Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2107.06364 · Physical review. B./Physical review. B

Scaling theory of few-particle delocalization

Louk Rademaker

Published Dec 6, 2021• 0 citations
arXivHTMLPDF

Abstract

We develop a scaling theory of interaction-induced delocalization of few-particle states in disordered quantum systems. In the absence of interactions, all single-particle states are localized in $d<3$, while in $d\ensuremath{\ge}3$ there is a critical disorder below which states are delocalized. We hypothesize that such a delocalization transition occurs for $n$-particle bound states in $d$ dimensions when $d+n\ensuremath{\ge}4$. Exact calculations of disorder-averaged $n$-particle Green's functions support our hypothesis. In particular, we show that three-particle states in $d=1$ with nearest-neighbor repulsion will delocalize with ${W}_{c}\ensuremath{\approx}1.4t$ and with localization length critical exponent $\ensuremath{\nu}=1.5\ifmmode\pm\else\textpm\fi{}0.3$. The delocalization transition can be understood by means of a mapping onto a noninteracting problem with symplectic symmetry. We discuss the importance of this result for many-body delocalization, and how few-body delocalization can be probed in cold atom experiments.

DOI
10.1103/physrevb.104.214204

Related papers

Marginal Anderson localization and many-body delocalization

Rahul Nandkishore, Andrew C. Potter

We consider $d$-dimensional systems which are localized in the absence of interactions, but whose single-particle localization length diverges near a discrete set of (single-particle) energies, with critical exponent $\ensuremath{\nu}$. This class includes disordered systems with intrinsic or symmetry protected topological bands, such as disordered integer quantum Hall insulators. We show that such marginally localized systems exhibit anomalous properties intermediate between localized and extended, including vanishing dc conductivity but subdiffusive dynamics, and fractal entanglement (an entanglement entropy with a scaling intermediate between area and volume law). We investigate the stability of marginal localization in the presence of interactions, and argue that arbitrarily weak short-range interactions trigger delocalization for partially filled bands at nonzero energy density if $\ensuremath{\nu}\ensuremath{\ge}1/d$. We use the Harris-Chayes bound $\ensuremath{\nu}\ensuremath{\ge}2/d$ to conclude that marginal localization is generically unstable in the presence of interactions. Our results suggest the impossibility of stabilizing quantized Hall conductance at nonzero energy density.

arXiv:1406.0847•Physical Review B•Nov 11, 2014•65 citations

Delocalization transition via supersymmetry in one dimension

Leon Balents, Matthew P. A. Fisher

We use supersymmetric (SUSY) methods to study the delocalization transition at zero energy in a one-dimensional tight-binding model of spinless fermions with particle-hole symmetric disorder. Like the McCoy-Wu random transverse-field Ising model to which it is related, the fermionic problem displays two different correlation lengths for typical and mean correlations. Using the SUSY technique, mean correlators are obtained as quantum-mechanical expectation values for a U(2$|$1,1) ``superspin.'' In the scaling limit, this quantum mechanics is closely related to a $0+1$-dimensional Liouville theory, allowing an interpretation of the results in terms of simple properties of the zero-energy wave functions. Our primary results are the exact two-parameter scaling functions for the mean single-particle Green's functions. We also show how the Liouville quantum-mechanics approach can be extended to obtain the full set of multifractal scaling exponents $\ensuremath{\tau}(q)$, $y(q)$ at criticality. A thorough understanding of the unusual features of the present theory may be useful in applying SUSY to other delocalization transitions.

arXiv:cond-mat/9706069•Physical review. B, Condensed matter•Nov 15, 1997•125 citations

Universal properties of single-particle excitations across the many-body localization transition

Atanu Jana, V. Ravi Chandra, Arti Garg

Understanding the nature of the transition from the delocalized to the many-body localized (MBL) phase is an important unresolved issue. To probe the nature of the MBL transition, we investigate the universal properties of single-particle excitations produced in highly excited many-body eigenstates of a disordered interacting quantum many-body system. In a class of one-dimensional spinless fermionic models with random disorder, we study the finite-size scaling of the ratio of typical to average values of the single-particle local density of states and the scattering rates across the MBL transition. Our results indicate that the MBL transition in this class of one-dimensional models of spinless fermions is continuous in nature. For various ranges of interactions in the system, the critical exponent $\ensuremath{\nu}$ with which the correlation length $\ensuremath{\xi}$ diverges at the transition point ${W}_{c}, \ensuremath{\xi}\ensuremath{\sim}|W\ensuremath{-}{W}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}$, satisfies the Chayes-Chayes-Fisher-Spencer (CCFS) bound $\ensuremath{\nu}\ensuremath{\ge}2/d$ where $d$ is the physical dimension of the system. We also discuss why the critical exponent obtained from finite-size scaling of the conventional diagnostic of many-body localization, the level spacing ratio, strongly violates the CCFS bound while the single-particle density of states and scattering rates are consistent with the CCFS criterion.

arXiv:2212.08480•Physical review. B./Physical review. B•Jun 26, 2024•5 citations

Interplay disorder-interaction in one-dimensional quantum models

Jean‐Louis Pichard, Dietmar Weinmann A, Samuel De Toro Arias +4 more

We show that the crossover from the weak interaction limit towards the strong interaction limit may be accompanied by a delocalization effect in one dimensional disordered quantum models. The spin degrees of freedom are frozen and the spatial wave functions remain symmetric or antisymmetric when the strength U of a short range interaction is varied. The study concerns the excited states for two interacting particles and the ground state for a finite density of carriers. First, for two particles in a chain of length L, we establish a duality transformation mapping the behavior at weak U onto the behavior at strong U. For intermediate U, the mixing of the one body states and the interaction induced delocalization effect are maximum. Furthermore, if L ≈︂ L1 (the one particle localization length), the system becomes weakly chaotic with critical spectral statistics. This weak chaos is related to the multifractality of the interaction matrix. For two particles starting close to each other, localization is reached in two steps. Before the time t1 necessary to propagate over L1 , U de-favors the propagation. On the contrary, U favors a very slow delocalization after t1 , characterized by a log(t) spreading of the center of mass. Similarly, the curvatures of the energy levels with respect to an enclosed magnetic flux decrease as a function of U for L < L1 and increase for L > L1 . The changes of the curvatures can be described by a conductance-like single scaling parameter. Second, using the density renormalization group algorithm, we have studied the ground state energy of a finite density of spinless fermions and its change under twisted boundary conditions. For a large disorder, a charge reorganization is induced by the interaction: When the system becomes instable between the inhomogeneous configuration driven by the random potential (Anderson insulator) and the homogeneous one driven by repulsive interactions (Mott insulator), the ground state sensitivity can be enhanced by orders of magnitude. In contrast, no enhancement occurs at weaker disorder, when there are many particles on a scale L1.

Annalen der Physik•Nov 1, 1998•2 citations

Many-Body Delocalization in Strongly Disordered System with Long-Range Interactions: Finite Size Scaling

Alexander L. Burin

Many-body localization in a disordered system of interacting spins coupled by the long-range interaction $1/R^α$ is investigated combining analytical theory considering resonant interactions and a finite size scaling of exact numerical solutions with a number of spins $N$. The numerical results for a one-dimensional system are consistent with the general expectations of analytical theory for $d$-dimensional system including the absence of localization in the infinite system at $α<2d$ and a universal scaling of a critical energy disordering $W_{c} \propto N^{\frac{2d-α}{d}}$. %The finite size effect on the interaction stimulated delocalization of energy in the ensemble of interacting two level systems in amorphous solids at low temperature is discussed.

arXiv:1409.7990•Sep 29, 2014•1 citation

Detecting delocalization-localization transitions from full density distributions

Miroslav Hopjan, Giuliano Orso, Fabian Heidrich‐Meisner

Characterizing the delocalization transition in closed quantum systems with a many-body localized phase is a key open question in the field of nonequilibrium physics. We exploit the fact that localization of particles as realized in Anderson localization and standard many-body localization (MBL) implies Fock-space localization in single-particle basis sets characterized by a real-space index. Using a recently introduced quantitative measure for Fock-space localization computed from the density distributions, the occupation distance, we systematically study its scaling behavior across delocalization transitions and identify critical points from scaling collapses of numerical data. Excellent agreement with literature results is found for the critical disorder strengths of noninteracting fermions, such as the one-dimensional Aubry-Andr\'e model and the three-dimensional Anderson model. We observe a distinctively different scaling behavior in the case of interacting fermions with random disorder consistent with a Kosterlitz-Thouless transition. Finally, we use our measure to extract the transition point as a function of filling for interacting fermions.

arXiv:2105.10584•Physical review. B./Physical review. B•Dec 6, 2021•25 citations
We develop a scaling theory of interaction-induced delocalization of few-particle states in disordered quantum systems. In the absence of interactions, all single-particle states are localized in d<3d<3d<3, while in d\ensuremath≥3d\ensuremath{\ge}3d\ensuremath≥3 there is a critical disorder below which states are delocalized. We hypothesize that such a delocalization transition occurs for nnn-particle bound states in ddd dimensions when d+n\ensuremath≥4d+n\ensuremath{\ge}4d+n\ensuremath≥4. Exact calculations of disorder-averaged nnn-particle Green's functions support our hypothesis. In particular, we show that three-particle states in d=1d=1d=1 with nearest-neighbor repulsion will delocalize with Wc\ensuremath≈1.4t{W}_{c}\ensuremath{\approx}1.4tWc​\ensuremath≈1.4t and with localization length critical exponent \ensuremathν=1.5\ifmmode±\else\textpm\fi0.3\ensuremath{\nu}=1.5\ifmmode\pm\else\textpm\fi{}0.3\ensuremathν=1.5\ifmmode±\else\textpm\fi0.3. The delocalization transition can be understood by means of a mapping onto a noninteracting problem with symplectic symmetry. We discuss the importance of this result for many-body delocalization, and how few-body delocalization can be probed in cold atom experiments.
We consider ddd-dimensional systems which are localized in the absence of interactions, but whose single-particle localization length diverges near a discrete set of (single-particle) energies, with critical exponent $\ensuremath{\nu}$. This class includes disordered systems with intrinsic or symmetry protected topological bands, such as disordered integer quantum Hall insulators. We show that such marginally localized systems exhibit anomalous properties intermediate between localized and extended, including vanishing dc conductivity but subdiffusive dynamics, and fractal entanglement (an entanglement entropy with a scaling intermediate between area and volume law). We investigate the stability of marginal localization in the presence of interactions, and argue that arbitrarily weak short-range interactions trigger delocalization for partially filled bands at nonzero energy density if $\ensuremath{\nu}\ensuremath{\ge}1/d$. We use the Harris-Chayes bound $\ensuremath{\nu}\ensuremath{\ge}2/d$ to conclude that marginal localization is generically unstable in the presence of interactions. Our results suggest the impossibility of stabilizing quantized Hall conductance at nonzero energy density.
We use supersymmetric (SUSY) methods to study the delocalization transition at zero energy in a one-dimensional tight-binding model of spinless fermions with particle-hole symmetric disorder. Like the McCoy-Wu random transverse-field Ising model to which it is related, the fermionic problem displays two different correlation lengths for typical and mean correlations. Using the SUSY technique, mean correlators are obtained as quantum-mechanical expectation values for a U(2$|$1,1) ``superspin.'' In the scaling limit, this quantum mechanics is closely related to a 0+10+10+1-dimensional Liouville theory, allowing an interpretation of the results in terms of simple properties of the zero-energy wave functions. Our primary results are the exact two-parameter scaling functions for the mean single-particle Green's functions. We also show how the Liouville quantum-mechanics approach can be extended to obtain the full set of multifractal scaling exponents $\ensuremath{\tau}(q)$, y(q)y(q)y(q) at criticality. A thorough understanding of the unusual features of the present theory may be useful in applying SUSY to other delocalization transitions.
Understanding the nature of the transition from the delocalized to the many-body localized (MBL) phase is an important unresolved issue. To probe the nature of the MBL transition, we investigate the universal properties of single-particle excitations produced in highly excited many-body eigenstates of a disordered interacting quantum many-body system. In a class of one-dimensional spinless fermionic models with random disorder, we study the finite-size scaling of the ratio of typical to average values of the single-particle local density of states and the scattering rates across the MBL transition. Our results indicate that the MBL transition in this class of one-dimensional models of spinless fermions is continuous in nature. For various ranges of interactions in the system, the critical exponent $\ensuremath{\nu}$ with which the correlation length $\ensuremath{\xi}$ diverges at the transition point ${W}_{c}, \ensuremath{\xi}\ensuremath{\sim}|W\ensuremath{-}{W}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}$, satisfies the Chayes-Chayes-Fisher-Spencer (CCFS) bound $\ensuremath{\nu}\ensuremath{\ge}2/d$ where ddd is the physical dimension of the system. We also discuss why the critical exponent obtained from finite-size scaling of the conventional diagnostic of many-body localization, the level spacing ratio, strongly violates the CCFS bound while the single-particle density of states and scattering rates are consistent with the CCFS criterion.
We show that the crossover from the weak interaction limit towards the strong interaction limit may be accompanied by a delocalization effect in one dimensional disordered quantum models. The spin degrees of freedom are frozen and the spatial wave functions remain symmetric or antisymmetric when the strength U of a short range interaction is varied. The study concerns the excited states for two interacting particles and the ground state for a finite density of carriers. First, for two particles in a chain of length L, we establish a duality transformation mapping the behavior at weak U onto the behavior at strong U. For intermediate U, the mixing of the one body states and the interaction induced delocalization effect are maximum. Furthermore, if L ≈︂ L1 (the one particle localization length), the system becomes weakly chaotic with critical spectral statistics. This weak chaos is related to the multifractality of the interaction matrix. For two particles starting close to each other, localization is reached in two steps. Before the time t1 necessary to propagate over L1 , U de-favors the propagation. On the contrary, U favors a very slow delocalization after t1 , characterized by a log(t) spreading of the center of mass. Similarly, the curvatures of the energy levels with respect to an enclosed magnetic flux decrease as a function of U for L < L1 and increase for L > L1 . The changes of the curvatures can be described by a conductance-like single scaling parameter. Second, using the density renormalization group algorithm, we have studied the ground state energy of a finite density of spinless fermions and its change under twisted boundary conditions. For a large disorder, a charge reorganization is induced by the interaction: When the system becomes instable between the inhomogeneous configuration driven by the random potential (Anderson insulator) and the homogeneous one driven by repulsive interactions (Mott insulator), the ground state sensitivity can be enhanced by orders of magnitude. In contrast, no enhancement occurs at weaker disorder, when there are many particles on a scale L1.
Many-body localization in a disordered system of interacting spins coupled by the long-range interaction 1/Rα1/R^α1/Rα is investigated combining analytical theory considering resonant interactions and a finite size scaling of exact numerical solutions with a number of spins NNN. The numerical results for a one-dimensional system are consistent with the general expectations of analytical theory for ddd-dimensional system including the absence of localization in the infinite system at α<2dα<2dα<2d and a universal scaling of a critical energy disordering Wc∝N2d−αdW_{c} \propto N^{\frac{2d-α}{d}}Wc​∝Nd2d−α​. %The finite size effect on the interaction stimulated delocalization of energy in the ensemble of interacting two level systems in amorphous solids at low temperature is discussed.
Characterizing the delocalization transition in closed quantum systems with a many-body localized phase is a key open question in the field of nonequilibrium physics. We exploit the fact that localization of particles as realized in Anderson localization and standard many-body localization (MBL) implies Fock-space localization in single-particle basis sets characterized by a real-space index. Using a recently introduced quantitative measure for Fock-space localization computed from the density distributions, the occupation distance, we systematically study its scaling behavior across delocalization transitions and identify critical points from scaling collapses of numerical data. Excellent agreement with literature results is found for the critical disorder strengths of noninteracting fermions, such as the one-dimensional Aubry-Andr\'e model and the three-dimensional Anderson model. We observe a distinctively different scaling behavior in the case of interacting fermions with random disorder consistent with a Kosterlitz-Thouless transition. Finally, we use our measure to extract the transition point as a function of filling for interacting fermions.