Quantum phase transitions in random spin systems

After reviewing the basics of the cavity method in classical systems, we show how its quantum version, with some appropriate approximation scheme, can be used to study a system of spins with random ferromagnetic interactions and a random transverse field. The quantum cavity equations describing the ferromagnetic-paramagnetic phase transition can be transformed into the well-known problem of a classical directed polymer in a random medium. The glass transition of this polymer problem translates into the existence of a 'Griffiths phase' close to the quantum phase transition of the quantum spin problem, where the physics is dominated by rare events. The physical behaviour of random transverse-field ferromagnets on the Bethe lattice is found to be very similar to that found in finite-dimensional systems, and the quantum cavity method gets back the known exact results of the one-dimensional problem.After reviewing the basics of the cavity method in classical systems, we show how its quantum version, with some appropriate approximation scheme, can be used to study a system of spins with random ferromagnetic interactions and a random transverse field. The quantum cavity equations describing the ferromagnetic-paramagnetic phase transition can be transformed into the well-known problem of a classical directed polymer in a random medium. The glass transition of this polymer problem translates into the existence of a 'Griffiths phase' close to the quantum phase transition of the quantum spin problem, where the physics is dominated by rare events. The physical behaviour of random transverse-field ferromagnets on the Bethe lattice is found to be very similar to that found in finite-dimensional systems, and the quantum cavity method gets back the known exact results of the one-dimensional problem.

arXiv:1009.3725Journal of Statistical Mechanics Theory and ExperimentJan 20, 201129 citations

Quantum Phase Transitions in Spin-Lattice Systems

MANFRED L. RISTIG, Khaled Abu Qasem, DAMIAN J. J. FARNELL

Jan 1, 20010 citations

Effect of Random Fluctuations on Quantum Spin-Glass Transitions at Zero Temperature

Kazutaka Takahashi, Yoshiki Matsuda

We study the effects of random fluctuations on quantum phase transitions by the energy gap analysis. For the infinite-ranged spin-glass models with a transverse field, we find that a strong sample-to-sample fluctuation effect leads to broad distributions of the energy gap. As a result, the linear, spin-glass, and nonlinear susceptibilities behave differently from each other. The power-law tail of the distribution implies a quantum Griffiths-like effect that could be observed in various random quantum systems. We also discuss the mechanisms of the phase transition in terms of the energy gap by comparing the Sherrington-Kirkpatrick model and random energy model, which demonstrate the difference between the continuous and discontinuous phase transitions.We study the effects of random fluctuations on quantum phase transitions by the energy gap analysis. For the infinite-ranged spin-glass models with a transverse field, we find that a strong sample-to-sample fluctuation effect leads to broad distributions of the energy gap. As a result, the linear, spin-glass, and nonlinear susceptibilities behave differently from each other. The power-law tail of the distribution implies a quantum Griffiths-like effect that could be observed in various random quantum systems. We also discuss the mechanisms of the phase transition in terms of the energy gap by comparing the Sherrington-Kirkpatrick model and random energy model, which demonstrate the difference between the continuous and discontinuous phase transitions.

arXiv:0911.0995Journal of the Physical Society of JapanApr 12, 20106 citations

Spin reduction transition in spin- 32 random Heisenberg chains

Gil Refael, Stefan K. Kehrein, Daniel S. Fisher

Random spin- $\frac{3}{2}$ antiferromagnetic Heisenberg chains are investigated using an asymptotically exact renormalization group. Randomness is found to induce a quantum phase transition between two random-singlet phases. In the strong randomness phase the effective spins at low energies are ${S}_{\mathrm{eff}}=\frac{3}{2},$ while in the weak randomness phase the effective spins are ${S}_{\mathrm{eff}}=\frac{1}{2}.$ Separating them is a quantum critical point near which there is a nontrivial mixture of spin- $\frac{1}{2},$ spin- $1,$ and spin- $\frac{3}{2}$ effective spins at low temperatures.Random spin-$\frac{3}{2}$ antiferromagnetic Heisenberg chains are investigated using an asymptotically exact renormalization group. Randomness is found to induce a quantum phase transition between two random-singlet phases. In the strong randomness phase the effective spins at low energies are ${S}_{\mathrm{eff}}=\frac{3}{2},$ while in the weak randomness phase the effective spins are ${S}_{\mathrm{eff}}=\frac{1}{2}.$ Separating them is a quantum critical point near which there is a nontrivial mixture of spin-$\frac{1}{2},$ spin-$1,$ and spin-$\frac{3}{2}$ effective spins at low temperatures.

arXiv:cond-mat/0111295Physical review. B, Condensed matterAug 2, 200251 citations

Random spin-

\frac{3}{2}

antiferromagnetic Heisenberg chains are investigated using an asymptotically exact renormalization group. Randomness is found to induce a quantum phase transition between two random-singlet phases. In the strong randomness phase the effective spins at low energies are

{S}_{\mathrm{eff}}=\frac{3}{2},

while in the weak randomness phase the effective spins are

{S}_{\mathrm{eff}}=\frac{1}{2}.

Separating them is a quantum critical point near which there is a nontrivial mixture of spin-

\frac{1}{2},

spin-

1,

and spin-

\frac{3}{2}

effective spins at low temperatures.