Boson Vortex duality. Abstract

Transcription

1 Boson Vortex duality Subir Sachdev Department of Physics, Harvard University, Cambridge, Massachusetts, 0238, USA and Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada (Dated: March 5, 206) Abstract We describe boson-vortex duality for bosons at integer filling in lattices in one and two spatial dimenions.

2 I. BOSONS AT INTEGER FILLING IN ONE DIMENSION From the previous chapter, we can work with a relativistic theory of bosons. We write the boson field B e i,andfocusonlyonthefluctuationsofthephase in discrete square lattice model of spacetime. So we are interested in the Euclidean partition function = Y i where the action is the Y model 2 0 d i e S () S = K hi,ji cos( i j ). (2) Our treatment below follows classic JKKN paper []. It is convenient to write this is a lattice gauge theory notation S = K i,µ cos( µ i ), (3) where µ extends over x,, the two directions of spacetime. Here µ defines a discrete lattice derivative with µf(x i ) f(x i +ˆµ) f(x i ), with ˆµ avectorofunitlength. Now we introduce the Villain representation e K( which is clearly valid for large K. cos( )) n= e K( 2 n)2 /2 We will use it for all values of K: this is OK because the right-hand-side preserves an essential feature for all K periodicity in. Then we can write the partition function as with = m iµ S = K 2 Y i 2 0 (4) d i 2 e S (5) ( µ i 2 m iµ ) 2, (6) i,µ where the m iµ are independent integers on all the links of the square lattice. Now we need the exact Fourier series representation of a periodic function of e K( 2 n)2 /2 = p e p2 /() ip. (7) 2 K n= p= Note that both sides of the equation are invariant under! +2. Then (5) canberewritten as (ignoring overall normalization constants) = p iµ Y i d i 2 e S (8)

3 with S = p 2 iµ + ip iµ µ i. (9) Again, the p iµ are an independent set of integers on the links of the square lattice. i,µ Now the advantage of (9) isthatalltheintegralsoverthe i factorize, and each i integral can be performed exactly. Each integral leads to a divergence-free constraint on the p iµ integers µp iµ =0. (0) We can solve this constraint by writing p iµ as the curl of another integer valued field, h, which resides on the sites,, oftheduallattice p iµ = µ h. () Then the partition function becomes that of a height or solid-on-solid (SOS) model = h e S (2) with S = ( µh ) 2 (3) This can also describe the statistical mechanics of a two-dimensional surface upon which atoms are being added discretely on the sites, andh is the height of the atomic surface. We are now almost at the final, dual, form of the original Y model in (3). We simply have to approximate the discrete height field h by a continuous field writing, for any function f(h). Formally, we can do this by f(h) = h= = = = d f( / ) ( h) p= p= d f( / ) h= d f( / )e 2ip d f( / )e (ln y)p2 +2ip " +2 p= # y p2 cos(2p ) (4) for y =. Wenowsubstitute(4) into(3), and then examine the situation for y! 0: this is the only unjustified approximation we make here. It is justified more carefully by JKKN using 3

4 renormalization group arguments, who show that all of the basic physics is apparent already at small y. Insuchalimitwecanwritetheexactresult(4) asapproximately h= f(h) d f( / )exp(2y cos(2 )) (5) Substituting (5) into(3), we find that our final dual theory of the Y model is the sine-gordon theory! with S sg = = Y ( µ ) 2 2y d e Ssg (6) cos(2 ) (7) The mapping between (3)and(7)isclearlytheparallelofthemappingwemetearlierinLuttinger liquid theory. A. Mappings of observables We begin with the boson current J µ. This can obtained by coupling the Y model to a fixed external gauge field A µ by replacing (3) by S = K i,µ cos( µ i A iµ ), (8) and defining the current J iµ = S A iµ. (9) Note that because the chemical potential of the bosons is i times the time-component of the vector potential (in Euclidean time), the boson density operator is i times J (this factor of i must be kept in mind in all Euclidean path integrals). Then carrying through the mappings above we find that the sine-gordon action is replaced by S sg = ( µ ) 2 2y So now we can identify the current operator in accord with our previous mapping in Luttinger liquid theory. cos(2 )+ i A µ µ (20) J iµ = i µ (2) Another useful observable is the vorticity. A little thought using the action(6) shows that we can identify v = µ µ m i (22) 4

5 as the integer vorticity on site. Soweextendtheaction(6) to S = K 2 ( µ i 2 m iµ ) 2 + i2 µ µ m i (23) i,µ Now every vortex/antivortex in the partition function at site appears with a factor of e ±i2. For the duality mapping we need an extended version of the identify (7) n= e K( 2 n)2 /2+i2 n = p 2 K p= e (p )2 /() i(p ), (24) which holds for any real,, andk. Thenwefindthat(7) maps to S sg = ( µ ) 2 2y cos(2 2 ) (25) We therefore observe that each factor e ±i2 appears with a factor of ye ±2i.Soweidentifyy with the vortex fugacity, and e ±2i as the vortex/anti-vortex operator. This is also in complete accord with our conclusions from Luttinger liquid theory. II. BOSONS AT INTEGER FILLING IN TWO DIMENSIONS The initial analysis in two dimensions tracks that in one dimension: everything in Section I until Eq. (0) also applies in two dimensions. However, the solution of the divergence-free condition now takes the form p iµ = µ h (26) where h µ is now an integer-valued field on the links of the dual lattice. Then, promoting h µ to a continuous field a µ /(2 ) (whichreplaces replaced by with S = = Y / in (5)), the sine-gordon theory in (6) and(7) is ( µ a ) 2 2y da µ e S (27) cos(a µ )) (28) Notice that the first terms has the form of Maxwell term in electrodynamics, and is invariant under gauge transformations. We can also make the second term gauge invariant by introducing an angular scalar field # on the links of the dual lattice. Then we obtain the action of scalar electrodynamics on a lattice, which is the the final form of the lattice dual theory: = Y Y d# j da µ e S qed (29) 5

6 with S qed = ( µ a ) 2 2y cos( µ# a µ )). (30) Clearly, we have not changed anything, apart from an overall constant in the path integral: we can absorb the # by a gauge transformation of a µ,andthenthe# integrals just yield a constant prefactor. A. Mapping of observables The mappings in Section IA have direct generalizations to two dimensions. By coupling the bosons to an external vector potential A µ,wefindthatthebosoncurrent operator is given by (replacing (2)) J iµ = i 2 µ a (3) So the boson current maps to the electromagnetic flux of the dual scalar QED theory. Similarly, we can now define the vorticity current by (replacing (22)) v µ = µ m i, (32) and then the analog of (25) inthepresenceofasource2 µ coupling to the vorticity current is S qed = ( µ a ) 2 2y cos( µ# a µ 2 µ )). (33) This tells us that the gauge-invariant current of the dual scalar field e i# is precisely equal to the vorticity current of the original boson theory. B. Universal continuum theory The connection described so far may appear specialized to particular lattice models of bosons at integer filling. However, it is possible to state the boson-vortex mapping in rather precise and universal times as an exact correspondence between two di erent field theories, as was first argued by Dasgupta and Halperin [2]. In direct boson perspective, we have already seen that the vicinity of the superfluid-insulator transition is described by a field theory for the complex field e i = D e S S = d 3 µ 2 + r 2 + u 4. (34) 6

7 In the dual vortex formulation, we can deduce a field theory from (33)forthecomplexfield e i# : = D Da µ e S apple S = d 3 x (@ µ ia µ ) 2 + s 2 + v 4 + ( a ) 2. (35) The precise claim is that the universal theory describing the phase transition in S, astheparameter r is tuned across a superfluid-insulator transition at r = r c,isidenticaltothetheorydescribing the phase transition in S as a function of the tuning parameter s. Akeyobservationisthatthe duality reverses the phases in which the fields are condensed; in particular the phases are Superfluid: InS : h i 6=0andr<r c. However, in S : h i =0ands>s c. Insulator: InS : h i =0andr>r c. However, in S : h i6=0ands<s c. We can also extend this precise duality to include the presence of an arbitrary spacetime dependent external gauge field A µ. In the boson theory, this couples minimally to, asexpected [A µ ]= D e S S = d 3 x h(@ µ ia µ ) 2 + r 2 + u i 2 4, (36) while in the vortex theory, the coupling follows from (3): [A µ ]= D Da µ e S apple S = d 3 x (@ µ ia µ ) 2 + s 2 + v ( a ) 2 + i 2 µ A a. (37) The last term is a mutual Chern-Simons term. The equivalence between (36) and(37) is reflected in the equality of their partition functions as functionals of A µ [A µ ]= [A µ ], (38) after a suitable normalization, and mappings between renormalized couplings away from the quantum critical point. This is a powerful non-perturbative connection between two strongly interacting quantum field theories, and holds even for large and spacetime-dependent A µ. It maps arbitrary multi-point correlators of the boson current to associated correlators of the electromagnetic flux in the vortex theory. In particular, by taking one derivative of both theories with respect to A µ, we have the operator µ = 2 a (39) between the boson and the vortex theories. Let us now consider the nature of the excitations in the superfluid and Mott insulating phases in turn. 7

8 . Superfluid In the boson theory, r<r c,the field is condensed. So the only low-energy excitation is the Nambu-Goldstone mode associated with the phase of. Wewrite e i,andthee ectivetheory for is S = s d 3 x (@ µ A µ ) 2 (40) 2 where s is the helicity modulus, and we have included the form of the coupling to the external field A µ by gauge invariance. In the vortex theory, s>s c,andso is uncondensed and gapped. Let us ignore the field to begin with. Then the only gapless fluctuations are associated with photon a µ,anditslowenergy e ective action is S = apple d 3 x ( 8 2 a ) 2 + i s 2 µ A a. (4) In 2+ dimensions, there is only one polarization of a gapless photon, and this corresponds precisely to the gapless scalar in the boson theory. Indeed, it is not di cult to prove that (40) and(4) are exactly equivalent, using a Hubbard-Stratanovich transformation (see the analysis below (46)), which is essential a continuous version of the discrete angle-integer transforms we have used in our duality analysis so far. Turning to gapped excitations, in the vortex theory we have particles and anti-particles. They interact via a long-range interaction mediated by the exchange of the gapless photon. For static vortices, this interaction has the form a Coulomb interaction ln(r) in 2+ dimensions. In the boson theory, this logarithmic interaction precisely computes the interaction between vortices in the superfluid phase. 2. Mott insulator The Mott insulator is simplest in the theory: there are gapped particle and anti-particle excitations, quanta of, whichcarrytotala µ charge Q =andq = respectively. These excitations only have short-range interactions (associated with u). The situation in the vortex theory is subtle, but ultimately yields the same set of excitations. The field is condensed. Consequently, by the Higgs mechanism, the a µ gauge field has a non-zero mass and has a gap - so there is no gapless photon mode, as expected. But where are the excitations with quantized charges Q = ±? Theseare vortices in vortices. In particular, S has solutions of its saddle point equations which are Abrikosov vortices. Because is condensed, any finite energy solution of must have the phase-winding of exactly match the line integral of a µ.inparticular,wecanlookfortime-independentvortexsaddlepointsolutionscenteredatthe origin in which (x) =f(x)e i#(x) (42) 8

9 in which the angle # winds by 2 upon encircling the origin. The saddle point equations show that f(x!0) x, while r s f(x!)= v. (43) Under these conditions, it is not di cult to show that finiteness of the energy requires I I dx i # = dx i a i = d 2 x i a j (44) on any contour far from the center of the vortex. As the phase # must be single-valued, we have from (39,44) that Q = 2 d 2 x i a j = ± (45) in the Abrikosov vortex/anti-vortex, as required. So the important conclusion is that the Q = ± particle and anti-particle excitations of the Mott insulator are the Abrikosov vortices and antivortices of the dual vortex theory. There is another way to run through the above vortices in vortices argument. Let us apply the mapping from (36) to(37) tothevortextheory. Inotherwords,letusmomentarilythinkof as the boson and a µ as an external source field. Then the boson-to-vortex mapping from (36) to (37) appliedto(37) yieldsatheoryforanewdualscalar e,andanewgaugefieldb µ controlled by the action S e = apple d 3 x (@ µ ib µ ) e 2 + er e 2 + eu 2 e 4 + e ( b ) 2 + i 2 µ a b ( a ) 2 + i 2 µ A a. (46) Now we can exactly perform the Gaussian integral over a µ : to keep issues of gauge invariance transparent, it is convenient to first decouple the Maxwell term using an auxilliary field P µ apple S e = d 3 x (@ µ ib µ ) e 2 + er e 2 + eu 2 e 4 + e ( b ) 2 + i 2 µ a b K 2 P 2 µ i µ P a + i 2 µ A a. (47) We can now perform the integral over a µ,andobtainadeltafunctionconstraintwhichsets P µ = b µ + A µ, (48) where is an arbitrary scalar corresponding to a gauge choice; so we have apple S e = d 3 x (@ µ ib µ ) e 2 + er e 2 + eu 2 e 4 + e ( b ) 2 + K 2 (b µ + A µ ) 2. (49) The last term in (49)impliesthattheb µ gauge field has been Higgsed to the value b µ = A µ +@ µ. Setting b µ to this value, we observe that can be gauged away, and then S e reduces to the original 9

10 boson theory in (36). So applying the particle-to-vortex duality to the vortex theory yields back the particle theory. [] J. V. José, L. P. Kadano, S. Kirkpatrick, and D. R. Nelson, Renormalization, vortices, and symmetry-breaking perturbations in the two-dimensional planar model, Phys. Rev. B 6, 27 (977). [2] C. Dasgupta and B. I. Halperin, Phase transition in a lattice model of superconductivity, Phys. Rev. Lett. 47, 556 (98). 0