arXiv subjects

Cecile Monthus

Publications and source records attributed to Cecile Monthus.

At least 73 records · Page 4Linked to original sources

arXiv2016

Strong Disorder Renewal Approach to DNA denaturation and wetting : typical and large deviation properties of the free energy

For the DNA denaturation transition in the presence of random contact energies, or equivalently the disordered wetting transition, we introduce a Strong Disorder Renewal Approach to construct the optimal contacts in each disordered sample of size $L$. The transition is found to be of infinite order, with a correlation length diverging with the essential singularity $\ln ξ(T) \propto |T-T_c |^{-1}$. In the critical region, we analyze the statistics over samples of the free-energy density $f_L$ and of the contact density, which is the order parameter of the transition. At the critical point, both decay as a power-law of the length $L$ but remain distributed, in agreement with the general phenomenon of lack of self-averaging at random critical points. We also obtain that for any real $q>0$, the moment $\overline{Z_L^q} $ of order $q$ of the partition function at the critical point is dominated by some exponentially rare samples displaying a finite free-energy density, i.e. by the large deviation sector of the probability distribution of the free-energy density.

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arXiv2016

Finite size scaling for the Many-Body-Localization Transition : finite-size-pseudo-critical points of individual eigenstates

To understand the finite-size-scaling properties of phases transitions in classical and quantum models in the presence of quenched disorder, it has proven to be fruitful to introduce the notion of a finite-size-pseudo-critical point in each disordered sample and to analyze its sample-to-sample fluctuations as a function of the size. For the Many-Body-Localization transition, where very strong eigenstate-to-eigenstate fluctuations have been numerically reported even within a given disordered sample at a given energy density [X. Yu, D. J. Luitz, B. K. Clark, arxiv:1606.01260 and V. Khemani, S. P. Lim, D. N. Sheng, D. A. Huse,arxiv:1607.05756], it seems thus useful to introduce the notion of a finite-size-pseudo-critical point for each individual eigenstate and to study its eigenstate-to-eigenstate fluctuations governed by the correlation length exponent $ν$. The scaling properties of critical eigenstates are also expected to appear much more clearly if one considers each eigenstate at its finite-size-pseudo-critical point, where it is 'truly critical', while standard averages over eigenstates and samples in the critical region actually see a mixture of states that are effectively either localized or delocalized.

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arXiv2016

Multifractality of eigenstates in the delocalized non-ergodic phase of some random matrix models : Wigner-Weisskopf approach

The delocalized non-ergodic phase existing in some random $N \times N$ matrix models is analyzed via the Wigner-Weisskopf approximation for the dynamics from an initial site $j_0$. The main output of this approach is the inverse $Γ_{j_0}(N)$ of the characteristic time to leave the state $j_0$ that provides some broadening $Γ_{j_0}(N) $ for the weights of the eigenvectors. In this framework, the localized phase corresponds to the region where the broadening $Γ_{j_0}(N) $ is smaller in scaling than the level spacing $Δ_{j_0}(N) \propto \frac{1}{N}$, while the delocalized non-ergodic phase corresponds to the region where the broadening $Γ_{j_0}(N) $ decays with $N$ but is bigger in scaling than the level spacing $Δ_{j_0}(N) $. Then the number $\frac{Γ_{j_0}(N)}{Δ_{j_0}(N)} $ of resonances grows only sub-extensively in $N$. This approach allows to recover the multifractal spectrum of the Generalized-Rosenzweig-Potter (GRP) Matrix model [V.E. Kravtsov, I.M. Khaymovich, E. Cuevas and M. Amini, New. J. Phys. 17, 122002 (2015)]. We then consider the Lévy generalization of the GRP Matrix model, where the off-diagonal matrix elements are drawn with an heavy-tailed distribution of Lévy index $1<μ<2$ : the dynamics is then governed by a stretched exponential of exponent $β=\frac{2 (μ-1)}μ$ and the multifractal properties of eigenstates are explicitly computed.

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arXiv2016

Many-Body-Localization Transition : sensitivity to twisted boundary conditions

For disordered interacting quantum systems, the sensitivity of the spectrum to twisted boundary conditions depending on an infinitesimal angle $ϕ$ can be used to analyze the Many-Body-Localization Transition. The sensitivity of the energy levels $E_n(ϕ)$ is measured by the level curvature $K_n=E_n"(0)$, or more precisely by the Thouless dimensionless curvature $k_n=K_n/Δ_n$, where $Δ_n$ is the level spacing that decays exponentially with the size $L$ of the system. For instance $Δ_n \propto 2^{-L}$ in the middle of the spectrum of quantum spin chains of $L$ spins, while the Drude weight $D_n=L K_n$ studied recently by M. Filippone, P.W. Brouwer, J. Eisert and F. von Oppen [arxiv:1606.07291v1] involves a different rescaling. The sensitivity of the eigenstates $\vert ψ_n(ϕ) > $ is characterized by the susceptibility $χ_n=-F_n"(0)$ of the fidelity $F_n =\vert < ψ_n(0) \vert ψ_n(ϕ) >\vert $. Both observables are distributed with probability distributions displaying power-law tails $P_β(k) \simeq A_β \vert k \vert^{-(2+β)} $ and $Q(χ) \simeq B_β χ^{-\frac{3+β}{2}} $, where $β$ is the level repulsion index taking the values $β^{GOE}=1$ in the ergodic phase and $β^{loc}=0$ in the localized phase. The amplitudes $A_β$ and $B_β$ of these two heavy tails are given by some moments of the off-diagonal matrix element of the local current operator between two nearby energy levels, whose probability distribution has been proposed as a criterion for the Many-Body-Localization transition by M. Serbyn, Z. Papic and D.A. Abanin [Phys. Rev. X 5, 041047 (2015)].

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arXiv2016

Localization transition in random Levy matrices : multifractality of eigenvectors in the localized phase and at criticality

For random Lévy matrices of size $N \times N$, where matrix elements are drawn with some heavy-tailed distribution $P(H_{ij}) \propto N^{-1} |H_{ij} |^{-1-μ}$ with $0<μ<2$ (infinite variance), there exists an extensive number of finite eigenvalues $E=O(1)$, while the maximal eigenvalue grows as $E_{max} \sim N^{\frac{1}μ}$. Here we study the localization properties of the corresponding eigenvectors via some strong disorder perturbative expansion that remains consistent within the localized phase and that yields their Inverse Participation Ratios (I.P.R.) $Y_q$ as a function of the continuous parameter $0 q_c$ but diverge in the region $0<q<q_c$ near the origin. In the region $1<μ<2$, only the sub-extensive fraction $N^{\frac{3}{2+μ}}$ of the biggest eigenvalues corresponding to the region $| E| \geq N^{\frac{(μ-1)}{μ(2+μ)}} $ remains localized, while the extensive number of other states of smaller energy are delocalized. For the extensive number of finite eigenvalues $E=O(1)$, the localization/delocalization transition thus takes place at the critical value $μ_c=1$ corresponding to Cauchy matrices : the Inverse Participation Ratios $Y_q$ of the corresponding critical eigenstates follow the Strong-Multifractality Spectrum characterized by the generalized fractal dimensions $D^{criti}(q) = \frac{1-2q}{1-q} θ(0 \leq q \leq \frac{1}{2})$, which has been found previously in various other Localization problems in spaces of effective infinite dimensionality.

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arXiv2016

Flow towards diagonalization for Many-Body-Localization models : adaptation of the Toda matrix differential flow to random quantum spin chains

The iterative methods to diagonalize matrices and many-body Hamiltonians can be reformulated as flows of Hamiltonians towards diagonalization driven by unitary transformations that preserve the spectrum. After a comparative overview of the various types of discrete flows (Jacobi, QR-algorithm) and differential flows (Toda, Wegner, White) that have been introduced in the past, we focus on the random XXZ chain with random fields in order to determine the best closed flow within a given subspace of running Hamiltonians. For the special case of the free-fermion random XX chain with random fields, the flow coincides with the Toda differential flow for tridiagonal matrices which is related to the classical integrable Toda chain and which can be seen as the continuous analog of the discrete QR-algorithm. For the random XXZ chain with random fields that displays a Many-Body-Localization transition, the present differential flow should be an interesting alternative to compare with the discrete flow that has been proposed recently to study the Many-Body-Localization properties in a model of interacting fermions (L. Rademaker and M. Ortuno, Phys. Rev. Lett. 116, 010404 (2016)).

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arXiv2016

Many-Body-Localization Transition : strong multifractality spectrum for matrix elements of local operators

For short-ranged disordered quantum models in one dimension, the Many-Body-Localization is analyzed via the adaptation to the Many-Body context [M. Serbyn, Z. Papic and D.A. Abanin, PRX 5, 041047 (2015)] of the Thouless point of view on the Anderson transition : the question is whether a local interaction between two long chains is able to reshuffle completely the eigenstates (Delocalized phase with a volume-law entanglement) or whether the hybridization between tensor states remains limited (Many-Body-Localized Phase with an area-law entanglement). The central object is thus the level of Hybridization induced by the matrix elements of local operators, as compared with the difference of diagonal energies. The multifractal analysis of these matrix elements of local operators is used to analyze the corresponding statistics of resonances. Our main conclusion is that the critical point is characterized by the Strong-Multifractality Spectrum $f(0 \leq α\leq 2)=\fracα{2}$, well known in the context of Anderson Localization in spaces of effective infinite dimensionality, where the size of the Hilbert space grows exponentially with the volume. Finally, the possibility of a delocalized non-ergodic phase near criticality is discussed.

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arXiv2016

Many Body Localization Transition in the strong disorder limit : entanglement entropy from the statistics of rare extensive resonances

The space of one-dimensional disordered interacting quantum models displaying a Many-Body-Localization Transition seems sufficiently rich to produce critical points with level statistics interpolating continuously between the Poisson statistics of the Localized phase and the Wigner-Dyson statistics of the Delocalized Phase. In this paper, we consider the strong disorder limit of the MBL transition, where the critical level statistics is close to the Poisson statistics. We analyse a one-dimensional quantum spin model, in order to determine the statistical properties of the rare extensive resonances that are needed to destabilize the MBL phase. At criticality, we find that the entanglement entropy can grow with an exponent $0<α< 1$ anywhere between the area law $α=0$ and the volume law $α=1$, as a function of the resonances properties, while the entanglement spectrum follows the strong multifractality statistics. In the MBL phase near criticality, we obtain the simple value $ν=1$ for the correlation length exponent. Independently of the strong disorder limit, we explain why for the Many-Body-Localization transition concerning individual eigenstates, the correlation length exponent $ν$ is not constrained by the usual Harris inequality $ν\geq 2/d$, so that there is no theoretical inconsistency with the best numerical measure $ν= 0.8 (3)$ obtained by D. J. Luitz, N. Laflorencie and F. Alet, Phys. Rev. B 91, 081103 (2015).

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arXiv2016

Level repulsion exponent $β$ for Many-Body Localization Transitions and for Anderson Localization Transitions via Dyson Brownian Motion

The generalization of the Dyson Brownian Motion approach of random matrices to Anderson Localization (AL) models [Chalker, Lerner and Smith PRL 77, 554 (1996)] and to Many-Body Localization (MBL) Hamiltonians [Serbyn and Moore arxiv:1508.07293] is revisited to extract the level repulsion exponent $β$, where $β=1$ in the delocalized phase governed by the Wigner-Dyson statistics, $β=0$ in the localized phase governed by the Poisson statistics, and $0<β_c<1$ at the critical point. The idea is that the Gaussian disorder variables $h_i$ are promoted to Gaussian stationary processes $h_i(t)$ in order to sample the disorder stationary distribution with some time correlation $τ$. The statistics of energy levels can be then studied via Langevin and Fokker-Planck equations. For the MBL quantum spin Hamiltonian with random fields $h_i$, we obtain $β=2q^{EA}_{n,n+1}(N)/q^{EA}_{n,n}(N) $ in terms of the Edwards-Anderson matrix $q^{EA}_{nm}(N) \equiv \frac{1}{N} \sum_{i=1}^N | < ϕ_n | σ_i^z | ϕ_m> |^2 $ for the same eigenstate $m=n$ and for consecutive eigenstates $m=n+1$. For the Anderson Localization tight-binding Hamiltonian with random on-site energies $h_i$, we find $β=2 Y_{n,n+1}(N)/(Y_{n,n}(N)-Y_{n,n+1}(N)) $ in terms of the Density Correlation matrix $Y_{nm}(N) \equiv \sum_{i=1}^N | < ϕ_n | i> |^2 | |^2 $ for consecutive eigenstates $m=n+1$, while the diagonal element $m=n$ corresponds to the Inverse Participation Ratio $Y_{nn}(N) \equiv \sum_{i=1}^N | < ϕ_n | i> |^4 $ of the eigenstate $| ϕ_n>$.

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arXiv2016

Real-space renormalization for the finite temperature statics and dynamics of the Dyson Long-Ranged Ferromagnetic and Spin-Glass models

The finite temperature dynamics of the Dyson hierarchical classical spins models is studied via real-space renormalization rules concerning the couplings and the relaxation times. For the ferromagnetic model involving Long-Ranged coupling $J(r) \propto r^{-1-σ}$ in the region $1/2<σ<1$ where there exists a non-mean-field-like thermal Ferromagnetic-Paramagnetic transition, the RG flows are explicitly solved: the characteristic relaxation time $τ(L)$ follows the critical power-law $τ(L)\propto L^{z_c(σ)} $ at the phase transition and the activated law $\ln τ(L)\propto L^ψ $ with $ψ=1-σ$ in the ferromagnetic phase. For the Spin-Glass model involving random Long-Ranged couplings of variance $\overline{J^2(r)} \propto r^{-2σ}$ in the region $2/3<σ<1$ where there exists a non-mean-field-like thermal SpinGlass-Paramagnetic transition, the coupled RG flows of the couplings and of the relaxation times are studied numerically : the relaxation time $τ(L)$ follows some power-law $τ(L)\propto L^{z_c(σ)} $ at criticality and the activated law $\ln τ(L)\propto L^ψ $ in the Spin-Glass phase with the dynamical exponent $ψ=1-σ=θ$ coinciding with the droplet exponent governing the flow of the couplings $J(L) \propto L^θ $.

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arXiv2016

Many-Body Localization : construction of the emergent local conserved operators via block real-space renormalization

A Fully Many-Body Localized (FMBL) quantum disordered system is characterized by the emergence of an extensive number of local conserved operators that prevents the relaxation towards thermal equilibrium. These local conserved operators can be seen as the building blocks of the whole set of eigenstates. In this paper, we propose to construct them explicitly via some block real-space renormalization. The principle is that each RG step diagonalizes the smallest remaining blocks and produces a conserved operator for each block. The final output for a chain of $N$ spins is a hierarchical organization of the $N$ conserved operators with $\left(\frac{\ln N}{\ln 2}\right)$ layers. The system-size nature of the conserved operators of the top layers is necessary to describe the possible long-ranged order of the excited eigenstates and the possible critical points between different FMBL phases. We discuss the similarities and the differences with the Strong Disorder RSRG-X method that generates the whole set of the $2^N$ eigenstates via a binary tree of $N$ layers. The approach is applied to the Long-Ranged Quantum Spin-Glass Ising model, where the constructed excited eigenstates are found to be exactly like ground states in another disorder realization, so that they can be either in the paramagnetic phase, in the spin-glass phase or critical.

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arXiv2015

Dyson Hierarchical Long-Ranged Quantum Spin-Glass via real-space renormalization

We consider the Dyson hierarchical version of the quantum Spin-Glass with random Gaussian couplings characterized by the power-law decaying variance $\overline{J^2(r)} \propto r^{-2σ}$ and a uniform transverse field $h$. The ground state is studied via real-space renormalization to characterize the spinglass-paramagnetic zero temperature quantum phase transition as a function of the control parameter $h$. In the spinglass phase $h<h_c$, the typical renormalized coupling grows with the length scale $L$ as the power-law $J_L^{typ}(h) \propto Υ(h) L^θ$ with the classical droplet exponent $θ=1-σ$, where the stiffness modulus vanishes at criticality $Υ(h) \propto (h_c-h)^μ $, whereas the typical renormalized transverse field decays exponentially $ h^{typ}_L(h) \propto e^{- \frac{L}ξ}$ where the correlation length diverges at the transition $ξ\propto (h_c-h)^{-ν}$. At the critical point $h=h_c$, the typical renormalized coupling $J_L^{typ}(h_c) $ and the typical renormalized transverse field $ h^{typ}_L(h_c)$ display the same power-law behavior $L^{-z}$ with a finite dynamical exponent $z$. The RG rules are applied numerically to chains containing $L=2^{12}=4096 $ spins in order to measure these critical exponents for various values of $σ$ in the region $1/2<σ<1$.

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arXiv2015

Star junctions and watermelons of pure or random quantum Ising chains : finite-size properties of the energy gap at criticality

We consider $M \geq 2$ pure or random quantum Ising chains of $N$ spins when they are coupled via a single star junction at their origins or when they are coupled via two star junctions at the their two ends leading to the watermelon geometry. The energy gap is studied via a sequential self-dual real-space renormalization procedure that can be explicitly solved in terms of Kesten variables containing the initial couplings and and the initial transverse fields. In the pure case at criticality, the gap is found to decay as a power-law $Δ_M \propto N^{-z(M)} $ with the dynamical exponent $z(M)=\frac{M}{2}$ for the single star junction (the case $M=2$ corresponds to $z=1$ for a single chain with free boundary conditions) and $z(M)=M-1$ for the watermelon (the case $M=2$ corresponds to $z=1$ for a single chain with periodic boundary conditions). In the random case at criticality, the gap follows the Infinite Disorder Fixed Point scaling $\ln Δ_M = -N^ψ g$ with the same activated exponent $ψ=\frac{1}{2}$ as the single chain corresponding to $M=2$, and where $g$ is an $O(1)$ random positive variable, whose distribution depends upon the number $M$ of chains and upon the geometry (star or watermelon).

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arXiv2015

Dyson hierarchical quantum ferromagnetic Ising chain with pure or random transverse fields

The Dyson hierarchical version of the quantum Ising chain with Long-Ranged power-law ferromagnetic couplings $J(r) \propto r^{-1-σ}$ and pure or random transverse fields is studied via real-space renormalization. For the pure case, the critical exponents are explicitly obtained as a function of the parameter $σ$, and are compared with previous results of other approaches. For the random case, the RG rules are numerically applied and the critical behaviors are compared with previous Strong Disorder Renormalization results.

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arXiv2015

Pure and Random Quantum Ising Chain : Shannon and Renyi entropies of the ground state via real space renormalization

The Shannon and the Renyi entropies of the ground state wavefunction in the pure and in the random quantum Ising chain are studied via the self-dual Fernandez-Pacheco real-space renormalization procedure. In particular, we analyze the critical behavior of the leading extensive term at the quantum phase transition : the derivative with respect to the control parameter is found to be logarithmically divergent in the pure case, and to display a cusp singularity in the random case. This cusp singularity for the random case is also derived via the Strong Disorder Renormalization approach.

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arXiv2015

Block Renormalization for quantum Ising models in dimension $d=2$ : applications to the pure and random ferromagnet, and to the spin-glass

For the quantum Ising chain, the self-dual block renormalization procedure of Fernandez-Pacheco [Phys. Rev. D 19, 3173 (1979)] is known to reproduce exactly the location of the zero-temperature critical point and the correlation length exponent $ν=1$. Recently, Miyazaki and Nishimori [Phys. Rev. E 87, 032154 (2013)] have proposed to study the disordered quantum Ising model in dimensions $d>1$ by applying the Fernandez-Pacheco procedure successively in each direction. To avoid the inequivalence of directions of their approach, we propose here an alternative procedure where the $d$ directions are treated on the same footing. For the pure model, this leads to the correlation length exponents $ν\simeq 0.625$ in $d=2$ (to be compared with the 3D classical Ising model exponent $ν\simeq 0.63$) and $ν\simeq 0.5018$ (to be compared with the 4D classical Ising model mean-field exponent $ν=1/2$). For the disordered model in dimension $d=2$, either ferromagnetic or spin-glass, the numerical application of the renormalization rules to samples of linear size $L=4096$ yields that the transition is governed by an Infinite Disorder Fixed Point, with the activated exponent $ψ\simeq 0.65$, the typical correlation exponent $ν_{typ} \simeq 0.44$ and the finite-size correlation exponent $ν_{FS} \simeq 1.25$. We discuss the similarities and differences with the Strong Disorder Renormalization results.

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arXiv2014

Fractal dimension of spin glasses interfaces in dimensions $d=2$ and $d=3$ via strong disorder renormalization at zero temperature

For Gaussian Spin Glasses in low dimensions, we introduce a simple Strong Disorder renormalization procedure at zero temperature. In each disordered sample, the difference between the ground states associated to Periodic and Anti-Periodic boundary conditions defines a system-size Domain Wall. The numerical study in dimensions $d=2$ (up to sizes $2048^2$) and $d=3$ (up to sizes $128^3$) yields fractal Domain Walls of dimensions $d_s(d=2) \simeq 1.27$ and $d_s(d=3) \simeq 2.55$ respectively.

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arXiv2014

Low-temperature dynamics of Long-Ranged Spin-Glasses : full hierarchy of relaxation times via real-space renormalization

We consider the long-ranged Ising spin-glass with random couplings decaying as a power-law of the distance, in the region of parameters where the spin-glass phase exists with a positive droplet exponent. For the Metropolis single-spin-flip dynamics near zero temperature, we construct via real-space renormalization the full hierarchy of relaxation times of the master equation for any given realization of the random couplings. We then analyze the probability distribution of dynamical barriers as a function of the spatial scale. This real-space renormalization procedure represents a simple explicit example of the droplet scaling theory, where the convergence towards local equilibrium on larger and larger scales is governed by a strong hierarchy of activated dynamical processes, with valleys within valleys.

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