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

Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

Role of quantum fluctuation in inducing ergodicity in the spin glass phase and its effect in quantum annealing

Sudip Mukherjee

Published Dec 4, 2022• 1 citation
Publisher

Abstract

We review the numerical studies on the critical behaviour of the quantum Sherrington-Kirkpatrick (SK) spin glass model, which indicate that a quantum critical behaviour is observed up to a low but non-zero value of temperature. We revisit the numerical investigations on the spin glass order parameter distributions, which identify a low temperature along with high transverse field spin glass phase where the order parameter distribution becomes a delta function in the thermodynamic limit indicating the restoration of replica symmetry and ergodic nature of the system. In the remaining spin glass phase associated with high temperature and low transverse field, the observed distribution is broad akin to the Parisi order parameter distribution. This essentially indicates the non-ergodic behaviour of the system. We further discuss the annealing dynamics studies on the quantum SK model. Such investigations reveal the system size independence of annealing time when the annealing paths go through the ergodic spin glass region. Interestingly, when such dynamics are performed in the non-ergodic spin glass phase the annealing time becomes an increasing function of the system size. Spin autocorrelation shows faster relaxation in the ergodic spin glass region compared with that found in the non-ergodic spin glass region. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.

DOI
10.1098/rsta.2021.0406

Related papers

On the Question of Ergodicity in Quantum Spin Glass Phase and Its Role in Quantum Annealing

Sudip Mukherjee, Bikas K Chakrabarti

We first review, following our earlier studies, the critical behavior of the quantum Sherrington-Kirkpatrick (SK) model at finite as well as at zero temperatures. Through the analysis of the Binder cumulant we determined the entire phase diagram of the model and from the scaling analysis of the numerical data we obtained the correlation length exponent. For both the critical Binder cumulant and the correlation length exponent, we observed a crossover from classical- to quantum-fluctuation-dominated values at a finite temperature. We studied the behavior of the order parameter distribution of the model in the glass phase (at finite and zero temperatures). Along with a classical-fluctuation-dominated nonergodic region (where the replica symmetry is broken), we also found a quantum-fluctuation-dominated low-temperature ergodic region in the spin glass phase. In this quantum-fluctuation-dominated region, the order parameter distribution has a narrow peak around its most probable value, eventually becoming a delta function in the infinite-system-size limit (indicating replica symmetry restoration or ergodicity in the system). We also found that the annealing time (to reach a very low energy level of the classical SK model) becomes practically system-size-independent when the annealing paths pass through this ergodic region. In contrast, when such paths pass through the nonergodic region, the convergence time grows rapidly with the system size. We present a new study of the autocorrelation of the spins in both ergodic and nonergodic regions. We found a significant increase in the relaxation time (and also a change in the relaxation behavior) in the classical-fluctuation-dominated (nonergodic) region compared with that in the quantum-fluctuation-dominated (ergodic) region of the spin glass phase.

arXiv:1809.01944•Journal of the Physical Society of Japan•Mar 1, 2019•9 citations

Search of the ground state(s) of spin glasses and quantum annealing

Sudip Mukherjee

We review our earlier studies on the order parameter distribution of the quantum Sherrington-Kirkpatrick (SK) model. Through Monte Carlo technique, we investigate the behavior of the order parameter distribution at finite temperatures. The zero temperature study of the spin glass order parameter distribution is made by the exact diagonalization method. We find in low-temperature (high-transverse-field) spin glass region, the tail (extended up to zero value of order parameter) and width of the order parameter distribution become zero in thermodynamic limit. Such observations clearly suggest the existence of a low-temperature (high-transverse-field) ergodic region. We also find in high-temperature (low-transverse-field) spin glass phase the order parameter distribution has nonzero value for all values of the order parameter even in infinite system size limit, which essentially indicates the nonergodic behavior of the system. We study the annealing dynamics by the paths which pass through both ergodic and nonergodic spin glass regions. We find the average annealing time becomes system size independent for the paths which pass through the quantum-fluctuation-dominated ergodic spin glass region. In contrast to that, the annealing time becomes strongly system size dependent for annealing down through the classical-fluctuation-dominated nonergodic spin glass region. We investigate the behavior of the spin autocorrelation in the spin glass phase. We observe that the decay rate of autocorrelation towards its equilibrium value is much faster in the ergodic region with respect to the nonergodic region of the spin glass phase.

arXiv:1905.01946•May 6, 2019•0 citations

Possible ergodic-nonergodic regions in the quantum Sherrington-Kirkpatrick spin glass model and quantum annealing

Sudip Mukherjee, Atanu Rajak, Bikas K. Chakrabarti

We explore the behavior of the order parameter distribution of the quantum Sherrington-Kirkpatrick model in the spin glass phase using Monte Carlo technique for the effective Suzuki-Trotter Hamiltonian at finite temperatures and that at zero temperature obtained using the exact diagonalization method. Our numerical results indicate the existence of a low- but finite-temperature quantum-fluctuation-dominated ergodic region along with the classical fluctuation-dominated high-temperature nonergodic region in the spin glass phase of the model. In the ergodic region, the order parameter distribution gets narrower around the most probable value of the order parameter as the system size increases. In the other region, the Parisi order distribution function has nonvanishing value everywhere in the thermodynamic limit, indicating nonergodicity. We also show that the average annealing time for convergence (to a low-energy level of the model, within a small error range) becomes system size independent for annealing down through the (quantum-fluctuation-dominated) ergodic region. It becomes strongly system size dependent for annealing through the nonergodic region. Possible finite-size scaling-type behavior for the extent of the ergodic region is also addressed.

arXiv:1706.01446•Physical review. E•Feb 28, 2018•15 citations

Stability of the quantum Sherrington-Kirkpatrick spin glass model

A. P. Young

I study in detail the quantum Sherrington-Kirkpatrick (SK) model, i.e., the infinite-range Ising spin glass in a transverse field, by solving numerically the effective one-dimensional model that the quantum SK model can be mapped to in the thermodynamic limit. I find that the replica symmetric solution is unstable down to zero temperature, in contrast to some previous claims, and so there is not only a line of transitions in the (longitudinal) field-temperature plane (the de Almeida-Thouless, AT, line) where replica symmetry is broken, but also a quantum de Almeida-Thouless (QuAT) line in the transverse field-longitudinal field plane at T=0. If the QuAT line also occurs in models with short-range interactions its presence might affect the performance of quantum annealers when solving spin glass-type problems with a bias (i.e., magnetic field).

arXiv:1707.07107•Physical review. E•Sep 11, 2017•29 citations

Ergodic and localized regions in quantum spin glasses on the Bethe lattice

Gianni Mossi, Antonello Scardicchio

By considering the quantum dynamics of a transverse-field Ising spin glass on the Bethe lattice, we find the existence of a many-body localized (MBL) region at small transverse field and low temperature. The region is located within the thermodynamic spin glass phase. Accordingly, we conjecture that quantum dynamics inside the glassy region is split into a small MBL region and a large delocalized (but not necessarily ergodic) region. This has implications for the analysis of the performance of quantum adiabatic algorithms.This article is part of the themed issue 'Breakdown of ergodicity in quantum systems: from solids to synthetic matter'.

arXiv:1703.03678•Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences•Oct 30, 2017•13 citations

Many-body localization-delocalization transition in the quantum Sherrington-Kirkpatrick model

Sudip Mukherjee, Sabyasachi Nag, Arti Garg

We analyze the many-body localization- (MBL) to-delocalization transition in the Sherrington-Kirkpatrick (SK) model of Ising spin glass in the presence of a transverse field $\mathrm{\ensuremath{\Gamma}}$. Based on energy-resolved analysis, which is of relevance for a closed quantum system, we show that the quantum SK model has many-body mobility edges separating the MBL phase, which is nonergodic and nonthermal, from the delocalized phase, which is ergodic and thermal. The range of the delocalized regime increases with an increase in the strength of $\mathrm{\ensuremath{\Gamma}}$, and eventually for $\mathrm{\ensuremath{\Gamma}}$ larger than ${\mathrm{\ensuremath{\Gamma}}}_{\text{CP}}$ the entire many-body spectrum is delocalized. We show that the Renyi entropy is almost independent of the system size in the MBL phase while the delocalized phase shows extensive Renyi entropy. We further obtain the spin-glass transition curve in the energy density $\ensuremath{\epsilon}\text{\ensuremath{-}}\mathrm{\ensuremath{\Gamma}}$ plane from the collapse of the eigenstate spin susceptibility. We demonstrate that in most of the parameter regime, the spin-glass transition occurs close to the MBL transition, indicating that the spin-glass phase is nonergodic and nonthermal while the paramagnetic phase is delocalized and thermal.

arXiv:1712.00204•Physical review. B./Physical review. B•Apr 18, 2018•17 citations
We review the numerical studies on the critical behaviour of the quantum Sherrington-Kirkpatrick (SK) spin glass model, which indicate that a quantum critical behaviour is observed up to a low but non-zero value of temperature. We revisit the numerical investigations on the spin glass order parameter distributions, which identify a low temperature along with high transverse field spin glass phase where the order parameter distribution becomes a delta function in the thermodynamic limit indicating the restoration of replica symmetry and ergodic nature of the system. In the remaining spin glass phase associated with high temperature and low transverse field, the observed distribution is broad akin to the Parisi order parameter distribution. This essentially indicates the non-ergodic behaviour of the system. We further discuss the annealing dynamics studies on the quantum SK model. Such investigations reveal the system size independence of annealing time when the annealing paths go through the ergodic spin glass region. Interestingly, when such dynamics are performed in the non-ergodic spin glass phase the annealing time becomes an increasing function of the system size. Spin autocorrelation shows faster relaxation in the ergodic spin glass region compared with that found in the non-ergodic spin glass region. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.
We first review, following our earlier studies, the critical behavior of the quantum Sherrington-Kirkpatrick (SK) model at finite as well as at zero temperatures. Through the analysis of the Binder cumulant we determined the entire phase diagram of the model and from the scaling analysis of the numerical data we obtained the correlation length exponent. For both the critical Binder cumulant and the correlation length exponent, we observed a crossover from classical- to quantum-fluctuation-dominated values at a finite temperature. We studied the behavior of the order parameter distribution of the model in the glass phase (at finite and zero temperatures). Along with a classical-fluctuation-dominated nonergodic region (where the replica symmetry is broken), we also found a quantum-fluctuation-dominated low-temperature ergodic region in the spin glass phase. In this quantum-fluctuation-dominated region, the order parameter distribution has a narrow peak around its most probable value, eventually becoming a delta function in the infinite-system-size limit (indicating replica symmetry restoration or ergodicity in the system). We also found that the annealing time (to reach a very low energy level of the classical SK model) becomes practically system-size-independent when the annealing paths pass through this ergodic region. In contrast, when such paths pass through the nonergodic region, the convergence time grows rapidly with the system size. We present a new study of the autocorrelation of the spins in both ergodic and nonergodic regions. We found a significant increase in the relaxation time (and also a change in the relaxation behavior) in the classical-fluctuation-dominated (nonergodic) region compared with that in the quantum-fluctuation-dominated (ergodic) region of the spin glass phase.
We review our earlier studies on the order parameter distribution of the quantum Sherrington-Kirkpatrick (SK) model. Through Monte Carlo technique, we investigate the behavior of the order parameter distribution at finite temperatures. The zero temperature study of the spin glass order parameter distribution is made by the exact diagonalization method. We find in low-temperature (high-transverse-field) spin glass region, the tail (extended up to zero value of order parameter) and width of the order parameter distribution become zero in thermodynamic limit. Such observations clearly suggest the existence of a low-temperature (high-transverse-field) ergodic region. We also find in high-temperature (low-transverse-field) spin glass phase the order parameter distribution has nonzero value for all values of the order parameter even in infinite system size limit, which essentially indicates the nonergodic behavior of the system. We study the annealing dynamics by the paths which pass through both ergodic and nonergodic spin glass regions. We find the average annealing time becomes system size independent for the paths which pass through the quantum-fluctuation-dominated ergodic spin glass region. In contrast to that, the annealing time becomes strongly system size dependent for annealing down through the classical-fluctuation-dominated nonergodic spin glass region. We investigate the behavior of the spin autocorrelation in the spin glass phase. We observe that the decay rate of autocorrelation towards its equilibrium value is much faster in the ergodic region with respect to the nonergodic region of the spin glass phase.
We explore the behavior of the order parameter distribution of the quantum Sherrington-Kirkpatrick model in the spin glass phase using Monte Carlo technique for the effective Suzuki-Trotter Hamiltonian at finite temperatures and that at zero temperature obtained using the exact diagonalization method. Our numerical results indicate the existence of a low- but finite-temperature quantum-fluctuation-dominated ergodic region along with the classical fluctuation-dominated high-temperature nonergodic region in the spin glass phase of the model. In the ergodic region, the order parameter distribution gets narrower around the most probable value of the order parameter as the system size increases. In the other region, the Parisi order distribution function has nonvanishing value everywhere in the thermodynamic limit, indicating nonergodicity. We also show that the average annealing time for convergence (to a low-energy level of the model, within a small error range) becomes system size independent for annealing down through the (quantum-fluctuation-dominated) ergodic region. It becomes strongly system size dependent for annealing through the nonergodic region. Possible finite-size scaling-type behavior for the extent of the ergodic region is also addressed.
I study in detail the quantum Sherrington-Kirkpatrick (SK) model, i.e., the infinite-range Ising spin glass in a transverse field, by solving numerically the effective one-dimensional model that the quantum SK model can be mapped to in the thermodynamic limit. I find that the replica symmetric solution is unstable down to zero temperature, in contrast to some previous claims, and so there is not only a line of transitions in the (longitudinal) field-temperature plane (the de Almeida-Thouless, AT, line) where replica symmetry is broken, but also a quantum de Almeida-Thouless (QuAT) line in the transverse field-longitudinal field plane at T=0. If the QuAT line also occurs in models with short-range interactions its presence might affect the performance of quantum annealers when solving spin glass-type problems with a bias (i.e., magnetic field).
By considering the quantum dynamics of a transverse-field Ising spin glass on the Bethe lattice, we find the existence of a many-body localized (MBL) region at small transverse field and low temperature. The region is located within the thermodynamic spin glass phase. Accordingly, we conjecture that quantum dynamics inside the glassy region is split into a small MBL region and a large delocalized (but not necessarily ergodic) region. This has implications for the analysis of the performance of quantum adiabatic algorithms.This article is part of the themed issue 'Breakdown of ergodicity in quantum systems: from solids to synthetic matter'.
We analyze the many-body localization- (MBL) to-delocalization transition in the Sherrington-Kirkpatrick (SK) model of Ising spin glass in the presence of a transverse field $\mathrm{\ensuremath{\Gamma}}$. Based on energy-resolved analysis, which is of relevance for a closed quantum system, we show that the quantum SK model has many-body mobility edges separating the MBL phase, which is nonergodic and nonthermal, from the delocalized phase, which is ergodic and thermal. The range of the delocalized regime increases with an increase in the strength of $\mathrm{\ensuremath{\Gamma}}$, and eventually for $\mathrm{\ensuremath{\Gamma}}$ larger than ${\mathrm{\ensuremath{\Gamma}}}_{\text{CP}}$ the entire many-body spectrum is delocalized. We show that the Renyi entropy is almost independent of the system size in the MBL phase while the delocalized phase shows extensive Renyi entropy. We further obtain the spin-glass transition curve in the energy density $\ensuremath{\epsilon}\text{\ensuremath{-}}\mathrm{\ensuremath{\Gamma}}$ plane from the collapse of the eigenstate spin susceptibility. We demonstrate that in most of the parameter regime, the spin-glass transition occurs close to the MBL transition, indicating that the spin-glass phase is nonergodic and nonthermal while the paramagnetic phase is delocalized and thermal.