14 research outputs found
Exact solution of free entropy for matrix-valued geometric Brownian motion with non-commutative matrices via the replica method
Geometric Brownian motion (GBM) is a standard model in stochastic
differential equations. In this study, we consider a matrix-valued GBM with
non-commutative matrices. Introduction of non-commutative matrices into the
matrix-valued GBM makes it difficult to obtain an exact solution because the
existence of noise terms prevents diagonalization. However, we show that the
replica method enables us to overcome this difficulty. We map the time
evolution operator of the matrix-valued GBM with non-commutative matrices into
the partition function of the isotropic Lipkin-Meshkov-Glick model used in
quantum spin systems. Then, solving the eigenvalue problem of the isotropic
Lipkin-Meshkov-Glick model, we obtain an analytical expression of the free
entropy. Numerical simulation is consistent with our analytical result. Thus,
our expression is the exact solution of the free entropy for the matrix-valued
GBM with non-commutative matrices.Comment: 9 pages, 4 figure
Mean-field theory is exact for Ising spin glass models with Kac potential in non-additive limit on Nishimori line
Recently, Mori [Phys. Rev. E 84, 031128 (2011)] has conjectured that the free
energy of Ising spin glass models with the Kac potential in the non-additive
limit, such as the power-law potential in the non-additive regime, is exactly
equal to that of the Sherrington-Kirkpatrick model in the thermodynamic limit.
In this study, we prove that his conjecture is true on the Nishimori line at
any temperature in any dimension. One of the key ingredients of the proof is
the use of the Gibbs-Bogoliubov inequality on the Nishimori line. We also
consider the case in which the probability distribution of the interaction is
symmetric, where his conjecture is true at any temperature in one dimension but
is an open problem in the low-temperature regime in two or more dimensions.Comment: 12pages, 0 figur
Gibbs-Bogoliubov inequality on Nishimori line
The Gibbs-Bogoliubov inequality states that the free energy of a system is
always lower than that calculated by a trial function. In this study, we show
that a counterpart of the Gibbs-Bogoliubov inequality holds on the Nishimori
line for Ising spin-glass models with Gaussian randomness. Our inequality
states that the quenched free energy of a system is always lower than that
calculated using a quenched trial function. The key component of the proof is
the convexity of the pressure function with
respect to the parameters along the Nishimori line, which differs from the
conventional convexity with respect to the inverse temperature. When our
inequality was applied to mean-field models, such as the
Sherrington-Kirkpatrick model and -spin model, the bound coincided with the
replica-symmetric solution indicating that the equality holds.Comment: 9 pages, 0 figur
Anomalous behavior of the energy gap in the one-dimensional quantum XY model
We re-examine the well-studied one dimensional spin-1/2 model to reveal
its nontrivial energy spectrum, in particular the energy gap between the ground
state and the first excited state. In the case of the isotropic model --
the model -- the gap behaves very irregularly as a function of the system
size at a second order transition point. This is in stark contrast to the usual
power-law decay of the gap and is reminiscent of the similar behavior at the
first order phase transition in the infinite-range quantum model. The gap
also shows nontrivial oscillatory behavior for the phase transitions in the
anisotropic model in the incommensurate phase. We observe a close relation
between this anomalous behavior of the gap and the correlation functions. These
results, those for the isotropic case in particular, are important from the
viewpoint of quantum annealing where the efficiency of computation is strongly
affected by the size dependence of the energy gap.Comment: 25 pages, 8 figures. arXiv admin note: substantial text overlap with
arXiv:1501.0292