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Physical Review B

Grand canonical finite size numerical approaches in one and two dimensions: Real space energy renormalization and edge state generation

Chisa Hotta, Satoshi Nishimoto, Naokazu Shibata

Published Mar 21, 2013• 55 citations
Publisher

Abstract

The grand canonical numerical analysis recently developed for quantum many-body systems on a finite cluster [C. Hotta and N. Shibata, Phys. Rev. B 86, 041108(R) (2012)] is the technique to efficiently obtain the physical quantities in an applied field. There, the observables are the continuous and real functions of fields, mimicking their thermodynamic limit, even when a small cluster is adopted. We develop a theory to explain the mechanism of this analysis based on the deformation of the Hamiltonian. The deformation spatially scales down the energy unit from the system center toward zero at the open edge sites, which introduces the renormalization of the energy levels in a way reminiscent of Wilson's numerical renormalization group. However, compared to Wilson's case, our deformation generates a number of far well-localized edge states near the chemical potential level, which are connected via a very small quantum fluctuation in $k$ space with the ``bulk'' states which spread at the center of the system. As a response to the applied field, the particles on the cluster are self-organized to tune the particle number of the bulk states to their thermodynamic limit by using the ``edges'' as a buffer. We demonstrate the present analysis in two-dimensional quantum spin systems on square and triangular lattices, and determine the smooth magnetization curve with a clear $\frac{1}{3}$ plateau structure in the latter.

DOI
10.1103/physrevb.87.115128

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The grand canonical numerical analysis recently developed for quantum many-body systems on a finite cluster [C. Hotta and N. Shibata, Phys. Rev. B 86, 041108(R) (2012)] is the technique to efficiently obtain the physical quantities in an applied field. There, the observables are the continuous and real functions of fields, mimicking their thermodynamic limit, even when a small cluster is adopted. We develop a theory to explain the mechanism of this analysis based on the deformation of the Hamiltonian. The deformation spatially scales down the energy unit from the system center toward zero at the open edge sites, which introduces the renormalization of the energy levels in a way reminiscent of Wilson's numerical renormalization group. However, compared to Wilson's case, our deformation generates a number of far well-localized edge states near the chemical potential level, which are connected via a very small quantum fluctuation in kkk space with the bulk'' states which spread at the center of the system. As a response to the applied field, the particles on the cluster are self-organized to tune the particle number of the bulk states to their thermodynamic limit by using the edges'' as a buffer. We demonstrate the present analysis in two-dimensional quantum spin systems on square and triangular lattices, and determine the smooth magnetization curve with a clear 13\frac{1}{3}31​ plateau structure in the latter.
We exploit a prescription to observe directly the physical properties of the thermodynamic limit in a continuously applied field in one-dimensional quantum finite lattice systems. By systematically scaling down the energy of the Hamiltonian of the open system from center toward both ends, one could adopt the edge sites with a negligibly small energy scale as the grand canonical small particle bath, and equilibrium states with noninteger arbitrary conserved numbers, e.g., electron numbers or sz{s}_{z}sz​, are realized in the main part of the system. This will enable the evaluation of response functions under a continuously varying external field in a small lattice without any fine-tuning or scaling of parameters while keeping the standard numerical accuracy. Demonstrations are given on quantum spin systems and on a Hubbard model by the density-matrix renormalization group.
Real-space renormalization-group techniques for quantum systems can be divided into two basic categories: those capable of representing correlations following a simple boundary (or area) law, and those which are not. I discuss the scaling of the accuracy of gapped systems in the latter case and analyze the resultant spatial anisotropy. It is apparent that particular points in the system, that are somehow ``central'' in the renormalization, have local quantities that are much closer to the exact results in the thermodynamic limit than the system-wide average. Numerical results from the tree-tensor network and tensor renormalization-group approaches for the two-dimensional transverse-field Ising model and three-dimensional classical Ising model, respectively, clearly demonstrate this effect.
A zero-temperature real-space renormalization-group method for quantum systems is applied to the S=12XYS=\frac{1}{2}\mathrm{XY}S=21​XY model with a ZZZ field with the Hamiltonian $H=\ensuremath{-}\frac{1}{2}J\ensuremath{\Sigma}{〈i, j〉}^{}[(1+\ensuremath{\gamma}){S}_{i}^{x}{S}_{j}^{x}+(1\ensuremath{-}\ensuremath{\gamma}){S}_{i}^{y}{S}_{j}^{y}]\ensuremath{-}h\ensuremath{\Sigma}{i}^{}{S}_{i}^{z}$ on quadratic, triangular, and hexagonal lattices in two dimensions (2D) and on a cubic lattice in 3D. In all the isotropic ($\ensuremath{\gamma}=0$) 2D cases the method gives no order in the ground state ∣0〉|0〉∣0〉 but changes in the correlation functions at ${(\frac{h}{J})}_{c}=\frac{1}{2}{z}^{\ensuremath{'}}$ [${z}^{\ensuremath{'}}$ is the number of nearest neighbors (NN)] are observed. An argument is presented that at (hJ)c{(\frac{h}{J})}_{c}(Jh​)c​ the method becomes exact in the limit of infinite-size spin blocks. It is suggested that the correlation functions $〈0|{{S}_{0}}^{\ensuremath{\alpha}}{{S}_{R}}^{\ensuremath{\alpha}}|0〉$ ($\ensuremath{\alpha}=x, y$) for large RRR have a power-law behavior $\ensuremath{\sim}{R}^{\ensuremath{-}\ensuremath{\eta}}$ for 0< ∣hJ∣<∣(hJ)c∣0<~|\frac{h}{J}|<|{(\frac{h}{J})}_{c}|0< ∣Jh​∣<∣(Jh​)c​∣ with $\ensuremath{\eta}\ensuremath{\approx}1.5$ for ∣hJ∣<(hJ)c∣|\frac{h}{J}|<{(\frac{h}{J})}_{c}|∣Jh​∣<(Jh​)c​∣ and $\ensuremath{\eta}=2$ at (hJ)c{(\frac{h}{J})}_{c}(Jh​)c​. The dynamical exponent zzz is obtained to be \ensuremath{\approx} 0.3 for $|\frac{h}{J}|<\frac{1}{2}{z}^{\ensuremath{'}}$ and at (hJ)c{(\frac{h}{J})}_{c}(Jh​)c​ close to 2. An indirect method of calculating the ground-state energy is developed, and the results are compared with the known estimates. For $\ensuremath{\gamma}\ensuremath{\ne}0$ and h=0h=0h=0 the $\ensuremath{\gamma}$ dependences of the order in the ground state ${\ensuremath{\mu}}^{x}(\ensuremath{\gamma})$ is calculated to be ${\ensuremath{\mu}}^{x}\ensuremath{\sim}{\ensuremath{\gamma}}^{0.21}$ for $\ensuremath{\gamma}\ensuremath{\rightarrow}0$ and h=0h=0h=0. In a 3D cubic lattice ($\ensuremath{\gamma}=0$) the analysis of recursion relations indicates the existence of order, in accord with recent exact theorems.
We present a quantum cluster solver for the spin- S Heisenberg model on a two-dimensional lattice. The formalism is based on the real-space renormalization procedure and uses the lattice point group-theoretical analysis and non-Abelian SU (2) spin symmetry technique. The exact diagonalization procedure is used twice at each renormalization group step. The method is applied to the spin-half antiferromagnet on a square lattice, and a calculation of local observables is demonstrated. A symmetry-based truncation procedure is suggested and verified numerically.
A formulation of numerical real-space renormalization groups for quantum many-body problems is presented and several algorithms utilizing this formulation are outlined. The methods are presented and demonstrated using S=1/2 and S=1 Heisenberg chains as test cases. The key idea of the formulation is that rather than keep the lowest-lying eigenstates of the Hamiltonian in forming a new effective Hamiltonian of a block of sites, one should keep the most significant eigenstates of the block density matrix, obtained from diagonalizing the Hamiltonian of a larger section of the lattice which includes the block. This approach is much more accurate than the standard approach; for example, energies for the S=1 Heisenberg chain can be obtained to an accuracy of at least ${10}^{\mathrm{\ensuremath{-}}9}$. The method can be applied to almost any one-dimensional quantum lattice system, and can provide a wide variety of static properties.
Abstract We study the temperature dependence of the thermodynamic properties of spin-1/2 antiferromagnets on two-dimensional lattices. Our analysis employs the sine-square deformation (SSD), in which a real-space envelope function is applied to the Hamiltonian so that the local energy scale is smoothly reduced to zero at the system boundaries. The quantum eigenstates of the SSD Hamiltonian exhibit bulk-like behavior near the system center, effectively mimicking the thermodynamic limit even in small finite-size calculations. Using these fictitious bulk states, we compute the energy density, specific heat, and magnetic susceptibility as functions of temperature. We find that both the triangular- and kagome-lattice antiferromagnets show either a shoulder or a pronounced double-peak structure in the low-temperature specific heat, whereas the kagome case particularly shows a strong enhancement of magnetic susceptibility down to the lowest temperature range. These direct comparisons, together with the square-lattice and one-dimensional cases, reveal that although both frustrated systems retain a substantial amount of entropy, the low-energy excitations below ∼ 0.5 J ∼0.5J\sim 0.5J∼0.5J of the kagome lattice are predominantly governed by the magnetic excited states, whereas not much for the triangular lattice.