text: 15 p p + i p p x-y x-y x+y x+y !" " " " !" !# !"" # # # !"" !# !" " " " !" # " " !" !" !" !"!# " " !" !" " " " !" !" # !" !# !# !# " # !" !" " !" " " !" " !" $%’ (cid:47)+& (cid:47)+& ;<<= .8#,95 ;<= ;<<<= 2233,,##//--11 2233,,##//--11 $ $ $ !$%$& $ $%$& ( (cid:54))+) $ * ,-. 1#236,3#6( !: y x FIG. 15: The phase diagram in the centre shows the mixing angle α in Eq. 6 versus n2 U and n ∆E in units of E . Three p p 0 rec phases arise (regions (I),(II), and (III)) separated by 2nd-order transitions (white dashed lines). In (I) and (III) only a single condensation point is populated such that striped order prevails. In the central region (II) both condensation points are populated and chiral order emerges. Reproduced from Ref. [57]. − n ∆E π cos(2α) = 0 − and δφ = . (5) g (ρ ρ ) 2 0 1 (cid:82) (cid:82) According to Fig. 10(b), we now specify the Bloch functions ψ±(r) in a typical plaquet√te to be com√posed of local s(r)-orbitals in the shallow wells and local p±(r)-orbitals in the deep wells, i.e. ψ±(r) = n s(r) + n p±(r) with (cid:82) | | | | s p normalized wave functions 1 = (cid:5) d2r s(r) 2 = (cid:5) d2r p±(r) 2. The fractional populations n and n of the shallow and (cid:82) s p deep wells, respectively, satisfy n +n = n . Assuming that s(r) and p±(r) have approximately disjoint supports, one − | s |p − | 0 | − ﬁnds g (ρ ρ ) = gn2 (cid:5) d2r( p (r) 2 p−(r) 2)2. In the harmonic approximation this yields g (ρ ρ ) = 3n2 U 0 1 2 p + ≡ | | 0 1 2 p p with the interaction per particle in the p-orbitals U g (cid:5) d2r p±(r) 4 and thus p 3 n ∆E cos(2α) = 0 . (6) 2 n2 U p p This result agrees with a more involved calculation assuming a three-band tight-binding model reported in Ref. [57]. A numerical mean-ﬁeld calculation based upon the Gross-Pitaevskii equation is found in Ref. [28]. Eq. 6 gives rise to the phase diagram plotted in Fig. 15 with respect to the two experimentally tunable parameters n ∆E and n2 U . 0 p p The latter can be readily tuned by varying n via adjustment of θ (cf. Eq. 1), even if U is ﬁxed. Three phases arise p p (regions (I),(II), and (III)), separated by 2nd order transitions (white dashed lines). In (I) and (III) only a single condensation point is populated such that striped order prevails. In the central region (II) both condensation points are populated and the wave function displays chiral order as already discussed in Fig. 14(b). The phase diagram shows that for small values of the interaction in the p-orbitals a small distortion of the C 4 symmetry of the lattice can yield a signiﬁcant depletion of one of the condensation points X±, while at large values of the interaction the system prefers equilibrated populations of X±. This property of the system has been directly (cid:104) − (cid:105) (cid:104) (cid:105) tested experimentally in Ref. [57]. In Fig. 16(a) the relative population diﬀerence n n− / n + n− of the + + condensation points X±, averaged for ﬁve measurements, is plotted versus the lattice distortion parameter n ∆E 0 (6#3.9,-3(# )*+ 0*& ( .-, */ / 1-(2#34*/563*67*(cid:95) ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: p

text: p

text: i

text: p

text: p

text: x-y

text: x-y

text: x+y

text: x+y

text: y

text: x

text: FIG.

text: The

text: phase

text: diagram

text: in

text: the

text: centre

text: shows

text: the

text: mixing

text: angle

text: α

text: in

text: Eq.

text: versus

text: U

text: and

text: n

text: ∆E

text: in

text: units

text: of

text: E

text: Three

text: p

text: p

text: rec

text: phases

text: arise

text: (regions

text: (I),(II),

text: and

text: (III))

text: separated

text: by

text: transitions

text: (white

text: dashed

text: lines).

text: In

text: (I)

text: and

text: (III)

text: only

text: a

text: single

text: condensation

text: point

text: is

text: populated

text: such

text: that

text: striped

text: order

text: prevails.

text: In

text: the

text: central

text: region

text: (II)

text: both

text: condensation

text: points

text: are

text: populated

text: and

text: chiral

text: order

text: emerges.

text: Reproduced

text: from

text: Ref.

text: n

text: ∆E

text: π

text: and

text: δφ

text: g

text: (ρ

text: ρ

text: According

text: to

text: Fig.

text: we

text: now

text: specify

text: the

text: Bloch

text: functions

text: ψ±(r)

text: in

text: a

text: typical

text: plaquet√te

text: to

text: be

text: com√posed

text: of

text: local

text: s(r)-orbitals

text: in

text: the

text: shallow

text: wells

text: and

text: local

text: p±(r)-orbitals

text: in

text: the

text: deep

text: wells,

text: i.e.

text: ψ±(r)

text: n

text: s(r)

text: n

text: p±(r)

text: with

text: s

text: p

text: normalized

text: wave

text: functions

text: s(r)

text: p±(r)

text: The

text: fractional

text: populations

text: n

text: and

text: n

text: of

text: the

text: shallow

text: and

text: s

text: p

text: deep

text: wells,

text: respectively,

text: satisfy

text: n

text: +n

text: n

text: Assuming

text: that

text: s(r)

text: and

text: p±(r)

text: have

text: approximately

text: disjoint

text: supports,

text: one

text: s

text: |p

text: ﬁnds

text: g

text: (ρ

text: ρ

text: p

text: (r)

text: p−(r)

text: In

text: the

text: harmonic

text: approximation

text: this

text: yields

text: g

text: (ρ

text: ρ

text: U

text: p

text: p

text: p

text: with

text: the

text: interaction

text: per

text: particle

text: in

text: the

text: p-orbitals

text: U

text: g

text: p±(r)

text: and

text: thus

text: p

text: n

text: ∆E

text: U

text: p

text: p

text: This

text: result

text: agrees

text: with

text: a

text: more

text: involved

text: calculation

text: assuming

text: a

text: three-band

text: tight-binding

text: model

text: reported

text: in

text: Ref.

text: A

text: numerical

text: mean-ﬁeld

text: calculation

text: based

text: upon

text: the

text: Gross-Pitaevskii

text: equation

text: is

text: found

text: in

text: Ref.

text: Eq.

text: gives

text: rise

text: to

text: the

text: phase

text: diagram

text: plotted

text: in

text: Fig.

text: with

text: respect

text: to

text: the

text: two

text: experimentally

text: tunable

text: parameters

text: n

text: ∆E

text: and

text: U

text: p

text: p

text: The

text: latter

text: can

text: be

text: readily

text: tuned

text: by

text: varying

text: n

text: via

text: adjustment

text: of

text: θ

text: (cf.

text: Eq.

text: even

text: if

text: U

text: is

text: ﬁxed.

text: Three

text: phases

text: arise

text: p

text: p

text: (regions

text: (I),(II),

text: and

text: (III)),

text: separated

text: by

text: order

text: transitions

text: (white

text: dashed

text: lines).

text: In

text: (I)

text: and

text: (III)

text: only

text: a

text: single

text: condensation

text: point

text: is

text: populated

text: such

text: that

text: striped

text: order

text: prevails.

text: In

text: the

text: central

text: region

text: (II)

text: both

text: condensation

text: points

text: are

text: populated

text: and

text: the

text: wave

text: function

text: displays

text: chiral

text: order

text: as

text: already

text: discussed

text: in

text: Fig.

text: The

text: phase

text: diagram

text: shows

text: that

text: for

text: small

text: values

text: of

text: the

text: interaction

text: in

text: the

text: p-orbitals

text: a

text: small

text: distortion

text: of

text: the

text: C

text: symmetry

text: of

text: the

text: lattice

text: can

text: yield

text: a

text: signiﬁcant

text: depletion

text: of

text: one

text: of

text: the

text: condensation

text: points

text: X±,

text: while

text: at

text: large

text: values

text: of

text: the

text: interaction

text: the

text: system

text: prefers

text: equilibrated

text: populations

text: of

text: X±.

text: This

text: property

text: of

text: the

text: system

text: has

text: been

text: directly

text: tested

text: experimentally

text: in

text: Ref.

text: In

text: Fig.

text: the

text: relative

text: population

text: diﬀerence

text: n

text: n−

text: n

text: n−

text: of

text: the

text: condensation

text: points

text: X±,

text: averaged

text: for

text: ﬁve

text: measurements,

text: is

text: plotted

text: versus

text: the

text: lattice

text: distortion

text: parameter

text: n

text: ∆E

text: ##LTFigure##

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text: 25 or of the lattice geometry should be possible. Only a few examples have been considered yet, which however clearly illustrate the wealth of new physics arising when orbital degrees of freedom come into play in optical lattice systems. Orbital degrees of freedom provide two most interesting consequences: directionality of tunneling and interactions, and orbital degeneracy. As a consequence of the latter, even small interactions can determine the ground state of the system. In our work with bosonic atoms in the second band of a bipartite square lattice, orbital degeneracy is the basis for the emergence of interaction-induced chiral superﬂuidity. An obvious next step is to also study orbital physics in optical lattices with fermions. Excited bands can come with interesting topological structure, which in addition to the physics associated with broken symmetries can give rise to exotic phenomena like, for example, topologically protected chiral edge states. Orbital optical lattices open up new avenues to experimentally study such phenomena also with bosons, which is not possible in electronic condensed matter. ACKNOWLEDGMENTS This work was partially supported by DFG-SFB 925 and the Hamburg Centre for Ultrafast Imaging (CUI). AH is grateful to W. Vincent Liu, Cristiane Morais Smith, Arun Paramekanti, Ludwig Mathey, and Claus Zimmermann for discussions and to the entire team of the Wilczek quantum center for the outstanding hospitality. [1] V. S. Letokhov, Zh. Eksp. Theor. Fiz. 7, 348 (1968). [2] P. S. Jessen and I. H. Deutsch, Adv. At. Mol. Opt. Phys. 37, 95 (1996). [3] C. Adams and E. Riis, Progress in Quantum Electronics 21,1 (1997). [4] H. Metcalf and P. van der Straten, Laser cooling and trapping (Springer, New York, 1999). [5] P. Verkerk, B. Lounis, C. Salomon, C. Cohen-Tannoudji, J.-Y. Courtois, G. Grynberg, Phys. Rev. Lett. 68, 3861 (1992). [6] P. S. Jessen, C. Gerz, P. D. Lett, W. D. Phillips, S. L. Rolston, R.J.C. Spreeuw, C. I. Westbrook, Phys. Rev. Lett. 69, 49 (1992). [7] A. Hemmerich and T. Ha¨nsch, Phys. Rev. Lett. 70, 410, (1993). [8] A. Hemmerich, C. Zimmermann, and T. Ha¨nsch, Europhys. Lett. 22, 89 (1993). [9] G. Grynberg, B. Lounis, P. Verkerk, J.-Y. Courtois, and C. Salomon, Phys. Rev. Lett. 70, 2249 (1993). [10] M. Weidemu¨ller, A. Hemmerich, A. Go¨rlitz, T. Esslinger, and T. H¨ansch, Phys. Rev. Lett. 75, 4583 (1995). [11] G. Birkl, M. Gatzke, I. H. Deutsch, S.L. Rolston, W. D. Phillips, Phys. Rev. Lett. 75, 2823 (1995). [12] H. F. Hess, Phys. Rev. B 34, 3476 (1986). [13] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269, 198-201 (1995). [14] B. P. Anderson and M. A. Kasevich, Science 282, 1686-1689 (1998). [15] M. Lewenstein et al., Adv. Phys. 56, 243 (2007). [16] A. Derevianko and H. Katori, Rev. Mod. Phys. 83, 331 (2011). [17] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Phys. Rev. B 40, 546 (1989). [18] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. 81, 3108 (1998). [19] M. Greiner, O. Mandel, T. Esslinger, T. W. Ha¨nsch, and I. Bloch, Nature 415, 39-44 (2002). [20] R. P. Feynman, Statistical Mechanics: A Set of lectures. Addison-Wesley Publishing Company (1972). [21] C. Wu, Mod. Phys. Lett. B 23, 1-24 (2009). [22] J. Dalibard, F. Gerbier, G. Juzeliu¯nas, and P. O¨hberg, Rev. Mod. Phys. 83 1523-43 (2011). [23] Y. Tokura and N. Nagaosa, Science 288, 462 (2000). [24] S. Maekawa et al., Physics of Transition Metal Oxides, Springer Series in Solid-State Sciences Vol. 144, Springer (2004). [25] Y. Maeno, H. Hashimoto, K. Yoshida, N. Nishizaki, T. Fujita, J. G. Bednorz, and F. Lichtenberg, Nature 372 532 (1994). [26] A. Isacsson and S. Girvin, Phys. Rev. A 72 053604 (2005). [27] W. V. Liu and C. Wu, Phys. Rev. A, 74 013607 (2006). [28] Z. Cai and C. Wu, Phys. Rev. A 84, 033635 (2011). [29] X. Li, E. Zhao, and W. V. Liu, Phys. Rev. A 83, 063626 (2011). [30] Z. Cai, L.-M. Duan, and C. Wu, Phys. Rev. A 86, 051601(R) (2012). [31] X. Li, Z. Zhang, and W. V. Liu, Phys. Rev. Lett. 108, 175302 (2012). [32] J.-P. Martikainen and J. Larson, Phys. Rev. A 86, 023611 (2012). [33] K. Sun, W. V. Liu, A. Hemmerich, and S. Das Sarma, Nature Physics 8, 67 (2012). [34] F. H´ebert et al., Phys. Rev. B 87, 224505 (2013). [35] B. Liu, X.-L.Yu, and W.-M. Liu, Phys. Rev. A 88, 063605 (2013). [36] F. Pinheiro, G.M. Bruun, J.-P. Martikainen, and J. Larson, Phys. Rev. Lett. 111, 205302, (2013). [37] X. Li, X., A. Paramekanti, A. Hemmerich, and W. V. Liu, Nature Communications 5, 3205 (2014). [38] T. Mu¨ller, S. Fo¨lling, A. Widera, and I. Bloch, Phys. Rev. Lett. 99, 200405 (2007). [39] J. Sebby-Strabley, M. Anderlini, P. S. Jessen, and J. V. Porto, Phys. Rev. A. 73, 033605 (2006). [40] S. F¨olling, S. Trotzky, P. Cheinet, M. Feld, R. Saers, A. Widera, T. Mu¨ller, and I. Bloch, Nature 448, 1029 (2007). [41] M. Anderlini, P. J. Lee, B. L. Brown, J. Sebby-Strabley, W. D. 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label: paragraph

text: or

text: of

text: the

text: lattice

text: geometry

text: should

text: be

text: possible.

text: Only

text: a

text: few

text: examples

text: have

text: been

text: considered

text: yet,

text: which

text: however

text: clearly

text: illustrate

text: the

text: wealth

text: of

text: new

text: physics

text: arising

text: when

text: orbital

text: degrees

text: of

text: freedom

text: come

text: into

text: play

text: in

text: optical

text: lattice

text: systems.

text: Orbital

text: degrees

text: of

text: freedom

text: provide

text: two

text: most

text: interesting

text: consequences:

text: directionality

text: of

text: tunneling

text: and

text: interactions,

text: and

text: orbital

text: degeneracy.

text: As

text: a

text: consequence

text: of

text: the

text: latter,

text: even

text: small

text: interactions

text: can

text: determine

text: the

text: ground

text: state

text: of

text: the

text: system.

text: In

text: our

text: work

text: with

text: bosonic

text: atoms

text: in

text: the

text: second

text: band

text: of

text: a

text: bipartite

text: square

text: lattice,

text: orbital

text: degeneracy

text: is

text: the

text: basis

text: for

text: the

text: emergence

text: of

text: interaction-induced

text: chiral

text: superﬂuidity.

text: An

text: obvious

text: next

text: step

text: is

text: to

text: also

text: study

text: orbital

text: physics

text: in

text: optical

text: lattices

text: with

text: fermions.

text: Excited

text: bands

text: can

text: come

text: with

text: interesting

text: topological

text: structure,

text: which

text: in

text: addition

text: to

text: the

text: physics

text: associated

text: with

text: broken

text: symmetries

text: can

text: give

text: rise

text: to

text: exotic

text: phenomena

text: like,

text: for

text: example,

text: topologically

text: protected

text: chiral

text: edge

text: states.

text: Orbital

text: optical

text: lattices

text: open

text: up

text: new

text: avenues

text: to

text: experimentally

text: study

text: such

text: phenomena

text: also

text: with

text: bosons,

text: which

text: is

text: not

text: possible

text: in

text: electronic

text: condensed

text: matter.

text: ACKNOWLEDGMENTS

text: This

text: work

text: was

text: partially

text: supported

text: by

text: DFG-SFB

text: and

text: the

text: Hamburg

text: Centre

text: for

text: Ultrafast

text: Imaging

text: (CUI).

text: AH

text: is

text: grateful

text: to

text: W.

text: Vincent

text: Liu,

text: Cristiane

text: Morais

text: Smith,

text: Arun

text: Paramekanti,

text: Ludwig

text: Mathey,

text: and

text: Claus

text: Zimmermann

text: for

text: discussions

text: and

text: to

text: the

text: entire

text: team

text: of

text: the

text: Wilczek

text: quantum

text: center

text: for

text: the

text: outstanding

text: hospitality.

text: V.

text: S.

text: Letokhov,

text: Zh.

text: Eksp.

text: Theor.

text: Fiz.

text: P.

text: S.

text: Jessen

text: and

text: I.

text: H.

text: Deutsch,

text: Adv.

text: At.

text: Mol.

text: Opt.

text: Phys.

text: C.

text: Adams

text: and

text: E.

text: Riis,

text: Progress

text: in

text: Quantum

text: Electronics

text: H.

text: Metcalf

text: and

text: P.

text: van

text: der

text: Straten,

text: Laser

text: cooling

text: and

text: trapping

text: (Springer,

text: New

text: York,

text: P.

text: Verkerk,

text: B.

text: Lounis,

text: C.

text: Salomon,

text: C.

text: Cohen-Tannoudji,

text: J.-Y.

text: Courtois,

text: G.

text: Grynberg,

text: Phys.

text: Rev.

text: Lett.

text: P.

text: S.

text: Jessen,

text: C.

text: Gerz,

text: P.

text: D.

text: Lett,

text: W.

text: D.

text: Phillips,

text: S.

text: L.

text: Rolston,

text: R.J.C.

text: Spreeuw,

text: C.

text: I.

text: Westbrook,

text: Phys.

text: Rev.

text: Lett.

text: A.

text: Hemmerich

text: and

text: T.

text: Ha¨nsch,

text: Phys.

text: Rev.

text: Lett.

text: A.

text: Hemmerich,

text: C.

text: Zimmermann,

text: and

text: T.

text: Ha¨nsch,

text: Europhys.

text: Lett.

text: G.

text: Grynberg,

text: B.

text: Lounis,

text: P.

text: Verkerk,

text: J.-Y.

text: Courtois,

text: and

text: C.

text: Salomon,

text: Phys.

text: Rev.

text: Lett.

text: M.

text: Weidemu¨ller,

text: A.

text: Hemmerich,

text: A.

text: Go¨rlitz,

text: T.

text: Esslinger,

text: and

text: T.

text: H¨ansch,

text: Phys.

text: Rev.

text: Lett.

text: G.

text: Birkl,

text: M.

text: Gatzke,

text: I.

text: H.

text: Deutsch,

text: S.L.

text: Rolston,

text: W.

text: D.

text: Phillips,

text: Phys.

text: Rev.

text: Lett.

text: H.

text: F.

text: Hess,

text: Phys.

text: Rev.

text: B

text: M.

text: H.

text: Anderson,

text: J.

text: R.

text: Ensher,

text: M.

text: R.

text: Matthews,

text: C.

text: E.

text: Wieman,

text: and

text: E.

text: A.

text: Cornell,

text: Science

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text: M.

text: Anderlini,

text: P.

text: S.

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text: V.

text: Porto,

text: Phys.

text: Rev.

text: A.

text: S.

text: F¨olling,

text: S.

text: Trotzky,

text: P.

text: Cheinet,

text: M.

text: Feld,

text: R.

text: Saers,

text: A.

text: Widera,

text: T.

text: Mu¨ller,

text: and

text: I.

text: Bloch,

text: Nature

text: M.

text: Anderlini,

text: P.

text: J.

text: Lee,

text: B.

text: L.

text: Brown,

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: where the variable x is not in the variable set Y . Indeed, SDDs properly contain OBDDs, and hence are at least as succinct as OBDDs, while preserving tractability of all key tasks that are tractable on OBDDs. For this reason, they have been used in a variety of applications in artiﬁcial intelligence and probabilistic reasoning, as reported, for instance, by [2, 9]. Not only SDDs are as tractable as OBDDs, but they also tend to be more succinct than OBDDs in practice; in fact, knowledge compilers often produce much smaller SDDs than OBDDs by heuristically leveraging the additional ﬂexibility of variable trees in SDDs with respect to variable orderings in OBDDs [4, 9]. Nonetheless, the basic theoretical question about the relative succinctness of OBDDs and SDDs has been open since Darwiche introduced SDDs [7, 13]: Are SDDs exponentially more succinct than OBDDs? The results in the literature did not even exclude the possibility for OBDDs to polynomially simulate SDDs [16], until recently Razgon proved a quasipolynomial separation [13]. The above question stands, though, as for instance OBDDs could still quasipolynomially simulate SDDs. Contribution. We prove in this article that SDDs are exponentially more succinct than OBDDs. Thus, in particular, OBDDs cannot quasipolynomially simulate SDDs. More precisely, we construct an inﬁnite family of boolean functions such that every member of the family has polynomial compressed SDD size but exponential OBDD size (Theorem 4). Compressed SDDs contain OBDDs,3 and are regarded as a natural SDD class because of their canonicity: two compressed SDDs computing the same function are syntactically equal up to syntactic manipulations preserving polynomial size [7]. The restriction to compressed SDDs makes our result stronger, because general SDDs are believed (despite not known) to be exponentially more succinct than compressed SDDs [2]. We separate compressed SDDs and OBDDs by a function, which we call the generalized hidden weighted bit function because, indeed, it contains the hidden weighted bit function (HWB) as a subfunction. HWB is perhaps the simplest function known to be hard on OBDDs [3]: it computes the subsets of { } 1, . . . , n having size i and containing the number i, for i = 1, . . . , n. It turns out that HWB itself has small (uncompressed) SDDs (Theorem 3), which immediately separates SDDs and OBDDs. The construction, a slight variation of which gives the compressed case (Lemma 1 and Lemma 2), is based on the following two observations. The ﬁrst observation is that HWB can be expressed as a sentential decision of the form (1) by distinguishing the following primes: • for i = 1, . . . , n, the subsets of size i containing the number i (each of these n primes is taken by (cid:62) HWB, so their subs will be equivalent to ); • − the empty subset, and the subsets of size i not containing the number i for i = 1, . . . , n 1 (none ⊥ of these n primes is taken by HWB, so their subs will be equivalent to ). The second observation is that each of the above primes has small OBDD size under any variable ordering (Proposition 2). With these two observations it is fairly straightforward to implement the hidden weighted bit function by a small (uncompressed) SDD (Theorem 3). A direct inspection of our construction allows to straightforwardly derive some facts about compression previously observed in the literature [2], namely that the SDD size may increase exponentially either by compressing SDDs over ﬁxed variable trees, or by conditioning (unboundedly many variables) over ﬁxed variable trees (see Section ). Organization. The article is organized as follows. In Section we present the technical background, culminating in the quasipolynomial separation of SDDs and OBDDs proved by Razgon (Theorem 1). In Section , we separate (uncompressed) SDDs and OBDDs by the hidden weighted bit function (Theorem 3) 3More precisely, compressed SDDs contain reduced OBDDs; see [15, Deﬁnition 1.3.2]. 2 ##LTLine##

text: where

text: the

text: variable

text: x

text: is

text: not

text: in

text: the

text: variable

text: set

text: Y

text: Indeed,

text: SDDs

text: properly

text: contain

text: OBDDs,

text: and

text: hence

text: are

text: at

text: least

text: as

text: succinct

text: as

text: OBDDs,

text: while

text: preserving

text: tractability

text: of

text: all

text: key

text: tasks

text: that

text: are

text: tractable

text: on

text: OBDDs.

text: For

text: this

text: reason,

text: they

text: have

text: been

text: used

text: in

text: a

text: variety

text: of

text: applications

text: in

text: artiﬁcial

text: intelligence

text: and

text: probabilistic

text: reasoning,

text: as

text: reported,

text: for

text: instance,

text: by

text: Not

text: only

text: SDDs

text: are

text: as

text: tractable

text: as

text: OBDDs,

text: but

text: they

text: also

text: tend

text: to

text: be

text: more

text: succinct

text: than

text: OBDDs

text: in

text: practice;

text: in

text: fact,

text: knowledge

text: compilers

text: often

text: produce

text: much

text: smaller

text: SDDs

text: than

text: OBDDs

text: by

text: heuristically

text: leveraging

text: the

text: additional

text: ﬂexibility

text: of

text: variable

text: trees

text: in

text: SDDs

text: with

text: respect

text: to

text: variable

text: orderings

text: in

text: OBDDs

text: Nonetheless,

text: the

text: basic

text: theoretical

text: question

text: about

text: the

text: relative

text: succinctness

text: of

text: OBDDs

text: and

text: SDDs

text: has

text: been

text: open

text: since

text: Darwiche

text: introduced

text: SDDs

text: Are

text: SDDs

text: exponentially

text: more

text: succinct

text: than

text: OBDDs?

text: The

text: results

text: in

text: the

text: literature

text: did

text: not

text: even

text: exclude

text: the

text: possibility

text: for

text: OBDDs

text: to

text: polynomially

text: simulate

text: SDDs

text: until

text: recently

text: Razgon

text: proved

text: a

text: quasipolynomial

text: separation

text: The

text: above

text: question

text: stands,

text: though,

text: as

text: for

text: instance

text: OBDDs

text: could

text: still

text: quasipolynomially

text: simulate

text: SDDs.

text: Contribution.

text: We

text: prove

text: in

text: this

text: article

text: that

text: SDDs

text: are

text: exponentially

text: more

text: succinct

text: than

text: OBDDs.

text: Thus,

text: in

text: particular,

text: OBDDs

text: cannot

text: quasipolynomially

text: simulate

text: SDDs.

text: More

text: precisely,

text: we

text: construct

text: an

text: inﬁnite

text: family

text: of

text: boolean

text: functions

text: such

text: that

text: every

text: member

text: of

text: the

text: family

text: has

text: polynomial

text: compressed

text: SDD

text: size

text: but

text: exponential

text: OBDD

text: size

text: (Theorem

text: Compressed

text: SDDs

text: contain

text: and

text: are

text: regarded

text: as

text: a

text: natural

text: SDD

text: class

text: because

text: of

text: their

text: canonicity:

text: two

text: compressed

text: SDDs

text: computing

text: the

text: same

text: function

text: are

text: syntactically

text: equal

text: up

text: to

text: syntactic

text: manipulations

text: preserving

text: polynomial

text: size

text: The

text: restriction

text: to

text: compressed

text: SDDs

text: makes

text: our

text: result

text: stronger,

text: because

text: general

text: SDDs

text: are

text: believed

text: (despite

text: not

text: known)

text: to

text: be

text: exponentially

text: more

text: succinct

text: than

text: compressed

text: SDDs

text: We

text: separate

text: compressed

text: SDDs

text: and

text: OBDDs

text: by

text: a

text: function,

text: which

text: we

text: call

text: the

text: generalized

text: hidden

text: weighted

text: bit

text: function

text: because,

text: indeed,

text: it

text: contains

text: the

text: hidden

text: weighted

text: bit

text: function

text: (HWB)

text: as

text: a

text: subfunction.

text: HWB

text: is

text: perhaps

text: the

text: simplest

text: function

text: known

text: to

text: be

text: hard

text: on

text: OBDDs

text: it

text: computes

text: the

text: subsets

text: of

text: n

text: having

text: size

text: i

text: and

text: containing

text: the

text: number

text: i,

text: for

text: i

text: n.

text: It

text: turns

text: out

text: that

text: HWB

text: itself

text: has

text: small

text: (uncompressed)

text: SDDs

text: (Theorem

text: which

text: immediately

text: separates

text: SDDs

text: and

text: OBDDs.

text: The

text: construction,

text: a

text: slight

text: variation

text: of

text: which

text: gives

text: the

text: compressed

text: case

text: (Lemma

text: and

text: Lemma

text: is

text: based

text: on

text: the

text: following

text: two

text: observations.

text: The

text: ﬁrst

text: observation

text: is

text: that

text: HWB

text: can

text: be

text: expressed

text: as

text: a

text: sentential

text: decision

text: of

text: the

text: form

text: by

text: distinguishing

text: the

text: following

text: primes:

text: for

text: i

text: n,

text: the

text: subsets

text: of

text: size

text: i

text: containing

text: the

text: number

text: i

text: (each

text: of

text: these

text: n

text: primes

text: is

text: taken

text: by

text: HWB,

text: so

text: their

text: subs

text: will

text: be

text: equivalent

text: to

text: the

text: empty

text: subset,

text: and

text: the

text: subsets

text: of

text: size

text: i

text: not

text: containing

text: the

text: number

text: i

text: for

text: i

text: n

text: (none

text: of

text: these

text: n

text: primes

text: is

text: taken

text: by

text: HWB,

text: so

text: their

text: subs

text: will

text: be

text: equivalent

text: to

text: The

text: second

text: observation

text: is

text: that

text: each

text: of

text: the

text: above

text: primes

text: has

text: small

text: OBDD

text: size

text: under

text: any

text: variable

text: ordering

text: (Proposition

text: With

text: these

text: two

text: observations

text: it

text: is

text: fairly

text: straightforward

text: to

text: implement

text: the

text: hidden

text: weighted

text: bit

text: function

text: by

text: a

text: small

text: (uncompressed)

text: SDD

text: (Theorem

text: A

text: direct

text: inspection

text: of

text: our

text: construction

text: allows

text: to

text: straightforwardly

text: derive

text: some

text: facts

text: about

text: compression

text: previously

text: observed

text: in

text: the

text: literature

text: namely

text: that

text: the

text: SDD

text: size

text: may

text: increase

text: exponentially

text: either

text: by

text: compressing

text: SDDs

text: over

text: ﬁxed

text: variable

text: trees,

text: or

text: by

text: conditioning

text: (unboundedly

text: many

text: variables)

text: over

text: ﬁxed

text: variable

text: trees

text: (see

text: Section

text: Organization.

text: The

text: article

text: is

text: organized

text: as

text: follows.

text: In

text: Section

text: we

text: present

text: the

text: technical

text: background,

text: culminating

text: in

text: the

text: quasipolynomial

text: separation

text: of

text: SDDs

text: and

text: OBDDs

text: proved

text: by

text: Razgon

text: (Theorem

text: In

text: Section

text: we

text: separate

text: (uncompressed)

text: SDDs

text: and

text: OBDDs

text: by

text: the

text: hidden

text: weighted

text: bit

text: function

text: (Theorem

text: precisely,

text: compressed

text: SDDs

text: contain

text: reduced

text: OBDDs;

text: see

text: Deﬁnition

text: ##LTLine##


text: and then modify the construction to separate compressed SDDs and OBDDs (Theorem 4). We discuss our results in Section . 2 Background We collect background notions and facts from the literature [5, 10, 7, 13]. Structured Deterministic NNFs. Let X be a ﬁnite set of variables. Let C be a boolean circuit ⊥ (cid:62) on input variables X, built using fanin 0 constant gates (labelled by or ), fanin 1 negation gates ¬ ∨ ∧ (labelled by ), and unbounded fanin disjunction and conjunction gates (labelled by and ). The unique sink node (outdegree 0) in the underlying directed acyclic graph (DAG) of C is called the output gate of C; source nodes (indegree 0) are called input gates, and are labelled by constants or variables in X; in particular, C is allowed to not read some of the variables in X, see Figure 1 (left). A boolean circuit C on variables X is in negation normal form, in short an NNF, if the gates labelled ¬ by have wires only from input gates. Without loss of generality we assume that NNFs have input ¬ gates labelled by constants or literals on variables in X (and no internal gates labelled by ). { } → { } As usual, an NNF C on input variables X computes a boolean function f : 0, 1 X 0, 1 ; in this ≡ (cid:48) case we also write C f. Two NNFs C and C on the same input variables are equivalent if they ≡ (cid:48) compute the same boolean function; again we write C C . The size of an NNF C, in symbols size(C), is the number of arcs in its underlying DAG. Let f be a L L L boolean function and let be a class of NNFs. The size of f relative to (or, in short, the size of f), L L denoted by (f), is equal to the minimum over the sizes of all circuits in computing f: L { ∈ L ≡ } (f) = min size(C): C , C f . Let C be an NNF on input variables X, and let g be a gate of C. We denote by C the subcircuit g (cid:87) of C having g as its output gate, that is, the circuit whose underlying DAG is the subgraph of the underlying DAG of C induced by the nodes having a directed path to g (labelled as in C). ∨ An NNF C on input variables X is deterministic if, for every -gate g in C, say of the form m g , i=1 i it holds that ∧ ≡ ⊥ C C gi gj ≤ ≤ ⊥ for all 1 i < j m, where we formally regard C , C , and as NNFs on input variables X. We NNF gi gj denote by the class of all deterministic NNFs. d Let Y be a ﬁnite nonempty set of variables. A variable tree (in short, a vtree) for the variable set Y is a rooted, full, ordered, binary tree T whose leaves correspond bijectively to Y ; indeed, we identify each leaf in T with the variable in Y it corresponds to. Let v be an internal node of the vtree T. We let v and v denote respectively the left and right child l r ⊆ of v, and T denote the subtree of T rooted at v. We also let Y Y denote (the variables corresponding v v to) the leaves of T ; clearly T is a vtree for the variable set Y . v v v Let C be an NNF on input variables X, and let T be a vtree for the variable set Y . ∧ We say that C respects T if the following holds. First, every -gate g in C has fanin exactly 2. ∧ Second, let g be an -gate in C having wires from gates h and h . Then there exists an internal node v 1 2 in T such that the input gates of the subcircuit C mention only variables in T and the input gates of h1 vl the subcircuit C mention only variables in T . In this case, we also say that g respects v. h2 vr Note that, in particular, the sets of variables mentioned by C and C are disjoint; it follows that h1 h2 C is decomposable [6]. Also note that, by deﬁnition, if an NNF reading all the variables in a set X is ⊆ structured by a vtree for the variable set Y , then X Y and the inclusion can be strict; see Figure 1. This feature is crucial in our construction (see, for instance, the proof of Theorem 3). NNF A structured NNF is an NNF respecting some vtree. See Figure 1. We denote by the class of s all structured NNFs. 3

text: and

text: then

text: modify

text: the

text: construction

text: to

text: separate

text: compressed

text: SDDs

text: and

text: OBDDs

text: (Theorem

text: We

text: discuss

text: our

text: results

text: in

text: Section

label: section

text: Background

text: We

text: collect

text: background

text: notions

text: and

text: facts

text: from

text: the

text: literature

text: Structured

text: Deterministic

text: NNFs.

text: Let

text: X

text: be

text: a

text: ﬁnite

text: set

text: of

text: variables.

text: Let

text: C

text: be

text: a

text: boolean

text: circuit

text: on

text: input

text: variables

text: X,

text: built

text: using

text: fanin

text: constant

text: gates

text: (labelled

text: by

text: or

text: fanin

text: negation

text: gates

text: (labelled

text: by

text: and

text: unbounded

text: fanin

text: disjunction

text: and

text: conjunction

text: gates

text: (labelled

text: by

text: and

text: The

text: unique

text: sink

text: node

text: (outdegree

text: in

text: the

text: underlying

text: directed

text: acyclic

text: graph

text: (DAG)

text: of

text: C

text: is

text: called

text: the

text: output

text: gate

text: of

text: C;

text: source

text: nodes

text: (indegree

text: are

text: called

text: input

text: gates,

text: and

text: are

text: labelled

text: by

text: constants

text: or

text: variables

text: in

text: X;

text: in

text: particular,

text: C

text: is

text: allowed

text: to

text: not

text: read

text: some

text: of

text: the

text: variables

text: in

text: X,

text: see

text: Figure

text: (left).

text: A

text: boolean

text: circuit

text: C

text: on

text: variables

text: X

text: is

text: in

text: negation

text: normal

text: form,

text: in

text: short

text: an

text: NNF,

text: if

text: the

text: gates

text: labelled

text: by

text: have

text: wires

text: only

text: from

text: input

text: gates.

text: Without

text: loss

text: of

text: generality

text: we

text: assume

text: that

text: NNFs

text: have

text: input

text: gates

text: labelled

text: by

text: constants

text: or

text: literals

text: on

text: variables

text: in

text: X

text: (and

text: no

text: internal

text: gates

text: labelled

text: by

text: As

text: usual,

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text: most two is wrong. Hence, every minimal vertex separator of H is of size at least 3. Now, our claim is to prove that there exists at least one minimal vertex separator of size 3. By Lemma 2, \{ } (cid:116)(cid:117) the degree of every vertex in V (P) v is 3. The graph H is 3-connected. l Theorem 1. For a tree T, the graph H obtained from Algorithm 1 is 3-connected. Further, H is obtained from T by augmenting a minimum set of edges. (cid:100) (cid:101) Proof. The lower bound for the tri-connectivity augmentation of trees is 2l1+l2 by Lemma 1. If the tree T 2 is a path, Algorithm 1 calls Algorithm 2, which converts the tree to a 3-connected graph H by augmenting (cid:100) (cid:101) exactly 2l1+l2 edges (by Lemma 4 and Lemma 6). If the tree T is a non-path tree, the Algorithm 1 calls 2 (cid:100) (cid:101) Algorithm 3, which converts the tree to a 3-connected graph H by augmenting exactly 2l1+l2 edges (by 2 Lemma 5 and Lemma 7). Thus, for a tree T the graph obtained from Algorithm 1 is 3-connected. Further, (cid:116)(cid:117) H is obtained by using a minimum connectivity augmentation set. Therefore, the claim follows. 2.3 Implementation and Analysis of the algorithm | | | | Let T be a tree with the vertex set V (T) such that V (T) = n, with the edge set E(T) such that E(T) = m and l be the number of leaves. The Algorithm Path Augmentation() takes O(1) time in Step 2, O(n) time for Steps 3-8 and O(1) time for Steps 9-12 of Algorithm 2. Thus, the Algorithm Path Augmentation() takes O(n) time. The Algorithm Non-Path Augmentation(): Since, the level ordering can be implemented by the data structure QUEUE, the Step 3 takes O(n) time, takes O(ln) time for Steps 4-7 of Algorithm 3 and O(n) time for the Step 8. We implement the data structure QUEUE for the Steps 8-16, which is used to keep track of marked and unmarked vertices in W. This process ends after visiting all vertices in T and it takes O(n2) time for Steps 10-16 and the algorithm takes O(n) time for Steps 18-38. In total, the Algorithm Non-Path Augmentation() takes O(n2) time. Therefore, Algorithm 1 takes O(n2) time. Thus, for a given tree, a minimum tri-connectivity augmentation set can be found in O(n2) time. 2.4 Trace of the Algorithm (Algorithm 2) Snapshot of Iteration-1 v v v v 1 v v v v v 7 1 v v v v v 7 2 3 4 5 6 2 3 4 5 6 Input Tree: Path of length 7 After augmenting the edge {v , v } 1 7 Snapshot of Iteration-2 Snapshot of Iteration-3 Snapshot of Iteration-4 v1 v2 v3 v4 v5 v6 v7 v1 v2 v3 v4 v5 v6 v7 v1 v2 v3 v4 v5 v6 v7 After augmenting the edge {v1 , v4 } After augmenting the edge {v2 , v4 } After augmenting the edge {v3 , v6 } Snapshot of Iteration-5 The resulting Tri-connected Graph Tree edges v v 1 v2 v3 v4 v5 v6 7 v v Augmented edges After augmenting the edge {v , v } 1 v v v v v 7 4 7 2 3 4 5 6 Fig. 1. Trace of Algorithm 2 when n = 7 8 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: 13 where v is the two-body interaction, deﬁned We begin by looking for the poles of the phys- 2 above. ical conductivity tensor, which correspond to ze- According to SH, the composite fermion con- ros of Det(ρ). To leading order in 1/p, these poles ductivity tensor, for a general value of p, can be are located at the zeros of Det(σcf), which would expressed in terms of an inﬁnite sum of terms in- yield dispersion minima at X = z , R = 0 where z n n volving associated Laguerre polynomials. It the is the n’th zero of the Bessel function J . Here, ∝1 limit of large p, one can employ a semiclassical however, we calculate the momenta ( X) at approximation, where the sums can be carried these dispersion minima to next order in 1/p and out, and one can write the conductivity tensor address the question of their PH symmetry near in closed form in terms of Bessel functions. For half-ﬁlling. − the moment, we shall employ this semiclassical To leading order in R and ∆X = X z , the σcf n approximation, and shall comment later on the tensor is given by corrections that would be expected if one were to − employ the full expressions for σˆcf(q, ω). J2(z ) 1 σcf = i 0 n pR, xx πz2 n J2(z ) 1. Semiclassical calculation of ρˆ σcf = 0 n p∆X, xy πz n − J2(z ) 1 J2(z ) p(∆X)2 The semiclass(cid:20)ical results of SH may be w(cid:21)ritten σcf = i 0 n pR + i 0 n , (77) (in units where e2/h = 1/2π) as yy πzn2J02(zn) π R 2pR −1 πR where the following Bessel function identities σcf = i + J (X)J− (X) , xx πX2 2 2sin(πR) R R were used to reach the above result: pR σxcfy = iσxcfx + Xsin(πR)JR+1(X)J−R(X), J0(z) = J1(cid:48)(z) + J1z(z), p σycfy = σxcfx + isin(πR)JR+1(X)J1−R(X), (75) ∂νJν=1(z) = π2 Y1(z) + J0z(z), ≡ ≡ ∗ | | 2 − where R ω/∆ω and X qR = 2p+1 qkF = = J (z)Y (z) J (z)Y (z). (78) | | ∗ c C B πz 1 0 0 1 2 p q (R is the cyclotron radius of the composite fekrFmionC), ∆ω = ∆B/m, and J (X) is the Bessel We are looking for values of R and ∆X that satisfy c ν function of the ﬁrst kind. The full resistivity is − given by the composition rule Det(ρˆσˆcf) = Det(1 4π(cid:15)ˆσcf) = 0. (79) − − ρ = (σcf) 1 4π(cid:15)ˆ. (76) Using Eq. (77), we ﬁnd the dispersion curve (cid:20) (cid:21) (cid:20) (cid:21) (cid:16) (cid:17) − 4(J2(z ) 1) 2 4J (z ) 2 z 2 − 0 n (pR)2 = 0 n p∆X + n + (1 J2(z )). (80) z2J (z ) z 4 0 n n 0 n n (cid:18) (cid:19) The dispersion minima are then given by which gives √ −z ∆X = 4np, (81) qn ∼ zn2|p|B 1 − 21p . (84) (cid:18) (cid:19) which means that at ν = p/(2p + 1), we have For p = p0 with p0 pos(cid:18)itive, we(cid:19)have ∆B = B/(2p0+ − 1 1) and √ X = z 1 . (82) n 4p ∼ z B − 1 ∼ z ∆B q n 1 n , (85) Since the composite fermion Fermi momentum k n 2p 2p B1/2 F 0 0 is determined solely by the ele(cid:18)ctron d(cid:19)ensity in the − − − while for p = (cid:18)p 1, w(cid:19)e have ∆B(cid:18)= B/((cid:19)2p + 1) HLR theory, we(cid:113)have 0 0 and | | | | √ √ 2 p q ∼ 2√q p 1 X = 1 + , (83) ∼ z B 1 ∼ z B − 1 22pp+B1 B 4p qn 2(pn0 + 1) 1 + 2p0 n2p0 1 2p0 , (86) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: v

text: is

text: the

text: two-body

text: interaction,

text: deﬁned

text: We

text: begin

text: by

text: looking

text: for

text: the

text: poles

text: of

text: the

text: phys-

text: above.

text: ical

text: conductivity

text: tensor,

text: which

text: correspond

text: to

text: ze-

text: According

text: to

text: SH,

text: the

text: composite

text: fermion

text: con-

text: ros

text: of

text: Det(ρ).

text: To

text: leading

text: order

text: in

text: these

text: poles

text: ductivity

text: tensor,

text: for

text: a

text: general

text: value

text: of

text: p,

text: can

text: be

text: are

text: located

text: at

text: the

text: zeros

text: of

text: Det(σcf),

text: which

text: would

text: expressed

text: in

text: terms

text: of

text: an

text: inﬁnite

text: sum

text: of

text: terms

text: in-

text: yield

text: dispersion

text: minima

text: at

text: X

text: z

text: R

text: where

text: z

text: n

text: n

text: volving

text: associated

text: Laguerre

text: polynomials.

text: It

text: the

text: is

text: the

text: n’th

text: zero

text: of

text: the

text: Bessel

text: function

text: J

text: Here,

text: ∝1

text: limit

text: of

text: large

text: p,

text: one

text: can

text: employ

text: a

text: semiclassical

text: however,

text: we

text: calculate

text: the

text: momenta

text: X)

text: at

text: approximation,

text: where

text: the

text: sums

text: can

text: be

text: carried

text: these

text: dispersion

text: minima

text: to

text: next

text: order

text: in

text: and

text: out,

text: and

text: one

text: can

text: write

text: the

text: conductivity

text: tensor

text: address

text: the

text: question

text: of

text: their

text: PH

text: symmetry

text: near

text: in

text: closed

text: form

text: in

text: terms

text: of

text: Bessel

text: functions.

text: For

text: half-ﬁlling.

text: the

text: moment,

text: we

text: shall

text: employ

text: this

text: semiclassical

text: To

text: leading

text: order

text: in

text: R

text: and

text: ∆X

text: X

text: z

text: the

text: σcf

text: n

text: approximation,

text: and

text: shall

text: comment

text: later

text: on

text: the

text: tensor

text: is

text: given

text: by

text: corrections

text: that

text: would

text: be

text: expected

text: if

text: one

text: were

text: to

text: employ

text: the

text: full

text: expressions

text: for

text: σˆcf(q,

text: ω).

text: σcf

text: i

text: n

text: pR,

text: xx

text: n

text: Semiclassical

text: calculation

text: of

text: ρˆ

text: σcf

text: n

text: p∆X,

text: xy

text: πz

text: n

text: The

text: results

text: of

text: SH

text: may

text: be

text: σcf

text: i

text: n

text: pR

text: i

text: n

text: (in

text: units

text: where

text: as

text: yy

text: π

text: R

text: −1

text: πR

text: where

text: the

text: following

text: Bessel

text: function

text: identities

text: σcf

text: i

text: J

text: (X)J−

text: (X)

text: xx

text: R

text: R

text: were

text: used

text: to

text: reach

text: the

text: above

text: result:

text: pR

text: σxcfy

text: iσxcfx

text: p

text: σycfy

text: σxcfx

text: π2

text: where

text: R

text: ω/∆ω

text: and

text: X

text: qR

text: qkF

text: J

text: (z)Y

text: (z)

text: J

text: (z)Y

text: (z).

text: c

text: C

text: B

text: πz

text: p

text: q

text: (R

text: is

text: the

text: cyclotron

text: radius

text: of

text: the

text: composite

text: fekrFmionC),

text: ∆ω

text: ∆B/m,

text: and

text: J

text: (X)

text: is

text: the

text: Bessel

text: We

text: are

text: looking

text: for

text: values

text: of

text: R

text: and

text: ∆X

text: that

text: satisfy

text: c

text: ν

text: function

text: of

text: the

text: ﬁrst

text: kind.

text: The

text: full

text: resistivity

text: is

text: given

text: by

text: the

text: composition

text: rule

text: Det(ρˆσˆcf)

text: ρ

text: (σcf)

text: Using

text: Eq.

text: we

text: ﬁnd

text: the

text: dispersion

text: curve

text: (z

text: z

text: n

text: n

text: p∆X

text: n

text: (z

text: z

text: n

text: n

text: n

text: n

text: The

text: dispersion

text: minima

text: are

text: then

text: given

text: by

text: which

text: gives

text: −z

text: ∆X

text: qn

text: which

text: means

text: that

text: at

text: ν

text: we

text: have

text: For

text: p

text: with

text: ∆B

text: and

text: X

text: z

text: n

text: z

text: B

text: z

text: ∆B

text: q

text: n

text: n

text: Since

text: the

text: composite

text: fermion

text: Fermi

text: momentum

text: k

text: n

text: F

text: is

text: determined

text: solely

text: by

text: the

text: in

text: the

text: while

text: for

text: p

text: have

text: HLR

text: theory,

text: and

text: p

text: q

text: p

text: X

text: z

text: B

text: z

text: B

text: B

text: qn

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 18 As in the HLR theory the presence of potential fermion conductivity tensor σˆDF(q, ω). Within the disorder will cause ﬂuctuations in electron den- semiclassical approximation, these zeros occur at sity, which will lead to ﬂuctuations in the eﬀec- ω = 0, if tive ﬁeld b(r) proportional to the self-consistent electric potential V (r). Potential ﬂuctuations do B1/2 q = z | |. (113) not lead to ﬂuctuations in a or in the eﬀective n n 2 p 0 DF electric ﬁeld e. Therefore, if the potential ﬂuctu- These values are clearly PH symmetric and are ations are statistically PH symmetric, so that all identical to the results obtained using HLR in odd moments of b are zero, the Dirac fermions will | | Subsection IV A 1, through order ∆B 2. This re- see a ﬁeld that is statistically time-reversal sym- sult for Dirac composite fermions was also ob- metric, and ρˆDF will be purely diagonal. There- | | tained in [58], to lowest order in ∆B , with careful fore, we recover σ = 1/4π as required by particle xy attention to interaction eﬀects. hole symmetry. As in the HLR case, the actual locations of the At a ﬁnite frequency ω, in the absence of impu- magnetoroton minima in the Dirac theory will be rities, in the RPA, the resistivity tensor for Dirac shifted from these values (by amounts small com- fermions is readily calculated to be pared to q ), and the frequency values will be − ∗ n ρˆDF = iωm /n (111) shifted from zero, due to interaction eﬀects and el to corrections to the semiclassical theory, but all ∗ where m = kF /vD. As this is purely diagonal, the such shifts should be symmetric in ∆B. ac Hall conductivity remains ﬁxed at the value Finally, we discuss properties of the Dirac La- required by PH symmetry. However, the diago- grangian (103) in the case where the Dirac mass nal conductivity σxx predicted by (109) does not mD is not set equal to zero, so the theory is agree with the result σxx(ω) = 0, which is required not explicitly PH symmetric. As was observed by Kohn→’s theorem in the limit where the electron by Son(cid:29)[1], in the non-relativistic limit, where mass m 0, and electrons are restricted to the m v k , the Dirac action reduces precisely D D F lowest Landau level. As remarked above, this can to the HLR action (4), after a redeﬁnition of the → be corrected, beyond the RPA, by including the gauge ﬁeld, (a a + A ). As we have seen, the µ µ µ eﬀects of Landau interaction parameters F±1. HLR theory and the massless Dirac theory give Using the Son-Dirac Lagrangian for mD = 0, identical results for long-wavelength low-energy one predicts that fractional quantized Hall states properties in the limit of ν = 1/2, so that PH sym- should occur when metry reappears in this case. We ﬁnd that there B is a similar emergent PH symmetry for interme- ∆B = , (112) diate values of m . Since the Lagrangian for the 2p D DF Dirac theory with ﬁnite m includes a Chern- D where p is half of an odd integer, either positive Simons term identical to that in the HLR theory, DF or negative. This condition is obviously PH sym- the relations between the composite fermion and metric and it is equivalent to the HLR prediction, the electronic response functions are identical in with the identiﬁcation p = p + 1/2. The shift in the two theories. The semiclassical theory for DF the choice of indexing reﬂects the presence of a the minima of the magnetoroton spectra take the Berry phase of π for the Dirac fermions at the same form as we found in Section III D above, Fermi energy. The energy gaps in the quantized which implies that the spectrum is, again, sym- | | Hall states are given, within RPA by Eq. (14) metric in ∆B, at least through order ∆B 2. Sim- ∗ with m replaced by m = k /v . As remarked ilarly, we ﬁnd that the Hall conductance in the F D previously, the gaps will obey PH symmetry pro- presence of impurities at ν = 1/2 is ﬁxed at 1/4π, vided that the velocity v is assumed to depend at least through order (1/l )2, under the same D cf on the magnetic ﬁeld, and not on the electron conditions that we assumed in the analysis of density, or more generally, if v is assumed to be HLR in Section III. D an even function of ∆B. An apparent diﬀerence between HLR and a According to PH symmetry, the magneto-exciton Dirac theory with ﬁnite m is that in the latter D spectra should also be independent of the sign of case the fermions near the Fermi energy have a ∆B. The positions of the magnetoroton minima non-zero Berry curvature. This Berry curvature may be found, to lowest order in ∆B, by tracking is the same as that which results from spin-orbit the dispersion of poles in the electrical conduc- coupling in a semiconductor, which, as we have tivity σˆ(q, ω), as was done in Subsection IV A 1 remarked, is responsible for side-jump contribu- above in the HLR picture. Taking into account tions to the anomalous Hall eﬀect in semiconduc- Eq. (109), we see that within the Dirac descrip- tor models. However, in the limit of scattering tion, poles in σˆ(q, ω) coincide with the occurrence wavevectors q much smaller than k , which we F of a zero in the determinant of the composite have assumed in our analysis, the matrix element ##LTLine## ##LTLine##

text: As

text: in

text: the

text: HLR

text: theory

text: the

text: presence

text: of

text: potential

text: fermion

text: conductivity

text: tensor

text: σˆDF(q,

text: ω).

text: Within

text: the

text: disorder

text: will

text: cause

text: ﬂuctuations

text: in

text: electron

text: den-

text: semiclassical

text: approximation,

text: these

text: zeros

text: occur

text: at

text: sity,

text: which

text: will

text: lead

text: to

text: ﬂuctuations

text: in

text: the

text: eﬀec-

text: ω

text: if

text: tive

text: ﬁeld

text: b(r)

text: proportional

text: to

text: the

text: self-consistent

text: electric

text: potential

text: V

text: (r).

text: Potential

text: ﬂuctuations

text: do

text: q

text: z

text: not

text: lead

text: to

text: ﬂuctuations

text: in

text: a

text: or

text: in

text: the

text: eﬀective

text: n

text: n

text: p

text: DF

text: electric

text: ﬁeld

text: e.

text: Therefore,

text: if

text: the

text: potential

text: ﬂuctu-

text: These

text: values

text: are

text: clearly

text: PH

text: symmetric

text: and

text: are

text: ations

text: are

text: statistically

text: PH

text: symmetric,

text: so

text: that

text: all

text: identical

text: to

text: the

text: results

text: obtained

text: using

text: HLR

text: in

text: odd

text: moments

text: of

text: b

text: are

text: zero,

text: the

text: Dirac

text: fermions

text: will

text: Subsection

text: IV

text: A

text: through

text: order

text: ∆B

text: This

text: re-

text: see

text: a

text: ﬁeld

text: that

text: is

text: statistically

text: time-reversal

text: sym-

text: sult

text: for

text: Dirac

text: composite

text: fermions

text: was

text: also

text: ob-

text: metric,

text: and

text: ρˆDF

text: will

text: be

text: purely

text: diagonal.

text: There-

text: tained

text: in

text: to

text: lowest

text: order

text: in

text: ∆B

text: with

text: careful

text: fore,

text: we

text: recover

text: σ

text: as

text: required

text: by

text: particle

text: xy

text: attention

text: to

text: interaction

text: eﬀects.

text: hole

text: symmetry.

text: As

text: in

text: the

text: HLR

text: case,

text: the

text: actual

text: locations

text: of

text: the

text: At

text: a

text: ﬁnite

text: frequency

text: ω,

text: in

text: the

text: absence

text: of

text: impu-

text: magnetoroton

text: minima

text: in

text: the

text: Dirac

text: theory

text: will

text: be

text: rities,

text: in

text: the

text: RPA,

text: the

text: resistivity

text: tensor

text: for

text: Dirac

text: shifted

text: from

text: these

text: values

text: (by

text: amounts

text: small

text: com-

text: fermions

text: is

text: readily

text: calculated

text: to

text: be

text: pared

text: to

text: q

text: and

text: the

text: frequency

text: values

text: will

text: be

text: n

text: ρˆDF

text: iωm

text: /n

text: shifted

text: from

text: zero,

text: due

text: to

text: interaction

text: eﬀects

text: and

text: el

text: to

text: corrections

text: to

text: the

text: semiclassical

text: theory,

text: but

text: all

text: where

text: m

text: kF

text: /vD.

text: As

text: this

text: is

text: purely

text: diagonal,

text: the

text: such

text: shifts

text: should

text: be

text: symmetric

text: in

text: ∆B.

text: ac

text: Hall

text: conductivity

text: remains

text: ﬁxed

text: at

text: the

text: value

text: Finally,

text: we

text: discuss

text: properties

text: of

text: the

text: Dirac

text: La-

text: required

text: by

text: PH

text: symmetry.

text: However,

text: the

text: diago-

text: grangian

text: in

text: the

text: case

text: where

text: the

text: Dirac

text: mass

text: nal

text: conductivity

text: σxx

text: predicted

text: by

text: does

text: not

text: mD

text: is

text: not

text: set

text: equal

text: to

text: zero,

text: so

text: the

text: theory

text: is

text: agree

text: with

text: the

text: result

text: σxx(ω)

text: which

text: is

text: required

text: not

text: explicitly

text: PH

text: symmetric.

text: As

text: was

text: observed

text: by

text: Kohn→’s

text: theorem

text: in

text: the

text: limit

text: where

text: the

text: electron

text: by

text: in

text: the

text: non-relativistic

text: limit,

text: where

text: mass

text: m

text: and

text: electrons

text: are

text: restricted

text: to

text: the

text: m

text: v

text: k

text: the

text: Dirac

text: action

text: reduces

text: precisely

text: D

text: D

text: F

text: lowest

text: Landau

text: level.

text: As

text: remarked

text: above,

text: this

text: can

text: to

text: the

text: HLR

text: action

text: after

text: a

text: redeﬁnition

text: of

text: the

text: be

text: corrected,

text: beyond

text: the

text: RPA,

text: by

text: including

text: the

text: gauge

text: ﬁeld,

text: (a

text: a

text: A

text: As

text: we

text: have

text: seen,

text: the

text: µ

text: µ

text: µ

text: eﬀects

text: of

text: Landau

text: interaction

text: parameters

text: HLR

text: theory

text: and

text: the

text: massless

text: Dirac

text: theory

text: give

text: Using

text: the

text: Son-Dirac

text: Lagrangian

text: for

text: mD

text: identical

text: results

text: for

text: long-wavelength

text: low-energy

text: one

text: predicts

text: that

text: fractional

text: quantized

text: Hall

text: states

text: properties

text: in

text: the

text: limit

text: of

text: ν

text: so

text: that

text: PH

text: sym-

text: should

text: occur

text: when

text: metry

text: reappears

text: in

text: this

text: case.

text: We

text: ﬁnd

text: that

text: there

text: B

text: is

text: a

text: similar

text: emergent

text: PH

text: symmetry

text: for

text: interme-

text: ∆B

text: diate

text: values

text: of

text: m

text: Since

text: the

text: Lagrangian

text: for

text: the

text: D

text: DF

text: Dirac

text: theory

text: with

text: ﬁnite

text: m

text: includes

text: a

text: Chern-

text: D

text: where

text: p

text: is

text: half

text: of

text: an

text: odd

text: integer,

text: either

text: positive

text: Simons

text: term

text: identical

text: to

text: that

text: in

text: the

text: HLR

text: theory,

text: DF

text: or

text: negative.

text: This

text: condition

text: is

text: obviously

text: PH

text: sym-

text: the

text: relations

text: between

text: the

text: composite

text: fermion

text: and

text: metric

text: and

text: it

text: is

text: equivalent

text: to

text: the

text: HLR

text: prediction,

text: the

text: electronic

text: response

text: functions

text: are

text: identical

text: in

text: with

text: the

text: identiﬁcation

text: p

text: p

text: The

text: shift

text: in

text: the

text: two

text: theories.

text: The

text: semiclassical

text: theory

text: for

text: DF

text: the

text: choice

text: of

text: indexing

text: reﬂects

text: the

text: presence

text: of

text: a

text: the

text: minima

text: of

text: the

text: magnetoroton

text: spectra

text: take

text: the

text: Berry

text: phase

text: of

text: π

text: for

text: the

text: Dirac

text: fermions

text: at

text: the

text: same

text: form

text: as

text: we

text: found

text: in

text: Section

text: III

text: D

text: above,

text: Fermi

text: energy.

text: The

text: energy

text: gaps

text: in

text: the

text: quantized

text: which

text: implies

text: that

text: the

text: spectrum

text: is,

text: again,

text: sym-

text: Hall

text: states

text: are

text: given,

text: within

text: RPA

text: by

text: Eq.

text: metric

text: in

text: ∆B,

text: at

text: least

text: through

text: order

text: ∆B

text: Sim-

text: with

text: m

text: replaced

text: by

text: m

text: k

text: /v

text: As

text: remarked

text: ilarly,

text: we

text: ﬁnd

text: that

text: the

text: Hall

text: conductance

text: in

text: the

text: F

text: D

text: previously,

text: the

text: gaps

text: will

text: obey

text: PH

text: symmetry

text: pro-

text: presence

text: of

text: impurities

text: at

text: ν

text: is

text: ﬁxed

text: at

text: vided

text: that

text: the

text: velocity

text: v

text: is

text: assumed

text: to

text: depend

text: at

text: least

text: through

text: order

text: under

text: the

text: same

text: D

text: cf

text: on

text: the

text: magnetic

text: ﬁeld,

text: and

text: not

text: on

text: the

text: electron

text: conditions

text: that

text: we

text: assumed

text: in

text: the

text: analysis

text: of

text: density,

text: or

text: more

text: generally,

text: if

text: v

text: is

text: assumed

text: to

text: be

text: HLR

text: in

text: Section

text: III.

text: D

text: an

text: even

text: function

text: of

text: ∆B.

text: An

text: apparent

text: diﬀerence

text: between

text: HLR

text: and

text: a

text: According

text: to

text: PH

text: symmetry,

text: the

text: magneto-exciton

text: Dirac

text: theory

text: with

text: ﬁnite

text: m

text: is

text: that

text: in

text: the

text: latter

text: D

text: spectra

text: should

text: also

text: be

text: independent

text: of

text: the

text: sign

text: of

text: case

text: the

text: fermions

text: near

text: the

text: Fermi

text: energy

text: have

text: a

text: ∆B.

text: The

text: positions

text: of

text: the

text: magnetoroton

text: minima

text: non-zero

text: Berry

text: curvature.

text: This

text: Berry

text: curvature

text: may

text: be

text: found,

text: to

text: lowest

text: order

text: in

text: ∆B,

text: by

text: tracking

text: is

text: the

text: same

text: as

text: that

text: which

text: results

text: from

text: spin-orbit

text: the

text: dispersion

text: of

text: poles

text: in

text: the

text: electrical

text: conduc-

text: coupling

text: in

text: a

text: semiconductor,

text: which,

text: as

text: we

text: have

text: tivity

text: σˆ(q,

text: ω),

text: as

text: was

text: done

text: in

text: Subsection

text: IV

text: A

text: remarked,

text: is

text: responsible

text: for

text: side-jump

text: contribu-

text: above

text: in

text: the

text: HLR

text: picture.

text: Taking

text: into

text: account

text: tions

text: to

text: the

text: anomalous

text: Hall

text: eﬀect

text: in

text: semiconduc-

text: Eq.

text: we

text: see

text: that

text: within

text: the

text: Dirac

text: descrip-

text: tor

text: models.

text: However,

text: in

text: the

text: limit

text: of

text: scattering

text: tion,

text: poles

text: in

text: σˆ(q,

text: ω)

text: coincide

text: with

text: the

text: occurrence

text: wavevectors

text: q

text: much

text: smaller

text: than

text: k

text: which

text: we

text: F

text: of

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text: zero

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text: determinant

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text: the

text: composite

text: have

text: assumed

text: in

text: our

text: analysis,

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text: matrix

text: element

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text: ##LTLine##