We study quantum SU( $M$ ) spins with all-to-all and random Heisenberg exchange interactions of root-mean-square strength $J$ . The $M\ensuremath{\rightarrow}\ensuremath{\infty}$ model has a quantum spin liquid ground state with the spinons obeying the equations of the Sachdev-Ye-Kitaev (SYK) model. Numerical studies of the SU(2) model with $S=1/2$ spins show spin glass order in the ground state, but also display SYK spin liquid behavior in the intermediate frequency spin spectrum. We employ a $1/M$ expansion to describe the crossover from fractionalized fermionic spinons to a confining spin glass state with weak spin glass order ${q}_{EA}$ . The SYK spin liquid behavior persists down to a frequency ${\ensuremath{\omega}}_{*}\ensuremath{\sim}J{q}_{EA}$, and for $\ensuremath{\omega}<{\ensuremath{\omega}}_{*}$, the spin spectral density is linear in $\ensuremath{\omega}$, thus quenching the extensive zero temperature entropy of the spin liquid. The linear $\ensuremath{\omega}$ spectrum is qualitatively similar to that obtained earlier using bosonic spinons for large ${q}_{EA}$ . We argue that the extensive SYK spin liquid entropy is transformed as $T\ensuremath{\rightarrow}0$ to an extensive complexity of the spin glass state. We comment on holographic connections of the confinement transition to the fragmentation of black holes with ${\mathrm{AdS}}_{2}$ horizons.We study quantum SU($M$) spins with all-to-all and random Heisenberg exchange interactions of root-mean-square strength $J$. The $M\ensuremath{\rightarrow}\ensuremath{\infty}$ model has a quantum spin liquid ground state with the spinons obeying the equations of the Sachdev-Ye-Kitaev (SYK) model. Numerical studies of the SU(2) model with $S=1/2$ spins show spin glass order in the ground state, but also display SYK spin liquid behavior in the intermediate frequency spin spectrum. We employ a $1/M$ expansion to describe the crossover from fractionalized fermionic spinons to a confining spin glass state with weak spin glass order ${q}_{EA}$. The SYK spin liquid behavior persists down to a frequency ${\ensuremath{\omega}}_{*}\ensuremath{\sim}J{q}_{EA}$, and for $\ensuremath{\omega}<{\ensuremath{\omega}}_{*}$, the spin spectral density is linear in $\ensuremath{\omega}$, thus quenching the extensive zero temperature entropy of the spin liquid. The linear $\ensuremath{\omega}$ spectrum is qualitatively similar to that obtained earlier using bosonic spinons for large ${q}_{EA}$. We argue that the extensive SYK spin liquid entropy is transformed as $T\ensuremath{\rightarrow}0$ to an extensive complexity of the spin glass state. We comment on holographic connections of the confinement transition to the fragmentation of black holes with ${\mathrm{AdS}}_{2}$ horizons.

arXiv:2110.00007Physical review. B./Physical review. BFeb 14, 202220 citations

Quantum spin glasses

Subir Sachdev A

Cambridge University Press eBooksApr 7, 20110 citations

QUANTUM SPIN GLASSES

R. N. Bhatt

Series on directions in condensed matter physicsDec 1, 19974 citations

Spin-liquid and spin-glass behavior in quantum spin models with all-to-all p -spin interactions

Shusei Wadashima, Yukitoshi Motome

Spin-liquid and spin-glass states represent two distinct phases of disordered quantum spin systems. These states are, in principle, distinguished by quantum-entangled fluctuations and spin freezing, but identifying each state and characterizing the transition between them remains challenging. Here, we systematically explore the relationship between the spin-liquid and spin-glass states using a model with all-to-all random interactions among $p$ spins, which interpolates between the Ising-like one-component, XY-like two-component, and isotropic three-component cases. By analyzing the system-size $N$ dependence of the Edwards-Anderson order parameter and the density of states, we identify the transition from the spin liquid to the spin glass for various values of $p$ . We show that the phase diagrams for different $p$ can be unified through a scaling with $N/{p}^{2}$ , revealing that increasing anisotropy in the interactions systematically suppresses the spin-liquid phase and extends the spin-glass regime. Furthermore, we examine the competition between multiple-spin interactions and anisotropy under an external magnetic field in the isotropic case, and find that the spin-liquid phase transitions into the spin-glass phase before entering a quantum paramagnetic phase. Our findings provide insights into quantum disordered phases and the transitions between them.Spin-liquid and spin-glass states represent two distinct phases of disordered quantum spin systems. These states are, in principle, distinguished by quantum-entangled fluctuations and spin freezing, but identifying each state and characterizing the transition between them remains challenging. Here, we systematically explore the relationship between the spin-liquid and spin-glass states using a model with all-to-all random interactions among $p$ spins, which interpolates between the Ising-like one-component, XY-like two-component, and isotropic three-component cases. By analyzing the system-size $N$ dependence of the Edwards-Anderson order parameter and the density of states, we identify the transition from the spin liquid to the spin glass for various values of $p$. We show that the phase diagrams for different $p$ can be unified through a scaling with $N/{p}^{2}$, revealing that increasing anisotropy in the interactions systematically suppresses the spin-liquid phase and extends the spin-glass regime. Furthermore, we examine the competition between multiple-spin interactions and anisotropy under an external magnetic field in the isotropic case, and find that the spin-liquid phase transitions into the spin-glass phase before entering a quantum paramagnetic phase. Our findings provide insights into quantum disordered phases and the transitions between them.

arXiv:2508.08154Physical review. B./Physical review. BNov 4, 20250 citations

Quantum Spin Glasses

Heiko Rieger

Lecture notes in physicsNov 16, 20050 citations

We study quantum SU(

M

) spins with all-to-all and random Heisenberg exchange interactions of root-mean-square strength

J

. The $M\ensuremath{\rightarrow}\ensuremath{\infty}$ model has a quantum spin liquid ground state with the spinons obeying the equations of the Sachdev-Ye-Kitaev (SYK) model. Numerical studies of the SU(2) model with

S=1/2

spins show spin glass order in the ground state, but also display SYK spin liquid behavior in the intermediate frequency spin spectrum. We employ a

1/M

expansion to describe the crossover from fractionalized fermionic spinons to a confining spin glass state with weak spin glass order

{q}_{EA}

. The SYK spin liquid behavior persists down to a frequency ${\ensuremath{\omega}}_{*}\ensuremath{\sim}J{q}_{EA}$, and for $\ensuremath{\omega}<{\ensuremath{\omega}}_{*}$, the spin spectral density is linear in $\ensuremath{\omega}$, thus quenching the extensive zero temperature entropy of the spin liquid. The linear $\ensuremath{\omega}$ spectrum is qualitatively similar to that obtained earlier using bosonic spinons for large

{q}_{EA}

. We argue that the extensive SYK spin liquid entropy is transformed as $T\ensuremath{\rightarrow}0$ to an extensive complexity of the spin glass state. We comment on holographic connections of the confinement transition to the fragmentation of black holes with

{\mathrm{AdS}}_{2}

horizons.

p

spins, which interpolates between the Ising-like one-component, XY-like two-component, and isotropic three-component cases. By analyzing the system-size

N

dependence of the Edwards-Anderson order parameter and the density of states, we identify the transition from the spin liquid to the spin glass for various values of

p

. We show that the phase diagrams for different

p

can be unified through a scaling with

N/{p}^{2}

, revealing that increasing anisotropy in the interactions systematically suppresses the spin-liquid phase and extends the spin-glass regime. Furthermore, we examine the competition between multiple-spin interactions and anisotropy under an external magnetic field in the isotropic case, and find that the spin-liquid phase transitions into the spin-glass phase before entering a quantum paramagnetic phase. Our findings provide insights into quantum disordered phases and the transitions between them.