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A simple, general and practically exact method is developed for the equilibrium properties of the macroscopic physical systems with translational symmetry. Applied to the Ising model in two and three dimension, a modest calculation gives the spontaneous magnetization and the specific heat to less than 1% error.PACS numbers: 05.50.+q, 64.60.Cn, 75.10.Hk To calculate the equilibrium properties of a given physical system will probably be the most basic theoretical task. In 1944, Onsager [1] calculated the partition function of the two dimensional Ising model analytically and demonstrated the power of the exact solution, followed by the exact spontaneous magnetization by Yang [2]. Since then some remarkable progresses have been made in two-dimensional classical and one-dimensional quantum systems analytically [3][4][5][6], and by numerical methods such as DMRG (density matrix renormalization group) [7]. However, reflecting the three-dimensionality of our world, much needed knowledge is for three dimension, and several thousand papers have been said written on the three dimensional Ising model. Among others, a tremendous effort has been paid to the precise determination of the critical temperature T c in association with the concept of universality near the critical point [8]. Owing to the concerted effort of RG-finite size scaling-Monte Carlo (MC) method, T c is known as precise as 4.5114(1) [9], as well as the spontaneous magnetization [10]. The effort of high and low temperature expansions by the diagrammatic method for T c is also remarkable [11]. Yet another noteworthy recent progress is along the line of DMRG [12].We here present a new method for the equilibrium properties of the macroscopic physical systems with translational symmetry. Our method is purely algebraic, works directly on a ∞ x ∞ x ∞ lattice and, other than seeking a convergence in entanglement space (see below), it does not invoke any other notions such as numerical RG, nor make any approximations. We have calculated the spontaneous magnetization and the specific heat in two and three dimensional Ising models to less than 1% error. Note that without the help of RG, the 1% precision is what one can typically expect from exact methods such as Bethe Ansatz [5,6]. The new method is simple, general and computationally efficient. In fact, our 30 minutes calculation already gives fairly precise results as demonstrated below.Let us consider the ferromagnetic Ising model on a square or simple cubic lattice,where σ i takes + or -1 and the summation is over the nearest neighbor pairs. The partition function is given by Z = Tr[exp(−βH)], where Tr means to take trace over the 2 N spin configurations with N being the number of lattice sites and β = 1/kT . We follow the following steps for the two dimensional case.First, note that the local pair-density matrix can be written as exp(βσ i σ j ) = cosh(β) + sinh(β)σ i σ j . This is the simplest case of more general statement that any local density matrix when regarded as a real symmetric matrix can be written b...

The entanglement perturbation theory is developed to calculate the excitation spectrum in one dimension. Applied to the spin-1 2 antiferromagnetic Heisenberg model, it reproduces the des CloiseauxPearson Bethe ansatz result. As for spin-1, the spin-triplet magnon spectrum has been determined for the first time for the entire Brillouin zone, including the Haldane gap at k = π.PACS numbers: 71.10. Li, 02.90.+p, 71.10.Fd, 75.10.Jm The importance of elementary excitations in condensed matter systems may be best understood in the superfluid 4 He. The Tisza two-fluid model with the experimentally found phonon-roton spectrum explains fundamental properties of the superfluid 4 He [1]. Feynman's effort then to explain the roton spectrum is well known [2]. From the theorem of Bloch-Floquet, the elementary excitation with momentum k for a translationally invariant Hamiltonian H is written aswhere |g is the ground state and the summation over l extends over the entire lattice sites. The O l is a local cluster operator to be determined for a given Hamiltonian.In spite of a simplicity and validity of the expression (1), not much progress has been made along this line since the days of Feynman. The Heisenberg antiferromagnet (HA) described by the Hamiltonianis probably the best studied system concerning the excitation spectrum. In particular, Haldane conjectured in 1983 that the half-odd integer and integer spins might behave essentially differently [3], which together with a field theoretic prediction of Affleck [4] for a logarithmic correction to the power-law behavior in the spin-spin correlation function in the spin-1 2 case, triggered an intensive study of HA ranging from the exact diagonalization [5,6] and Monte Carlo [7,8] to DMRG (density matrix renormalization group) [9][10][11]. These studies along with the Bethe ansatz solution for the spin-1 2 case [12] lead to a confirmation of the both claims. Concerning the elementary excitation for the entire Brillouin zone, however, Takahashi's two attempts following the Feynman variational method for 4 He and a projector-Monte Carlo method were the only studies [13,14]. And none of the previous studies gave a serious consideration to the expression (1).In this Letter, we analyze (1) exactly for the HA (2) with periodic boundary conditions by the recently developed entanglement perturbation theory (EPT). EPT is a novel many-body method which takes into account correlations systematically. Its mathematical implementation is singular value decomposition (SVD), intuitively divide and conquer. EPT has addressed so far classical statistical mechanics [15], 1D quantum ground states [16] and 2D quantum ground states [17]. We here address the elementary excitation in one dimension. By EPT, we are not only free from a negative sign problem which is inherent to MC for quantum spins and fermions, but can also handle an order of magnitude larger systems than MC and DMRG. The key of the success lies in our ability of calculating the ground state |g precisely and most importantly in an u...

A simple, general and practically exact method is developed to calculate the ground states of 1D macroscopic quantum systems with translational symmetry. Applied to the Hubbard model, a modest calculation reproduces the Bethe Ansatz results.PACS numbers: 71.10. Fd, 71.27.+a, 75.10.Lp Since the very beginning of the quantum theory, to solve the Schrödinger equation for macroscopic quantum systems has been one of the main tasks of theoretical physics. It would not be an exaggeration to say that, due to lack of such methods, a considerable effort of theoretical physicists has been devoted to the development of a variety of perturbative and approximate methods and numerical simulations. But a desire for powerful non-perturbative methods has grown stronger over the last couple of decades with the list of phenomena played by strongly correlated electrons getting longer, particularly since the discovery of high temperature superconductivity in copper oxides [1]. While we have seen a considerable progress in rigorous treatment of quantum 1D and classical 2D systems over the last several decades [2-9], these rigorous methods are not flexible enough to solve non-integrable models in one dimension, nor, most probably, generalizable to higher dimensions. On the other hand, the method of NRG (numerical renormalization group), particularly DMRG (density matrix RG) has seen a remarkable success first in quantum 1D systems [10] and then in finite Fermi systems, competing well with the conventional quantum chemistry calculations [11]. More recently, the notion of entanglement from quantum information theory [12] helped a further progress in NRG towards the finite temperature as well as dynamical quantities [13][14][15].In a recent article, we have developed a simple, general and practically exact method to calculate statistical mechanical properties of macroscopic classical systems with translational symmetry up to three dimensions [16]. We here extend this method to solve the Schrödinger equation for 1D quantum ground states with translational symmetry. As a benchmark model for this development, we consider the Hubbard model. Just like our recent work on the 3D Ising model, our method is purely algebraic and other than seeking a convergence in entanglement space, it does not employ any other notions such as NRG, nor make any approximations. Our results for the ground state energy and the local magnetic moment in the 1D Hubbard model agree with the known exact results by Bethe Ansatz [8,17]. An important difference of the present method from the Bethe Ansatz, however, should be emphasized: the new method is not rigorous but mathematically much simpler, general and therefore readily applicable to any quantum spins, fermions and bosons. This is a reflection of the fact that our recent method for the Ising model is applicable to any classical statistical systems with translational symmetry. Yet another but probably the most significant remark here is that the success in 1D Hubbard model should constitute an essential ingredient ...

A recently developed numerical method, entanglement perturbation theory (EPT), is used to study the antiferromagnetic Heisenberg spin chains with z-axis anisotropy λ and magnetic field B. To demonstrate the accuracy, we first apply EPT to the isotropic spin-1 2 antiferromagnetic Heisenberg model, and find that EPT successfully reproduces the exact Bethe Ansatz results for the ground state energy, the local magnetization, and the spin correlation functions (Bethe ansatz result is available for the first 7 lattice separations). In particular, EPT confirms for the first time the asymptotic behavior of the spin correlation functions predicted by the conformal field theory, which realizes only for lattice separations larger than 1000. Next, turning on the z-axis anisotropy and the magnetic field, the 2-spin and 4-spin correlation functions are calculated, and the results are compared with those obtained by Bosonization and density matrix renormalization group methods. Finally, for the spin-1 antiferromagnetic Heisenberg model, the ground state phase diagram in λ space is determined with help of the Roomany-Wyld RG finite-size-scaling. The results are in good agreement with those obtained by the level-spectroscopy method.