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:2306.09228

Low energy excitations of vector spin glasses

Flavio Nicoletti

Published Jun 15, 2023• 0 citations
arXivHTMLPDF

Abstract

The work of this thesis concerns the problem of linear low energy excitations of vector spin glass models. An analytical and numerical study is carried out, considering a fully connected random-field Heisenberg model at zero temperature, a fully-connected vector p-spin glass model and a sparse random-field Heisenberg model. We test these models against the low temperature behavior of finite dimensional glassy systems, in particular we show that they posses phases where the density of states is gapless with quasi-localised modes. In the case of the sparse model, we show that the density of states follows a quartic law at low frequency, consistently with several recent measures of this quantity that can be found in the literature of computer glasses. In all the three models, the spin glass transition is characterised in terms of the behavior of the softest excitations. We found that in the fully connected models the zero temperature spin glass transition in a field is a delocalisation transition of the softest modes. In the sparse case, a weaker form of delocalisation appears at the transition. These results broaden our understanding of the zero temperature critical point, by showing how spin glass ordering affects the way the system responds to small magnetic perturbations.

DOI
10.48550/arxiv.2306.09228

Related papers

Soft modes in vector spin glass models on sparse random graphs

Silvio Franz, Cosimo Lupo, Flavio Nicoletti +2 more

The authors study here low-energy linear excitations in XY and Heisenberg spin glasses at zero temperature, considering systems defined on sparse random regular graphs. The localization properties of the soft modes are linked to the spin glass transition in a field. The authors discover a new kind of low-energy excitation, called a ``Concentrated and Delocalized Low Energy Mode'' (CDLEM), given that this excitation is restricted to a sub-extensive number of vertices spread over the largest possible distances.

arXiv:2409.10312•Physical review. B./Physical review. B•Jan 23, 2025•5 citations

Spin Glasses: Criticality and Energy Landscapes

Marco Baity‐Jesi

This thesis addresses the surprising features of zero-temperature statics and dynamics of several spin glass models, including correlations between soft spins that arise spontaneously during avalanches, and the discovery of localized states that involve the presence of two-level systems. It also presents the only detailed historiographical research on the spin glass theory. Despite the extreme simplicity of their definition, spin glasses display a wide variety of non-trivial behaviors that are not yet fully understood. In this thesis the author sheds light on some of these, focusing on both the search for phase transitions under perturbations of Hamiltonians and the zero-temperature properties and responses to external stimuli. After introducing spin glasses and useful concepts on phase transitions and numerics, the results of two massive Monte Carlo campaigns on three-dimensional systems are presented: The first of these examines the de Almeida Thouless transition, and proposes a new finite-size scaling ansatz, which accelerates the convergence to the thermodynamic limit. The second reconstructs the phase diagram of the Heisenberg spin glass with random exchange anisotropy

•Jun 28, 2016•2 citations

Dynamic critical behavior of random spin chains in a field

Iveta R. Pimentel, Robin B. Stinchcombe

The dynamic critical behavior of one-dimensional random spin systems in a field at zero temperature is investigated by a generalized transfer-matrix scaling technique. Three spin models are considered: Heisenberg in a longitudinal field, XY in a transverse field, and Ising in a transverse field, with either spin-glass or random-field disorder. Near the transition, i.e., for small values of the reduced field \ensuremath{\Delta}=(H-${H}_{c}$), where ${H}_{c}$ is the critical field, one finds that the dispersion relation for the low-frequency \ensuremath{\omega}, small-wave-vector k spin-wave excitations takes the scaling form \ensuremath{\omega}=${k}^{z}$f(\ensuremath{\Delta}/${k}^{\mathrm{cphi}}$), and exact results are given for the dynamic exponent z and crossover exponent cphi. This form contains a crossover of the frequency \ensuremath{\omega} between two different asymptotic behaviors: ${k}^{z}$ for \ensuremath{\Delta}\ensuremath{\ll}${k}^{\mathrm{cphi}}$ and ${h}^{\ensuremath{\delta}}$ for \ensuremath{\Delta}\ensuremath{\gg}${k}^{\mathrm{cphi}}$, where the field exponent is \ensuremath{\delta}=z/cphi. In the case of the Heisenberg systems the spin-glass disorder gives rise to the nontrivial dynamic exponent z=(3/2, whereas in the random-field case it is the exponent associated to the field that becomes nontrivial, taking the value (4/3. For the transverse XY systems the random-field disorder implies nontrivial values for both the dynamic and field exponents, (3/2 and (3/4, respectively, whereas the spin-glass disorder does not affect the dynamics of the system, which behaves like a pure transverse XY ferromagnet. Finally, for the transverse Ising systems neither the random field nor the spin-glass disorder affects the dynamics, which is the same as for a pure transverse Ising ferromagnet.

Physical review. B, Condensed matter•Sep 1, 1989•9 citations

Superfluidity and localization in Bosonic glasses

Xiao Yu

Bosonic excitations within long-range ordered, but strongly inhomogeneous phases have been studied in quite some detail. My thesis focuses instead on understanding the insulating, localized phase of disordered bosonic systems. In particular I study localization properties of strongly interacting bosons and spin systems in a disorder potential at zero temperature. I focus on simple, prototypical spin models (Ising model and XY model) in random fields on a Cayley tree with large connectivity. Regarding the nature of the quantum phase transition in strong disorder I find the following results: i) With a uniformly distributed disorder non-extensive excitations in the disordered phase are all localized. ii)Moreover, I find that the order arises due to a collective condensation, which is qualitatively distinct from a Bose Einstein condensation of single particle excitations into a delocalized state. In particular, in non-frustrated Bose glasses, I do not find evidence for a boson mobility edge in the Bose glass. These results are qualitatively different from claims in the recent literatures . Considering that (many body) localization of bosons is a kind of quantum glass transition, it is an interesting question to ask what phenomena occur, if the ingredients for more conventional (classical) glassy physics are added to a disordered bosons system, namely: random, frustrated interactions between the bosons. One can still think about such a system as bosons in a disordered potential, where the disordered potential is, at least partly, self-generated by random frustrated interactions between the bosons. This question takes us to the study of another type of disordered systems: glassy systems. Those are typically characterized by low temperature phases with an inhomogeneous density or magnetization pattern, which is extremely long-lived due to the occurrence of non-trivial ergodicity breaking. I study a solvable model of hard core bosons (pseudospins) subject to disorder and frustrating interactions. This solvable model provides insight into the possibility of coexistence of super uidity and glassy density order, as well as into the nature of the coexistence phase (the superglass). In particular, for the considered mean field model I prove the existence of a superglass phase. This complements the numerical evidence for such phases provided by quantum Monte Carlo investigations in finite dimensions and on random graphs. Those were, however, limited to finite temperature, and could thus not fully elucidate the structure of the phases at T = 0. In contrast, my analytical approach allows one to understand the quantum phase transition between glassy superfluid and insulator, and the non-trivial role played by glassy correlations. When the frustrated interactions are strong enough, the superfluid order may be destroyed. As I will show in a mean field model, this happens within the glass phase of the system, where a disorder induced superfluid-insulator phase transition takes place to give way to a frustrated Bose glass. The glassy background on top of which this happens leads to many interesting phenomena which seem not to have been noticed before. To understand the nature of the glassy superfluid-insulator quantum phase transition at zero temperature and the transport properties on the insulating, Bose glass side of the transition is the goal of the third part of my thesis. To address the above questions, I studied an exactly solvable model of a glassy superfluid-insulator quantum phase transition on a Bethe lattice geometry with high connectivity. My main results can be summarized as follows: i) I found that the superfluid-insulator transition is shifted to stronger hopping. This is a result of the pseudo gap in the density of states of the glass state, which tends to strongly disfavor the onset of superfluidity. ii) In the glassy insulator, the discrete local energy levels become broadened due to the quantum fluctuations.The level-broadening process appears as a phase transition which has strong similarities with an Anderson localization transition, and has implications on many body localization. By using the locator expansion for bosons I found that, the glassy insulator has a finite mobility edge for the bosonic excitations, which, however, does not close upon approaching the SI quantum phase transition point. This finding helps to understand the nature of the superfluid-to-frustrated Bose glass transition: the superfluid emerges as a collective phase ordering phenomenon at zero temperature, and not as a condensation in to a single particle delocalized state, in contrast to opposite predictions in the recent literatures. The existence of a mobility edge in the insulator suggests the possibility of phononless, activated transport in the bosonic insulator, which might be a candidate explanation for the experimentally seen activated transport, which has remained a mystery for a long time.

•Nov 20, 2012•0 citations

Dynamical Response of Quantum Spin-Glass Models at T=0

Liliana Arrachea, Marcelo J. Rozenberg

We study the behavior of two archetypal quantum spin glasses at $T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ by exact diagonalization techniques: the random Ising model in a transverse field and the random Heisenberg model. The behavior of the dynamical spin response is obtained in the spin-glass ordered phase. In both models it is gapless and has the general form ${\ensuremath{\chi}}^{\ensuremath{'}\ensuremath{'}}(\ensuremath{\omega})\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}q\ensuremath{\delta}(\ensuremath{\omega})+{\ensuremath{\chi}}_{\mathrm{reg}}^{\ensuremath{'}\ensuremath{'}}(\ensuremath{\omega})$, with ${\ensuremath{\chi}}_{\mathrm{reg}}^{\ensuremath{'}\ensuremath{'}}(\ensuremath{\omega})\ensuremath{\sim}\ensuremath{\omega}$ for the Ising and ${\ensuremath{\chi}}_{\mathrm{reg}}^{\ensuremath{'}\ensuremath{'}}(\ensuremath{\omega})\ensuremath{\sim}\mathrm{const}$ for the Heisenberg, at low frequencies. The method provides new insight to the physical nature of the low-lying excitations.

Physical Review Letters•May 28, 2001•29 citations

Low Temperature Phase of the m-component Spin Glass

A. Braun

We investigate the low temperature phase of a vector spin glass in the limit of large component numbers $m$ of the spins, where this model is expected to be replica symmetric. On a Bethe lattice, a random graph with fixed and finite connectivity, we employ a self-consistent effective-field approach to the cavity method to describe equilibrium states of the system in terms of the configuration of cavity fields, which are reduced mean fields onto each spin. We make use of an iterative algorithm to find such equilibrium states for finite system sizes $N$ and observe a phase transition, which is indicated by an order parameter describing the preferred direction each spin points into in terms of the configuration of cavity, resp. mean local fields. On single realizations of Bethe lattices the transition is sharp and accompanied by the Generalized-Bose-Einstein condensation, in which the cavity, resp. local fields of the spins condense into a subspace of the available $m$-dimensional space for large $m$. The dimension of this subspace, $m_{\rm eff}(T)$ depends on the temperature. Is either one or two right at the critical temperature and increases below. Furthermore, for fixed temperature it is expected to scale with the system size $N$ with an exponent $\mu_m$ which depends on the connectivity. Averaging over many realizations of disorder we find $\overline{m_{\rm eff}}(T_c)=2$ and $\overline{m_{\rm eff}}(T)\sim N^{\mu_m}$. Even more, the cavity fields are subject to rotations in the low temperature phase, i.e. there is no freezing in time as generally expected. In case of $m_{\rm eff}(T_c)=2$ the whole configuration rotates at the critical temperature about the same angle and the same axis of rotation corresponding to a single droplet. In addition, we find replica symmetry breaking below the large-$m$ limit, i.e. for $m_{\rm eff}(T) > m$. These results are supported by results of parallel tempering Monte Carlo simulations. Replica symmetry breaking is further investigated by an analytical method in the large connectivity limit, where we compare the free energy in terms of spin-spin-correlations of different replicas calculated with a replica symmetric and a one-step broken replica symmetry ansatz. In the large-$m$ limit, we find that both solutions coexist for finite connectivities and that replica symmetry holds for full connectivity up to corrections in second order around the saddle point for $m\to\infty$. In the last chapter we calculate the sample-to-sample fluctuations of the free energy of the SK model of the large-$m$ limit. We derive an exact connection between these fluctuations and bond chaos. Making use of a replica calculation for the {\it large deviations} and a scaling ansatz for the {\it small deviations}, we quantify bond chaos, i.e. calculate the probability of finding a given link overlap for two systems with possibly different sets of bonds. With this, we find the sample-to-sample fluctuations to scale with the number of spins, $N$, with an exponent $\mu$ in between $1/5$ and $3/10$.

•Jan 1, 2011•0 citations
The work of this thesis concerns the problem of linear low energy excitations of vector spin glass models. An analytical and numerical study is carried out, considering a fully connected random-field Heisenberg model at zero temperature, a fully-connected vector p-spin glass model and a sparse random-field Heisenberg model. We test these models against the low temperature behavior of finite dimensional glassy systems, in particular we show that they posses phases where the density of states is gapless with quasi-localised modes. In the case of the sparse model, we show that the density of states follows a quartic law at low frequency, consistently with several recent measures of this quantity that can be found in the literature of computer glasses. In all the three models, the spin glass transition is characterised in terms of the behavior of the softest excitations. We found that in the fully connected models the zero temperature spin glass transition in a field is a delocalisation transition of the softest modes. In the sparse case, a weaker form of delocalisation appears at the transition. These results broaden our understanding of the zero temperature critical point, by showing how spin glass ordering affects the way the system responds to small magnetic perturbations.
The authors study here low-energy linear excitations in XY and Heisenberg spin glasses at zero temperature, considering systems defined on sparse random regular graphs. The localization properties of the soft modes are linked to the spin glass transition in a field. The authors discover a new kind of low-energy excitation, called a ``Concentrated and Delocalized Low Energy Mode'' (CDLEM), given that this excitation is restricted to a sub-extensive number of vertices spread over the largest possible distances.
This thesis addresses the surprising features of zero-temperature statics and dynamics of several spin glass models, including correlations between soft spins that arise spontaneously during avalanches, and the discovery of localized states that involve the presence of two-level systems. It also presents the only detailed historiographical research on the spin glass theory. Despite the extreme simplicity of their definition, spin glasses display a wide variety of non-trivial behaviors that are not yet fully understood. In this thesis the author sheds light on some of these, focusing on both the search for phase transitions under perturbations of Hamiltonians and the zero-temperature properties and responses to external stimuli. After introducing spin glasses and useful concepts on phase transitions and numerics, the results of two massive Monte Carlo campaigns on three-dimensional systems are presented: The first of these examines the de Almeida Thouless transition, and proposes a new finite-size scaling ansatz, which accelerates the convergence to the thermodynamic limit. The second reconstructs the phase diagram of the Heisenberg spin glass with random exchange anisotropy
The dynamic critical behavior of one-dimensional random spin systems in a field at zero temperature is investigated by a generalized transfer-matrix scaling technique. Three spin models are considered: Heisenberg in a longitudinal field, XY in a transverse field, and Ising in a transverse field, with either spin-glass or random-field disorder. Near the transition, i.e., for small values of the reduced field \ensuremath{\Delta}=(H-Hc{H}_{c}Hc​), where Hc{H}_{c}Hc​ is the critical field, one finds that the dispersion relation for the low-frequency \ensuremath{\omega}, small-wave-vector k spin-wave excitations takes the scaling form \ensuremath{\omega}=kz{k}^{z}kzf(\ensuremath{\Delta}/kcphi{k}^{\mathrm{cphi}}kcphi), and exact results are given for the dynamic exponent z and crossover exponent cphi. This form contains a crossover of the frequency \ensuremath{\omega} between two different asymptotic behaviors: kz{k}^{z}kz for \ensuremath{\Delta}\ensuremath{\ll}kcphi{k}^{\mathrm{cphi}}kcphi and ${h}^{\ensuremath{\delta}}$ for \ensuremath{\Delta}\ensuremath{\gg}kcphi{k}^{\mathrm{cphi}}kcphi, where the field exponent is \ensuremath{\delta}=z/cphi. In the case of the Heisenberg systems the spin-glass disorder gives rise to the nontrivial dynamic exponent z=(3/2, whereas in the random-field case it is the exponent associated to the field that becomes nontrivial, taking the value (4/3. For the transverse XY systems the random-field disorder implies nontrivial values for both the dynamic and field exponents, (3/2 and (3/4, respectively, whereas the spin-glass disorder does not affect the dynamics of the system, which behaves like a pure transverse XY ferromagnet. Finally, for the transverse Ising systems neither the random field nor the spin-glass disorder affects the dynamics, which is the same as for a pure transverse Ising ferromagnet.
Bosonic excitations within long-range ordered, but strongly inhomogeneous phases have been studied in quite some detail. My thesis focuses instead on understanding the insulating, localized phase of disordered bosonic systems. In particular I study localization properties of strongly interacting bosons and spin systems in a disorder potential at zero temperature. I focus on simple, prototypical spin models (Ising model and XY model) in random fields on a Cayley tree with large connectivity. Regarding the nature of the quantum phase transition in strong disorder I find the following results: i) With a uniformly distributed disorder non-extensive excitations in the disordered phase are all localized. ii)Moreover, I find that the order arises due to a collective condensation, which is qualitatively distinct from a Bose Einstein condensation of single particle excitations into a delocalized state. In particular, in non-frustrated Bose glasses, I do not find evidence for a boson mobility edge in the Bose glass. These results are qualitatively different from claims in the recent literatures . Considering that (many body) localization of bosons is a kind of quantum glass transition, it is an interesting question to ask what phenomena occur, if the ingredients for more conventional (classical) glassy physics are added to a disordered bosons system, namely: random, frustrated interactions between the bosons. One can still think about such a system as bosons in a disordered potential, where the disordered potential is, at least partly, self-generated by random frustrated interactions between the bosons. This question takes us to the study of another type of disordered systems: glassy systems. Those are typically characterized by low temperature phases with an inhomogeneous density or magnetization pattern, which is extremely long-lived due to the occurrence of non-trivial ergodicity breaking. I study a solvable model of hard core bosons (pseudospins) subject to disorder and frustrating interactions. This solvable model provides insight into the possibility of coexistence of super uidity and glassy density order, as well as into the nature of the coexistence phase (the superglass). In particular, for the considered mean field model I prove the existence of a superglass phase. This complements the numerical evidence for such phases provided by quantum Monte Carlo investigations in finite dimensions and on random graphs. Those were, however, limited to finite temperature, and could thus not fully elucidate the structure of the phases at T = 0. In contrast, my analytical approach allows one to understand the quantum phase transition between glassy superfluid and insulator, and the non-trivial role played by glassy correlations. When the frustrated interactions are strong enough, the superfluid order may be destroyed. As I will show in a mean field model, this happens within the glass phase of the system, where a disorder induced superfluid-insulator phase transition takes place to give way to a frustrated Bose glass. The glassy background on top of which this happens leads to many interesting phenomena which seem not to have been noticed before. To understand the nature of the glassy superfluid-insulator quantum phase transition at zero temperature and the transport properties on the insulating, Bose glass side of the transition is the goal of the third part of my thesis. To address the above questions, I studied an exactly solvable model of a glassy superfluid-insulator quantum phase transition on a Bethe lattice geometry with high connectivity. My main results can be summarized as follows: i) I found that the superfluid-insulator transition is shifted to stronger hopping. This is a result of the pseudo gap in the density of states of the glass state, which tends to strongly disfavor the onset of superfluidity. ii) In the glassy insulator, the discrete local energy levels become broadened due to the quantum fluctuations.The level-broadening process appears as a phase transition which has strong similarities with an Anderson localization transition, and has implications on many body localization. By using the locator expansion for bosons I found that, the glassy insulator has a finite mobility edge for the bosonic excitations, which, however, does not close upon approaching the SI quantum phase transition point. This finding helps to understand the nature of the superfluid-to-frustrated Bose glass transition: the superfluid emerges as a collective phase ordering phenomenon at zero temperature, and not as a condensation in to a single particle delocalized state, in contrast to opposite predictions in the recent literatures. The existence of a mobility edge in the insulator suggests the possibility of phononless, activated transport in the bosonic insulator, which might be a candidate explanation for the experimentally seen activated transport, which has remained a mystery for a long time.
We study the behavior of two archetypal quantum spin glasses at T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 by exact diagonalization techniques: the random Ising model in a transverse field and the random Heisenberg model. The behavior of the dynamical spin response is obtained in the spin-glass ordered phase. In both models it is gapless and has the general form ${\ensuremath{\chi}}^{\ensuremath{'}\ensuremath{'}}(\ensuremath{\omega})\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}q\ensuremath{\delta}(\ensuremath{\omega})+{\ensuremath{\chi}}_{\mathrm{reg}}^{\ensuremath{'}\ensuremath{'}}(\ensuremath{\omega})$, with ${\ensuremath{\chi}}_{\mathrm{reg}}^{\ensuremath{'}\ensuremath{'}}(\ensuremath{\omega})\ensuremath{\sim}\ensuremath{\omega}$ for the Ising and ${\ensuremath{\chi}}_{\mathrm{reg}}^{\ensuremath{'}\ensuremath{'}}(\ensuremath{\omega})\ensuremath{\sim}\mathrm{const}$ for the Heisenberg, at low frequencies. The method provides new insight to the physical nature of the low-lying excitations.
We investigate the low temperature phase of a vector spin glass in the limit of large component numbers mmm of the spins, where this model is expected to be replica symmetric. On a Bethe lattice, a random graph with fixed and finite connectivity, we employ a self-consistent effective-field approach to the cavity method to describe equilibrium states of the system in terms of the configuration of cavity fields, which are reduced mean fields onto each spin. We make use of an iterative algorithm to find such equilibrium states for finite system sizes NNN and observe a phase transition, which is indicated by an order parameter describing the preferred direction each spin points into in terms of the configuration of cavity, resp. mean local fields. On single realizations of Bethe lattices the transition is sharp and accompanied by the Generalized-Bose-Einstein condensation, in which the cavity, resp. local fields of the spins condense into a subspace of the available mmm-dimensional space for large mmm. The dimension of this subspace, meff(T)m_{\rm eff}(T)meff​(T) depends on the temperature. Is either one or two right at the critical temperature and increases below. Furthermore, for fixed temperature it is expected to scale with the system size NNN with an exponent μm\mu_mμm​ which depends on the connectivity. Averaging over many realizations of disorder we find meff‾(Tc)=2\overline{m_{\rm eff}}(T_c)=2meff​​(Tc​)=2 and meff‾(T)∼Nμm\overline{m_{\rm eff}}(T)\sim N^{\mu_m}meff​​(T)∼Nμm​. Even more, the cavity fields are subject to rotations in the low temperature phase, i.e. there is no freezing in time as generally expected. In case of meff(Tc)=2m_{\rm eff}(T_c)=2meff​(Tc​)=2 the whole configuration rotates at the critical temperature about the same angle and the same axis of rotation corresponding to a single droplet. In addition, we find replica symmetry breaking below the large-mmm limit, i.e. for meff(T)>mm_{\rm eff}(T) > mmeff​(T)>m. These results are supported by results of parallel tempering Monte Carlo simulations. Replica symmetry breaking is further investigated by an analytical method in the large connectivity limit, where we compare the free energy in terms of spin-spin-correlations of different replicas calculated with a replica symmetric and a one-step broken replica symmetry ansatz. In the large-mmm limit, we find that both solutions coexist for finite connectivities and that replica symmetry holds for full connectivity up to corrections in second order around the saddle point for m→∞m\to\inftym→∞. In the last chapter we calculate the sample-to-sample fluctuations of the free energy of the SK model of the large-mmm limit. We derive an exact connection between these fluctuations and bond chaos. Making use of a replica calculation for the {\it large deviations} and a scaling ansatz for the {\it small deviations}, we quantify bond chaos, i.e. calculate the probability of finding a given link overlap for two systems with possibly different sets of bonds. With this, we find the sample-to-sample fluctuations to scale with the number of spins, NNN, with an exponent μ\muμ in between 1/51/51/5 and 3/103/103/10.