Quantum-Number Projected Path-Integral
Renormalization Group method
Symmetry
plays a crucial role in understanding quantum many-body systems. For example,
the Hubbard model preserves total spin, total momentum, and some geometrical symmetries on a lattice, by which we can classify low-energy
eigenstates and can elucidate the nature of
low-energy phenomena like excitation spectra and spectroscopic properties.
For
strongly correlated electron systems, numerical approaches are an indispensable
tool for their studies and various methods such as quantum Monte Carlo methods
and Density Matrix Renormalization Group method have been presented, while
except exact diagonalization method whose tractable
system size is severely limited, these methods cannot fully take an advantage
of symmetry and excitation spectra have not been well explored.
Recently we have developed
a quantum-number projection technique [1], by which advantage of symmetry can
be fully extracted. For instance, we can handle wave functions with definite
and exact symmetries and lowest energy state with different symmetries can be
treated in the same footing of ground state. Moreover we have presented its
implementation [1] to the path-integral renormalization group (PIRG) method [2]
and succeeded in extremely enlarging the feasibility of the PIRG.
The quantum-number projection method can pick up a
component with required symmetries from symmetry broken wave functions. In the
Hubbard-type models, symmetries such as spin and momentum have a significant role in the low-energy states,
while explicit construction of symmetry imposed wave function is quite
complicated for a large number of electrons. However, the quantum-number
projection technique enables us to easily handle symmetry-imposed wave function
in a compact and numerically tractable form by integral or summation [1]. For
instance, for spin symmetry, we can consider spin projection, which is
represented by one-dimensional integral over Euler's angle in spin space. The
present spin projection is a counterpart of angular momentum projection, which
is often used in nuclear many-body problem. The momentum projection is also
simply given from the superposition of spatially translated basis functions.
Geometrical symmetry on a lattice such as the inversion and rotation symmetries
is also considered in the similar way. By this method we can exactly treat spin,
momentum and other quantum numbers to investigate the Hubbard-type
models.
Moreover this method is
well harmonized with the PIRG [2]. In this method, the wave function
is expressed by a linear combination of L basis states which are deliberately
optimized by the auxiliary-field Quantum Monte Carlo technique [2], while the
symmetries are not retained in each basis state in general because the numerically
manageable number of the basis states, L, is limited. Therefore
by applying the quantum-number projection onto each basis state, we
can implement it into the PIRG. Consequently we can exactly treat the symmetry
and extract the state with specified quantum numbers by the PIRG. Concerning
its procedure for quantum-number projection and PIRG process, we have
proposed two ways of implementing quantum-number projection into the PIRG [1].
One way is to carry out quantum-number projection afterwards for the already
obtained PIRG wave function (PIRG+QP). Other is to carry out the PIRG by using
quantum-number projected basis states (QP-PIRG). By both methods, the ground
state can be efficiently extracted by specifying the quantum number with higher
accuracy than the PIRG without projection.
Comparing
these two methods, the latter method can give more accurate eigenenergy
than the former does.
By this extension of
the PIRG, we have made following essential progresses: (1)
Precision of the numerical calculation is substantially improved. (2) The
quantum number of the ground state is exactly determined. (3) The extended PIRG
by the quantum-number projection can handle excited states and their
spectroscopic properties in addition to the ground state.
As a numerical
demonstration, we investigate the standard Hubbard model on two-dimensional
square lattice. We can obtain an excited state with S=1 and k=(p, p) in addition to the
ground state with S=0 and k=(0,0) for the 6x6 half-filled lattice with U/t=4 as
shown in Fig. 1. The ground state energy by the QP-PIRG is very close to one by
the
Recently by
using the PIRG method, the non-magnetic insulator (NMI) phase has been found
near the Mott transition for relatively large t' [4].
As the present
extended PIRG can explicitly handle quantum numbers, it will be useful to
clarify its nature. It is also worth noting that the present quantum-number
projection technique and the PIRG method work well irrespective of the details
of the considered system. Therefore the present approach will work for other
systems in addition to Hubbard-type models.
References
[1] T. Mizusaki and M.Imada, Phys. Rev. B69, 125110 (2004).
[2] M. Imada and T. Kashima, J.
Phys. Soc. Jpn. 69, 2723 (2000);
T. Kashima and M. Imada,
J. Phys. Soc. Jpn. 70, 2287 (2001).
[3] M. Imada, T. Mizusaki,
[4] T. Kashima and M. Imada, J. Phys. Soc. Jpn., 70,3052 (2001); H. Morita,
Y. Noda and
M. Imada, Phys. Rev. Lett.,
89, 176803 (2002).
Authors
Takahiro Mizusaki(a) and Masatoshi Imada
(a)
Fig. 1:
We demonstrate
the efficiency of the QP-PIRG [1] in the case of the 2D half-filled Hubbard
model with 6 by 6 square lattice and the periodic boundary condition. The
parameters are at t=1, t'=0 and U=4. The energy is obtained by the extrapolation
of the energy to the zero energy variance [2]. The extrapolations of the ground
state (S=0, k=(0,0)) and first excited state (S=1,k=(p,p))
energies are shown. Although exact ground state energy is not available, the
ground-state energy of