aa r X i v : . [ h e p - t h ] N ov Non-hermitian bosonization
V.K. SazonovInstitute of Physics, Department of Theoretical Physics, University ofGraz, Universit¨atsplatz 5, A-8010 Graz, AustriaAugust 20, 2018
Abstract
A method for the bosonization of complex actions is presented.Together with the convergent perturbation theory it provides a con-ceptually new way for bypassing fermion sign problems.
Bosonization of fermionic field theories is one of the most interesting prob-lems of physics. Initially it was considered within the framework of (1 + 1)-dimensional models [1–7], where there is no spin and therefore bosons andfermions are similar to each other. The bosonization in (1 + 1)-dimensionsis a powerful tool in quantum field theory and condensed matter physics.Its extension to higher dimensions is important for the understanding ofthe underlying relations between bosonic and fermionic degrees of freedom,as well as for developing new computational methods. Several significantresults were obtained for the (2 + 1)-dimensional case [8–12]. The most gen-eral approaches valid for any dimension d were suggested by L¨uscher [13]and Slavnov [14–17]. Both of these methods rely on the utilization of aux-iliary ( d + 1)-dimensional bosonic matter fields . However, the applicationof these techniques is restricted only to real (non-complex) actions with aneven number of fermionic flavors. The bosonization of complex actions wasan open problem for the decades. Here we construct a bosonization forcomplex actions with an even number of flavors. Therefore, we establish In the L¨uscher multi-boson approach the sum over the bosonic flavors in the action isequivalent to an extra spatial dimension.
For definiteness we consider the lattice discretization of the path integral.The generalization to the continuum case is given at the end of Section 3.Consider the partition sum of two flavors of fermions ¯ ψ i,x , ψ i,x , i = 1 , U Z = Y i =1 , x =1 ..N Z [ d ¯ ψ i,x ][ dψ i,x ] Z [ dU x ] e − S G + P x,y,i ¯ ψ x,i K x,y ψ y,i , (1)where N is the d -dimensional lattice volume. Extension of the followingsteps to other even numbers of flavors is trivial. K x,y is the kernel of thefermion action. We represent it as K x,y = A x,y + H x,y , where A x,y is anti-hermitian and H x,y is hermitian. Inserting unity with an auxiliary integra-tion and the delta function we change the term P y,i ¯ ψ i,x H x,y ψ i,y to a newbosonic variable f x Z = Y i =1 , x =1 ..N Z [ d ¯ ψ i,x ][ dψ i,x ] Z [ dU x ] Z ∞−∞ [ df x ] e − S G + P x (cid:0)(cid:2) P y,i ¯ ψ i,x A x,y ψ i,y (cid:3) + f x (cid:1) δ (cid:0) f x − X y,i ¯ ψ i,x H x,y ψ i,y (cid:1) . (2)Representing the delta function as δ ( a − b ) = 12 π lim λ → Z ∞−∞ dh e ih ( a − b ) − λ ( h + a ) , we rewrite partition sum as Z = 1(2 π ) N lim λ → Y i =1 , x =1 ..N Z [ d ¯ ψ i,x ][ dψ i,x ] Z [ dU x ] Z ∞−∞ [ df x ] Z ∞−∞ [ dh x ] e − S G + P x (cid:0)(cid:2) P y,i ¯ ψ i,x ( A x,y − ih n H x,y ) ψ i,y (cid:3) + f x + ih x f x − λ ( h x + f x ) (cid:1) . (3)Integration over the fermion fields gives the squared determinant of the anti-hermitian operatordet( A x,y − ih x H x,y ) ≡ det(( iB ) ) x,y = ( − N det( B ) x,y , (4)2here B x,y = − i ( A x,y − ih x H x,y ) is hermitian. The factor ( − N is notsignificant, since it cancels in all vacuum expectation values of observables.The transformations (2), (3) and definition (4) map the non-hermitianpart of the fermion determinant to the auxiliary bosonic fields. Therefore,the initial problem reduces to the bosonization of the determinant of a her-mitian matrix. The utilization of the hermitization for the Slavnov bosonization procedureis straightforward. Following [14], we writedet( B ) x,y = lim α → ,b → Z [ dφ k,x ][ dφ † k,x ] Z [ dχ k,x ][ dχ † k,x ] e a b P x P nk = − n +1 P x (cid:2) α φ † k +1 ,x − φ † k,xb φ k,x − P y [ φ † k,x ( B ) x,y φ k,y ] − i √ L ( φ † k,x χ x + χ † x φ k,x ) (cid:3) . (5)Here φ k,x , φ † k,x are d +1 dimensional bosonic fields, carrying the same indicesas the initial fermions. The fields χ k,x , χ † k,x are d -dimensional and implementthe non-local constraint n X k = − n +1 φ k ( x ) = n X k = − n +1 φ † k ( x ) = 0 . (6)Then, the partition function is given by the expression Z = 1(2 π ) N lim λ → lim α → ,b → Z [ dφ k,x ][ dφ † k,x ] Z [ dU x ] Z [ dχ x ][ dχ † x ] Z ∞−∞ [ df x ] Z ∞−∞ [ dh x ] e − S G + P x (cid:2) f x + ih x f x − λ ( h x + f x ) (cid:3) e a b P x P nk = − n +1 (cid:2) α φ † k +1 ,x − φ † k,xb φ k,x + P y [ φ k,x ( A x,y − ih x H x,y ) φ † k,y ] − i √ L ( φ † k,x χ x + χ † x φ k,x ) (cid:3) . (7)Here the order of limits is crucial for the convergence of the integrals andthe limit λ → Z = 1(2 π ) N lim λ → lim α → Z Dφ ( x, τ ) Dφ † ( x, τ ) Z DA µ ( x ) Z Dχ ( x ) Dχ † ( x ) Z Df ( x ) Z Dh ( x ) e − S G [ A µ ]+ R d d x (cid:2) f ( x )+ ih ( x ) f ( x ) − λ ( h ( x )+ f ( x )) (cid:3) · e R dτ R d d x (cid:2) α∂ τ φ † ( x,τ ) φ ( x,τ )+ (cid:0) R d d y φ ( x,τ )[ A ( x,y ) − ih ( x ) H ( x,y )] φ † ( y,τ ) (cid:1)(cid:3) · e − i R dτ R d d x ( φ † ( x,τ ) χ ( x )+ χ † ( x ) φ ( x,τ )) , (8)3here S G [ A µ ] denotes the continuum gauge action. In the L¨uscher approach the matrix ( B ) x,y must be bounded, but this is notthe case. To bound ( B ) x,y , we borrow the damping factor e − λ/ P x h x fromthe delta function and represent it as a determinant of a diagonal matrix( T ) x,y = diag { e − λh / , ..., e − λh N / } x,y . (9)Then, the determinant of ( B ) x,y is substituted by e − λ/ P x h x det( B ) x,y = det( T B ) x,y = det( T BB T T ) x,y ≡ det( Q ) x,y . (10)The matrix Q x,y = ( T B ) x,y is bounded. We normalize it as e Q x,y ≡ Q x,y /Q max ( λ ),where Q max ( λ ) is the largest eigenvalue of Q x,y . The determinant det( e Q ) x,y ∈ (0; 1] and can be inverted with n flavors of bosonic fieldsdet( Q ) x,y = Q Nmax ( λ ) det( e Q ) x,y = Q Nmax ( λ ) lim n →∞ Y k =1 ..n ; x =1 ..N Z [ dφ k,x ][ dφ † k,x ] e − P x P nk =1 (cid:2) | P y [( e Q x,y − µ k δ x,y ) φ k,y ] | + ν k | φ k,x | (cid:3) , (11)where µ k , ν k are real constants [13]. The partition function can be expressedas Z = 1(2 π ) N lim λ → Q Nmax ( λ ) lim n →∞ Z [ dφ k,x ][ dφ † k,x ] Z [ dU x ] Z ∞−∞ [ df x ] Z ∞−∞ [ dh x ] e − S G + P x (cid:2) f x + ih x f x − λ ( h x + f x ) (cid:3) e − P x P nk =1 (cid:2) | P y [[ e − λh x/ ( h x H x,y − iA x,y ) /Q max ( λ ) − µ k δ x,y ] φ k,y ] | + ν k | φ k,x | (cid:3) . (12)As in the previous section, the limits in (12) are not interchangeable. The presented hermitization procedure is applicable to any fermionic the-ory, which may be reduced to the fermion determinant, i.e. theories withan action that is bilinear in ψ and ¯ ψ . Therefore, we have constructed a4osonization of complex fermionic actions with even number of flavors andproved the fundamental relation between the d -dimensional fermions and( d + 1)-dimensional bosons.The suggested bosonization is highly interesting also from a computa-tional point of view. One of the promising approaches to calculations on thelattice and path integrals is the convergent perturbation theory [18–22]. Itwas initially formulated only for purely bosonic models and the extension totheories containing fermions can be now obtained using bosonization. Forlattice QED with an even number of flavors at zero chemical potential thiswas done in [23]. The current work extends the applicability of the conver-gent perturbation theory and opens a new approach to lattice theories witha complex action problem. I acknowledge A. Alexandrova and C. Gattringer for discussions and forreading the manuscript. This work was supported by the Austrian ScienceFund FWF Grant Nr. I 1452-N27.
References [1] S Coleman.
Phys. Rev. D , 11:2088–2097, Apr 1975.[2] A. Polyakov and P.B. Wiegmann.
Physics Letters B , 131(13):121 – 126,1983.[3] A.M. Polyakov and P.B. Wiegmann.
Physics Letters B , 141(34):223 –228, 1984.[4] Edward Witten.
Communications in Mathematical Physics , 92(4):455–472, 1984.[5] R.E. Gamboa Sarav, F.A. Schaposnik, and J.E. Solomin.
NuclearPhysics B , 185(1):239 – 253, 1981.[6] K. Furuya, R.E.Gamboa Sarav, and F.A. Schaposnik.
Nuclear PhysicsB , 208(1):159 – 181, 1982.[7] C. M. Na´on.
Phys. Rev. D , 31:2035–2044, Apr 1985.[8] A. Kovner and P.S. Kurzepa.
International Journal of Modern PhysicsA , 09(27):4669–4700, 1994. 59] Gordon W. Semenoff.
Phys. Rev. Lett. , 61:517–520, Aug 1988.[10] M. L¨uscher.
Nuclear Physics B , 326(3):557 – 582, 1989.[11] E.C. Marino.
Physics Letters B , 263(1):63 – 68, 1991.[12] Luis Huerta and Jorge Zanelli.
Phys. Rev. Lett. , 71:3622–3624, Nov1993.[13] M. L¨uscher.
Nuclear Physics B , 418(3):637–648, 1994.[14] A. A. Slavnov.
Phys.Lett. B , 366:253–260, 1996.[15] A. A. Slavnov.
Phys.Lett. B , 388:147–153, 1996.[16] A. A. Slavnov.
Phys.Lett. B , 469:155–160, 1999.[17] A. A. Slavnov.
Nuclear Physics B - Proceedings Supplements , 88:210–214, 2000.[18] V.V. Belokurov, Yu.P. Solov’ev, and E.T. Shavgulidze.
Theoretical andMathematical Physics , 109(1):1287–1293, 1996.[19] V.V. Belokurov, Yu.P. Solov’ev, and E.T. Shavgulidze.
Theoretical andMathematical Physics , 109(1):1294–1301, 1996.[20] V.V. Belokurov, Yu.P. Solov’ev, and E.T. Shavgulidze.
Russian Math-ematical Surveys , 52(2):392, 1997.[21] V.V. Belokurov, Yu.P. Solov’ev, and E.T. Shavgulidze.
Fundament. iprikl. matem. , pages 693–713, 1997.[22] V.V. Belokurov, Yu.P. Solov’ev, and E.T. Shavgulidze.