Localization phenomena due to random potential, namely the Anderson localization (AL), is at the heart position in modern condensed matter physics.^[1–3] When including the effect of interaction, many-body localization (MBL) emerges and has spurred intensive studies on disordered quantum many-body systems.^[4–7] Interestingly, if local conservation, e.g., Z₂ gauge symmetry, exists in Hamiltonian, AL/MBL can exist without external quenched disorder, thus certain translation-invariant quantum systems can exhibit AL or MBL.^[8–11] However, existing examples of disorder-free AL and MBL are still rare, in spite of general interests on their relation to quantum thermalization, lattice gauge field, topological order and novel quantum liquid.^[9,11–15]

Recently, we have revisited a modified Kondo lattice model, namely the Ising–Kondo lattice (IKL),^[16] which is shown to reduce to fermions moving on static potential problem due to local conservation of localized moments, thus admits a solution by classical Monte Carlo (MC) simulation.^[17] On square lattice at half-filling, Fermi liquid (FL), Mott insulator (MI) and Néel antiferromagnetic insulator (NAI) are established (see Fig. 1). When doping is introduced, spin-stripe physics emerges with competing magnetic ordered states, similar to the t–J model and f-electron materials.^[18,19] As emphasized by Antipov et al. in the context of the Falicov–Kimball model,^[8,20] since the weight of static potential satisfies Boltzmann distribution, and if temperature is high enough, the probability distribution of all configurations of static potential tends to be equal, thus may realize binary random potential distribution. When such an intrinsic random potential is active, AL of fermions appears without external quenched disorder.

In this paper, we explore the possibility of accessing AL in the IKL model on square lattice. To simplify the discussion and to meet our previous work, here we focus on the half-filling case though doping the half-filled system does not involve any technical difficulty (an example on a doped system is given in Appendix D). By inspecting density of state (DOS), inverse participation ratio (IPR) and temperature-dependent resistance, we will show that the AL phase emerges in intermediate coupling regime between metallic FL and insulating MI above antiferromagnetic critical temperature (see Fig. 1). The occurrence of AL results from quenched disorder, formed by the conservative localized moments at each site. Interestingly, we find a many-body wavefunction, which captures elements in all three paramagnetic phases and is used to compute their entanglement entropy. In light of these findings, the disorder-free AL phenomena could exist in more generic translation-invariant quantum many-body systems.

The rest of this paper is organized as follows: In Section 2, the IKL model is introduced and explained. In Section 3, MC simulation is performed and observables such as DOS, IPR and temperature-dependent resistance are computed. Analysis on MC data suggests the appearance of AL in intermediate coupling regime where thermal fluctuation destroys magnetic long-ranged order. A wavefunction is constructed and the entanglement entropy is evaluated. Section 4 gives a summary and a brief discussion on AL in generic quantum many-body systems.

The IKL model on square lattice at half-filling is defined as follows:^[17]

$$ \begin{eqnarray}\begin{array}{lll}\hat{H} & = & -t\displaystyle \sum _{\langle i,j\rangle,\sigma }{\hat{c}}_{i\sigma }^{\dagger }{\hat{c}}_{j\sigma }+\displaystyle \frac{J}{2}\displaystyle \sum _{j\sigma }{\hat{S}}_{j}^{z}{\hat{c}}_{j\sigma }^{\dagger }{\hat{\sigma }^{z}}{\hat{c}}_{j\sigma },\end{array}\end{eqnarray}$$ (1)

In Ref. [17], we have observed that f-electron’s spin/localized moment at each site is conservative since [S^jz,H^]=0. Therefore, taking the eigenstates of spin S^jz as bases, the Hamiltonian (1) is automatically reduced to a free fermion moving on effective static potential {q_j}

$$ \begin{eqnarray}\hat{H}(q)=-t\displaystyle \sum _{\langle i,j\rangle,\sigma }{\hat{c}}_{i\sigma }^{\dagger }{\hat{c}}_{j\sigma }+\displaystyle \sum _{j\sigma }\displaystyle \frac{J}{4}{q}_{j}{\hat{c}}_{j\sigma }^{\dagger }{\hat{\sigma }^{z}}{\hat{c}}_{j\sigma }.\end{eqnarray}$$ (2)

A careful reader may note that isotropic Kondo lattice with ferromagnetic coupling has been studied by classical MC simulation.^[28,29] The algorithm of MC in our model is similar to those models, while they considered the ferromagnetic coupling (we consider antiferromagnetic coupling), thus their resultant phase diagram is rather different from ours. Technically, in their model, the local spin is approximated with classical vector, then the MC simulation is carried out by sampling classical configurations of those vectors. In contrast, our model is the Ising version of Kondo lattice models, and the local spin is the quantum Ising spin. When we choose their eigenstates, such a quantum spin is exactly transformed to classical spin and MC is performed straightforwardly. Therefore, our model is exactly simulated by MC while the ferromagnetic isotropic Kondo lattice is approximately simulated.

In terms of MC, we have determined the finite temperature phase diagram in Fig. 1. In addition to well-established FL, MI and NAI in previous work, interestingly we find an AL phase in intermediate coupling regime at high T. There is no transition but crossover from FL to AL and AL to MI. Since the former three phases have been clearly studied,^[17] in this work, we focus on the AL phase.

To characterize the AL phase from FL or MI, we study the DOS, IPR and temperature-dependent resistance of conduction electrons.^[3,8] The DOS of c-fermion N(ω) is evaluated from

$$ \begin{eqnarray}N(\omega )\simeq \displaystyle \frac{1}{{N}_{{\rm{m}}}{N}_{{\rm{s}}}}\displaystyle \sum _{\{{q}_{j}\}}\displaystyle \sum _{n\sigma }\delta (\omega -{E}_{n\sigma }(q)).\end{eqnarray}$$ (3)

The IPR measures tendency of localization. Here the energy/frequency-dependent IPR is used,

$$ \begin{eqnarray}{\rm{IPR}}(\omega )\simeq \displaystyle \frac{1}{{N}_{{\rm{m}}}{N}_{{\rm{s}}}}\displaystyle \sum _{\{{q}_{j}\}}\displaystyle \sum _{n\sigma }\displaystyle \sum _{j}\delta (\omega -{E}_{n\sigma }(q)){({\phi }_{n\sigma }^{j})}^{4}.\end{eqnarray}$$ (4)

The temperature-dependent resistance is related to static conductance σ_dc as ρ = 1 / σ_dc, which reads

$$ \begin{eqnarray}{\sigma }_{{\rm{dc}}}=\displaystyle \frac{\pi {t}^{2}{e}^{2}}{{N}_{{\rm{m}}}}\displaystyle \sum _{\{{q}_{j}\}}\displaystyle \int {\rm{d}}\omega \displaystyle \frac{-\partial {f}_{{\rm{F}}}(\omega )}{\partial \omega }{\Phi }^{q}(\omega ).\end{eqnarray}$$ (5)

Now, from MC calculation of these quantities, e.g., Figs. 2–4, we find an AL phase in intermediate coupling, beside well-established FL and MI at high T regime. In Fig. 2, AL has finite N(0) though its strength is heavily suppressed and looks like a pseudogap. The reason is that due to the preformed local antiferromagnetic order, the band gap begins to form at low T. When increasing temperature, excitation of localized moments appears and it acts like impurity scattering center in the well-formed antiferromagnetic background. Then, the conduction electron scatters from such impurity and contributes impurity bound state, which fills in the band gap.^[30] At high T, the long-ranged antiferromagnetic order melts but the band gap survives due to remaining local antiferromagnetic order. After considering impurity bound states, N(0) in AL is finite and a pseudogap-like behavior appears.

In Fig. 3, we note that IPR of FL satisfies the expected inverse-volume law while AL and MI have saturated IPR around ω = 0. Additionally, extrapolation of IPR at ω = 0 into infinite system size indicates that the localization length in FL is infinite while AL and MI only have finite localization length.

The T-dependent resistance of conduction electron is shown in Fig. 4, and the crossover from FL to AL and MI is clearly demonstrated. In both AL and MI, the insulating behaviors appear before the formation of insulating NAI (T > T_c), suggesting that they are insulators driven by correlation and thermal fluctuation.

As found by MC simulation, the AL phase appears in intermediate coupling regime above the magnetic long-ranged ordered state. If T is high enough, the effective Boltzmann weight ρ(q) for given configuration of static potential/Ising spin should be equal. Thus, c-fermion feels an effective potential, which works as binary random potential (recall that q_j = ±1 has two values). Averaging over ρ(q) leads to a disorder average for c-fermion and the AL phase is realized.

Technically, the above statement means that

$$ \begin{eqnarray}{\langle \hat{O}\rangle }_{T\to \infty }\simeq \displaystyle \frac{1}{{N}_{{\rm{m}}}}\displaystyle \sum _{\{{q}_{j}\}}({\hat{O}}^{q}(q)+\langle \langle {\hat{O}}^{c}(q)\rangle \rangle ),\end{eqnarray}$$ (6)

To justify the above argument, we show DOS and IPR in Fig. 5 using Eq. (6), where both DOS and IPR agree with ones in AL phase in Figs. 2(b) and 3(b) in intermediate coupling regime (J / t = 8). Thus, it is suggested again that AL phase in our model results from effective random potential formed by localized moment. As a matter of fact, if we consider weak and strong coupling case, their DOS and IPR will be similar to the counterpart in Figs. 2 and 3 as well, thus all three states at high-T are stable in the T = ∞ limit.

Motivated by arguments in the last subsection and Eq. (6), we write down the following many-body state, which approximates FL, AL and MI,

$$ \begin{eqnarray}|\varPsi \rangle =\displaystyle \frac{1}{\sqrt{{N}_{{\rm{m}}}}}\displaystyle \sum _{\{{q}_{j}\}}|{q}_{1},{q}_{2},\ldots,{q}_{{N}_{{\rm{s}}}}\rangle \otimes |\psi \rangle \end{eqnarray}$$ (7)

In the context of AL, both quenched disorder and annealed disorder are static disorder. In literature, the difference between these two kinds of disorder is that, by definition, quenched disorder corresponds to a time-independent probability distribution function (PDF)^[8,31–34]

$$ \begin{eqnarray}P(\{{q}_{1},{q}_{2},\ldots,{q}_{{N}_{{\rm{s}}}}\})=\displaystyle \prod _{i=1}^{{N}_{{\rm{s}}}}P({q}_{i}),\end{eqnarray}$$ (8)

In our model, the quenched disorder means that the configuration of local moments is randomly chosen at every site, where P(q_i) cannot fluctuate in time. We have realized the quenched disorder at the infinite temperature (T = ∞) situation. A quenched disordered many-body wavefunction is constructed with N_m randomly chosen configurations of local moments. Here each P(q_i) is the same and time-independent, thus leads to a global constant-type P({q₁,q₂,…,q_Ns}) as well. Under this constant-type PDF, the many-body wavefunction could be regarded as quenched disordered state.^[8]

As mentioned above, with this many-body wavefunction we can cover the main results of MC simulation at high temperature. It suggests that the quenched disordered state do capture the main physics of IKL in the high temperature region. Therefore, it is reasonable to attribute the existence of disorder-free AL to the intrinsic quenched disorder of local moments.

The entanglement entropy S_EE is used to characterize the universal quantum correlation in many-body state. For our model, we calculate S_EE for each slater determinant state |ψ〉 in the given Ising configuration {q_j} as follows.^[35]

We consider the open boundary condition and use |ψ〉 to compute the equal-time correlation function gij,σq as gij,σq=〈ψ|c^iσ†c^jσ|ψ〉. Dividing our system into A and B parts, a reduced correlation function g˜ij,σq is constructed as

 1

Firstly, S_EE decreases monotonically from small-J to large-J regime, agreeing with the increased localization tendency. Another interesting feature is that when J / t ≃ 12, S_EE / L_c collapses into a single line, thus indicating a crossover from AL to MI at T = ∞. More close inspection on S_EE shows a linear-dependence on L_c in the MI regime, which is the well-known area-law for S_EE.^[36] For the FL regime, its S_EE deviates from the area law with a logarithmic correction.^[37,38] A fitting in FL gives S_EE ≃ 0.275L_c ln L_c + 5.05 for J / t = 2. As for AL, it qualitatively obeys the area law though a small deviation exists due to inevitable mixing with delocalized states.

In conclusion, we have established an AL phase in a modified Kondo lattice model without external disorder potential. The presence of AL results from quenched disorder, formed by conservative localized moment at each site and is a stable phase even at infinite temperature. A many-body wavefunction is constructed to understand AL, FL and MI. Their entanglement entropy is computed and the area law is violated in FL. In light of these findings, we recall that the interacting many-electron system can be rewritten as free electrons moving on fluctuated background field after the Hubbard–Stratonovich transformation.^[39,40] As noted by Antipov et al.,^[8] if dynamics of the background field is frozen, it can act as a random potential to scatter electrons. Then, AL or even MBL phases could be observed in generic many-body Hamiltonian. It will be interesting to see whether the isotropic Kondo lattice (transverse Kondo coupling is included), which is the standard model for heavy fermion study, will support the presence of those localized states of matter.