Transfer-matrix density-matrix renormalization-group theory for thermodynamics of one-dimensional quantum systems. Xiaoqun Wang. Institut Romand de ...
PHYSICAL REVIEW B
VOLUME 56, NUMBER 9
1 SEPTEMBER 1997-I
Transfer-matrix density-matrix renormalization-group theory for thermodynamics of one-dimensional quantum systems Xiaoqun Wang Institut Romand de Recherche Nume´rique en Physique des Materiaux, IN-Ecublens, CH-1015 Lausanne, Switzerland
Tao Xiang Interdisciplinary Research Centre in Superconductivity, The University of Cambridge, Cambridge, CH3 0HE, United Kingdom ~Received 19 March 1997! The transfer-matrix density-matrix renormalization-group method for one-dimensional quantum lattice systems has been developed by considering the symmetry property of the transfer matrix and introducing the asymmetric reduced density matrix. We have evaluated a number of thermodynamic quantities of the anisotropic spin-1/2 Heisenberg model using this method and found that the results agree very accurately with the exact ones. The relative errors for the spin susceptibility are less than 1023 down to T50.01J with 80 states kept. @S0163-1829~97!02034-1#
The density-matrix renormalization group ~DMRG!1 is a powerful numerical method for studying the ground and lowlying state properties of low-dimensional lattice models. It has been applied successfully to a number of strongly correlated systems at zero temperature in one dimension ~1D!2–6 as well as two dimensions ~2D!.7–9 In 1995, Nishino pointed out that as the partition function of a 2D classical system is determined only by the maximum eigenvalue of a transfer matrix in the thermodynamic limit, the DMRG idea can be extended to find this extreme eigenvalue and the corresponding eigenstate. Subsequently one can study thermodynamic properties. He calculated the specific heat of the 2D Ising model using this so-called transfer-matrix DMRG method and found that the result agrees very accurately with the exact solution.10 Recently, Bursill, Xiang, and Gehring have developed a transfer-matrix DMRG algorithm for 1D quantum systems.11 They tested the method on the dimerized spin-1/2 XY model and obtained encouraging results. However, a proper implementation of the transfer-matrix DMRG for studying the thermodynamic properties of 1D quantum systems, especially at low temperature, remains challenging. In this paper, we report our recent progress on the development of the transfer-matrix DMRG method for 1D quantum systems. We have considered the symmetry properties of the quantum transfer matrix12,13 and introduced an asymmetric reduced density matrix which optimizes truncated basis states. This study allows us to achieve significantly accurate results at low temperature and greatly enhances the applicability of the method. For the S51/2 Heisenberg antiferromagnetic chain, we found that the relative error for the spin susceptibility is of the order of 1024 down to the temperature T50.01J (J is the exchange constant! with 80 states kept. This size of error and the value of the temperature are smaller than those for typical quantum Monte Carlo results as well as the thermodynamic DMRG results.14 The thermodynamic DMRG method14 cannot treat accurately a physical system where the correlated length diverges ~for example the low-temperature region of the S51/2 Heisenberg model! because of the finite-size effect. However, there is no such problem in the transfer-matrix DMRG method 0163-1829/97/56~9!/5061~4!/$10.00
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since this method deals directly with an infinite lattice system. We demonstrate our method using the 1D anisotropic spin-1/2 Heisenberg antiferromagnetic model N
ˆ5 H
(i hˆ i ;
y hˆ i 5J @ Sˆ xi Sˆ xi11 1Sˆ iy Sˆ i11 1DSˆ zi Sˆ zi11 # .
~1!
We shall set J51. To apply the DMRG idea, we use the ˆ is Trotter formula to decompose the partition function. H ˆ ˆ separated into two parts, H o and H e , containing those terms with i being odd or even, respectively. The partition function is represented in terms of the quantum transfer matrix TM : 12,13 ˆ
ˆ
Z5 lim Tr@ e2 e H o e2 e H e # M 5 lim TrT N/2 M , e →0
e →0
~2!
where e 5 b /M , b 51/T, and M is the Trotter number. The elements of the asymmetric matrix TM are determined by the product of 2M local transfer matrices
^ s 31 ••• s 32M u TM u s 11 ••• s 12M & M
5
( k51 ) t ~ s 32k21 s 32ku s 22k21 s 22k ! 2
$sk %
3 t ~ s 22k s 22k11 u s 12k s 12k11 ! ,
~3!
2 1 1 2 2 1 1 t ( s 22M s 2M with 11 u s 2M s 2M 11 )5 t ( s 2M s 1 u s 2M s 1 ), imposing periodic boundary conditions in the Trotter direction. The local transfer matrix is given by i11 i i i11 i ˆ i i11 t ( s i11 k s k11 u s k s k11 )5 ^ s k11 ,s k11 u exp(2ehi)usk,sk & where s ik 5(21) i1k s ik and u s ik & is an eigenstate of Sˆ zi : Sˆ zi u s ik & 5s ik u s ik & . The basis u s ik & ^ u s ik11 & is used to represent t and to construct the corresponding Trotter space. The superscripts and subscripts in TM and t represent the coordinates of spins in the real and Trotter space, respectively.
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© 1997 The American Physical Society
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FIG. 1. Configurations of the superblocks: ~a! M 5odd and ~b! M 5even. The left and right transfer matrices are connected, via the summation over states s 91 and s 92 , to form a periodic time-slice chain for TM .
The local Hamiltonian hˆ i conserves the total spin at sites i i11 i and i11, i.e., s ik 1s i11 k 5s k11 1s k11 . In the Trotter space this conservation law can be expressed as i11 s ik 1 s ik11 5 s i11 This means that k 1 s k11 . i11 i i t ( s i11 k s k11 u s k s k11 ) is block diagonal for each value of i i s k 1 s k11 . It turns out that the total sum of s ik at site i is , and TM is block diconserved in TM , i.e., ( k s ik 5 ( k s i11 k agonal according to the value of ( k s ik . For Eq. ~1!, it can be further proved that the maximum eigenvalue of TM occurs in the ( k s ik 50 sub-block.15 Therefore only the ( k s ik 50 subblock in TM is considered in our DMRG iterations. This consideration allows us to keep more basis states in the truncation of the basis set and to save computer CPU time. In the limit N→`, one can study the thermodynamic
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properties using the maximum eigenvalue l and corresponding left ^ c L u and right u c R & eigenvectors of the transfer matrix TM . The free energy is determined purely by l, F52(1/2b )lnl. From the derivatives of F one can in principle calculate the internal energy U, the magnetization M z , the specific heat C v , the spin susceptibility x , and other quantities. However, as it is difficult to evaluate accurately a derivative of a function in numerical calculations, we find that it is better to evaluate U and M z directly from ^ c L u and u c R & , and then calculate C v and x from the derivative of U ˆ & T /N and M z , respectively. For instance, U5 ^ H 5 ^ hˆ 1 & T 5 ^ c L u Th 1 u c R & /l, where ^& T is the thermal average with respect to the thermodynamic density matrix ˆ )/Z and ^ c L u c R & 51 is assumed hereafter. The r th5exp(2bH definition of Th 1 is similar to that of TM subject to the decomposition. Its matrix elements can be obtained from the right-hand side of Eq. ~3! by replacing t ( s 21 s 22 u s 11 s 12 ) with t h 1 ( s 21 s 22 u s 11 s 12 )5 ^ s 12 s 22 u hˆ 1 exp(2thˆ1)us11s21&. Similarly, one can find out the relation between the magnetization M z 5 ^ ( i Sˆ zi & T /N and the maximum eigenvectors of TM . In our calculation, we fix e and increase the chain length 2M . For each M , the temperature T51/e M . As M is small, one can find l, ^ c L u and u c R & exactly. For large M we extend the DMRG idea to approximately but accurately find l, ^ c L u and u c R & for a periodic time-slice chain. Figure 1 shows two configurations of the superblocks of TM . The superblock consists of two blocks, which we call renormalized blocks, in the dashed frames and two time slices. The system contains a renormalized block and one slice. The rest is thus its environment. We use n s and n e to label the basis states of the renormalized blocks in the system and the environment, respectively. The states of two time slices are represented by s 1 and s 2 . The elements of the right transfer matrix is denoted by To ( s 91 ,n s , s 92 ; s 1 ,n s , s 2 ) or 8 Te ( s 81 ,n s8 , s 92 ; s 91 ,n s , s 2 ) if M 5odd or even. The left transfer matrix can be obtained by transposing the right one. Therefore the superblock’s TM is given by
FIG. 2. The specific heat C v (T) for both D50 and D51. The solid curve is the exact results. Circles and pluses are the transfer matrix DMRG results with the asymmetric and symmetric density matrix, respectively. e 50.05 and m580 are used in the transfer-matrix DMRG calculations. Inset: a polynomial fit for the lowtemperature C v .
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TM ~ n 8e , s 82 , s 81 ,n s8 ;n e , s 2 , s 1 ,n s ! 5
5
(
s 91 , s 92
(
s 19 , s 29
To ~ s 91 ,n e , s 92 ; s 81 ,n 8e , s 82 ! To ~ s 91 ,n s8 , s 92 ; s 1 ,n s , s 2 !
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for M 5odd, ~4!
Te ~ s 2 ,n e , s 92 ; s 91 ,n 8e , s 82 ! Te ~ s 81 ,n s8 , s 92 ; s 91 ,n s , s 2 !
for M 5even.
To grow the chain, we have the following recursive relations: Te ~ s 81 , ˜ n s8 , s 92 ; s 91 , ˜ n s ,s2!5
t ~ s 81 , s 8 u s 91 , s 9 ! To ~ s 9 ,n s8 , s 92 ; s ,n s , s 2 ! , ( s
To ~ s 91 , ˜ n s8 , s 92 ; s 1 , ˜ n s ,s2!5
t ~ s 91 , s 9 u s 1 , s ! Te ~ s 8 ,n s8 , s 92 ; s 9 ,n s , s 2 ! , ( s
9
9
~5!
which enlarge the renormalized blocks by one slice with u ˜ n s & 5 u s & ^ u n s & as the corresponding Trotter space and u s & as the states of a spin added. Initially, for 2M 54, Te ( s 81 , s 8 , s 92 ; s 91 , s , s 2 )5 ( s 9 t ( s 81 , s 8 u s 91 , s 9 ) t ( s 9 , s 92 u s , s 2 ). As the number of states in u ˜ n s & exceeds m, Te ( s 81 , ˜ n s8 , s 92 ; s 91 , ˜ n s , s 2 ) and To ( s 91 , ˜ n s8 , s 92 ; s 1 , ˜ n s , s 2 ) are renormalized by Te ~ s 81 ,n s8 , s 92 ; s 91 ,n s , s 2 ! 5
To ~ s 91 ,n s8 , s 92 ; s 1 ,n s , s 2 ! 5
(
˜ n s8 ˜ ns
(
˜ n s8 ˜ ns
O l ~ n s8 , ˜ n s8 ! Te ~ s 81 , ˜ n s8 , s 92 ; s 91 , ˜ n s , s 2 ! O r~ ˜ n s ,n s ! , ~6! O l ~ n s8 , ˜ n s8 ! To ~ s 91 , ˜ n s8 , s 92 ; s 1 , ˜ n s , s 2 ! O r~ ˜ n s ,n s ! ,
where the transformation matrices O l,r are constructed by m truncated basis states from a reduced density matrix discussed below. Other operators such as hˆ 1 can be renormalized by Eq. ~6! with t h 1 instead of t in Eq. ~5!. We compute the maximum eigenvalue l and the corresponding right eigenvector u c R & of TM using a projection method. Iterating u c K & 5TM u c K21 & , we reach u c R & 5 u c K & and TM u c R & 5l u c R & for sufficient large K. u c 0 & is an arbitrary trial vector which is not orthogonal to u c R & . In our calculations, we find that the value of K needed for producing an eigenvalue with a relative error less than 10216 is generally less than 20, but it increases with increasing M . The left eigenvector u c L & can be calculated similarly. However, for the systems as we study here, the wave function of ^ c L u can be directly read out from the wave function of u c R & : c L (n s , s 2 , s 1 ,n e )5 c R (n e , s 2 , s 1 ,n s ) by constructing the superblocks with a reflection symmetry as involved in Eq. ~4!. A density matrix for the whole system ~i.e., superblock! N/2 can be defined as r 5T N/2 M /TrT M . This is a generalization of the thermodynamic density matrix r th in the Trotter space. We form the reduced density matrix for the augmented renormalized block by performing a partial trace on r for the states of the environment
r s5
Trn e s 2 T N/2 M TrT N/2 M
.
~7!
In the thermodynamic limit, r s 5Trn e s 2 u c R &^ c L u , thus the matrix element of r s is given by
r s~ ˜ n s8 , ˜ n s!5
c L~ ˜ n e ,˜ n s8 ! c R ~ ˜ n e ,˜ n s!, ( ˜ n e
~8!
with u ˜ n & 5 u s & ^ u n & . r s is an asymmetric matrix since L c u Þ( u c R & ) † . The eigenvalue of r s gives the probability of ^ the corresponding eigenstates onto which the system is projected as the response to its environment. ( u c R &^ c R u which is used to define the density matrix for the augmented system block in Ref. 11 is not a true projection operator for the maximum eigenvectors of TM .) The transformation matrices O l,r in Eq. ~6! are thus built up by using m left or right eigenvectors of r s corresponding to m most probable eigenvalues. Systematic errors come from two sources. One is the finiteness of e , and the other is the truncation of basis set in the DMRG iterations. The first type of error is generally very small and in principle it can be further reduced by doing an extrapolation with respect to e 2 .16,17 The error due to the truncation is difficult to estimate. A lower bound for this type of error is given by the truncation error p m 512 ( i m w i , where w i (i51, . . . ,m) are the m largest eigenvalues of r s . We found that p m is generally less than 1025 when m516 and decreases rapidly with increasing m for the spin 1/2 system. Figure 2 shows the results for the specific heat C v (T) down to T50.02 with m580 and e 50.05 for D50,1 cases. C v is obtained from the first derivative of U. For the XY model (D50!, we find that the relative errors are less than 1025 down to T50.02 for U and less than 1023 down to T50.03 for C v compared with the exact results.16 For the isotropic antiferromagnetic Heisenberg model (D51!, the precision of the results is similar. The maximum value of C v is 0.3515 at T50.47. At low temperature C v varies linearly with T. The coefficient of the T term is shown to be 2 u /(3sinu) with u 5cos21D.18 By fitting our results with a
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FIG. 3. The spin susceptibility x (T) for both D50 and D51. The solid curves and circles, respectively, are the exact results ~Ref. 19! and the transfer-matrix DMRG results with e 50.05 and m580. Inset compares the transfer-matrix DMRG results for m532 ~diamonds! and 80 ~circles! with the Bethe ansatz results in the lowtemperature regime for D51.
polynomial up to seventh order in T for 0.03&T
