Spherical Deformation for One-dimensional Quantum Systems

Transcription

1 1 Spherical Deformation for One-dimensional Quantum Systems Andrej Gendiar 1,, Roman Krcmar 1, and Tomotoshi ishino,3 arxiv: v [cond-mat.str-el] 30 Mar Institute of Electrical Engineering, Slovak Academy of Sciences, Dúbravská cesta 9, SK , Bratislava, Slovakia Institute for Theoretical Physics C, RWTH University Aachen, D-5056 Aachen, Germany 3 Department of Physics, Graduate School of Science, Kobe University, Kobe , Japan System size dependence of the ground-state energy E is considered for -site onedimensional 1D) quantum systems with open boundary condition, where the interaction strength decreases towards the both ends of the system. It is shown that the finite size correction to the energy per site E / lim E / is of the order of 1/, when the reduction factor of the interaction is sinusoidal. We discuss the origin of this fast convergence from the view point of the spherical geometry. 1. Introduction A purpose of numerical studies in condensed matter physics is to obtain bulk properties of systems in the thermodynamic limit. In principle numerical methods are applicable to systems with finite degrees of freedom, and therefore occasionally it is impossible to treat infinite system directly. A way of estimating the thermodynamic limit is to study finite size systems, and subtract the finite-size corrections by means of extrapolation with respect to the system size. 1),) As an example of extensive functions, which is essential for bulk properties, we consider the ground state energy E of -site one-dimensional 1D) quantum systems. In this article we focus on the convergence of energy per site E / with respect to the system size. In order to clarify the discussion, we specify the form of lattice Hamiltonian Ĥ = ĥ l,l+1 + ĝ l, 1.1) l l which contains on-site terms ĝ l and nearest neighbor interactions ĥl,l+1. We assume that the operator form of ĥl,l+1 and ĝ l are independent to the site index l. It is possible to include ĝ l into ĥl,l+1 by the redefinition ĥ l,l+1 + ĝl + ĝ l+1 ĥl,l+1, 1.) and therefore we group ĝ l with ĥl,l+1 as shown in the left hand side if it is convenient. A typical example of such Ĥ is the spin Hamiltonian of the Heisenberg chain Ĥ = J l Ŝ l Ŝl+1 B l Ŝ Z l, 1.3) typeset using PTPTEX.cls Ver.0.9

2 A. Gendiar, R. Krcmar, and T. ishino where Ŝl represents the spin operator at l-th site, and ŜZ l its Z-component. The parameters J and B are, respectively, the neighboring interaction strength and the external magnetic field. In this case ĥl,l+1 and ĝ l are, respectively, J Ŝl Ŝl+1 and BŜZ l. If the chain is infinitely long, the Hamiltonian Ĥ is translational invariant, and the ground state Ψ 0 is uniform when there is no symmetry breaking. For example, the bond-energy J Ŝl Ŝl+1 = Ψ 0 J Ŝl Ŝl+1 Ψ 0 of the integer-spin Heisenberg chain is independent on l. This homogeneous property of the system is violated if only a part of the interactions ĥ1,, ĥ,3,..., and ĥ, is present, and the rest does not exist. In other words, if we consider an -site open boundary system defined by the Hamiltonian Ĥ Open = ĥ l,l+1 + ĝ l, 1.4) the ground state Ψ 0 is normally non-uniform. As a result the expectation values ĥl,l+1 and ĝ l are position dependent, especially near the boundary of the system. As a result, the ground state energy E of this -site system is normally not proportional to the system size. Such a finite size effect is non-trivial when the system is gapless, as observed in the S = 1/ Heisenberg spin chain. 18) In case that we are interested in the bulk property of the system, it is better to reduce the boundary effect as rapidly as possible. For this purpose Vekić and White introduced a sort of smoothing factor A l to the Hamiltonian Ĥ Smooth = A l ĥ l,l+1 + ĝl + ĝ ) l+1, 1.5) where A l is almost unity deep inside the system and decays to zero near the both boundaries of the system. 3) The factor A l is adjusted so that the boundary effect disappears rapidly with respect to the distance from the boundary. A simplest parametrization is to reduce only A 1 and A from unity, leaving other factors equal to unity. This simple choice of A l is often used for calculations of the Haldane gap. 4) As an alternative approach, Ueda and ishino recently introduced the hyperbolic deformation, which is characterized by the non-uniform Hamiltonian Ĥ Hyp. = cosh [ λ l +1 )] ĥ l,l+1 + ĝl + ĝ ) l+1, 1.6) where λ is a small positive constant of the order of ),) As long as the form of the Hamiltonian is concerned, Ĥ Hyp. can be regarded as a special case of Ĥ Smooth in Eq. 1 5) with A l = cosh [ λ l +1 )]. But in the scheme of hyperbolic deformation, the factor cosh [ λ l +1 )] is an increasing function of l + 1)/, and therefore the boundary effect is in principle enhanced. This enhancement works uniformly for most of the lattice sites, and the expectation value

3 Spherical Deformations for 1D Quantum Systems 3 h l,l+1 = Ψ 0 h l,l+1 Ψ 0 for the ground state Ψ 0 becomes nearly independent on l for most of the bonds. After obtaining the expectation value h l,l+1 at the center of the system for several values of the deformation parameter λ, one can perform an extrapolation towards λ = 0 to get the energy per site of the undeformed system. Such an extrapolation is possible since the hyperbolic deformation has an effect of decreasing the correlation length of the system. The hyperbolically deformed system is closely related to classical lattice models on the hyperbolic plane with a constant and negative curvature. 5),6),7),8),9),10),11),1),13),14),15),16) In this article we imagine the case of a positive constant curvature, where the classical lattice models are on a sphere. The corresponding quantum Hamiltonian can be written as Ĥ Sph. = sin π l 1 ) ĥ + ĝl + ĝ ) l+1 l,l+1, 1.7) where A l = sin [ π l 1 ) /) ] decreases to zero toward the system boundary. We call such a modification of the bond strength as the spherical deformation. We analyze the ground state Ψ 0 and the ground-state energy E of this deformed Hamiltonian for the case of free Fermion on the lattice. We find that the difference E / lim E /, 1.8) which is the finite size correction to the energy per site E /, is of the order of 1/. This scaling with respect to is the same as that observed for the system with periodic boundary conditions, described by the Hamiltonian Ĥ Periodic = ĥ l,l+1 + ĝl + ĝ ) l+1 + ĥ,1 + ĝ + ĝ ) ) Structure of this article is as follows. In the next section we introduce a spinless Fermion system on 1D lattice. For tutorial purpose, the finite size effect is reviewed for systems with open and periodic boundary conditions. In Sec. 3 we show our numerical results obtained from the diagonalization of the spherically deformed Hamiltonian ĤSph. in Eq. 1 7). In Sec. 4 we consider geometrical meaning of the spherical deformation by way of the Trotter decomposition applied to the deformed Hamiltonian. We also consider a continuous limit, where the lattice spacing becomes zero. We summarize the obtained results in the last section.. Energy corrections in the free fermion system As an example of 1D quantum systems, we consider the spinless Fermions on the 1D lattice, where the Hamiltonian is defined as Ĥ = t l ) ĉ µ l ĉ lĉl,.1)

4 4 A. Gendiar, R. Krcmar, and T. ishino where t and µ are, respectively, the hopping parameter and the chemical potential. For simplicity we set µ = 0 and treat the half-filled state when µ is not explicitly shown. As a preparation for the spherical deformation, let us observe the ground state properties of the above Hamiltonian, when open or periodic boundary conditions are imposed for finite size systems. First we consider the -site system with open boundary condition when µ = 0, where the Hamiltonian is written as Ĥ O = t ) ĉ..) Since there is no interaction, the one-particle eigenstate ψ m represented by the wave function mπl 0 ĉ l ψ m = ψ m l) = sin.3) is essential for the ground-state analysis, where m is the integer within the range 1 m. The corresponding one-particle energy is ε m = t cos mπ + 1,.4) and the ground-state energy at half-filling is obtained by summing up all the negative eigenvalues. Assuming that is even, the ground-state energy is obtained as / EO = m=1 [ ] ε m = t 1 sin 1 π/ + 1.5) after a short calculation. Expanding the r.h.s. with respect to, one finds the asymptotic form EO π t + t ) π 1..6) Compared with the energy per site in the thermodynamic limit EO lim = 1 π π/ 0 t cos k dk = π t,.7) it is shown that the finite size correction to the energy per site or even to the energy per bond) is of the order of 1/. The -dependence of the energy correction changes if we impose the periodic boundary conditions, where the Hamiltonian is given by Ĥ P = t ) ) ĉ t ĉ ĉ1 1ĉ..8) In this case, the one-particle wave function is the plane wave [ ] 1 ψ m l) = exp mπl 1) i,.9)

5 Spherical Deformations for 1D Quantum Systems 5 where m is the integer that satisfies / + 1 < m /. The corresponding one-particle energy is ε m = t cos mπ..10) If is a multiple of four, the ground state energy at half-filling is calculated as /4 EP = m= /4+1 Thus, the finite size correction to the energy per site EP π ) t = t cot π + t π ε m = t cot π..11) πt 3.1) is of the order of 1/. As it is verified in the above calculations, the finite size correction to the energy per site E / decreases more rapidly for the system with periodic boundary condition than that with open boundary condition. Regardless of this fact, the open boundary systems are often chosen in numerical studies by the density matrix renormalization group DMRG) method 18),17),19),0) because of the simplicity in numerical calculation. It should be noted that for those systems that exhibits incommensurate modulation, the open boundary condition is more appropriate than the periodic boundary condition. Thus, it will be convenient if there is a way of decreasing the finite size correction to E / as rapid as 1/ also in the open boundary systems. 3. Spherical deformation Bond Stren gth Fig. 1. Bond strength of the spherically deformed open-boundary system of the size = 1. In this section we consider the -site open boundary system described by the Hamiltonian Ĥ S = t sin π l ) 1 ) ĉ. 3.1)

6 6 A. Gendiar, R. Krcmar, and T. ishino Compared with the undeformed Hamiltonian ĤO in Eq. ), the strength of the hopping term is scaled by the factor A l = sin [ π l 1 ] ) /), which decreases towards the system boundary as shown in Fig. 1. For a geometrical reason which we discuss in the next section, we call the modification from ĤO to ĤS as the spherical deformation. Let us consider the dependence of the ground-state energy at halffilling. So far we have not obtained the analytic form of the one-particle wave function ψ m and the corresponding eigenvalue ε m for the deformed Hamiltonian ĤS. Thus, we calculate them numerically by diagonalizing Ĥ S in the one-particle subspace. We then obtain the expectation value ĉ and the ground state energy E S at half-filling. In the following numerical calculations, we set t as the unit of the energy. c l > + + < c cl+1 l + c l t -t cos π l / 399) l Fig.. The circles shows the expectation value ĉ +ĉ of the spherically deformed lattice Fermion model when = 400. For comparison, we also plot the same expectation value for the undeformed case by the cross marks. Figure shows the bond correlation functions ĉ of the ground state which is obtained by diagonalizing the deformed Hamiltonian ĤS when = 400. For comparison, we also show the same quantity obtained by the undeformed Hamiltonian ĤO of the same system size. As it is observed, the spherical deformation suppresses the position dependence in ĉ ; the ground state of Ĥ S is more uniform than that of Ĥ O. Thus, one expects that the ground state energy, which is the sum of negative one-particle eigenvalues / ES = m=1 ε m = t sin π l 1 is nearly proportional to the sum of the bond strength ) B = sin π l 1 ) ĉ, 3.) = sin 1 π ). 3.3)

7 Spherical Deformations for 1D Quantum Systems 7 It is also expected that the ratio ES /B rapidly converges to t/π, which is the expectation value ĉ in the thermodynamic limit. Figure 3 shows ES /B and EP / with respect to 1/. As it is shown, both quantities are proportional to 1/. Thus we conclude that the finite size energy correction to the energy per site is of the order of 1/ under the spherical deformation E S / B and EP / E S / B E P / Fig. 3. The finite size corrections to the energy per site, where crosses show ES /B and the open circles EP /. We have considered the half-filled cases where µ = 0. Away from half-filling we must include µ into the deformed Hamiltonian as ) ) Ĥ S = sin π l 1 lĉl +1 t ĉ t ĉ µĉ. 3.4) There is another way of including on-site terms, which is represented by the Hamiltonian Ĥ S = t sin π l ) 1 ) ĉ µ sin π l 1 ) ĉ lĉl, 3.5) where both ĤS and Ĥ S give the same thermodynamic limit. 4. Geometrical interpretation There is a D classical system behind a 1D quantum system, where the relation is called the quantum-classical correspondence. We show that spherically deformed Hamiltonian ĤS corresponds to a classical system on a sphere. We first consider the quantum-classical correspondence by way of the Trotter decomposition. 3),4) Let us divide ĤS in Eq. 3 1) into two parts Ĥ S = l=even A l ĥ l,l+1 + A l ĥ l,l+1 = Ĥ1 + Ĥ, 4.1) l=odd

8 8 A. Gendiar, R. Krcmar, and T. ishino ) where we have used the notation h l,l+1 = t ĉ, and where the deformation factor is given by A l = sin [ π l 1 ) /) ]. The imaginary time evolution of amount of β is then expressed by the boost operator e βĥs. By applying the Trotter decomposition to e βĥs, we obtain e βĥ S = e βĥ S /M ) M e βĥ 1 /M e βĥ /M ) M = e βĥ 1 e βĥ ) M, 4.) where M is the Trotter number 3),4) and β = β/m. Looking at the structure of infinitesimal time boost by Ĥ1 e βĥ1 = exp β ) A l ĥ l,l+1 = exp ) βa l ) ĥl,l+1, 4.3) we find that the quantity l=even l=even τ l = β A l 4.4) plays the role of the rescaled imaginary time. We can treat e βĥ in the same manner. It is possible to interpret τ l as a kind of proper time at the position l. Such interpretation leads us to an inhomogeneous boost operation on a sphere as shown in Fig. 4. This is the reason why we have used the term spherical deformation. Boost Fig. 4. Imaginary time boost on a sphere. Since the surface of the sphere is equivalent everywhere, it is natural to expect that the ground state of the spherically deformed Hamiltonian is approximately uniform. If the system size is finite, this uniformity is slightly violated as shown in Fig. because of the introduction of finite lattice spacing. Also, because of the positive curvature, the local quantities are slightly modified from those in the thermodynamic limit, where the curvature disappears. Such a curvature effect is a source of the finite size corrections shown in Fig. 3. It is possible to show the correspondence with the spherical geometry by taking the continuous limit to the lattice Hamiltonian ĤS, when µ is nearly equal to t.

9 Consider a one-particle state Spherical Deformations for 1D Quantum Systems 9 ψt) = ψ l t)ĉ l 0, 4.5) where the time evolution of the wave function ψ l t) is described by the Schrödinger equation i t ψ l = t sin π l ) 1 ψ l+1 t sin π l ) 3 ψ l 1 µ sin π l 1 ) ψ l 4.6) under the Hamiltonian in Eq. 3 5). ote that we have used the letter t for the time, in order to clarify the distinction from the hopping parameter t. Introducing the notation f l = sin [π l 1) /)], we can rewrite the above equation by use of differentials i t ψ l = t f ψ l+ 1 l+1 t f l 1 ψ l 1 µ f l ψ l 4.7) [ ) ) ] ) = t f l+ 1 ψl+1 ψ l fl 1 ψl ψ l 1 µ f l + t f l+ 1 + t f l 1 ψ l, which can be further transformed as i t ψ l = t ) [ψl+1 ) )] f l+ 1 + f l 1 ψ l ψl ψ l 1 t ) [ψl+1 ) )] f l+ 1 f l 1 ψ l + ψl ψ l 1 t ) f l+ 1 + f l 1 ψ l µ f l ψ l. 4.8) ow we introduce the lattice constant a = πr/ 1) where R is the diameter of the sphere. We also introduce the coordinate x = al, which satisfies 0 < x < πr. Using these notations we rewrite ψ l t) as ψx = al,t), and f l as fx) = sin x/r = sinθ, where θ = x/r is the angle from the north pole. The continuous limit can be taken by increasing the number of sites, where a is the decreasing function of. Simultaneously we increase t so that the relation a t = /m always holds, where m is the particle mass, and the Dirac constant. To prevent the divergence in the potential term, we adjust µ so that µ + t = V is satisfied, where V is a finite constant. Using these parametrizations, we obtain the Schrödinger equation in continuous space i ψx,t) = t m x [ fx) ] x ψx,t) + V fx)ψx,t). 4.9) This equation is derived from the Lagrangian L = i ψ x,t) [ t ψx,t) + fx) ψ ] x,t) ψx, t) + V ψ x,t)ψx,t), m x x 4.10)

10 10 A. Gendiar, R. Krcmar, and T. ishino where introduction of proper time τx,t) that satisfies dt = 1 fx) dτx,t) draws the following Lagrangian [ L = fx) i ψ x,τ) ψ ] x,τ) ψx,τ) ψx,τ) + + V ψ x,τ)ψx,τ), τ m x x 4.11) in the x-τ space. The action S is then written as S = sin x [ R dτdx i ψ τ ψ + ψ ] ψ m x x + V ψ ψ, 4.1) where we have used the relation fx) = sin x/r = sin θ. As it is seen, fx)dτdx plays the role of the integral measure. ote that the continuous limit for the field operator ĉ l ˆψx) can be taken in the same manner as Eqs. 4 6)-4 11) using the correspondence Eq. 4 5). 5. Conclusions and discussions We have investigated the ground state of the spherically deformed 1D free Fermion system at half-filling. The finite size correction to the energy per site is of the order of 1/. The reason for such rapid convergence is qualitatively explained by the quantum-classical correspondence, where the spherically deformed Hamiltonians essentially correspond to classical fields on a sphere. An interest in the spherical deformation is in the dynamical property. We conjecture that a moving one-particle wave packet on the deformed lattice oscillates nearly harmonically as a consequence of the circulation on the sphere. The oscillation may be also explained by a continuous refraction caused by a slower dynamics near the both ends of the system. If one is interested in the estimation of the excitation gap, the spherical deformation is not appropriate. This is because weak bonds near the system boundary induce spurious low-energy excitations. For this purpose, the hyperbolic deformation is more appropriate. 1),) Acknowledgements The authors thank to U. Schollwöck for stimulating discussions and encouragement. T.. is grateful to K. Okunishi for valuable discussions about deformations. This work is partially supported by Slovak Agency for Science and Research grant APVV , APVV-VVCE , QUTE, and VEGA grant o. 1/0633/09 A.G. and R.K.) as well as partially by a Grant-in-Aid for Scientific Research from Japanese Ministry of Education, Culture, Sports, Science and Technology T.. and A.G.). A.G. acknowledges support of the Alexander von Humboldt foundation. References 1) M.E Fisher in Proc. Int. School of Physics Enrico Fermi 51 M.S. Green Ed.) Academic Press, ew York, 1971) 1.

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