We study the random transverse field Ising model on a finite Cayley tree. This enables us to probe key questions arising in other important disordered quantum systems, in particular the Anderson transition and the problem of dirty bosons on the Cayley tree, or the emergence of non-ergodic properties in such systems. We numerically investigate this problem building on the cavity mean-field method complemented by state-of-the art finite-size scaling analysis. Our numerics agree very well with analytical results based on an analogy with the traveling wave problem of a branching random walk in the presence of an absorbing wall. Critical properties and finite-size corrections for the zero-temperature paramagnetic-ferromagnetic transition are studied both for constant (independent of the system volume) and algebraically vanishing (scaling as an inverse power law with the system volume) boundary conditions. In the later case, we reveal a regime which is reminiscent of the non-ergodic delocalized phase observed in other systems, thus shedding some light on critical issues in the context of disordered quantum systems, such as Anderson transitions, the many-body localization or disordered bosons in infinite dimensions.
We study the random transverse field Ising model on a finite Cayley tree. This enables us to probe key questions arising in other important disordered quantum systems, in particular the Anderson transition and the problem of dirty bosons on the Cayley tree, or the emergence of non-ergodic properties in such systems. We numerically investigate this problem building on the cavity mean-field method complemented by state-of-the art finite-size scaling analysis. Our numerics agree very well with analytical results based on an analogy with the traveling wave problem of a branching random walk in the presence of an absorbing wall. Critical properties and finite-size corrections for the zero-temperature paramagnetic-ferromagnetic transition are studied both for constant (independent of the system volume) and algebraically vanishing (scaling as an inverse power law with the system volume) boundary conditions. In the later case, we reveal a regime which is reminiscent of the non-ergodic delocalized phase observed in other systems, thus shedding some light on critical issues in the context of disordered quantum systems, such as Anderson transitions, the many-body localization or disordered bosons in infinite dimensions.
A disordered quantum system can be driven, e.g., by increasing disorder, through the Anderson transition between a delocalized and localized phase. Recently, Anderson localization on treelike graphs has attracted much attention in view of its connections with problems of many-body localization. One of the central questions in this context is that of ergodicity of eigenstates. The authors of this paper perform a numerical study of Anderson transition on random regular graphs (RRGs) with diagonal disorder. The problem can be described as a tight-binding model on a lattice with N sites that is locally a tree with constant connectivity. Focusing on the delocalized side of the transition, they show that the data can be interpreted in terms of the finite-size crossover from a small ($N\ensuremath{\ll}{N}_{c}$) to a large ($N\ensuremath{\gg}{N}_{c}$) system, where Nc is the correlation volume diverging exponentially at the transition. A distinct feature of this crossover is a nonmonotonicity of the spectral and wave-function statistics, which is related to properties of the critical phase in the studied model and renders the finite-size analysis highly nontrivial. The results support an analytical prediction that states in the delocalized phase (and at $N\ensuremath{\gg}{N}_{c}$) are ergodic.
Building on large-scale quantum Monte Carlo simulations, we investigate the zero-temperature phase diagram of hard-core bosons in a random potential on site-centered Cayley trees with branching number K=2. In order to follow how the Bose-Einstein condensate (BEC) is affected by the disorder, we focus on both the zero-momentum density, probing the quantum coherence, and the one-body density matrix (1BDM) whose largest eigenvalue monitors the off-diagonal long-range order. We further study its associated eigenstate which brings useful information about the real-space properties of this leading eigenmode. Upon increasing randomness, we find that the system undergoes a quantum phase transition at finite disorder strength between a long-range ordered BEC state, fully ergodic at large scale, and a new disordered Bose glass regime showing conventional localization for the coherence fraction while the 1BDM displays a nontrivial algebraic vanishing BEC density together with a nonergodic occupation in real space. These peculiar properties can be analytically captured by a simple phenomenological description on the Cayley tree which provides a physical picture of the Bose glass regime.
For the quantum Ising model with ferromagnetic random couplings J i , j > 0 and random transverse fields h i > 0 at zero temperature in finite dimensions d > 1, we consider the lowest order contributions in perturbation theory in ( J i , j / h i ) to obtain some information on the statistics of various observables in the disordered phase. We find that the two-point correlation scales as , where ξ typ is the typical correlation length, u is a random variable and ω coincides with the droplet exponent ω DP ( D = d − 1) of the directed polymer with D = ( d − 1) transverse directions. Our main conclusions are as follows. (i) Whenever ω > 0, the quantum model is governed by an infinite-disorder fixed point: there are two distinct correlation length exponents related by ν typ = (1 − ω)ν av ; the distribution of the local susceptibility χ loc presents the power-law tail P (χ loc ) ∼ 1/χ 1 + μ loc , where μ vanishes as ξ −ω av so that the averaged local susceptibility diverges in a finite neighborhood 0 < μ < 1 before criticality (Griffiths phase); the dynamical exponent z diverges near criticality as z = d /μ ∼ ξ ω av . (ii) In dimensions d ⩽ 3, any infinitesimal disorder flows toward this infinite-disorder fixed point with ω( d ) > 0 (for instance ω( d = 2) = 1/3 and ω( d = 3) ∼ 0.24). (iii) In finite dimensions d > 3, a finite disorder strength is necessary to flow toward the infinite-disorder fixed point with ω( d ) > 0 (for instance ω( d = 4) ≃ 0.19), whereas a finite-disorder fixed point remains possible for a small enough disorder strength. For the Cayley tree of effective dimension d = ∞, where ω = 0, we discuss the similarities and differences with the case of finite dimensions.
Strong disorder renormalization for the random transverse field Ising model leads to a complicated topology of surviving clusters as soon as d > 1. Even if one starts from a Cayley tree, the network of surviving renormalized clusters will contain loops, so that no analytical solution can be obtained. Here we introduce a modified procedure called 'boundary strong disorder renormalization' that preserves the tree structure, so that one can write simple recursions with respect to the number of generations. We first show that this modified procedure allows one to recover exactly most of the critical exponents for the one-dimensional chain. After this important check, we study the RG equations for the quantum Ising model on a Cayley tree with a uniform ferromagnetic coupling J and random transverse fields with support [ h min , h max ]. We find the following picture: (i) for J > h max , only bonds are decimated, so that the whole tree is a quantum ferromagnetic cluster; (ii) for J < h min , only sites are decimated, so that no quantum ferromagnetic cluster is formed, and the ferromagnetic coupling to the boundary coincides with the partition function of a directed polymer model in a random medium; (iii) for h min < J < h max , both sites and bonds can be decimated, and the quantum ferromagnetic clusters can either remain finite (the physics is then similar to (ii), with a quantitative mapping to a modified directed polymer model) or an infinite quantum ferromagnetic cluster appears. We find that the quantum transition can be of two types: (a) either the quantum transition takes place in the region where quantum ferromagnetic clusters remain finite, and the singularity of the ferromagnetic coupling to the boundary involves the typical correlation length exponent ν typ = 1; (b) or the quantum transition takes place at the point where an extensive quantum ferromagnetic cluster appears, with a correlation length exponent ν ≃ 0.75.
We investigate quantum criticality in the transverse-field Ising model on finite Cayley trees and demonstrate that their extensive boundary fundamentally changes the nature of the quantum phase transition compared with the infinite, boundary-free Bethe lattice. Using tree tensor-network simulations for systems with up to N=12286 sites, we identify a boundary-driven continuous transition at hxc≈1.76, distinct from the mean-field Bethe-lattice transition at hxc≈2.23. We characterize this transition through finite-size scaling of the magnetization and magnetic susceptibility and extract a consistent set of critical exponents. By introducing next-nearest-neighbor interactions, we demonstrate the robustness of the extracted critical exponents, providing numerical argument for the Ising boundary universality class on the Cayley tree. Furthermore, we show that the physics of the boundary-free Bethe lattice can be efficiently recovered on a finite Cayley tree by fine-tuning a local boundary field. The resulting \textit{effective Bethe line}, whose existence is tied to the mean-field nature of the bulk transition, effectively screens the extensive boundary and restores both the critical point and the mean-field critical scaling of the infinite Bethe lattice.
The 'cavity-mean-field' approximation developed for the random transverse field Ising model on the Cayley tree (Ioffe and Mézard 2010 Phys. Rev. Lett. 105 037001) has been found to reproduce the known exact result for the surface magnetization in d = 1 (Dimitrova and Mézard 2011 J. Stat. Mech. P01020). In the present paper, we propose to extend these ideas in finite dimensions d > 1 via a non-linear transfer approach for the surface magnetization. In the disordered phase, the linearization of the transfer equations corresponds to the transfer matrix for a directed polymer in a random medium of transverse dimension D = d − 1, in agreement with the leading order perturbative scaling analysis (Monthus and Garel 2011 arXiv: 1110.3145 ). We present numerical results of the non-linear transfer approach in dimensions d = 2 and 3. In both cases, we find that the critical point is governed by infinite disorder scaling. In particular, exactly at criticality, the one-point surface magnetization scales as , where ω c ( d ) coincides with the droplet exponent ω DP ( D = d − 1) of the corresponding directed polymer model, with ω c ( d = 2) = 1/3 and . The distribution P ( v ) of the positive random variable v of order O(1) presents a power-law singularity near the origin with a ( d = 2, 3) > 0, so all moments of the surface magnetization are governed by the same power-law decay with x s = ω c (1 + a ) independently of the order k .