grant Agreement No. 641277).


Appendix A Supplemental Material:
Remnants of Anderson localization in pre-thermalization induced by white noise

We consider the average, over different realizations of the noise, of the stochastic unitary dynamics (Eq.(5) in the main text)


The result in Ref. [25] gives an exact result for the following mean values, provided the noise is Gaussian:


where is the two point correlation function of the noise resolved in time and space, and where we have used the hermiticity of . Substituting the latter results in Eq. (11), we find

We now need to evaluate the response function occurring in Eq. (A). We assume that at the initial time, the system and the noises are uncorrelated, and following Ref. [26], we first formally integrate Eq. (11) in time, and then take a functional derivative with respect to and , finding


The variational derivative satisfy the same Liouvillian equation of the stochastic density matrix, , therefore we can write


where .

Rewriting Eq. (14) as


and substituting in Eq. (A) we end up with

which for the kind of delta-correlated Markovian noise considered in our work, see Eq. (3) in the main text, yields directly Eq. (6) of the main text, which we use to simulate the non-equilibrium dynamics of the model.

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