Chemistry Reference
In-Depth Information
smaller than with other methods. Using this approach, Young and Rieger con-
firmed numerically the analytical result
22,23
that the quantum critical point in
the random transverse-field Ising chain is of exotic infinite-randomness type.
The investigation of quantum phase transitions in disordered systems has
benefited in recent years from the strong-disorder renormalization group that
was introduced originally by Ma, Dasgupta, and Hu.
24
The basic idea of their
method is to successively integrate out local high-energy degrees of freedom in
perturbation theory. The quality of this method improves with increasing dis-
order strength, in contrast to many other techniques; and it becomes asympto-
tically exact at infinite-randomness critical points (where the effective disorder
strength diverges). This approach has been applied to a variety of classical and
quantum disordered systems, ranging from quantum spin chains to chemical
reaction-diffusion models with disorder. A recent review can be found in
Ref. 134. In one space dimension, the strong-disorder renormalization group
can often be solved analytically in closed form, as is the case for the random
transverse-field Ising chain
22,23
1
2
antiferromagnetic
Heisenberg chain as examples.
24,135
In higher dimensions, or for more compli-
cated Hamiltonians, the method can only be implemented numerically. For
instance, Montrunich et al.
19
studied the quantum phase transition in the
two-dimensional random transverse-field Ising model. In analogy with the
one-dimensional case,
22,23
they found an infinite randomness critical point,
but the critical exponents take different values. Schehr and Rieger
136
studied
the interplay between dissipation, quantum fluctuations, and disorder in the
random transverse-field Ising chain coupled to dissipative baths. In agreement
with theoretical predictions,
30,31
they found that the dissipation freezes the
quantum dynamics of large, locally ordered clusters, which then dominate
the low-energy behavior. This leads to a smearing of the quantum phase tran-
sition.
30
Let us also mention a class of methods that are not numerically exact but
have greatly fostered our understanding of quantum many-particle systems:
the dynamical mean-field theory (DMFT). Its development started with the
pioneering work of Metzner and Vollhardt
137
on the Hubbard model in infi-
nite dimensions. The basic idea behind this approach is a natural generaliza-
tion of the classical mean-field theories to quantum problems: The quantum
many-particle Hamiltonian is reduced to a quantum impurity problem
coupled to one or several self-consistent baths.
138
This impurity problem is
then solved self-consistently, either by approximate analytical methods or by
numerical methods. In contrast to classical mean-field theories such as
Hartree-Fock, the DMFT contains the full local quantum dynamics. (This
means that the DMFT suppresses spatial fluctuations but keeps the local imag-
inary time fluctuations.) DMFT methods have now been applied to a wide
variety of problems ranging from model Hamiltonians of strongly correlated
electrons to complex materials. For instance, the DMFT was instrumental in
understanding the Mott metal-insulator phase transition in the Hubbard
or
the random
S
ΒΌ
