Flavor-twisted boundary condition for simulations of quantum many-body systems
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Flavor-twisted boundary condition for simulations of quantum many-body systems
Wei-Guo Yin ∗ and Wei Ku Condensed Matter Physics and Materials Science Department,Brookhaven National Laboratory, Upton, New York 11973, U.S.A. (Dated: Received December 5, 2018)We present an approximative simulation method for quantum many-body systems based on coarsegraining the space of the momentum transferred between interacting particles, which leads to effec-tive Hamiltonians of reduced size with the flavor-twisted boundary condition. A rapid, accurate,and fast convergent computation of the ground-state energy is demonstrated on the spin- quantumantiferromagnet of any dimension by employing only two sites. The method is expected to be usefulfor future simulations and quick estimates on other strongly correlated systems. PACS numbers: 02.70.-c, 05.30.Fk, 75.10.Jm
Understanding a quantum many-body system is fundamentally challenging because of the exponential growth of thenumber of states with the system size. To estimate the physical properties of an intractably large system, a commonpractice is to extrapolate the results for several significantly reduced sizes, together with the periodic boundarycondition (PBC). A different yet complementary approach is to continuously twist the boundary conditions for onesolvable size. The typical flavor-independent version of the latter has been seen in solid state physics, while theflavor version has been used in quantum chromodynamics.
Here we apply the flavor-twisted boundary conditions(FTBC) to the spin- quantum antiferromagnet, one of the basic models in solid state physics. We show that theground-state energy can be accurately calculated using only two sites.We begin with an explicit derivation of FTBC, which is necessary for systematical studies of this and relatedmethods. Let us consider the connection between a large system of size L and a small one of size l , both withPBC. (To distinguish them, the upper-case and lower-case letters are used thoroughly for the large and small lattices,respectively.) The large system is described by the following general Hamiltonian in the second quantized languageand in the real space representation, H L = X I , J ,σ ( t σ IJ C † I σ C J σ + h.c. ) + X I , J ,σ J ′ , I ′ ,σ ′ U σ,σ ′ IJ ′ JI ′ C † I σ C † J ′ σ ′ C I ′ σ ′ C J σ , (1)where C I σ denote the quantum operator that annihilates a particle with flavor σ at site I . L = Q dα =1 L α is thetotal number of the lattice sites in a d -dimensional space with L α being the site number in the α -dimension. Thethermodynamic limit is reached when all L α → ∞ .There also exists a reciprocal space where the counterpart of the site I is the momentum point K . Imposing PBCto the real space discretizes the K space as follows K α = 2 πL α M, M = 0 , , , . . . , L α − large and small in the realspace to those of fine and coarse in the K grid, respectively. In the reciprocal space, H L = X K σ ε σ K C † K σ C K σ + 1 L X Q { X K σ, K ′ σ ′ U σ,σ ′ K , K ′ , Q C † K + Q ,σ C † K ′ − Q ,σ ′ C K ′ ,σ ′ C K ,σ } (3)where C K σ annihilates a particle with momentum K and spin σ and is given by C K σ = 1 √ L X I e − i K · R I C I σ , (4)where R I is the coordinates of the I -th site. The bare energy dispersion and the interaction function are ε σ K = X R J − R I t σ IJ e i K · ( R J − R I ) . (5) U σ,σ ′ K , K ′ , Q = X R J − R I R I ′ − R J ′ R J ′ − R I U σ,σ ′ IJ ′ JI ′ e i K · ( R J − R I ) e i K ′ · ( R I ′ − R J ′ ) e i Q · ( R J ′ − R I ) . (6) Q ss K ’ K ’ - QKK - Q s ’ s ’ q ss K ’ + k ’ K’ +( k ’ - q ) K + k K +( k - q ) s ’ s ’ ~ ~~ ~ (a) (b) FIG. 1: Illustration of the momentum transfer due to the particle-particle interaction: (a) original and (b) coarsened.
Eq. (3) appears to describe two particles with K and K ′ , respectively, interact with internal momentum transfer Q , as diagrammed in Fig. 1(a). Since the Q points are to be integrated [c.f. P Q {· · · } in Eq. (3)], here comes a well-known numerical trick: the summation (integration) over a fine grid may be well approximated by that over a rathercoarse grid. This numerical recipe receives particular support in the quantum many-body systems of interest, wherea small number of Q are far more important than the others, the so-called Q -mode resonance, e.g., Q = ( π, π, · · · , π )in antiferromagnetic correlation. The Q -mode resonance implies dramatic response to small stimuli such as changingtemperature, applied fields, pressure, doping, etc, which is important to determining the functionalities of the system.Therefore, it may be a good approximation to coarsen the Q grid as long as the important Q points are included.To find a way to coarsen the Q grid such that the resulting Hamiltonian can be readily transformed back to thereal space (where the parameters are often short-ranged, meriting for numerical computation), we consider the smallsystem of size l with PBC whose momenta k are given by: k α = 2 πl α m, m = 0 , , , . . . , l α − . (7)where l = Π dα =1 l α ≪ L is the total site number of the smaller lattice. Let the small system be commensurate withthe large one, that is, all the k points can be found in the K grid of the large system. Then, any momentum in thefine K grid can be rewritten as, K = ˜ K + k , K ′ = ˜ K ′ + k ′ , Q = ˜ Q + q , (8)where k ′ and q are in the k grid defined in Eq. (7); the L/l ‘twists’ ˜ K (likewise ˜ K ′ or ˜ Q ) consist of a subset of theoriginal K points that fill the interstitial region of the k points nearby the origin. For example, for d = 1,˜ K α ∈ ( − πl α , πl α ] . (9)As another example, l = 2 for the square lattice means that k = (0 ,
0) and ( π, π ), and that ˜ K cover half of the original K space satisfying cos( ˜ K x ) + cos( ˜ K y ) ≥ H L in terms of the real-space l -site lattice by introducing a new set of particle annihilationoperators, d k σ ; ˜ K = C ˜ K + k σ = C K σ , (10) d i σ ; ˜ K = 1 √ l X k e i k · r i d k σ ; ˜ K , (11)where r i is the coordinates of the i -th site of this l -site lattice, satisfying X k e i k · ( r j − r i ) = lδ i , j . (12)By using Eqs. (7-12), the original Hamiltonian for the L -site system, Eq. (1) or Eq. (3), can be transformed to a setof l -site subsystems described by the following new Hamiltonian in the real space representation H L = X ˜ K σ X i , j [ e i ˜ K · ( r j − r i ) t σ ij d † i σ ; ˜ K d j σ ; ˜ K + h.c. ] + lL X ˜ Q X ˜ K σ, ˜ K ′ σ ′ X i , i ′ , j , j ′ n e i ˜ K · ( r j − r i ) e i ˜ K ′ · ( r i ′ − r j ′ ) e i ˜ Q · ( r j ′ − r i ) U σ,σ ′ ij ′ ji ′ d † i σ ; ˜ K + ˜ Q d † j ′ σ ′ ; ˜ K ′ − ˜ Q d i ′ σ ′ ; ˜ K ′ d j σ ; ˜ K o . (13)This establishes the exact transformation between the large and the small, subject to the linear size of the smallsystem being not shorter than the range of any Hamiltonian parameters of Eq. (1). Approximation 1.—
Now we coarse the Q grid with ˜ Q ≡ . (14)This means that only the q points, whose total number is l , are retained as the internal momentum transfer. Then, K + q = ˜ K + k + q = ˜ K + k ′ . That is, any K point is scattered to the sum of the same twist and another point in { q } . Thus, a particle with a given ˜ K can be scattered by the particle-particle interaction into only l points in the K momentum space, instead of L points , as illustrated in Fig. 1(b). Eq. (13) is rewritten as H L ≃ lL X ˜ K σ, ˜ K ′ σ ′ H l ( ˜ K σ, ˜ K ′ σ ′ ) , (15)where the subsystems are H l ( ˜ K σ, ˜ K ′ σ ′ ) = X i , j [ e i ˜ K · ( r j − r i ) t σ ij d † i σ ; ˜ K d j σ ; ˜ K + h.c. ] ++ X i , j [ e i ˜ K ′ · ( r j − r i ) t σ ′ ij d † i σ ′ ; ˜ K ′ d j σ ′ ; ˜ K ′ + h.c. ] ++ X i , j , j ′ , i ′ e i ˜ K · ( r j − r i ) e i ˜ K ′ · ( r i ′ − r j ′ ) U σ,σ ′ ij ′ ji ′ d † i σ ; ˜ K d † j ′ σ ′ ; ˜ K ′ d i ′ σ ′ ; ˜ K ′ d j σ ; ˜ K . (16)Combined with Eq. (15), Eq. (16) for each pair of ˜ K and ˜ K ′ is not independent of another pair. For example, the { ˜ K , ˜ K ′ } subsystem and the { ˜ K , ˜ K ′′ } subsystem share the same momentum points with ˜ K . Therefore, all the ( L/l ) subsystems of size l are connected and one still has to deal with a case of size L . To simplify the calculations to size l , a further approximation is needed. Approximation 2 (FTBC).—
A simple approximation is to treat H l ( ˜ K σ, ˜ K ′ σ ′ ) independently. Thus, to estimate theexpectation value of an observable ˆ O L in the large system is to calculatetrace(ˆ ρ L ˆ O L ) ≃ X ˜ K σ, ˜ K ′ σ ′ trace(ˆ ρ l ; ˜ K σ, ˜ K ′ σ ′ ˆ O l ; ˜ K σ, ˜ K ′ σ ′ ) , (17)where ˆ ρ L denotes the density matrix for the large system and ˆ ρ l ; ˜ K σ, ˜ K ′ σ ′ denotes the density matrix for the isolatedsmall system with the twists ˜ K and ˜ K ′ , as determined by Eq. (16). ˆ O l ; ˜ K σ, ˜ K ′ σ ′ is the transformed observable followingEqs. (10)-(12) and Eq. (14). For example, the ground state energy is approximated by h Φ L | X ˜ K σ, ˜ K ′ σ ′ H l ( ˜ K σ, ˜ K ′ σ ′ ) | Φ L i ≃ X ˜ K σ, ˜ K ′ σ ′ h φ l ; ˜ K σ, ˜ K ′ σ ′ | H l ( ˜ K σ, ˜ K ′ σ ′ ) | φ l ; ˜ K σ, ˜ K ′ σ ′ i , (18)where | Φ L i denotes the ground-state wave function for the large system and | φ l ; ˜ K σ, ˜ K ′ σ ′ i denotes a wave function forthe isolated small system with the twists ˜ K and ˜ K ′ .This approximation is actually equivalent to FTBC (Ref. 7) as explained below. Comparing Eq. (16) with Eq. (1),one finds that they are very similar in form—only differ in the parameters by a phase factor e i ˜ K · ( r i − r j ) associatedwith the momentum twist ˜ K . Actually, solving Eq. (16) is equivalent to solving the following Hamiltonian for the l -site lattice H l ( ˜ K σ, ˜ K ′ σ ′ ) = X i , j ( t σ ij d † i σ ; ˜ K d j σ ; ˜ K + h.c. ) + X i , j ( t σ ′ ij d † i σ ′ ; ˜ K ′ d j σ ′ ; ˜ K ′ + h.c. )+ X i , j , j ′ , i ′ U σ,σ ′ ii ′ jj ′ d † i σ ; ˜ K d † i ′ σ ′ ; ˜ K ′ d j ′ σ ′ ; ˜ K ′ d j σ ; ˜ K (19)together with the boundary conditions such that translating its many-body wavefunction | Ψ l i along the α -th dimension l α steps yield e i ( N ˜ K α + N ˜ K ′ α ) l α | Ψ l i where N and N are the numbers of the two flavors of particles associated withthe two twists ˜ K and ˜ K ′ , respectively.With FTBC, the problem of solving the original Hamiltonian for a L -site system is reduced to solving ( L/l ) Hamiltonians for l -site subsystems, each corresponding to a given pair of ˜ K and ˜ K ′ . The computational benefitis substantial, since the number of states grows exponentially with the system size. For the Hubbard model as anexample, the calculation load is reduced from O (4 L L ) to O [( L/l ) × l l ]. In addition, the ( L/l ) prefactor is fullyparallelizable and it can be readily handled by using the same integration trick of replacing a fine-grid sum witha coarse-grid sum. Since FTBC is reached at the level of Hamiltonian and interpreted as boundary conditions, ithas least limitation and fully compatible with other many-body approaches, e.g., Lanczos exact diagonalization andquantum Monte Carlo methods. Approximation 3.—
The typical twisted boundary conditions (TBC) can be reached from Eqs. (15) and (16) byfurther taking the ˜ K = ˜ K ′ approximation, lL X ˜ K σ, ˜ K ′ σ ′ H l ( ˜ K σ, ˜ K ′ σ ′ ) ≈ X ˜ K σ,σ ′ H l ( ˜ K σ, ˜ K σ ′ ) . (20)With TBC, each subsystem has only one momentum twist ˜ K . Commonly, the one-site or two-site ( r i ′ = r i and r j ′ = r j ) interactions are the leading interaction terms. Then, the twisted phases of the interaction terms in Eq. (16)cancel, which renders the interaction terms for l to be of the same form as those for L . Thus, the approximation byTBC allows the continuous sampling of the momentum space for one-particle excitations, but it prohibits the samefor two-particle excitations. To compare, FTBC allows both in principle. It could be expected that FTBC is moreaccurate, as illustrated below.To complete, PBC is an additional approximation of TBC ( ˜ K = ˜ K ′ = 0). FTBC with one twist zero fixed ( ˜ K ′ = 0,referred to as FTBC0) was studied before.To illustrate all the above points, we use FTBC to estimate the ground state energy of the spin-1/2 Heisenbergquantum antiferromagnet of any dimension. Here the flavor is the spin of electrons, consisting of ↑ and ↓ . The resultsare compared with those obtained from using the other boundary conditions and linear spin-wave theory (LSW). The spin-1/2 antiferromagnetic Heisenberg model is given by H L = J X h I , J i [ S z I S z J + 12 ( S + I S − J + S − I S + J )] , (21)where the spin operator S + , − ,z I = P µν C † I µ σ + , − ,zµν C I ν with σ + , − ,zµν being the Pauli matrix elements. P h I , J i runs overnearest neighbors. It is not only the basic account of antiferromagnetism but also the ground zero of understandinghigh-temperature superconductivity in copper oxides. Two interesting states have been intensively considered inthe literature: the long-range ordered N´eel state and the resonating-valence-bond (RVB) state.
The concept ofRVB is based on the fact that the minimum bond energy ( − . J ) is realized in a two-site system, much lower thanthe bond energy ( − . J ) of the classic N´eel state; the valence-bond between two nearest-neighbor spins is arguablykey to understanding low-dimensional quantum antiferromagnets. However, the 2-site dimer breaks bonding to spinson the other sites in an extended system. For example, the total energy from the dimers is − . J per bond for d = 1 and − . J per bond for d = 2, much higher than the exact result − . J for d = 1 and the numericalresult − . J for d = 2, respectively. It is argued that the resonating (i.e., superposition of the degenerate statesof different dimer configurations) could lower the energy substantially. But an accurate estimation of the groundstate energy was achieved only when long-distance spin dimers were also included. With FTBC, after coarsening the Q mesh, one obtains H l ( ˜ K ↑ , ˜ K ↓ ) = J X h i , j i { s z i s z j + 12 [ e i ( ˜ K ↑ − ˜ K ↓ ) · ( r j − r i ) s + i s − j + h.c. ] } , (22)where s + i = d † i ↑ ; ˜ K ↑ d i ↓ ; ˜ K ↓ , s − i = d † i ↓ ; ˜ K ↓ d i ↑ ; ˜ K ↑ and s z i = ( d † i ↑ ; ˜ K ↑ d i ↑ ; ˜ K ↑ − d † i ↓ ; ˜ K ↓ d i ↓ ; ˜ K ↓ ). The spin-exchange terms stronglydepend on the double twists, ˜ K ↑ and ˜ K ↓ , for the spin-up and spin-down electrons, respectively. In comparison, theresult with TBC is H l ( ˜ K ↑ , ˜ K ↓ ) = J X i , j { s z i s z j + 12 ( s + i s − j + s − i s + j ) } , (23) TABLE I: The ground-state energy per bond (in unit of J ) of the spin-1/2 Heisenberg quantum antiferromagnet in thethermodynamic limit for the linear chain ( d = 1), the square lattice ( d = 2), and the body-centered-cubic lattice ( d = 3). Theresults from using FTBC, FTBC0, TBC or PBC are calculated with l = 2.Methods d = 1 d = 2 d = 3 d = ∞ exact/best known − . a − . b − . c − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . a Ref. 14,15. b Ref. 16. c Ref. 18. independent of twists, the same as with PBC in this case.The accuracy and convergency of FTBC are tested with the bond energy E ( L ; l ) = 1 n bond (cid:18) lL (cid:19) X ˜ K ↑ , ˜ K ↓ h H l ( ˜ K ↑ , ˜ K ↓ ) i , (24)where n bond is the number of the nearest-neighbor AF bonds in the small system. Let us first take the most dramaticapproximation, viz: l = 2 (note n bond = 1), the fundamental of the RVB state. With PBC, the q mesh contains onlytwo points: (0 , , . . . ,
0) and ( π, π, . . . , π ) corresponding to a spin singlet and a triplet, respectively; the energy of the q = 0 state is − . J . By using FTBC to continuously and smoothly twist the energies of the subsystems, the bondenergy becomes E ( L ; l = 2) = J (cid:18) L (cid:19) X ˜ K ↑ , ˜ K ↓ [ − − e i ( ˜ K ↑ − ˜ K ↓ ) · ( r − r ) ]= − J − J d Y α =1 π ) π Z − π d ˜ K α ↑ π Z − π d ˜ K α ↓ cos( ˜ K α ↑ /
2) cos( ˜ K α ↓ / − J − J (cid:18) π (cid:19) d for L → ∞ . (25)The results are listed in Table I, together with those obtained from using FTBC0, TBC, and PBC as well as theLSW results. The bond energy in the thermodynamic limit and for any dimension is accurately reproduced using thesimple Eq. (25) for l = 2.Next, the convergency of the bond energy E ( L → ∞ ; l ) with respect to l is presented in Fig. 2(a). The energyof each subsystem is calculated with the Lanczos exact diagonalization method and a 5-point Gaussian integrationin the twist momentum space, which is enough for the 6-digit accuracy. It is obvious that the estimate from FTBCconverges to the exact solution − J ln 2 + J/ L → ∞ fromfinite-size scaling of the PBC or TBC results. Finally, E ( L ; 2) and E ( L ; 4) for a number of L are plotted in Fig. 2(b)and compared with the exact solutions for L in order to show the fast convergence of FTBC with respect to the sizeof the simulated system. Overall, the errors for l = 4 are smaller than for l = 2 (0 .
003 versus 0 .
009 for L = 24). For l = 2, the small deviations from the exact solutions for L ≥ − . . /L . with χ = 1 . × − . This means that the error grows rather slowly as L increases away from l . These resultsindicate that a large-scale feature of a quantum antiferromagnet could be captured at the length scale of one latticeconstant with FTBC.Finally, it is worth mentioning that in the above derivation of FTBC, we have revealed a less approximated approach,i.e., Eq. (14) alone. It is interesting to explore this approach and compare it with FTBC. Also, the explicit formulationof FTBC could facilitate to devise other approximations with lighter computational load. These studies are beyondthe scope of the present work and will be published elsewhere.Summarizing, based on coarse graining the space of the momentum transferred between interacting particles,we have derived the flavor-twisted boundary condition for simulating quantum many-body systems with effectiveHamiltonians of reduced size. A rapid, accurate, and fast convergent computation of the ground-state energy is
16 18 20 22 24-0.447-0.446-0.445-0.444-0.443-0.442 l (a) FTBC TBC E n e r gy ( J ) (b) L FTBC, l = 2 FTBC, l = 4 exact FIG. 2: (a) Bond energy of the one-dimensional spin-1/2 Heisenberg antiferromagnet for L → ∞ estimated from using FTBC(diamonds) and TBC or PBC (circles) for a number of l . The horizontal solid line denotes the exact solution ( − J ln 2 + J/ L estimated from using FTBC for l = 2 and 4 (triangles), respectively, compared with the exact solutions (circles). demonstrated on the spin- quantum antiferromagnet of any dimension by employing only two sites. The method isexpected to be useful for future simulations and quick estimates on other strongly correlated systems.W.Y. is grateful to P. D. Johnson and T. Valla for collaborations that stimulated this work. This researchutilized resources at the New York Center for Computational Sciences at Stony Brook University/Brookhaven NationalLaboratory which is supported by the U.S. Department of Energy under Contract No. DE-AC02-98CH10886 and bythe State of New York. ∗ Corresponding author. To whom all correspondence should be addressed. E-mail: [email protected] E. Dagotto, Rev. Mod. Phys. , 763 (1994). N. W. Ashcroft and N. D. Mermin,
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