Non-Hermiticity induces localization: good and bad resonances in power-law random banded matrices

The power-law random banded matrix (PLRBM) is a paradigmatic ensemble to study the Anderson localization transition (AT). In $d$-dimension the PLRBM are random matrices with algebraic decaying off-diagonal elements $H_{\vec{n}\vec{m}}\sim 1/|\vec{n}-\vec{m}|^\alpha$, having AT at $\alpha=d$. In this work, we investigate the fate of the PLRBM to non-Hermiticity. We consider the case where the random on-site diagonal potential takes complex values, mimicking an open system, subject to random gain-loss terms. We provide an analytical understanding of the model by generalizing the Anderson-Levitov resonance counting technique to the non-Hermitian case. This generalization identifies two competing mechanisms due to non-Hermiticity: one favoring localization and the other delocalization. The competition between the two gives rise to AT at $d/2\le \alpha\le d$. The value of the critical $\alpha$ depends on the strength of the on-site potential, reminiscent of Hermitian disordered short-range models in $d>2$. Within the localized phase, the wave functions are algebraically localized with an exponent $\alpha$ even for $\alpha

Introduction -Random matrix theory (RMT) is a resounding resource to tackle problems in several contexts, ranging from nuclear physics to number theory [1][2][3][4].In disordered quantum systems, RMT has been widely used in Anderson localization (AL) and quantum chaos [3][4][5][6][7].For instance, the Anderson transition (AT) has been successfully studied using the so-called power-law random banded matrix (PLRBM) ensemble [8] and its cousin ultrametric matrices (UM) [9] as remarkable examples.In d dimensions, the PLRBM are matrices with random elements decaying algebraically with the distance, H nm ∼ 1 |n−m| α .The consideration of the PLRBM in terms of resonance counting, resolved over spatial distance [10], renormalization group (RG) [11], and nonlinear sigma model [8], finding AT, at α AT = d, separating ergodic and localized phases, has boosted the field of AT far beyond this paradigmatic example.That makes the PLRBM a fantastic toolbox for understanding universal properties in disordered quantum systems.In addition, recently the cousin UM model, having the same phase diagram and the properties, has found a direct application to the explanation of the avalanche mechanism of many-body localization (MBL) instability [12].
Motivated by the above studies, in this work, we take the route of studying AL in nH systems via their spatial resonance structure.In particular, we investigate a nH deformation of the PLRBM model.We break the Hermiticity by adding a complex on-site disorder but leaving Hermitian off-diagonal elements.Several works have considered this type of nH [28-30, 36, 38-42], and this choice mimics local random gain-loss terms.
We show that Hermiticity-breaking terms change the phase diagram of the PLRBM, see Fig.  8) and (c) the hybridization of levels, Eq. ( 10), may lead to the attraction between levels (shaded region), while the hopping term increases (green arrows).
and depends on the strength of the on-site disorder W .
To understand this counter-intuitive phenomenon, we generalize the Anderson-Levitov resonance counting and hybridization of resonant-level pairs, like in the standard RG [10,11].We uncover the competition of two mechanisms due to nH, favoring localization and delocalization.The complex-valued diagonal potential, unlike the real-valued one, Fig. 2(a), forms an effectively two-dimensional distribution [41], Fig. 2(b), which increases the level spacing parametrically in system size and suppresses the number of resonances.However, nH also induces a unique attraction between resonant levels, hybridizing them, Fig. 2(c), which forms the so-called "bad" resonances.The level attraction, caused by the nH terms, breaks the central assumption of spatial RG and dramatically changes the phase diagram.As a result, nH drives a new kind of transition that depends on the strength of on-site potential, like in short-range models in d > 2. In addition, for the nH PLRBM, we show that the localized phase has an algebraic nature also for α ≤ d, which is forbidden in its Hermitian counterpart.Finally, we show that our results find immediate application in dynamical phase transitions, driven by the competition between unitary time evolution and projective measurements, the so-called measurement-induced phase transitions.Indeed, the non-Hermitian Anderson transition is carried over to a Floquet version of our system, in which the non-unitary part play a role of a weak measurement.
Model & Methods -nH PLRBM in d dimensions reads as where ε n and j mn = j ⋆ nm are independent complexvalued box-distributed random variables with zero mean and the amplitudes m is a radial vector of a lattice in Z d , W is the onsite disorder strength, α -the power of the off-diagonal power-law decay, and b is the bandwidth of the decay.
For simplicity, we restrict our consideration to d ≤ 2, where short-range models (α → ∞) are localized for any amount of disorder and take b ≲ 1.The important gainloss nH is provided by the complex-valued diagonal term ϵ n ∈ C. Furthermore, we consider an non-unitary Floquet version of Eq. ( 1), which in second-quantization is given by U = e −iT mn jnmc † m cn e −T n εnc † n cn , with a period T , set to be 1 and creation (annihilation) fermionic operators c † n 's (c n 's).The non-unitary part is given by Re ε n , which corresponds to weak-measurements [24,32,33,43].Two considerations are in order, the Floquet model is non-interacting and therefore preserves Gaussianity, allowing us to inspect its single-particle modes to understand its phase diagram, and the Anderson transition for Floquet Hermitian model happens at α AT = 1 [44].
The Hermitian PLRBM, as well as UM model [9], hosts AT at α = d, independent of the parameters W and b.For α < d, the wave function are extended and ergodic,at the transition, the eigenstates show multifractality, weak for b ≫ 1 and strong for b ≪ 1 [6].In the localized phase (α > d), the wave functions are power-law localized, and the wave function decay rate coincides with α.The transition is understood using the so-called Anderson-Levitov resonance counting [10] with an RG approach [11].Its starting point is the perturbative locator expansion which converges, if most of the site pairs m and n are not in resonance, jmn εn−εm < 1.To find AT, one should count the number N res,n of resonant pairs jmn εn−εm > 1 for each ψ n state.For N res,n ≲ O(N 0 ) the perturbation theory is stable, and the state is localized, otherwise, for N res,n ≳ O(N c>0 ) the state delocalizes.At each ith step of the spatial RG [10,11], when |m − n| ≡ R is in a ring one can count the number of resonances by comparing the mean-level spacing in the d-dimensional ring [45] δ with the hopping term j mn ∼ 1/R α and progressively increase the distance at each step.Thus, for all α > d and large enough distances d) is the radius beyond which the resonances are absent.We can estimate the number of sites resonant to n as ∼ (1/W ) d/(α−d) .For these resonant sites, one must use degenerate perturbation theory.However, the hybridization of these resonance pairs does not produce other resonance pairs [11].As a result, for α > d the number of resonant pairs is bounded with system size.
The number of such resonances at a distance R grows as leading to This argument shows that for α < d N res,n diverges with system size N .Therefore AT in the Hermitian PLRBM happens at α AT = d independently off W nor b.Non-Hermitian case -We generalize the Anderson-Levitov resonance counting to the nH case.Following the same steps as in the Hermitian one, we subdivide the space in concentric rings around a lattice point n, Eq. ( 4), and compare the hopping term j R ∼ 1/R α with the mean level spacing δ R .The complex potential, Eq. (2a), has a 2d nature [42] and the mean level spacing is given by the root of the ratio between the entire available area ∼ W 2 and the number of sites in the ring ∼ R d , see Fig. 2(b), thus it decays parametrically slower than in the Hermitian case in Eq. ( 5).Focusing only on the increase of the level spacing, one might conclude that the system is more localized compared to the Hermitian one, and the AT should be at α AT = d/2.However, as we observe numerically in Figs. 1 and 3, the transition happens at W -dependent value, d/2 ≤ α AT (W ) ≤ d.We show that this is due to the level hybridization, which, in contrast to the Hermitian case, may enhance delocalization by level attraction.For clarity, let's consider the structure of a single resonance for both real and complex ε n and Hermitian hopping j nm = j * mn , because as we show below, the hybridization depends only on the mutual phase of j mn j nm and (ε n − ε m ) 2 .
In the resonance condition |ε m − ε n | < |j nm | for two sites n and m, following the degenerate perturbation theory, one should diagonalize a 2 × 2 matrix leading to the "renormalized" on-site energies For the Hermitian case, with (ε m − ε m ) 2 ≥ 0, we have level repulsion, ε leading to the convergent RG for α > α AT = d.
In the nH case, the inequality in Eq. ( 11) is not always satisfied.For example, at We refer to this level of attraction as "bad" resonances.It is this new phenomenon of level attraction, Fig. 2(c), which competes with the enhanced mean level spacing δ nH R and breaks the RG steps at d/2 < α < α AT (W ) as soon as the "bad" resonance number is significantly large.These "bad" resonances appear with a finite (but small) probability per eigenstate, which we next calculate.
We parameterize the hopping |j mn | = 2W J and diagonal elements with real parameters J, X, Y , x, y and rewrite the condition, Eq. ( 12) in general, as For a fixed J one can straightforwardly calculate the usual-resonance probability P res , defined as the conditional integral J 2 > (x 2 + y 2 ) over the parameters X, Y , x, and y, which is present also in the Hermitian case.The "bad"-resonance one P bad , present only in nH case, we calculate via the integral over (14) [46] The number of resonances is given by the summation over the distances R and the integration over the random-amplitude |s m,m+R | ≡ s distribution (being box for Re s mn and Im s mn in Eq. (2b), with s ≤ 1): For W > 1/2, and d = 1, one obtains the average number of the usual and "bad" resonances per eigenstate as [47] N 16) where ζ x is the zeta function, which converges at x > 1.
As a result, both N res and N bad are finite and Nindependent at α > d/2.If one neglects the contribution of "bad" resonances, Eq. ( 16) implies that the localization happens at α > d/2.However, "bad" resonances break down the entire RG analysis by enabling an avalanche of higher-order resonances.Therefore, localization in the nH case is achieved only at N bad ≪ 1 per eigenstate or at N bad • N ≪ N resonances in the entire system.From the numerical results, we find the transition at which immediately makes the power-law exponent α c (W ) to be strongly dependent on W via Eq.( 17).The average N bad and N res are generically non-integer, meaning that from realization to realization and from eigenstate to eigenstate these integers fluctuate.
Numerical results -We restrict our numerics to d = 1, minimizing finite-size effects.To understand the existence of the AT, we study the spectral statistics of the model, Eq. (1).In the extended phase, we expect the level statistics to be close to that of the Ginibre ensemble [48], a nH RMT.In the localized phase, the energy spectrum should be 2d Poisson.To separate these two limits, we study the complex gap ratio [49] where {Z n } is the spectrum of H, which is, in general, complex, Z n ∈ C. Z N N n and Z N N N n are the nearest neighbor (NN) and the next-nearest neighbor (NNN) of Z n with respect to the Euclidean distance in C, respectively.Decomposing r C n = r n e iθn , we analyze {r n } and {θ n }, separately.For Ginibre RMT − cos θ ≈ 0.229 and r ≈ 0.738 [49], while in the localized phase, we have − cos θ = 0 and r = 2/3.The overline indicates the average over disorder and energy spectrum.We have checked that no mobility edge, absent in PLRBM, appears in its non-Hermitian version.shows that β ≈ α.(b) D r q vs α for several small q.The dashed lines show the analytical prediction D r q = (2αq − 1)/(q − 1).
Figures 3(a) show r and −cos θ as a function of α a fixed W = 4, respectively.For small α, both quantities approach their RMT prediction (red dashed lines) with increasing N , while at large α, they tend to the Poisson values.An abrupt crossover between these limits appears around α c ≈ 0.72 ≤ α AT = 1, providing evidence of the existence of an AT.To further support the above observation, we compute the inverse participation ratio IP R r q and extract a fractal exponent D r q via its scaling with the system size Here ψ r n is the right eigenvector with eigenvalue Z n [50].0 ≤ D r q ≤ 1 measures the spread of the wave functions.In the ergodic phase, D r q → 1, while in the power-law localized phase, the behavior of D r q depends on the power β of the decay of the wave function ψ(m) ∼ 1/|m| β : D r q = 0 for q > 1/(2β), while for q < 1/(2β) the fractal exponent is D r q = (2αq − 1)/(q − 1).Varying q in D r q , we can probe if the phase is power-law localized.As shown in Fig. 3 (b), D r 2 ≈ 1 for small α < 0.85 indicates ergodicity.For larger α, we observe that D r 2 → 0. Importantly, D r 2 → 0 also at α c < α < d = 1.Now, to understand the dependence of the critical point on the onsite-disorder strength, we tune W . Figure 3(c)-(d) shows r, −cos θ and D r 2 at α = 0.85 < α AT versus W .The system-size N increase tends the curves in Fig. 3 (c)-(d) to their ergodic values at W ≤ 1.85, while we observe localization at larger W .The dependence of the critical point d/2 ≤ α c ≤ d on the disorder strength W is in a good agreement with the theoretical prediction of Eqs. ( 17), (18), see Fig. 1 and Fig. 3.In particular, we have that α c → 1/2 as 1/W 2 at large W , Fig. 1.
We tested our finding also in the Floquet case, see the inset of Fig. 3(b).As a result, the non-unitary Floquet model undergoes a localization-delocalization transition at α c (W ) ≤ α AT .Using the relation between entanglement entropy and transport properties for Gaussian state, we predict the existence of measurement-induced transition in entanglement in that model.
Conclusion -In this work, we inspect the fate of the Anderson transition in a nH PLRBM model with random gain and loss in d dimensions.We provide numerical and analytical evidence of the existence of AT, which happens at a smaller power α than in the Hermitian counterpart.
We generalize the Anderson-Levitov resonance counting to the nH case to elaborate an analytical understanding of the system.This technique reveals the emergence of two competing mechanisms: the parametric enhancement of the level spacing, enhancing localization, and the emerging "bad" resonances, favoring delocalization.The competition between the two leads to the Anderson transition at d/2 ≤ α c (W ) ≤ d.The critical point depends on the on-site-disorder amplitude W , similar to short-range models in d > 2. This result should be compared to the Hermitian case, where the transition happens at α = d for any W .We also show that in the localized phase, the wave-function envelopes decay algebraically from their maxima.The spatial decay coincides with the one of the PLRBM hopping term, also for α c (W ) ≤ α ≤ d.As a result, nH systems might also host localized algebraic phases with a decay rate lower than d, which is forbidden in Hermitian systems.
Together with the UM, nH in PLRBM opens the direct applications to the avalanche theory of many-body delocalization, suggested in [12] for the Hermitian case, and should stabilize the MBL phase beyond its Hermitian limit.[42] Using the parametrization (13) and the left condition in (14), for the box-distributed Re ε n and Im ε n in the interval and the j-resolved probability of usual resonances with J = |j|/(2W ).Note here that the choice of the real and imaginary amplitudes to be the same,  22).Indeed, as soon as both real and imaginary parts do not scale with the system size, the qualitative results are the same: the localization transition is shifted to smaller values of the power α and the critical line α c (W R , W I ), unlike the Hermitian case, depends on the disorder amplitudes W R , W I .
The plot of the above formula ( 22) is given in Fig. 5 (blue line) of the main text.One can see that the probability of usual resonances monotonically increases towards 1 with increasing hopping term J = |j mn |/(2W ) with respect to the disorder amplitude W .
The corresponding probability P 0 res (j) for the Hermitian case of Y ≡ y ≡ 0 as The probability of "bad" resonances is given by the conditions ( 12) and ( 14)and takes the form Both numbers of usual and "bad" resonances in the range J > 1 (i.e. at W > 1/2 for all R ≥ 1 and s < 1) (25) take the following form where ζ x is a zeta function, which converges at x > 1.
In order to find the moments ⟨s q ⟩, 2 ≤ q ≤ 4, one should calculate its distribution.For the box distribution of | Re j| and | Im j|, Eq. (2b), with the real s R and imaginary s I parts of s = s R + is I in the interval |s R |, |s I | < 1/ √ 2, one can find the one of s as The corresponding moments are given by After the substitution of ⟨s q ⟩ to the above expressions (26), one obtains coincide with the approximate expressions ( 16) and (17) in the main text.The expressions for W < 1/2 are straightforward to calculate from ( 25), (22), and ( 24), but have quite cumbersome analytical expressions.Like in the Hermitian case at α < d, where the presence of the extensive number of resonances N res per eigenstate breaks down the spatial renormalization group [11], in the nH case the presence of a small fraction of "bad" resonances per eigenstate does this job.
Indeed, the spatial renormalization group [11] is based on the assumption that one can hybridize the resonance pairs one-by-one, going from the strongest ones (at smaller distances) to the weaker ones (at larger distances) and never coming back to the same (previously resonant) pair, due to its hybridization.
In the nH "bad" resonances the situation is drastically different as each of such resonances attracts the levels even closer via the hybridization.This opens the possibility of the recurring resonance pairs at later stages of the renormalization group and, thus, to the avalanche of resonances.In order to avoid having such avalanche effects, one needs to have the number of "bad" resonances per eigenstate small enough.We estimate this number numerically in the main text.

Numerical results
In this section, we show further numerical data to support our claims.In particular, in the main text, we consider the inverse participation (IPR) ratio for the right eigenvector.Now, we consider the more generally defined as where ψ r n and ψ l n are the right-and left-eigenvectors.D 2 is the respective fractal exponent

FIG. 1 .
FIG.1.Phase diagram of the non-Hermitian PLRBM with random gain-loss terms in d = 1 dimension, showing the averaged spectrum gap ratio r vs the algebraic-decay rate α of the hopping terms and the diagonal disorder amplitude W . r = 2/3 for localized systems and r ≈ 0.738 for ergodic ones.For the Hermitian case, the transition happens at α = 1 for any W (vertical blue line).The dashed line is the analytical prediction for AT.

1 .FIG. 2 .
FIG. 2. Structure of resonances.(a) The level spacing in the Hermitian case, Eq. (5), is determined by the ratio of the energy interval W and the number of lattice sites NR ∼ R d at the distance R from a certain point.(b) In the non-Hermitian case, the complex (2d) distribution of εn leads to the mean area per level AR ≃ W 2 /R d , Eq. (8) and (c) the hybridization of levels, Eq. (10), may lead to the attraction between levels (shaded region), while the hopping term increases (green arrows).
is arbitrary and does not change much.The difference between the amplitudes of real | Re ε n | < W R and imaginary | Im ε n | < W I parts of disorder makes only quantitative difference in the phase diagram, which is straightforwardly seen from the above Eq.(

Figures 6 (
Figures 6(a)-(b) show IP R 2 and D 2 .These panels should be compared to Figs. 7(c)-(d), where we computed for the same value of W and b the right IP R r 2 and D r 2 .We do not observe any substantial changes and D r 2 ≈ D 2 .