Schrieffer-Wolff transformation of Anderson Models
aa r X i v : . [ c ond - m a t . s t r- e l ] D ec Schrieffer-Wolff transformation of Anderson Models
Rukhsan Ul Haq
Theoretical Sciences Unit, Jawaharlal Nehru Center for Advanced Scientific Research
Sachin Satish Bharadwaj ∗ Theoretical Sciences Unit, Jawaharlal Nehru Center for Advanced Scientific ResearchDepartment of Mechanical Engineering, RV College of EngineeringTheoretical Physics and Mathematics,Centre for Fundamental Research and Creative Education, Bangalore India
Abstract
Schrieffer-Wolff transformation is one of the very important transformations in the study of quantummany body physics. It is used to arrive at the low energy effective hamiltonian of Quantum many-bodyhamiltonians, which are not generally analytically tractable. In this paper we give a pedagogical review ofthis transformation for Anderson Impurity model(SIAM) and its lattice generlaization called, the PeriodicAnderson Model(PAM). ∗ [email protected] . INTRODUCTION In the study of quantum many-body physics, the hamiltonians are in general, very difficult tobe treated analytically. Thus, one tries to understand the low energy spectrum of the under-lying physics by computing, effective hamiltonians of the quantum many-body system. One ofthe prominent examples of this approach is the Kondo model, which is essentially the low energyeffective hamiltonian of the Anderson Impurity model(SIAM). The Kondo model which capturesthe low energy physics of Anderson Impurity model, is quite helpful in understanding the Kondophysics itself which is a many-body pheneomena. Schrieffer-Wolff(SW) transformation was his-torically used to get the Kondo model from SIAM. Schrieffer-Wolff transformation is a methodto arrive at effective hamiltonians and has been extensively used both in quantum mechanicsand quantum many-body physics. SW transformation can also be used to diagonalize a givenhamiltonain, as it forms a unitary transformation, yet it may not lead to a full-fledged diagonal-ization; rather, in many cases, it just reduces the hamiltonian to its corresponding band diagonalform. Yet another use of SW transformation is, to view the formalism as a degenerate pertur-bation theory. Historically SW transformation was used to solve many important problems andhas been given different names such as Frohlich transformation in the electron-phonon problem,Foldy-Wouthuysen transformation in relativistic Quantum mechanics and k.p perturbation theoryin semiconductor physics. SW transformation was generalized by Wegner, as the Flow equationmethod, which aids in circumventing the divergences from vanishing denominators, and also hasbeen quite a successful method in various other problems governing non-equilibrium physics aswell.The Schrieffer-Wolff transformation, as mentioned earlier, is basically a unitary transformation.It chooses a unitary operator to obtain the transformed hamiltonian. H ′ = U † HU (1) H ′ = e − S He S (2)= H + 12 [ S, H v ] + 13 [ S, [ S, H v ]] + ... (3)where, S is called generator of the SW transformation. Usually H can be written as H = H + H v , where H is the diagonal part and H v is the off-diagonal part of the hamiltonian. The generator S is chosen in such a way that it cancels the off-diagonal part to the first order, so that :[ S, H ] = − H v (4)The effective hamiltonian to the second order is given by H eff = H + 12 [ S, H v ] (5)1he most crucial step in the transformation is to compute the generator of the transformationand insofar the literature reveals no method to get the generator directly from the hamiltonian.In this work, we have come up with an explicit method to calculate the generator of SW trans-formation. There are other methods to get the effective hamiltonian without recourse to unitarytransformation; rather the method is based on projection operators(or Hubbard operators). Inthis paper we present the SW Transformation of Anderson models in a pedagogical style.
2. SCHRIEFFER WOLFF TRANSFORMATION OF SIAM
The generator of SWT for SIAM is given by S = X kσ ( A k + B k n d ¯ σ ) V k ( c † kσ d σ − d † σ c kσ ) (6)where A k and B k are given by A k = 1 ǫ d − ǫ k (7) B k = 1 ǫ d − ǫ k + U − ǫ d − ǫ k (8)To do the SWT we will have to calculate the following commutator:[ S, H v ] = "X kσ ( A k + B k n d ¯ σ ) V k ( c † kσ d σ − d † σ c kσ ) , X k ′ σ ′ V k ′ (cid:16) c † k ′ σ ′ d σ ′ + d † σ ′ c k ′ σ ′ (cid:17) (9)Commutators are calculated as described below: hP kσ A k V k (cid:16) c † kσ d σ − d † σ c kσ (cid:17) P k ′ σ ′ V k ′ (cid:16) c † k ′ σ ′ d σ ′ + d † σ ′ c k ′ σ ′ (cid:17)i = P kk ′ σ A k V k V k ′ (cid:16) c † kσ c k ′ σ + h.c. (cid:17) − P kσ A k V k (cid:0) d † σ d σ + h.c. (cid:1) (10) "X kσ B k V k n d ¯ σ (cid:16) c † kσ d σ − d † σ c kσ (cid:17) , X k ′ σ ′ V k ′ (cid:16) c † k ′ σ ′ d σ ′ + d † σ ′ c k ′ σ ′ (cid:17) = X kσ X k ′ σ ′ B k V k V k ′ h n d ¯ σ c † kσ d σ , c † k ′ σ ′ d σ ′ + d † σ ′ c k ′ σ ′ i − (11) h n d ¯ σ d † σ c kσ , c † k ′ σ ′ d σ ′ + d † σ ′ c k ′ σ ′ i = X kσ X k ′ σ ′ V k V k ′ h n d ¯ σ c † kσ d σ , c † k ′ σ ′ d σ ′ i + h n d ¯ σ c † kσ d σ , d † σ ′ c k ′ σ ′ i − h n d ¯ σ d † σ c kσ , c † k ′ σ ′ d σ ′ i − h n d ¯ σ d † σ c kσ , d † σ ′ c k ′ σ ′ i ]) (12)2o the commutator [ S, H v ] is given by:[ S, H v ] = P kk ′ σ A k V k V k ′ ( c † kσ c k ′ σ + h.c ) − X kσ A k V k ( d † σ d σ + h.c. ) − P kσ B k V k ( n d ¯ σ d † σ + h.c. ) + X kk ′ σ B k V k V k ′ ( d † ¯ σ c k ′ ¯ σ c † kσ d σ + h.c ) − P kk ′ σ B k V k V k ′ ( c † k ′ ¯ σ d ¯ σ c † kσ d σ + h.c. ) + X kk ′ σ B k V k V k ′ ( c † kσ c k ′ σ n d ¯ σ + h.c. ) (13)To show that we have got the Kondo exchange term in the above commutator, we will writeexchange term in terms of creation and annihilation operators.Ψ † k = c † k ↑ c † k ↓ ! Ψ k = c k ↑ c k ↓ ! Ψ † d = d †↑ d †↓ ! Ψ d = d ↑ d ↓ ! (14)4 (cid:16) Ψ † k S Ψ k ′ (cid:17) (cid:16) Ψ † d S Ψ d (cid:17) = Ψ † k Ψ k ′ ( σ.σ )Ψ † d Ψ d = Ψ † k Ψ k ′ ( σ.σ )Ψ † d Ψ d = (Ψ † k σ z Ψ k )(Ψ † d σ z Ψ d ) + 12 (cid:16) (Ψ † k σ + Ψ k )(Ψ † d σ − Ψ d ) + (Ψ † k σ − Ψ k )(Ψ † d σ + Ψ d ) (cid:17) (15)Now calculating the individual terms, we get :(Ψ † k σ z Ψ k ′ )(Ψ † σ z Ψ d ) = "(cid:16) c † k ↑ c † k ↓ (cid:17) − ! c k ′ ↑ c k ′ ↓ ! d †↑ d †↓ (cid:17) − ! d ↑ d ↓ ! c † k ↑ c † k ↓ (cid:17) c k ′ ↑ − c k ′ ↓ ! d †↑ d †↓ (cid:17) d ↑ − d ↓ ! c † k ↑ c † k ↓ (cid:17) c k ′ ↑ − c k ′ ↓ ! d †↑ d †↓ (cid:17) d ↑ − d ↓ ! = (cid:16) c † k ↑ c k ′ ↑ − c † k ↓ c k ′ ↓ (cid:17) (cid:16) d †↑ d ↑ − d ↓ d ↓ (cid:17) = X σ (cid:16) c † kσ c k ′ σ (cid:17) ( n dσ − n d ¯ σ ) (16) (cid:16) Ψ † k σ z Ψ k ′ (cid:17) (cid:16) Ψ † d σ z Ψ d (cid:17) = X σ c † kσ c k ′ σ ( n dσ − n d ¯ σ ) (17)3 Ψ † k σ + Ψ k ′ (cid:17) (cid:16) Ψ † d σ − Ψ d (cid:17) = "(cid:16) c † k ↑ c † k ↓ (cid:17) ! c k ′ ↑ c k ′ ↓ ! d †↑ d †↓ (cid:17) ! d ↑ d ↓ ! c † k ↑ c † k ↓ (cid:17) c k ′ ↓ ! d †↑ d †↓ (cid:17) d ↑ ! = c † k ↑ c k ′ ↓ d †↓ d ↑ (18) (cid:16) Ψ † k σ + Ψ k ′ (cid:17) (cid:16) Ψ † d σ − Ψ d (cid:17) = c † k ↑ c k ′ ↓ d †↓ d ↑ (19)Similarly one can calculate (cid:16) Ψ † k σ + Ψ k ′ (cid:17) (cid:16) Ψ † d σ − Ψ d (cid:17) = c † k ↓ c k ′ ↑ d †↑ d ↓ (20) (cid:16) Ψ † k σ Ψ k ′ (cid:17) (cid:16) Ψ † d σ Ψ d (cid:17) = X σ c † kσ c k ′ σ ( n dσ − n d ¯ σ ) + 12 (cid:16) c † k ↑ c k ↓ d †↓ d ↑ + h.c (cid:17) (21)4 (cid:16) Ψ † k S Ψ k ′ (cid:17) (cid:16) Ψ † d S Ψ d (cid:17) = X σ (cid:16) c † kσ c k ′ σ ( n dσ − n d ¯ σ ) + 2 c † kσ c k ′ ¯ σ d † ¯ σ d σ (cid:17) (22)Note that in the commutator above, we got the term : X kk ′ σ B k V k V k ′ (cid:16) c † kσ c k ′ σ n d ¯ σ − c † kσ c k ′ ¯ σ d † ¯ σ d σ (cid:17) + h.c (23)We shall show that this term gives the Kondo exchange term= X kk ′ σ B k V k V k ′ (cid:18) c † kσ c k ′ σ ( n dσ + n d ¯ σ c † kσ c k ′ σ ( n d ¯ σ − n dσ − c † kσ c k ′ ¯ σ d † ¯ σ d σ (cid:19) + h.c. = X kk ′ σ B k V k V k ′ (cid:18) − (cid:16) c † kσ c k ′ σ ( n dσ − n d ¯ σ ) + c † kσ c k ′ ¯ σ d † ¯ σ d σ (cid:17) + c † kσ c k ′ σ ( n dσ + n d ¯ σ (cid:19) + h.c. (24)So the first two terms give the exchange term with an additional term generated as well:= − X kk ′ J kk ′ (cid:16) Ψ † k S Ψ k ′ (cid:17) (cid:16) Ψ † d S Ψ d (cid:17) + X kk ′ σ B k V k V k ′ (cid:18) c † kσ c k ′ σ ( n dσ + n d ¯ σ (cid:19) + h.c (25)4he other terms that also get generated are given below: H dir = X kk ′ σ (cid:18) A k V k V k ′ + B k V k V k ′ n dσ + n d ¯ σ (cid:19) c † kσ c k ′ σ + h.c. (26) H hop = − X kσ V k ( A k + B k n d ¯ σ ) n dσ + h.c. (27) H ch = X kk ′ σ B k V k V k ′ (cid:16) c † k ¯ σ d ¯ σ c † k ′ σ d σ (cid:17) + h.c. (28)
3. SCHRIEFFER WOLFF TRANSFORMATION OF PAM
The generator of SW transformation for PAM is given by: S = X kσi ( A k + B k n iσ ) V k ( c † kσ f iσ e − ikR i − f † iσ c kσ e ikR i ) (29)where A k and B k are given by A k = 1 ǫ k − ǫ f (30) B k = 1 ǫ k − ǫ f − U − ǫ k − ǫ f (31)To carry out the SW transformation we have to calculate following commutator [ S, H v ] = "X kσi ( A k + B k n i ¯ σ ) V k ( c † kσ f iσ e − ikR i − f † iσ c kσ e ikR i ) , X k ′ jσ ′ V k ′ ( c † k ′ σ ′ f jσ ′ e − ik ′ R j + f † jσ ′ c k ′ σ ′ e ik ′ R j (32)The commutators are calculated in the following way: "X kiσ A k V k c † kσ f iσ e ikR i , X k ′ jσ ′ V k ′ c † k ′ σ ′ f jσ ′ e − ik ′ R j = X kiσ X k ′ jσ ′ A k V k V k ′ e − ikR i e − ik ′ R j h c † kσ f iσ , c † k ′ σ ′ f jσ ′ i (33)=0 (34)Similarly "X kiσ A k V k e ikR i f † iσ c kσ , X jk ′ σ ′ V k ′ e − ik ′ R j f † jσ ′ c k ′ σ ′ = 0 (35)5 X kiσ A k V k e − ikR i c † kσ f iσ , X k ′ jσ ′ V k ′ e ik ′ R j f † jσ ′ c k ′ σ ′ = X kiσ X k ′ jσ ′ A k V k ′ V k e − ikR i e ik ′ R j h c † kσ f iσ , f † jσ ′ c k ′ σ ′ i = X kiσ X k ′ σ ′ A k V k ′ V k e i ( k ′ − k ) R i c † kσ c k ′ σ − X ijkσ A k V k e ik ( R j − R i ) f † jσ f iσ (36)Hermitian conjugate of above commutator is given by: "X ikσ A k V k f † iσ c kσ e ikR i , X jk ′ σ ′ c † k ′ σ ′ f jσ ′ e − ik ′ R j = X kiσ X jk ′ σ ′ A k V k V k ′ e ikR i e − ik ′ R j h f † iσ c kσ , c † k ′ σ ′ f jσ ′ i = X ijkσ A k V k e ik ( R i − R j ) f † iσ f jσ + X kk ′ iσ A k V k V k ′ e i ( k − k ′ ) R i c † k ′ σ c kσ (37) "X kiσ B k V k n i ¯ σ c † kσ f iσ e − ikR i , X k ′ jσ ′ V k ′ c † k ′ σ ′ f jσ ′ e − ik ′ R j = X kiσ X jk ′ σ ′ V k V k ′ e − ikR i e − ik ′ R j h n i ¯ σ c † kσ f iσ , c † k ′ σ ′ f jσ ′ i = X kk ′ iσ B k V k V k ′ e − i ( k + k ′ ) R i c † k ′ ¯ σ c † kσ f i ¯ σ f iσ (38) "X kiσ B k V k n i ¯ σ f † iσ c kσ e ikR i , X jk ′ σ ′ V k ′ f † jσ ′ c k ′ σ ′ e ik ′ R j = X ikσ X jk ′ σ ′ B k V k V k ′ e ikR i e ik ′ R j h n i ¯ σ f † iσ c kσ , f † jσ ′ c k ′ σ ′ i = X k ′ kiσ B k V k V k ′ e i ( k + k ′ ) R i f † i ¯ σ f † iσ c k ′ ¯ σ c kσ (39)6 X ikσ B k V k n i ¯ σ c † kσ f iσ e − ikR i , X jk ′ σ ′ V k ′ f † jσ ′ c k ′ σ ′ e ik ′ R j = X ikσ X jk ′ σ ′ B k V k V k ′ e − ikR i e ik ′ R j h n i ¯ σ c † kσ f iσ , f † jσ ′ c k ′ σ ′ i = X ikk ′ σ B k V k V k ′ e i ( k ′ − k ) R i n i ¯ σ c † kσ c k ′ σ − X ijkσ B k V k e ik ( R j − R i ) n i ¯ σ f † jσ f iσ + X ikk ′ σ B k V k V k ′ e i ( k ′ − k ) R i f † i ¯ σ c k ¯ σ c † kσ f iσ (40) "X ikσ B k V k n i ¯ σ f † iσ c kσ e ikR i , X jk ′ σ ′ V k ′ c † k ′ σ ′ f jσ ′ e − ik ′ R j = X ikσ X jk ′ σ ′ B k V k V k ′ e ikR i e − ik ′ R j h n i ¯ σ f † iσ c kσ , c † k ′ σ ′ f jσ ′ i = X ikk ′ σ B k V k V k ′ e i ( k − k ′ ) R i n i ¯ σ c † k ′ σ c kσ − X ijkσ B k V k e ik ( R i − R j ) n i ¯ σ f † iσ f jσ + X ikk ′ σ e i ( k − k ′ ) R i c † k ′ ¯ σ f i ¯ σ f † iσ c kσ (41)Combining all these commutators, we get the final commutator as : "X kσi ( A k + B k n i ¯ σ ) V k ( c † kσ f iσ e − ikR i − f † iσ c kσ e ikR i ) , X k ′ jσ ′ V k ′ ( c † k ′ σ ′ f jσ ′ e − ik ′ R j + f † jσ ′ c k ′ σ ′ e ik ′ R j = X kiσ X k ′ j ′ σ ′ ( A k V k V k ′ e i ( k ′ − k ) R i c † kσ c k ′ σ + h.c. ) − X ijkσ ( A k V k e ik ( R j − R i ) f † jσ f iσ + h.c. )+ X kk ′ iσ ( B k V k V k ′ e i ( k + k ′ ) R i c † k ′ ¯ σ c kσ f i ¯ σ f iσ + h.c. ) + X ikk ′ σ B k V k V k ′ e i ( k ′ − k ) R i n i ¯ σ c † kσ c k ′ σ − X ijkσ ( B k V k e ik ( R j − R i ) n i ¯ σ f † jσ f iσ + h.c. ) + X ikk ′ σ ( B k V k V k ′ e i ( k ′ − k ) R i f † iσ c k ′ ¯ σ c † kσ f iσ + h.c. ) (42)7e see that there are many terms that comes out of SW transformation. One important termwhich gets produced, is Kondo exchange term. Proceeding in a similar way as was done for SIAM,we obtain the Kondo exchange term from the following terms of the commutator [ S, H v ]. X kk ′ iσ B k V k V k ′ e i ( k − k ′ ) R i (cid:16) n i ¯ σ c † k ′ σ + c † k ′ ¯ σ f i ¯ σ f † iσ c kσ (cid:17) = X kk ′ iσ B k V k V k ′ e i ( k − k ′ ) R i ( 12 ( n iσ + n i ¯ σ ) c † k ′ σ c kσ −
12 ( n iσ − n i ¯ σ ) c † k ′ σ c kσ + c † k ′ ¯ σ f i ¯ σ f † iσ c kσ )= X kk ′ iσ B k V k V k ′ e i ( k − k ′ ) R i (cid:18)
12 ( n iσ + n i ¯ σ ) c † k ′ σ c kσ (cid:19) − X kk ′ i B k V k V k ′ (cid:16) Ψ † k ′ S Ψ k (cid:17) (cid:16) Ψ † f S Ψ f (cid:17) (43)The exchange term comes from the second term of the above equation H ex = X kk ′ i J k ′ k e − i ( k ′ − k ) R i (cid:16) Ψ † k ′ S Ψ k (cid:17) (cid:16) Ψ † f S Ψ f (cid:17) (44)where J k ′ k is given by J k ′ k = V k V k ′ ( − ǫ k − ǫ f − U − ǫ k ′ − ǫ f − U + 1 ǫ k − ǫ f + 1 ǫ k ′ − ǫ f ) (45) H dir = X kk ′ iσ A k V k V k ′ e − i ( k ′ − k ) R i c † k ′ σ c kσ + (cid:18) B k V k V k ′
12 ( n iσ + n i ¯ σ ) c † k ′ σ c kσ (cid:19) (46) H dir = X kk ′ iσ (cid:18) W kk ′ − J kk ′ ( n iσ + n i ¯ σ ) e − i ( k ′ − k ) R i c † k ′ σ c kσ (cid:19) (47)where W kk ′ is given by W kk ′ = 12 V k ′ V k (cid:18) ǫ k − ǫ f + 1 ǫ k ′ − ǫ f (cid:19) (48) H hop = − X ijkσ B k V k ( e ik ( R i − R j ) n i ¯ σ f † iσ f jσ − e ik ( R j − R i ) n i ¯ σ f † jσ f iσ ) − X ijkσ A k V k e ik ( R j − R i ) f † jσ f iσ = − X ijkσ W kk ′ − J kk ′ ( n i ¯ σ + n j ¯ σ ) e − ik ( R i − R j ) f † jσ f iσ (49) H ch = X ikk ′ σ B k V k V k ′ e − i ( k + k ′ ) R i c † k ′ ¯ σ c † kσ f i ¯ σ f iσ + h.c. (50)8 . CONCLUSION Schrieffer-Wolff transformation is a very important transformation both in Quantum Mechanicsand Quantum many-body physics.It is used to get the effective hamiltonian of a given hamiltonianby integrating out high energy degrees of freedom. It is also used for the diagonalization of thehamiltonians. In quantum mechanics, it is the method to perform the degenerate perturbationtheory calculations. In this paper, we have presented the detailed calculations of Schrieffer-Wolfftransformation of Anderson Models in a pedagogical manner. SW transformation can be carriedout for other models as well, following the method given in this paper.
Acknowledgments
The authors acknowledge DST for financial support and JNCASR for conducive research environ-ment. [1] J.R.Schrieffer and P.A.Wolff,Relation between the Anderson and Kondo Hamiltonians,Phys.Rev.149,2,1966[2] Max Wagner,
Unitary transformations in Solid state physics ,North-Holland Physics PublishingHouse(1986)[3] Sergey Bravyi, David P. DiVincenzo, and Daniel Loss, Schrieffer-Wolff transformations for quantummany-body systems Annals of Physics , 2793 (2011)[4] Theory of Anderson impurity Model:The Schrieffer-Wolff transformation reexamined,Annals ofPhysics 252,1-32(1996)[5] A.C.Hewson,
The Kondo Problem to Heavy Fermions
Cambridge University Press 2003[6] Philip Phillips,
Advanced Solid State Physics
WewtView Press 2003[7] Piers Coleman,”Introduction to heavy fermion systems”[8] R.Chan and M.Gulasci ”An Exact Schrieffer-Wolff transformation”[9] F. Wegner, Ann. Phys. (Leipzig) , 77 (1994)[10] S.D. Glazek and K.G. Wilson, Phys. Rev. D , 5863 (1993)[11] P.Fazekas Lecture Notes on Electron Correlation and Magnetism ,World Scientific(1999),World Scientific(1999)