text: ∗ Spin Liquid Condensate of Spinful Bosons Biao Lian1 and Shoucheng Zhang1 (Dated: May 26, 2020) We introduce the concept of a bosonic spin liquid condensate (SLC), where spinful bosons in a lattice form a zero-temperature spin disordered charge condensate that preserves the spin rotation symmetry, but breaks the U(1) symmetry due to a spinless order parameter with charge one. It has an energy gap to all the spin excitations. We show that such SLC states can be realized in a system ≥ of spin S 2 bosons. In particular, we analyze the SLC phase diagram in the spin 2 case using a mean-ﬁeld variational wave function method. We show there is a direct analogy between the SLC and the resonating-valence-bond (RVB) state. Pursuing new states of matter is always one of the prin- We calculate explicitly the SLC phase diagram for spin 2 cipal aims of condensed matter physics. Compared to via a mean-ﬁeld variational wave function method, and fermions which have various novel phases, bosons appear study the spin excitations and the Goldstone mode of the to be somewhat trivial. The zero temperature phases of state. A lot more physics in SLC is awaiting exploration. bosons traditionally fall into three classes: the Mott insu- In a Mott insulator where interaction dominates, every lator (MI), the Bose glass, and the Bose-Einstein conden- site is in the lowest on-site state with a deﬁnite parti- sate (BEC) on a single-particle state [1]. For example, cle number n. As the hopping between sites increases, the spinful bosons in ultra-cold atom systems [2] usually a coherent superposition of several on-site states with condense onto a particular spinor state, forming a spinor diﬀerent particle numbers is preferred to gain the kinetic BEC that spontaneously breaks both the U(1) and the energy, which is the BEC state. For spin S bosons, the spin rotation SU(2) symmetries [3–11]. Whether bosons coherent superposition usually induces a non-vanishing (cid:104) (cid:105) can form other zero temperature phases such as a conden- spinor order parameter ϕ = ψ , where ψ is the m i,m i,m sate not on a single particle state (non-SPS), is an inter- boson ﬁeld operator, and one obtains a spinor BEC esting question that has been discussed extensively[12]. breaking the spin rotation symmetry. However, suppose In particular, a condensate of spinful bosons is a non- the lowest two on-site states are spin singlet states close SPS condensate if it breaks only the U(1) symmetry but to each other in energy, while the energies of all the preserves the spin rotation SU(2) symmetry [12]. So far, other states are much higher. When the hopping is not however, the attempts along this direction never come too large, the system may prefer a superposition of the with a rigorous example. Early proposed states like the two singlet states only, forming a spin singlet condensate (cid:104) (cid:105) spin-paired condensate of spin 1 bosons [13] are shown with ψ = 0 ensured by the spin rotation symmetry. i,m to be unstable “Shr¨odinger cat” states with no spin gap It is nonetheless possible to construct a non-vanishing in the thermodynamic limit [12, 14], though they may order parameter that carries a U(1) charge and total spin be favored at a ﬁnite temperature [15, 16]. Later works zero (see Eq. (2)). also discussed the possibility of stabilizing such conden- To ﬁnd such on-site state spectrums explicitly, we ex- sates with interactions and lattices, but gave no deﬁnite amine the spin Bose Hubbard model widely used for de- answer [17–19]. Looking for non-SPS condensates with scribing spin S bosons in a lattice. The Hamiltonian spin rotational invariance is nonetheless interesting, and H = HI + Ht can be written in(cid:34)the following t(cid:35)wo parts: is in some sense reminiscent of realizing spin liquids in (cid:88) (cid:88) (cid:88) fermionic systems. Such condensates may give a lot of nontrivial physics such as the spin-charge separation, in H = −µ(cid:88) nˆ(cid:16) + 1 S U(cid:17) PˆS (i) , analogy to those in spin liquids [20]. I i 2 2J 2J i i J=0 (1) In this letter, we introduce the concept of a bosonic − † H = tψ ψ + h.c. , spin liquid condensate (SLC), which is a robust non-SPS t (cid:104) (cid:105) i,m j,m ij ,m spin disordered charge condensate at zero temperature. It is deﬁned as a state of spinful bosons that preserves the ≥ where µ is the chemical potential, U 0 is the spin rotation symmetry with a spin gap, but has a spon- 2J on-site Hubbard interaction energy between two bosons taneously broken U(1) symmetry due to a locally deﬁned of total spin 2J, and t is the nearest hopping ampli- spinless order parameter that carries a U(1) charge. We (cid:80) ≥ tude. ψ is the boson ﬁeld operator, where i and show a way to construct an SLC for spin S 2 bosons in i,m j label the lattice sites, m denotes the spin z compo- a lattice. The SLC has a direct analogy to the resonat- † nent. nˆ = ψ (cid:80)ψ is the particle number on site ing valence bond (RVB) state for cuprate superconduc- i m i,m i,m P tors proposed by Anderson [21, 22], where bosons are free i. The non-negative projection operator ˆS (i) is de- P A† A 2AJ to move with their spins conﬁned in short range RVBs. ﬁned as ˆS (i) = (i) (i) with (i) = 2J m 2Jm 2Jm 2Jm ##LTLine##

text: Spin

label: title

text: Liquid

text: Condensate

text: of

text: Spinful

text: Bosons

text: Biao

label: author

text: and

text: Shoucheng

text: (Dated:

text: May

label: date

text: We

text: introduce

text: the

text: concept

text: of

text: a

text: bosonic

text: spin

text: liquid

text: condensate

text: (SLC),

text: where

text: spinful

text: bosons

text: in

text: a

text: lattice

text: form

text: a

text: zero-temperature

text: spin

text: disordered

text: charge

text: condensate

text: that

text: preserves

text: the

text: spin

text: rotation

text: symmetry,

text: but

text: breaks

text: the

text: symmetry

text: due

text: to

text: a

text: spinless

text: order

text: parameter

text: with

text: charge

text: one.

text: It

text: has

text: an

text: energy

text: gap

text: to

text: all

text: the

text: spin

text: excitations.

text: We

text: show

text: that

text: such

text: SLC

text: states

text: can

text: be

text: realized

text: in

text: a

text: system

text: of

text: spin

text: S

text: bosons.

text: In

text: particular,

text: we

text: analyze

text: the

text: SLC

text: phase

text: diagram

text: in

text: the

text: spin

text: case

text: using

text: a

text: mean-ﬁeld

text: variational

text: wave

text: function

text: method.

text: We

text: show

text: there

text: is

text: a

text: direct

text: analogy

text: between

text: the

text: SLC

text: and

text: the

text: resonating-valence-bond

text: (RVB)

text: state.

text: Pursuing

text: new

text: states

text: of

text: matter

text: is

text: always

text: one

text: of

text: the

text: prin-

text: We

text: calculate

text: explicitly

text: the

text: SLC

text: phase

text: diagram

text: for

text: spin

text: cipal

text: aims

text: of

text: condensed

text: matter

text: physics.

text: Compared

text: to

text: via

text: a

text: mean-ﬁeld

text: variational

text: wave

text: function

text: method,

text: and

text: fermions

text: which

text: have

text: various

text: novel

text: phases,

text: bosons

text: appear

text: study

text: the

text: spin

text: excitations

text: and

text: the

text: Goldstone

text: mode

text: of

text: the

text: to

text: be

text: somewhat

text: trivial.

text: The

text: zero

text: temperature

text: phases

text: of

text: state.

text: A

text: lot

text: more

text: physics

text: in

text: SLC

text: is

text: awaiting

text: exploration.

text: bosons

text: traditionally

text: fall

text: into

text: three

text: classes:

text: the

text: Mott

text: insu-

text: In

text: a

text: Mott

text: insulator

text: where

text: interaction

text: dominates,

text: every

text: lator

text: (MI),

text: the

text: Bose

text: glass,

text: and

text: the

text: Bose-Einstein

text: conden-

text: site

text: is

text: in

text: the

text: lowest

text: on-site

text: state

text: with

text: a

text: deﬁnite

text: parti-

text: sate

text: (BEC)

text: on

text: a

text: single-particle

text: state

text: For

text: example,

text: cle

text: number

text: n.

text: As

text: the

text: hopping

text: between

text: sites

text: increases,

text: the

text: spinful

text: bosons

text: in

text: ultra-cold

text: atom

text: systems

text: usually

text: a

text: coherent

text: superposition

text: of

text: several

text: on-site

text: states

text: with

text: condense

text: onto

text: a

text: particular

text: spinor

text: state,

text: forming

text: a

text: spinor

text: diﬀerent

text: particle

text: numbers

text: is

text: preferred

text: to

text: gain

text: the

text: kinetic

text: BEC

text: that

text: spontaneously

text: breaks

text: both

text: the

text: and

text: the

text: energy,

text: which

text: is

text: the

text: BEC

text: state.

text: For

text: spin

text: S

text: bosons,

text: the

text: spin

text: rotation

text: symmetries

text: Whether

text: bosons

text: coherent

text: superposition

text: usually

text: induces

text: a

text: non-vanishing

text: can

text: form

text: other

text: zero

text: temperature

text: phases

text: such

text: as

text: a

text: conden-

text: spinor

text: order

text: parameter

text: ϕ

text: ψ

text: where

text: ψ

text: is

text: the

text: m

text: i,m

text: i,m

text: sate

text: not

text: on

text: a

text: single

text: particle

text: state

text: (non-SPS),

text: is

text: an

text: inter-

text: boson

text: ﬁeld

text: operator,

text: and

text: one

text: obtains

text: a

text: spinor

text: BEC

text: esting

text: question

text: that

text: has

text: been

text: discussed

text: breaking

text: the

text: spin

text: rotation

text: symmetry.

text: However,

text: suppose

text: In

text: particular,

text: a

text: condensate

text: of

text: spinful

text: bosons

text: is

text: a

text: non-

text: the

text: lowest

text: two

text: on-site

text: states

text: are

text: spin

text: singlet

text: states

text: close

text: SPS

text: condensate

text: if

text: it

text: breaks

text: only

text: the

text: symmetry

text: but

text: to

text: each

text: other

text: in

text: energy,

text: while

text: the

text: energies

text: of

text: all

text: the

text: preserves

text: the

text: spin

text: rotation

text: symmetry

text: So

text: far,

text: other

text: states

text: are

text: much

text: higher.

text: When

text: the

text: hopping

text: is

text: not

text: however,

text: the

text: attempts

text: along

text: this

text: direction

text: never

text: come

text: too

text: large,

text: the

text: system

text: may

text: prefer

text: a

text: superposition

text: of

text: the

text: with

text: a

text: rigorous

text: example.

text: Early

text: proposed

text: states

text: like

text: the

text: two

text: singlet

text: states

text: only,

text: forming

text: a

text: spin

text: singlet

text: condensate

text: spin-paired

text: condensate

text: of

text: spin

text: bosons

text: are

text: shown

text: with

text: ψ

text: ensured

text: by

text: the

text: spin

text: rotation

text: symmetry.

text: i,m

text: to

text: be

text: unstable

text: “Shr¨odinger

text: cat”

text: states

text: with

text: no

text: spin

text: gap

text: It

text: is

text: nonetheless

text: possible

text: to

text: construct

text: a

text: non-vanishing

text: in

text: the

text: thermodynamic

text: limit

text: though

text: they

text: may

text: order

text: parameter

text: that

text: carries

text: a

text: charge

text: and

text: total

text: spin

text: be

text: favored

text: at

text: a

text: ﬁnite

text: temperature

text: Later

text: works

text: zero

text: (see

text: Eq.

text: also

text: discussed

text: the

text: possibility

text: of

text: stabilizing

text: such

text: conden-

text: To

text: ﬁnd

text: such

text: on-site

text: state

text: spectrums

text: explicitly,

text: we

text: ex-

text: sates

text: with

text: interactions

text: and

text: lattices,

text: but

text: gave

text: no

text: deﬁnite

text: amine

text: the

text: spin

text: Bose

text: Hubbard

text: model

text: widely

text: used

text: for

text: de-

text: answer

text: Looking

text: for

text: non-SPS

text: condensates

text: with

text: scribing

text: spin

text: S

text: bosons

text: in

text: a

text: lattice.

text: The

text: Hamiltonian

text: spin

text: rotational

text: invariance

text: is

text: nonetheless

text: interesting,

text: and

text: H

text: HI

text: Ht

text: can

text: be

text: written

text: following

text: parts:

text: is

text: in

text: some

text: sense

text: reminiscent

text: of

text: realizing

text: spin

text: liquids

text: in

text: fermionic

text: systems.

text: Such

text: condensates

text: may

text: give

text: a

text: lot

text: of

text: nontrivial

text: physics

text: such

text: as

text: the

text: spin-charge

text: separation,

text: in

text: H

text: S

text: PˆS

text: (i)

text: analogy

text: to

text: those

text: in

text: spin

text: liquids

text: I

text: i

text: i

text: i

text: In

text: this

text: letter,

text: we

text: introduce

text: the

text: concept

text: of

text: a

text: bosonic

text: H

text: tψ

text: ψ

text: h.c.

text: spin

text: liquid

text: condensate

text: (SLC),

text: which

text: is

text: a

text: robust

text: non-SPS

text: t

text: i,m

text: j,m

text: ij

text: ,m

text: spin

text: disordered

text: charge

text: condensate

text: at

text: zero

text: temperature.

text: It

text: is

text: deﬁned

text: as

text: a

text: state

text: of

text: spinful

text: bosons

text: that

text: preserves

text: the

text: where

text: µ

text: is

text: the

text: chemical

text: potential,

text: U

text: is

text: the

text: spin

text: rotation

text: symmetry

text: with

text: a

text: spin

text: gap,

text: but

text: has

text: a

text: spon-

text: on-site

text: Hubbard

text: interaction

text: energy

text: between

text: two

text: bosons

text: taneously

text: broken

text: symmetry

text: due

text: to

text: a

text: locally

text: deﬁned

text: of

text: total

text: spin

text: and

text: t

text: is

text: the

text: nearest

text: hopping

text: ampli-

text: spinless

text: order

text: parameter

text: that

text: carries

text: a

text: charge.

text: We

text: tude.

text: ψ

text: is

text: the

text: boson

text: ﬁeld

text: operator,

text: where

text: i

text: and

text: show

text: a

text: way

text: to

text: construct

text: an

text: SLC

text: for

text: spin

text: S

text: bosons

text: in

text: i,m

text: j

text: label

text: the

text: lattice

text: sites,

text: m

text: denotes

text: the

text: spin

text: z

text: compo-

text: a

text: lattice.

text: The

text: SLC

text: has

text: a

text: direct

text: analogy

text: to

text: the

text: resonat-

text: nent.

text: nˆ

text: ψ

text: is

text: the

text: particle

text: number

text: on

text: site

text: ing

text: valence

text: bond

text: (RVB)

text: state

text: for

text: cuprate

text: superconduc-

text: i

text: m

text: i,m

text: i,m

text: P

text: tors

text: proposed

text: by

text: Anderson

text: where

text: bosons

text: are

text: free

text: i.

text: The

text: non-negative

text: projection

text: operator

text: ˆS

text: (i)

text: is

text: de-

text: P

text: A†

text: A

text: to

text: move

text: with

text: their

text: spins

text: conﬁned

text: in

text: short

text: range

text: RVBs.

text: ﬁned

text: as

text: ˆS

text: (i)

text: (i)

text: (i)

text: with

text: (i)

text: m

text: ##LTLine##


text: 2 Then, for the SLC ground state to be favored, the tran- | (cid:105) sition amplitude between the two on-site states 2, 0, 0 | (cid:105) i and 3, 0, 0 must be large enough. The transition am- i plitude originates from the hopping energy H . How- | (cid:105) | (cid:105) t ever, the tunneling from 2, 0, 0 to 3, 0, 0 cannot be i i achieved by a single hop, since a single hop will change a singlet state to a state with total spin l = 2. For the transition to occur, the site must hop with nearby sites for at least 3 times, during which the site must go to two intermediate states with energy of order E , as is a shown in Fig. 1. In perturbation theory, this mechanism gives us a eﬀective hopping amplitude between the singlet dimer state and the trimer state t = t3/E2. When the | eff | ∼ a hopping amplitude increases to t /∆ 1, the system FIG. 1. The on-site energy levels computed for U = 3U = eff − − 4 2 would prefer a coherent superposition of the dimer and 30U0 and µ = µ0 ∆, where µ0 = 3U2 U0. The x and y | | (cid:28) axes are the particle number n and the energy E in units of trimer states.|On th| e(cid:28)other hand, the condition ∆ Ea U0 respectively. Enl represents the energy of the on-site state ensures that t/Ea 1, so the superposition with any with particle number n and total spin l. The arrows show other states is not yet favorable, and the system will form | (cid:105) | (cid:105) the transition process from 2, 0, 0 i to 3, 0, 0 i via three time an SLC. We can write down a spin-rotationally invariant h(cid:80)oppings, which must overcome an activation energy Ea. operator λˆ [23]: i (cid:88) (cid:104) | (cid:48) − (cid:48)(cid:105) (cid:104) | (cid:48) − (cid:48)m(cid:105) (cid:48) 2J, m S, m ; S, m m ψi,m(cid:48)ψi,m−m(cid:48), where 2J, m S, m ; S, m 2 † m is the Clebsch-Gordan coeﬃcient. The on-site state λˆi = Dm(cid:48)mψi,m(cid:48)ψi,mψi,m+m(cid:48) (2) energy spectrum can be obtained by diagonalizing HI. m,m(cid:48)=−2 For spin S = 1 bosons, we ﬁnd it impossible to have − (cid:48)(cid:104) | − − both the lowest two on-site states be spin singlets (see where the coeﬃcient D (cid:48) = ( 1)m+m 0, 0 2, m (cid:48) (cid:48)(cid:105)(cid:104) (cid:48)| m m(cid:48) (cid:105) supplementary material [23]). However, this is possible m ; 2, m + m 2, m + m 2, m ; 2, m , and deﬁne an order ≥ (cid:104) (cid:105) for bosons with spin S 2. parameter λ = λˆ . It can be shown that i We study spin 2 bosons in this letter to show how an − SLC can be realized. First, we need to ﬁnd a parame- [Sˆ , λˆ ] = 0 , [nˆ , λˆ ] = λˆ , i i i i i ter regime where the lowest two on-site states are sin- glet states. Generally, the on-site states can be labeled where Sˆ is the total spin of site i, hence λˆ is a charge 1 | (cid:105) i i as n, l, m, γ , where n is the particle number, l is the operator. The minimal ﬂux of a superﬂuid vortex is thus i total spin of the n particles, m is the z-component of 2π. The deﬁnition of λˆ again indicates the signiﬁcance i the total spin, and γ is an additional quantum number of hopping 3 times in the SLC. The existence of the or- ∝ (cid:104) †(cid:105) [23, 24]. In our discussion, we shall omit the label γ, since der parameter λ ψ ψ ψ also shows that there is an (cid:80)i i i all the states involved are distinguishable through their oﬀ diagonal long range order (ODLRO) in the 3-particle n, l, m labels [23]. The minimal two singlet states of spin density matrix ρ(3). The leading eigenvalue of ρ(3) can be | (cid:105) ∼ (cid:104) † (cid:105) ≈ | | ∼ O 2 bosons are the dimer state 2, 0, 0 and the trimer state estimated as r λˆ λˆ N λ 2 (N), where | (cid:105) i 3 j i j S 3, 0, 0 , consisting of two and three bosons respectively N and N are the total number of sites and bosons re- i S [25, 26]. We ﬁnd when U < U < (36U + 49U )/85 and spectively. However, there is no ODLRO in ρ(1) or ρ(2) − 0 2 4 0 µ = µ = 3U U , the two singlet states become de- [23, 27]. This implies that any boson participating in this 0 2 0 generate and have the lowest on-site energy. To see this condensate is “dressed”: Its spin is fully screened by a explicitly, we plot the on-site state energy spectrum for local virtual particle-hole pair, while its charge remains − U = 3U = 30U and µ = µ ∆ in Fig. 1. The on-site unchanged. This is a key diﬀerence between SLC and the 4 2 0 | (cid:105)0 energy of each state n, l, m is denoted by E . An easy spin-paired charge 2 condensate in Ref. [13], where the −i nl calculation shows that E E = ∆. For later conve- ODLRO arises in ρ(1) [12]. 30 20 nience, we deﬁne E as the energy diﬀerence between the To conﬁrm the existence of the SLC phase, we have a third and the lowest energy levels, as is shown in Fig. 1. proposed a mean-ﬁeld variational wave function for the | | (cid:28) The desired regime is then ∆ E . ground state of the system: a (cid:89) (cid:104) (cid:88) (cid:16) (cid:17) (cid:105) (cid:89) (cid:16) (cid:17) | (cid:105) † × | (cid:105) | (cid:105) SLC = Sym u + vψ ψ + h.c. α 2, 0, 0 + β 3, 0, 0 , (3) i,m j,m i i (cid:104) (cid:105) ij m i ##LTFigure## ##LTLine## ##LTLine##

label: paragraph

text: Then,

text: for

text: the

text: SLC

text: ground

text: state

text: to

text: be

text: favored,

text: the

text: tran-

text: sition

text: amplitude

text: between

text: the

text: two

text: on-site

text: states

text: i

text: and

text: must

text: be

text: large

text: enough.

text: The

text: transition

text: am-

text: i

text: plitude

text: originates

text: from

text: the

text: hopping

text: energy

text: H

text: How-

text: t

text: ever,

text: the

text: tunneling

text: from

text: to

text: cannot

text: be

text: i

text: i

text: achieved

text: by

text: a

text: single

text: hop,

text: since

text: a

text: single

text: hop

text: will

text: change

text: a

text: singlet

text: state

text: to

text: a

text: state

text: with

text: total

text: spin

text: l

text: For

text: the

text: transition

text: to

text: occur,

text: the

text: site

text: must

text: hop

text: with

text: nearby

text: sites

text: for

text: at

text: least

text: times,

text: during

text: which

text: the

text: site

text: must

text: go

text: to

text: two

text: intermediate

text: states

text: with

text: energy

text: of

text: order

text: E

text: as

text: is

text: a

text: shown

text: in

text: Fig.

text: In

text: perturbation

text: theory,

text: this

text: mechanism

text: gives

text: us

text: a

text: eﬀective

text: hopping

text: amplitude

text: between

text: the

text: singlet

text: dimer

text: state

text: and

text: the

text: trimer

text: state

text: t

text: When

text: the

text: eff

text: a

text: hopping

text: amplitude

text: increases

text: to

text: t

text: the

text: system

text: FIG.

text: The

text: on-site

text: energy

text: levels

text: computed

text: for

text: U

text: eff

text: would

text: prefer

text: a

text: coherent

text: superposition

text: of

text: the

text: dimer

text: and

text: and

text: µ

text: µ0

text: where

text: µ0

text: The

text: x

text: and

text: y

text: axes

text: are

text: the

text: particle

text: number

text: n

text: and

text: the

text: energy

text: E

text: in

text: units

text: of

text: trimer

text: states.|On

text: th|

text: hand,

text: the

text: condition

text: Ea

text: respectively.

text: Enl

text: represents

text: the

text: energy

text: of

text: the

text: on-site

text: state

text: ensures

text: that

text: t/Ea

text: so

text: the

text: superposition

text: with

text: any

text: with

text: particle

text: number

text: n

text: and

text: total

text: spin

text: l.

text: The

text: arrows

text: show

text: other

text: states

text: is

text: not

text: yet

text: favorable,

text: and

text: the

text: system

text: will

text: form

text: the

text: transition

text: process

text: from

text: i

text: to

text: i

text: via

text: three

text: time

text: an

text: SLC.

text: We

text: can

text: write

text: down

text: a

text: spin-rotationally

text: invariant

text: which

text: must

text: overcome

text: an

text: activation

text: energy

text: Ea.

text: operator

text: λˆ

text: i

text: m

text: S,

text: m

text: S,

text: m

text: m

text: where

text: m

text: S,

text: m

text: S,

text: m

text: m

text: is

text: the

text: Clebsch-Gordan

text: coeﬃcient.

text: The

text: on-site

text: state

text: λˆi

text: energy

text: spectrum

text: can

text: be

text: obtained

text: by

text: diagonalizing

text: HI.

text: For

text: spin

text: S

text: bosons,

text: we

text: ﬁnd

text: it

text: impossible

text: to

text: have

text: both

text: the

text: lowest

text: two

text: on-site

text: states

text: be

text: spin

text: singlets

text: (see

text: where

text: the

text: coeﬃcient

text: D

text: m

text: m

text: supplementary

text: material

text: However,

text: this

text: is

text: possible

text: m

text: m

text: m

text: m

text: m

text: m

text: m

text: and

text: deﬁne

text: an

text: order

text: for

text: bosons

text: with

text: spin

text: S

text: parameter

text: λ

text: λˆ

text: It

text: can

text: be

text: shown

text: that

text: i

text: We

text: study

text: spin

text: bosons

text: in

text: this

text: letter

text: to

text: show

text: how

text: an

text: SLC

text: can

text: be

text: realized.

text: First,

text: we

text: need

text: to

text: ﬁnd

text: a

text: parame-

text: [Sˆ

text: λˆ

text: [nˆ

text: λˆ

text: λˆ

text: i

text: i

text: i

text: i

text: i

text: ter

text: regime

text: where

text: the

text: lowest

text: two

text: on-site

text: states

text: are

text: sin-

text: glet

text: states.

text: Generally,

text: the

text: on-site

text: states

text: can

text: be

text: labeled

text: where

text: Sˆ

text: is

text: the

text: total

text: spin

text: of

text: site

text: i,

text: hence

text: λˆ

text: is

text: a

text: charge

text: i

text: i

text: as

text: n,

text: l,

text: m,

text: γ

text: where

text: n

text: is

text: the

text: particle

text: number,

text: l

text: is

text: the

text: operator.

text: The

text: minimal

text: ﬂux

text: of

text: a

text: superﬂuid

text: vortex

text: is

text: thus

text: i

text: total

text: spin

text: of

text: the

text: n

text: particles,

text: m

text: is

text: the

text: z-component

text: of

text: The

text: deﬁnition

text: of

text: λˆ

text: again

text: indicates

text: the

text: signiﬁcance

text: i

text: the

text: total

text: spin,

text: and

text: γ

text: is

text: an

text: additional

text: quantum

text: number

text: of

text: hopping

text: times

text: in

text: the

text: SLC.

text: The

text: existence

text: of

text: the

text: or-

text: In

text: our

text: discussion,

text: we

text: shall

text: omit

text: the

text: label

text: γ,

text: since

text: der

text: parameter

text: λ

text: ψ

text: ψ

text: ψ

text: also

text: shows

text: that

text: there

text: is

text: an

text: i

text: i

text: all

text: the

text: states

text: involved

text: are

text: distinguishable

text: through

text: their

text: oﬀ

text: diagonal

text: long

text: range

text: order

text: (ODLRO)

text: in

text: the

text: n,

text: l,

text: m

text: labels

text: The

text: minimal

text: two

text: singlet

text: states

text: of

text: spin

text: density

text: matrix

text: The

text: leading

text: eigenvalue

text: of

text: can

text: be

text: O

text: bosons

text: are

text: the

text: dimer

text: state

text: and

text: the

text: trimer

text: state

text: estimated

text: as

text: r

text: λˆ

text: λˆ

text: N

text: λ

text: (N),

text: where

text: i

text: j

text: i

text: j

text: S

text: consisting

text: of

text: two

text: and

text: three

text: bosons

text: respectively

text: N

text: and

text: N

text: are

text: the

text: total

text: number

text: of

text: sites

text: and

text: bosons

text: re-

text: i

text: S

text: We

text: ﬁnd

text: when

text: U

text: U

text: and

text: spectively.

text: However,

text: there

text: is

text: no

text: ODLRO

text: in

text: or

text: µ

text: µ

text: U

text: the

text: two

text: singlet

text: states

text: become

text: de-

text: This

text: implies

text: that

text: any

text: boson

text: participating

text: in

text: this

text: generate

text: and

text: have

text: the

text: lowest

text: on-site

text: energy.

text: To

text: see

text: this

text: condensate

text: is

text: “dressed”:

text: Its

text: spin

text: is

text: fully

text: screened

text: by

text: a

text: explicitly,

text: we

text: plot

text: the

text: on-site

text: state

text: energy

text: spectrum

text: for

text: local

text: virtual

text: particle-hole

text: pair,

text: while

text: its

text: charge

text: remains

text: U

text: and

text: µ

text: µ

text: in

text: Fig.

text: The

text: on-site

text: unchanged.

text: This

text: is

text: a

text: key

text: diﬀerence

text: between

text: SLC

text: and

text: the

text: energy

text: of

text: each

text: state

text: n,

text: l,

text: m

text: is

text: denoted

text: by

text: E

text: An

text: easy

text: spin-paired

text: charge

text: condensate

text: in

text: Ref.

text: where

text: the

text: −i

text: nl

text: calculation

text: shows

text: that

text: E

text: E

text: For

text: later

text: conve-

text: ODLRO

text: arises

text: in

text: nience,

text: we

text: deﬁne

text: E

text: as

text: the

text: energy

text: diﬀerence

text: between

text: the

text: To

text: conﬁrm

text: the

text: existence

text: of

text: the

text: SLC

text: phase,

text: we

text: have

text: a

text: third

text: and

text: the

text: lowest

text: energy

text: levels,

text: as

text: is

text: shown

text: in

text: Fig.

text: proposed

text: a

text: mean-ﬁeld

text: variational

text: wave

text: function

text: for

text: the

text: The

text: desired

text: regime

text: is

text: then

text: E

text: ground

text: state

text: of

text: the

text: system:

text: a

text: SLC

text: Sym

text: u

text: vψ

text: ψ

text: h.c.

text: α

text: β

text: i,m

text: j,m

text: i

text: i

text: ij

text: m

text: i

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: 10 (cid:104) (cid:105) −zt (cid:18)∗ ∗ · 2α2 + 3β(cid:19)2 ·7(cid:32)α2 + 8(cid:33)β2 (cid:32) (cid:33) (cid:32) (cid:33)  H /N = (cid:88)(u + u )(v + v )5 t S 2 5  √5 √ √  − | | 2 2 2 2 2 3 2 2 3 2 2 3 2 zt v 2α2β2 5 − − + + (30) m1 m2 m1 m2 5 5 5 m1,m2 −2 − 36 | | − = ztRe(u)Re(v)(2 + β2)(7 + β2) zt v 2(1 β2)β2 . 5 5 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) Putting the two parts of energy together, we have E | | − 2 − 36 | | − 1 | | = u 2β2∆ ztRe(u)Re(v) 2 + β2 7 + β2 zt v 2 1 β2 β2 + z v 2V β2 , (31) G 5 5 2 ≤ ≤ where 0 β2 1 is imposed. Obviously, for the energy to be the lowest, both u and v should be real and positive, as is assumed then. Now we optimize the variational energy, and derive the phase boundary of the SLC phase. By Eq. (25) in the E above we can eliminate u and write as a function of β2 and v. The minimum energy in the SLC phase is achieved G when 1 ∂E −4 (cid:20) (cid:16) (cid:17) − (cid:16)72 (cid:17) − (cid:21) 0 = G = (2 + β2)(7 + β2)β2∆ + zV (β2) zt(1 β2)β2 v ∂v 5 5 − 2 u(cid:16) (cid:17) −(cid:16) 4(cid:17) (cid:20)v (cid:21) zt (2 + β2)(7 + β2) (2 + β2)2(7 + β2)2 , (32) 5 v 25 u E 1 ∂ − 2 v 1 v 2 5 ∂V (β2) − − − 0 = G = ∆ zt (9 + 2β2) + z 2(9β2 + 2β4)∆ 36t(1 2β2) . u2 ∂(β2) 5 u 5 u 2 ∂(β2) If the system is right on the SLC phase boundary, the above conditions should give exactly β2 = 0 or β2 = 1. By (cid:16) (cid:17) (cid:16) (cid:17) setting β2 = 0, one obtains the phase boundary between SLC and the dimer MI, given by (cid:16) (cid:17) 28(cid:16) (cid:17)v 2 v − (cid:16) (cid:17) t + (G + G ) t = 0 , 5 u 32 12 u (33) − 18 v 1 v 2 − − − 36 v 2 ∆ zt + z (16G 7G + 21G 12G ) zt = 0 . 5 u 5 u 42 32 22 12 5 u (cid:16) (cid:17) (cid:16) (cid:17) For β2 = 1, the phase boundary between SLC and the trimer MI is given by (cid:16) (cid:17) 48(cid:16) (cid:17)v 2 − v − (cid:16) (cid:17) t + (G + G 2∆) t = 0 , 5 u 42 22 u (34) − 22 v 1 v 2 − − − 36 v 2 ∆ zt + z (32G 21G + 27G 16G 22∆) + zt = 0 . 5 u 5 u 42 32 22 12 5 u To ﬁnd the explicit expression of the phase boundaries, one has to eliminate (v/u) from Eqs. (33) and (34). In | | the elimination, one should ensure (v/u) > 0 for the result to be physical. One can further verify that 0 < β2 < 1 in between the two phase boundaries. In Fig. 2 in the main text, we have plotted explicitly the phase boundaries | | for U = 3U = 30U and z = 4. In particular in Fig. 4 below, we have calculated how the value of β 2 varies with 4 2 0 respect to the chemical potential µ for zt/U = 5.5. It can be explicitly seen that the system undergoes the phase 0 transitions from dimer MI to SLC and then to trimer MI. (cid:28) ∝ In the limit t/E 1, one ﬁnds in both cases v/u t/E , so the phase boundaries have the limiting form: a a zt2 ± zt3 ∆ + a± = b2± , (35) E E2 a a where a± and b± are coeﬃcients depending on the interactions U only. As is deﬁned before, ∆ can be further − 2J replaced by µ µ. 0 Determination of the phase boundary of spinor BEC ##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## ##LTLine##

label: equation

text: −zt

text: H

text: /N

text: u

text: )(v

text: v

text: t

text: S

text: √5

text: zt

text: v

text: −2

text: zt

text: v

text: Putting

text: the

text: two

text: parts

text: of

text: energy

text: together,

text: we

text: have

text: E

text: u

text: 2β2∆

text: ztRe(u)Re(v)

text: β2

text: β2

text: zt

text: v

text: β2

text: β2

text: z

text: v

text: β2

text: G

text: where

text: β2

text: is

text: imposed.

text: Obviously,

text: for

text: the

text: energy

text: to

text: be

text: the

text: lowest,

text: both

text: u

text: and

text: v

text: should

text: be

text: real

text: and

text: positive,

text: as

text: is

text: assumed

text: then.

text: Now

text: we

text: optimize

text: the

text: variational

text: energy,

text: and

text: derive

text: the

text: phase

text: boundary

text: of

text: the

text: SLC

text: phase.

text: By

text: Eq.

text: in

text: the

text: E

text: above

text: we

text: can

text: eliminate

text: u

text: and

text: write

text: as

text: a

text: function

text: of

text: β2

text: and

text: v.

text: The

text: minimum

text: energy

text: in

text: the

text: SLC

text: phase

text: is

text: achieved

text: G

text: when

text: ∂E

text: −4

text: G

text: β2)β2∆

text: zV

text: v

text: ∂v

text: zt

text: v

text: u

text: E

text: v

text: v

text: ∂V

text: G

text: zt

text: z

text: u

text: u

text: If

text: the

text: system

text: is

text: right

text: on

text: the

text: SLC

text: phase

text: boundary,

text: the

text: above

text: conditions

text: should

text: give

text: exactly

text: β2

text: or

text: β2

text: By

text: setting

text: β2

text: one

text: obtains

text: the

text: phase

text: boundary

text: between

text: SLC

text: and

text: the

text: dimer

text: MI,

text: given

text: by

text: v

text: t

text: (G

text: G

text: t

text: u

text: u

text: v

text: v

text: v

text: zt

text: z

text: zt

text: u

text: u

text: u

text: For

text: β2

text: the

text: phase

text: boundary

text: between

text: SLC

text: and

text: the

text: trimer

text: MI

text: is

text: given

text: by

text: v

text: t

text: (G

text: G

text: t

text: u

text: u

text: v

text: v

text: v

text: zt

text: z

text: zt

text: u

text: u

text: u

text: To

text: ﬁnd

text: the

text: explicit

text: expression

text: of

text: the

text: phase

text: boundaries,

text: one

text: has

text: to

text: eliminate

text: (v/u)

text: from

text: Eqs.

text: and

text: In

text: the

text: elimination,

text: one

text: should

text: ensure

text: (v/u)

text: for

text: the

text: result

text: to

text: be

text: physical.

text: One

text: can

text: further

text: verify

text: that

text: β2

text: in

text: between

text: the

text: two

text: phase

text: boundaries.

text: In

text: Fig.

text: in

text: the

text: main

text: text,

text: we

text: have

text: plotted

text: explicitly

text: the

text: phase

text: boundaries

text: for

text: U

text: and

text: z

text: In

text: particular

text: in

text: Fig.

text: below,

text: we

text: have

text: calculated

text: how

text: the

text: value

text: of

text: β

text: varies

text: with

text: respect

text: to

text: the

text: chemical

text: potential

text: µ

text: for

text: zt/U

text: It

text: can

text: be

text: explicitly

text: seen

text: that

text: the

text: system

text: undergoes

text: the

text: phase

text: transitions

text: from

text: dimer

text: MI

text: to

text: SLC

text: and

text: then

text: to

text: trimer

text: MI.

text: In

text: the

text: limit

text: t/E

text: one

text: ﬁnds

text: in

text: both

text: cases

text: v/u

text: t/E

text: so

text: the

text: phase

text: boundaries

text: have

text: the

text: limiting

text: form:

text: a

text: a

text: a±

text: E

text: a

text: a

text: where

text: a±

text: and

text: b±

text: are

text: coeﬃcients

text: depending

text: on

text: the

text: interactions

text: U

text: only.

text: As

text: is

text: deﬁned

text: before,

text: can

text: be

text: further

text: replaced

text: by

text: µ

text: µ.

text: Determination

text: of

text: the

text: phase

text: boundary

text: of

text: spinor

text: BEC

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: On the validity of angular momentum 19 Mu-Tao Wang Department of Mathematics, Columbia University, New York, NY 10027 e-mail: mtwang@math.columbia.edu Shing-Tung Yau Department of Mathematics, Harvard University, Cambridge, MA 02138 e-mail: yau@math.harvard.edu

text: On

text: the

text: validity

text: of

text: angular

text: momentum

text: Mu-Tao

text: Wang

text: Department

text: of

text: Mathematics,

text: Columbia

text: University,

text: New

text: York,

text: NY

text: e-mail:

text: mtwang@math.columbia.edu

label: footer

text: Shing-Tung

text: Yau

text: Department

text: of

text: Mathematics,

text: Harvard

text: University,

text: Cambridge,

text: MA

text: e-mail:

text: yau@math.harvard.edu


text: 10 P.-N. Chen, L.-H. Huang, M.-T. Wang and S.-T. Yau Continuing the c(cid:16)alculation of terms in πra of (4.1), (cid:17)we arrive at − − − − f r 1f + (f + r 2∆˜ f + 2r 1f )f f ra a rr r r a − − − − − − = A (p 1)rp 1 + (p(p 1)A + ∆˜ A)rp 2 + 2pArp 2 (pArp 1)(A rp) a a − − − = A (p 1)rp 1 + (p(p + 1)A + ∆˜ A)pAA r3p 3. a a (cid:3) Combining the above computations, we obtain the desired statement. Proposition 4.4. Let t = rpA(ua) be a hypersurface in the Minkowski space- time. Suppose p = 1. Then th(cid:90)e angular mo(cid:16)men(cid:104)tum integral is(cid:105)ﬁ(cid:17)nite and 3 ∂ − ∂ − 1 ∗ − |∇ | J(xi xj ) = (cid:15)ij x˜l d dA A∆˜ A ˜ A 2 dµ . (4.6) ∂xj ∂xi 24π l S2 S2 Proof. By Proposition 2.2, the terms in (4.3) which are closed one-forms on S2 do not contribute to the angular momentum integral. By setting p = 1 (cid:3)3 and letting r go to inﬁnity, we prove the identity. Our goal is to ﬁnd a function A on S2 so that (4.6) is not zero for some ∈ { } i, j 1, 2, 3 . ∈ { } (cid:54) (cid:90) (cid:90) Lemma 4.5. Let i, j 1, 2, 3 , i = j. Let p, q be positive even integers. We have the following inductive formulae − p(p 1) (cid:90) − (x˜i)p(x˜j)q dµ = (x˜i)p 2(x˜j)q dµ S2 (p + q)(p + q + 1) S2 S2 − S2 (cid:90) q(q 1) − + (x˜i)p(x˜j)q 2 dµ , (p + q)(p + q + 1) S2 S2 (cid:90) 4π (x˜i)qdµ = , S2 1 + q S2 (cid:90) 1 (x˜i)2(x˜j)q dµ = 4π, S2 (q + 3)(q + 1) S2 − − · · · (p 1)(p 3) 3 4π ≥ (x˜i)p(x˜j)q dµ = · · · − − for p 4. S2 (q + 1)(q + 3) (q + p 3) (q + p 1)(q + p + 1) S2 ∇ Proof. Denote by ∆˜ and ˜ the Laplace operator and gradient with respect to the standard metric on S2. Then − ∇ · ∇ − ∆˜ x˜i = 2x˜i and ˜ x˜i ˜ x˜j = δij x˜ix˜j. (4.7) Therefore, − − − − ∆˜ ((x˜i)p(x˜j)q) =p(p 1)(x˜i)p 2(x˜j)q + q(q 1)(x˜i)p(x˜j)q 2 − (p + q)(p + q + 1)(x˜i)p(x˜j)q. Integrating by parts, we obtain the ﬁrst formula. The remaning equations (cid:3) follow by induction. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: P.-N.

text: Chen,

text: L.-H.

text: Huang,

text: M.-T.

text: Wang

text: and

text: S.-T.

text: Yau

text: Continuing

text: the

text: of

text: terms

text: in

text: πra

text: of

text: arrive

text: at

text: f

text: r

text: (f

text: r

text: 2∆˜

text: f

text: )f

text: f

text: ra

text: a

text: rr

text: r

text: r

text: a

text: A

text: (p

text: (p(p

text: A)rp

text: (pArp

text: rp)

text: a

text: a

text: A

text: (p

text: (p(p

text: A)pAA

text: a

text: a

text: Combining

text: the

text: above

text: computations,

text: we

text: obtain

text: the

text: desired

text: statement.

text: Proposition

text: Let

text: t

text: rpA(ua)

text: be

text: a

text: hypersurface

text: in

text: the

text: Minkowski

text: space-

text: time.

text: Suppose

text: p

text: Then

text: angular

text: integral

text: and

text: J(xi

text: xj

text: x˜l

text: d

text: dA

text: A∆˜

text: A

text: A

text: dµ

text: ∂xj

text: ∂xi

text: l

text: Proof.

text: By

text: Proposition

text: the

text: terms

text: in

text: which

text: are

text: closed

text: one-forms

text: on

text: do

text: not

text: contribute

text: to

text: the

text: angular

text: momentum

text: integral.

text: By

text: setting

text: p

text: and

text: letting

text: r

text: go

text: to

text: inﬁnity,

text: we

text: prove

text: the

text: identity.

text: Our

text: goal

text: is

text: to

text: ﬁnd

text: a

text: function

text: A

text: on

text: so

text: that

text: is

text: not

text: zero

text: for

text: some

text: i,

text: j

text: Lemma

text: Let

text: i,

text: j

text: i

text: j.

text: Let

text: p,

text: q

text: be

text: positive

text: even

text: integers.

text: We

text: have

text: the

text: following

text: inductive

text: formulae

text: p(p

text: (x˜i)p(x˜j)q

text: dµ

text: (x˜i)p

text: dµ

text: (p

text: q)(p

text: q

text: q(q

text: (x˜i)p(x˜j)q

text: dµ

text: (p

text: q)(p

text: q

text: 4π

text: (x˜i)qdµ

text: q

text: dµ

text: (q

text: (p

text: 4π

text: (x˜i)p(x˜j)q

text: dµ

text: for

text: p

text: (q

text: (q

text: p

text: (q

text: p

text: p

text: Proof.

text: Denote

text: by

text: and

text: the

text: Laplace

text: operator

text: and

text: gradient

text: with

text: respect

text: to

text: the

text: standard

text: metric

text: on

text: Then

text: x˜i

text: and

text: x˜i

text: x˜j

text: δij

text: x˜ix˜j.

text: Therefore,

text: ((x˜i)p(x˜j)q)

text: =p(p

text: q(q

text: (p

text: q)(p

text: q

text: Integrating

text: by

text: parts,

text: we

text: obtain

text: the

text: ﬁrst

text: formula.

text: The

text: remaning

text: equations

text: follow

text: by

text: induction.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: degeneracy. These results are in agreement with the discussion in Section . 7.2 Degenerate LGSS models Many dynamical systems are most naturally modeled as degenerate in the sense that the transition kernel of the state-process does not admit any density with respect to a dominating measure. It is problematic to use (particle-ﬁlter- based) backward sampling methods for these models, owing to the fact that the backward kernel of the state process will also be degenerate. As a consequence, it is not possible to approximate the backward kernel using the forward ﬁlter particles. To illustrate how this diﬃculty can be remedied by a change of variables, conside(cid:18)r an L(cid:19)GSS(cid:18)model of th(cid:19)e(cid:18)form(cid:19) (cid:18) (cid:19) x A(cid:18) (cid:19)A x v ∼ N t+1 = 11 12 t + t , v (0, Q), (41a) z A A z 0 t t+1 21 22 t x ∼ N y = C t + e , e (0, R). (41b) t z t t t Since the Gaussian process noise enters only on the ﬁrst part of the state vector (or, equivalently, the process noise covariance matrix is rank deﬁcient) the state transition kernel is degenerate. However, for the same reason, the state component z is σ(x )-measurable and we can write z = z (x ). Therefore, t 1:t t t 1:t it is possible to rephrase (41) as a non-Markovian model with latent process { } given by x ≥ . t t 1 As a ﬁrst illustration, we simulate T = 200 samples from a fourth-order, − − ± single output system with poles at 0.65, 0.12, and 0.22 0.10i. We let dim(x ) = 1 and Q = R = 0.1. For simplicity, we assume that the system t | parameters are known and seek the joint smoothing distribution p(x y ). 1:T 1:T In the non-Markovian formulation it is possible to apply backward-simulation- based methods, such as PGAS and PGBS, as described in Section . The prob- lem, however, is that the non-Markovianity gives rise to an O(T2) computational complexity. To obtain more practical inference algorithms we employ the weight truncation strategy (39). First, we consider the coarse approximation (cid:96) = 1. We run PGAS and PGBS, both with N = 5 particles for 10 000 iterations (with the ﬁrst 1 000 discarded as burn-in). We then compute running means of the latent variables x and, 1:T from these, we compute the running root mean squared errors (RMSEs) (cid:15) n relative to the true posterior means (computed with a modiﬁed Bryson-Frazier smoother [47]). Hence, if no approximation would have been made, we would → expect (cid:15) 0, so any static error can be seen as the eﬀect of the truncation. The n results for ﬁve independent runs are shown in Figure 5. First, we note that both methods give accurate results. Still, the error for PGAS is signiﬁcantly lower than for PGBS. For further comparison, we also run an untruncated forward 3The “true” posterior mean is computed as the sample mean obtained from the ideal Gibbs sampler. 26 ##LTLine##

text: degeneracy.

text: These

text: results

text: are

text: in

text: agreement

text: with

text: the

text: discussion

text: in

text: Section

label: section

text: Degenerate

text: LGSS

text: models

text: Many

text: dynamical

text: systems

text: are

text: most

text: naturally

text: modeled

text: as

text: degenerate

text: in

text: the

text: sense

text: that

text: the

text: transition

text: kernel

text: of

text: the

text: state-process

text: does

text: not

text: admit

text: any

text: density

text: with

text: respect

text: to

text: a

text: dominating

text: measure.

text: It

text: is

text: problematic

text: to

text: use

text: (particle-ﬁlter-

text: based)

text: backward

text: sampling

text: methods

text: for

text: these

text: models,

text: owing

text: to

text: the

text: fact

text: that

text: the

text: backward

text: kernel

text: of

text: the

text: state

text: process

text: will

text: also

text: be

text: degenerate.

text: As

text: a

text: consequence,

text: it

text: is

text: not

text: possible

text: to

text: approximate

text: the

text: backward

text: kernel

text: using

text: the

text: forward

text: ﬁlter

text: particles.

text: To

text: illustrate

text: how

text: this

text: diﬃculty

text: can

text: be

text: remedied

text: by

text: a

text: change

text: of

text: variables,

text: an

text: of

text: x

text: x

text: v

text: N

text: t

text: t

text: v

text: Q),

text: z

text: A

text: A

text: z

text: t

text: t

text: x

text: N

text: y

text: C

text: t

text: e

text: e

text: R).

text: t

text: z

text: t

text: t

text: t

text: Since

text: the

text: Gaussian

text: process

text: noise

text: enters

text: only

text: on

text: the

text: ﬁrst

text: part

text: of

text: the

text: state

text: vector

text: (or,

text: equivalently,

text: the

text: process

text: noise

text: covariance

text: matrix

text: is

text: rank

text: deﬁcient)

text: the

text: state

text: transition

text: kernel

text: is

text: degenerate.

text: However,

text: for

text: the

text: same

text: reason,

text: the

text: state

text: component

text: z

text: is

text: σ(x

text: )-measurable

text: and

text: we

text: can

text: write

text: z

text: z

text: (x

text: Therefore,

text: t

text: t

text: t

text: it

text: is

text: possible

text: to

text: rephrase

text: as

text: a

text: non-Markovian

text: model

text: with

text: latent

text: process

text: given

text: by

text: x

text: t

text: t

text: As

text: a

text: ﬁrst

text: illustration,

text: we

text: simulate

text: T

text: samples

text: from

text: a

text: fourth-order,

text: single

text: output

text: system

text: with

text: poles

text: at

text: and

text: We

text: let

text: dim(x

text: and

text: Q

text: R

text: For

text: simplicity,

text: we

text: assume

text: that

text: the

text: system

text: t

text: parameters

text: are

text: known

text: and

text: seek

text: the

text: joint

text: smoothing

text: distribution

text: p(x

text: y

text: In

text: the

text: non-Markovian

text: formulation

text: it

text: is

text: possible

text: to

text: apply

text: backward-simulation-

text: based

text: methods,

text: such

text: as

text: PGAS

text: and

text: PGBS,

text: as

text: described

text: in

text: Section

text: The

text: prob-

text: lem,

text: however,

text: is

text: that

text: the

text: non-Markovianity

text: gives

text: rise

text: to

text: an

text: computational

text: complexity.

text: To

text: obtain

text: more

text: practical

text: inference

text: algorithms

text: we

text: employ

text: the

text: weight

text: truncation

text: strategy

text: First,

text: we

text: consider

text: the

text: coarse

text: approximation

text: We

text: run

text: PGAS

text: and

text: PGBS,

text: both

text: with

text: N

text: particles

text: for

text: iterations

text: (with

text: the

text: ﬁrst

text: discarded

text: as

text: burn-in).

text: We

text: then

text: compute

text: running

text: means

text: of

text: the

text: latent

text: variables

text: x

text: and,

text: from

text: these,

text: we

text: compute

text: the

text: running

text: root

text: mean

text: squared

text: errors

text: (RMSEs)

text: n

text: relative

text: to

text: the

text: true

text: posterior

text: means

text: (computed

text: with

text: a

text: modiﬁed

text: Bryson-Frazier

text: smoother

text: Hence,

text: if

text: no

text: approximation

text: would

text: have

text: been

text: made,

text: we

text: would

text: expect

text: so

text: any

text: static

text: error

text: can

text: be

text: seen

text: as

text: the

text: eﬀect

text: of

text: the

text: truncation.

text: The

text: n

text: results

text: for

text: ﬁve

text: independent

text: runs

text: are

text: shown

text: in

text: Figure

text: First,

text: we

text: note

text: that

text: both

text: methods

text: give

text: accurate

text: results.

text: Still,

text: the

text: error

text: for

text: PGAS

text: is

text: signiﬁcantly

text: lower

text: than

text: for

text: PGBS.

text: For

text: further

text: comparison,

text: we

text: also

text: run

text: an

text: untruncated

text: forward

text: “true”

text: posterior

text: mean

text: is

text: computed

text: as

text: the

text: sample

text: mean

text: obtained

text: from

text: the

text: ideal

text: Gibbs

text: sampler.

text: ##LTLine##


text: with N = 10 for 50 000 iterations (discarding the ﬁrst 10 000). For sampling the system parameters (γ, R , α, F), we use Metropolis-Hastings steps with a 0 Gaussian random walk proposal, tuned according to an initial trial run. For (ρ, σ2), we exploit the conjugacy of the normal-inverse-gamma prior to the like- lihood (43) and sample the variables from their true posterior. The innovation variables V are sampled from the PGAS kernel by Algorithm 2 (no truncation 1:T is used for the ancestor sampling weights). Since the latter step is the compu- tational bottleneck of the algorithm, we execute ten MH steps for θ, for each draw from the PGAS kernel. It is worth pointing out that, while the sampler eﬀectively targets the col- lapsed model (44), it is most straightforwardly implemented using the original state variables from (42). With x = (S , I , R )T we can simulate x k ∆k ∆k ∆k k+1 given x according to (42) which is used in the underlying particle ﬁlter. The k innovation variables V need only be taken into account for the ancestor sam- (cid:48) k pling step. Let V be the reference innovation trajectory. To compute the 1:T ancestor sampling weights (7) we need to evaluate the ratios, (cid:89) (cid:48) p ((V i − , V ), y ) ∝ T | (cid:48) θ 1:k 1 k:T 1:T g (y V i − , V ). (45) p (V i − , y − ) θ (cid:96) 1:k 1 k:(cid:96) θ 1:k 1 1:k 1 (cid:96)=k Using (43), the observation likelihood can be written as | (cid:48) N | { (cid:48) } P g (y V i − , V ) = (y ρ logit(I¯ xi − , V / ), σ2), (46) θ (cid:96) 1:k 1 k:(cid:96) (cid:96) (cid:96) k 1 k:(cid:96) { (cid:48) } where I xi − , V is obtained by simulating the system (42) from time − (cid:96) k 1 k:(cid:96) (cid:48) ∆(k 1) to time ∆(cid:96), initialized at xi − and using the innovation sequence V . k 1 k:(cid:96) Histograms representing the estimated posterior parameter distributions are shown in Figure 7. As can be seen, the true system parameters fall well within the credible regions. Finally, the identiﬁed model is used to make one-month- ahead predictions of the disease activity for the subsequent four years, as shown in Figure 8. The predictions are made by sub-sampling the Markov chain and, for each sample, running a particle ﬁlter on the validation data using 100 par- ticles. As can be seen, we obtain an accurate prediction of the disease activity, which falls within the estimated 95 % credibility intervals, one month in ad- vance. 8 Discussion PGAS is a novel approach to PMCMC that provides the statistician with an oﬀ-the-shelf class of Markov kernels which can be used to simulate, for instance, the typically high-dimensional and highly autocorrelated state trajectory in a state-space model. This opens up for using PGAS as a key component in dif- ferent inference algorithms, enabling both Bayesian and frequentist parameter inference as well as state inference. However, PGAS by no means limited to 30 ##LTLine##

text: with

text: N

text: for

text: iterations

text: (discarding

text: the

text: ﬁrst

text: For

text: sampling

text: the

text: system

text: parameters

text: (γ,

text: R

text: α,

text: F),

text: we

text: use

text: Metropolis-Hastings

text: steps

text: with

text: a

text: Gaussian

text: random

text: walk

text: proposal,

text: tuned

text: according

text: to

text: an

text: initial

text: trial

text: run.

text: For

text: (ρ,

text: we

text: exploit

text: the

text: conjugacy

text: of

text: the

text: normal-inverse-gamma

text: prior

text: to

text: the

text: like-

text: lihood

text: and

text: sample

text: the

text: variables

text: from

text: their

text: true

text: posterior.

text: The

text: innovation

text: variables

text: V

text: are

text: sampled

text: from

text: the

text: PGAS

text: kernel

text: by

text: Algorithm

text: (no

text: truncation

text: is

text: used

text: for

text: the

text: ancestor

text: sampling

text: weights).

text: Since

text: the

text: latter

text: step

text: is

text: the

text: compu-

text: tational

text: bottleneck

text: of

text: the

text: algorithm,

text: we

text: execute

text: ten

text: MH

text: steps

text: for

text: θ,

text: for

text: each

text: draw

text: from

text: the

text: PGAS

text: kernel.

text: It

text: is

text: worth

text: pointing

text: out

text: that,

text: while

text: the

text: sampler

text: eﬀectively

text: targets

text: the

text: col-

text: lapsed

text: model

text: it

text: is

text: most

text: straightforwardly

text: implemented

text: using

text: the

text: original

text: state

text: variables

text: from

text: With

text: x

text: (S

text: I

text: R

text: )T

text: we

text: can

text: simulate

text: x

text: k

text: ∆k

text: ∆k

text: ∆k

text: given

text: x

text: according

text: to

text: which

text: is

text: used

text: in

text: the

text: underlying

text: particle

text: ﬁlter.

text: The

text: k

text: innovation

text: variables

text: V

text: need

text: only

text: be

text: taken

text: into

text: account

text: for

text: the

text: ancestor

text: sam-

text: k

text: pling

text: step.

text: Let

text: V

text: be

text: the

text: reference

text: innovation

text: trajectory.

text: To

text: compute

text: the

text: ancestor

text: sampling

text: weights

text: we

text: need

text: to

text: evaluate

text: the

text: ratios,

text: p

text: ((V

text: i

text: V

text: y

text: T

text: θ

text: k:T

text: g

text: (y

text: V

text: i

text: V

text: p

text: (V

text: i

text: y

text: θ

text: θ

text: Using

text: the

text: observation

text: likelihood

text: can

text: be

text: written

text: as

text: N

text: P

text: g

text: (y

text: V

text: i

text: V

text: (y

text: ρ

text: logit(I¯

text: xi

text: V

text: θ

text: k

text: where

text: I

text: xi

text: V

text: is

text: obtained

text: by

text: simulating

text: the

text: system

text: from

text: time

text: k

text: ∆(k

text: to

text: time

text: initialized

text: at

text: xi

text: and

text: using

text: the

text: innovation

text: sequence

text: V

text: k

text: Histograms

text: representing

text: the

text: estimated

text: posterior

text: parameter

text: distributions

text: are

text: shown

text: in

text: Figure

text: As

text: can

text: be

text: seen,

text: the

text: true

text: system

text: parameters

text: fall

text: well

text: within

text: the

text: credible

text: regions.

text: Finally,

text: the

text: identiﬁed

text: model

text: is

text: used

text: to

text: make

text: one-month-

text: ahead

text: predictions

text: of

text: the

text: disease

text: activity

text: for

text: the

text: subsequent

text: four

text: years,

text: as

text: shown

text: in

text: Figure

text: The

text: predictions

text: are

text: made

text: by

text: sub-sampling

text: the

text: Markov

text: chain

text: and,

text: for

text: each

text: sample,

text: running

text: a

text: particle

text: ﬁlter

text: on

text: the

text: validation

text: data

text: using

text: par-

text: ticles.

text: As

text: can

text: be

text: seen,

text: we

text: obtain

text: an

text: accurate

text: prediction

text: of

text: the

text: disease

text: activity,

text: which

text: falls

text: within

text: the

text: estimated

text: credibility

text: intervals,

text: one

text: month

text: in

text: ad-

text: vance.

text: Discussion

text: PGAS

text: is

text: a

text: novel

text: approach

text: to

text: PMCMC

text: that

text: provides

text: the

text: statistician

text: with

text: an

text: oﬀ-the-shelf

text: class

text: of

text: Markov

text: kernels

text: which

text: can

text: be

text: used

text: to

text: simulate,

text: for

text: instance,

text: the

text: typically

text: high-dimensional

text: and

text: highly

text: autocorrelated

text: state

text: trajectory

text: in

text: a

text: state-space

text: model.

text: This

text: opens

text: up

text: for

text: using

text: PGAS

text: as

text: a

text: key

text: component

text: in

text: dif-

text: ferent

text: inference

text: algorithms,

text: enabling

text: both

text: Bayesian

text: and

text: frequentist

text: parameter

text: inference

text: as

text: well

text: as

text: state

text: inference.

text: However,

text: PGAS

text: by

text: no

text: means

text: limited

text: to

text: ##LTLine##


text: (cid:88) (cid:98) (cid:80) N wi− γN − (dx − ) = t 1 δ (dx − ). (2) θ,t 1 1:t 1 wl− xi1:t−1 1:t 1 i=1 l t 1 { } This particle system is propagated to time t by sampling ai, xi N inde- t t i=1 pendently from a proposal kernel, (cid:80) wa−t | M (a , x ) = t 1 r (x xat − ). (3) θ,t t t wl− θ,t t 1:t 1 l t 1 − Note that M depends on the complete particle system up to time t { }θ,t 1, xi − , wi− N , but for notational convenience we shall not make that 1:t 1 t 1 i=1 dependence explicit. Here, ai is the index of the ancestor particle of xi. In this t t formulation, the resampling step is implicit and corresponds to sampling these ancestor indices. When we write xi we refer to the ancestral path of particle 1:t xi. That is, the particle trajectory is deﬁned recursively as t xi = (xait − , xi). (4) 1:t 1:t 1 t Once we have generated N ancestor indices and particles from the proposal kernel (3), the particles are weighted according to wi = W (xi ) where the t θ,t 1:t weight function is given by γ (x ) W (x ) = θ,t 1:t | , (5) θ,t 1:t γ − (x − )r (x x − ) θ,t 1 1:t 1 θ,t t 1:t 1 ≥ for t 2. The procedure is initialized by sampling from a proposal density ∼ xi r (x ) and assigning importance weights wi = W (xi ) with W (x ) = 1 θ,1 1 1 θ,1 1 θ,1 1 γ (x )/r (x ). The SMC sampler is summarized in Algorithm 1. θ,1 1 θ,1 1 Algorithm 1 Sequential Monte Carlo (each step is for i = 1, . . . , N) ∼ 1: Draw xi1 rθ,1(x1). 2: Set w1i = Wθ,1(xi1). 3: for t = 2 to T do { } ∼ 4: Draw ait, xit Mθ,t(at, xt). 5: Set xi = (xait − , xi). 1:t 1:t 1 t 6: Set wti = Wθ,t(xi1:t). 7: end for It is interesting to note that the joint law of all the random variables gener- ated by Algorithm 1 can be written down explicitly. Let { } { } x = x1, . . . , xN and a = a1, . . . , aN , t t t t t t 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: N

text: wi−

text: γN

text: (dx

text: t

text: δ

text: (dx

text: θ,t

text: wl−

text: l

text: t

text: This

text: particle

text: system

text: is

text: propagated

text: to

text: time

text: t

text: by

text: sampling

text: ai,

text: xi

text: N

text: inde-

text: t

text: t

text: pendently

text: from

text: a

text: proposal

text: kernel,

text: wa−t

text: M

text: (a

text: x

text: t

text: r

text: (x

text: xat

text: θ,t

text: t

text: t

text: wl−

text: θ,t

text: t

text: l

text: t

text: Note

text: that

text: M

text: depends

text: on

text: the

text: complete

text: particle

text: system

text: up

text: to

text: time

text: t

text: }θ,t

text: xi

text: wi−

text: N

text: but

text: for

text: notational

text: convenience

text: we

text: shall

text: not

text: make

text: that

text: t

text: dependence

text: explicit.

text: Here,

text: ai

text: is

text: the

text: index

text: of

text: the

text: ancestor

text: particle

text: of

text: xi.

text: In

text: this

text: t

text: t

text: formulation,

text: the

text: resampling

text: step

text: is

text: implicit

text: and

text: corresponds

text: to

text: sampling

text: these

text: ancestor

text: indices.

text: When

text: we

text: write

text: xi

text: we

text: refer

text: to

text: the

text: ancestral

text: path

text: of

text: particle

text: xi.

text: That

text: is,

text: the

text: particle

text: trajectory

text: is

text: deﬁned

text: recursively

text: as

text: t

text: xi

text: (xait

text: xi).

text: t

text: Once

text: we

text: have

text: generated

text: N

text: ancestor

text: indices

text: and

text: particles

text: from

text: the

text: proposal

text: kernel

text: the

text: particles

text: are

text: weighted

text: according

text: to

text: wi

text: W

text: (xi

text: where

text: the

text: t

text: θ,t

text: weight

text: function

text: is

text: given

text: by

text: γ

text: (x

text: W

text: (x

text: θ,t

text: θ,t

text: γ

text: (x

text: )r

text: (x

text: x

text: θ,t

text: θ,t

text: t

text: for

text: t

text: The

text: procedure

text: is

text: initialized

text: by

text: sampling

text: from

text: a

text: proposal

text: density

text: xi

text: r

text: (x

text: and

text: assigning

text: importance

text: weights

text: wi

text: W

text: (xi

text: with

text: W

text: (x

text: γ

text: (x

text: )/r

text: (x

text: The

text: SMC

text: sampler

text: is

text: summarized

text: in

text: Algorithm

text: Algorithm

text: Sequential

text: Monte

text: Carlo

text: (each

text: step

text: is

text: for

text: i

text: N)

text: Draw

text: Set

text: for

text: t

text: to

text: T

text: do

text: Draw

text: ait,

text: xit

text: Mθ,t(at,

text: xt).

text: Set

text: xi

text: (xait

text: xi).

text: t

text: Set

text: wti

text: end

text: for

text: It

text: is

text: interesting

text: to

text: note

text: that

text: the

text: joint

text: law

text: of

text: all

text: the

text: random

text: variables

text: gener-

text: ated

text: by

text: Algorithm

text: can

text: be

text: written

text: down

text: explicitly.

text: Let

text: x

text: xN

text: and

text: a

text: aN

text: t

text: t

text: t

text: t

text: t

text: t

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: γ R0 2 3 4 9.6 10 10.4 α F 0.1 0.15 0.2 0 0.03 0.06 ρ σ 1.05 1.1 1.15 0.18 0.22 0.26 Figure 7: Posterior densities for the parameters of model (42)–(43). The true values are marked by vertical dashed lines. 2000 True 1500 1 month predictions 95 % credibility It 1000 500 0 10 20 30 40 50 60 70 80 90 Month Figure 8: Disease activity (number of infected individuals I ) over an eight year t period. The ﬁrst four years are used as estimation data, to ﬁnd the unknown parameters of the model. For the consecutive four years, one-month-ahead predictions are computed using the estimated model. 31 ##LTFigure## ##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## ##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## ##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## ##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##

text: γ

font: default

text: α

text: F

text: ρ

text: σ

text: Figure

text: Posterior

label: caption

text: densities

text: for

text: the

text: parameters

text: of

text: model

text: The

text: true

text: values

text: are

text: marked

text: by

text: vertical

text: dashed

text: lines.

text: True

text: month

text: predictions

text: credibility

text: It

text: Month

text: Figure

text: Disease

text: activity

text: (number

text: of

text: infected

text: individuals

text: I

text: over

text: an

text: eight

text: year

text: t

text: period.

text: The

text: ﬁrst

text: four

text: years

text: are

text: used

text: as

text: estimation

text: data,

text: to

text: ﬁnd

text: the

text: unknown

text: parameters

text: of

text: the

text: model.

text: For

text: the

text: consecutive

text: four

text: years,

text: one-month-ahead

text: predictions

text: are

text: computed

text: using

text: the

text: estimated

text: model.

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text: We return to this invariance property below, when it is shown to hold also for the proposed PGAS kernel. 3.2 Ancestor sampling (cid:48) As noted above, the PG algorithm keeps the reference trajectory x intact 1:T throughout the sampling procedure. While this results in a Markov kernel which leaves γ¯ invariant, it has been recognized that the mixing properties of this θ,T kernel can be very poor due to path degeneracy [5,11]. To address this fundamental problem we now turn to our new procedure, PGAS. The idea is to sample a new value for the index variable aN in an t ancestor sampling step. While this is a small modiﬁcation of the algorithm, the improvement in mixing can be quite considerable; see Section and the numerical evaluation in Section . The ancestor sampling step is implemented as follows. ≥ (cid:48) At time t 2, we consider the part of the reference trajectory x ranging t:T from the current time t to the ﬁnal time point T. The task is to artiﬁcially (cid:48) assign a history to this partial path. This is done by connecting x to one of { } t:T the particles xi − N . Recall that the ancestry of a particle is encoded via 1:t 1 i=1 the corresponding ancestor index. Hence, we can connect the partial reference { } path to one of the particles xi − N by assigning a value to the variable ∈ { } 1:t 1 i=1 aN 1, . . . , N . To do this, ﬁrst we compute the weights t (cid:101) (cid:48) (cid:44) γ ((xi − , x )) wi− | wi− θ,T 1:t 1 t:T (7) t 1 T t 1 γ − (xi − ) θ,t 1 1:t 1 (cid:48) for(cid:101)i = 1, . . . , N. Here, (xi − , x ) refers to the point in XT formed by 1:t 1 t:T P concatenating the two partial trajectories. Then, we sample aN with (aN = ∝ t t i) wi− | . The expression above can be understood as an application of t 1 T Bayes’ theorem, where the importance weight wi− is the prior probability of t 1 the particle xi − and the ratio between the target densities in (7) can be seen 1:t 1 (cid:48) as the likelihood that x originated from xi − . A formal argument for why t:T 1:t 1 (7) provides the correct ancestor sampling distribution, in order to retain the invariance properties of the kernel, is detailed in the proof of Theorem 1 in Section . The sampling procedure outlined above is summarized in Algorithm 2 and the family of PGAS kernels is formally deﬁned below. Note that the only dif- ference between PG and PGAS is on line 8 of Algorithm 2 (where, for PG, we would simply set aN = N). However, as we shall see, the eﬀect of this small t modiﬁcation on the mixing of the kernel is quite signiﬁcant. ≥ ∈ Deﬁnition 1 (PGAS kernels). For any N 1 and any θ Θ, Algorithm 2 (cid:48) maps x stochastically into x(cid:63) , thus implicitly deﬁning a Markov kernel PN 1:XT 1:T { ∈ } ≥ θ on (XT , T ). The family of Markov kernels PN : θ Θ , indexed by N 1, θ is referred to as the PGAS family of kernels. 6 ##LTLine##

text: We

text: return

text: to

text: this

text: invariance

text: property

text: below,

text: when

text: it

text: is

text: shown

text: to

text: hold

text: also

text: for

text: the

text: proposed

text: PGAS

text: kernel.

text: Ancestor

text: sampling

text: As

text: noted

text: above,

text: the

text: PG

text: algorithm

text: keeps

text: the

text: reference

text: trajectory

text: x

text: intact

text: throughout

text: the

text: sampling

text: procedure.

text: While

text: this

text: results

text: in

text: a

text: Markov

text: kernel

text: which

text: leaves

text: γ¯

text: invariant,

text: it

text: has

text: been

text: recognized

text: that

text: the

text: mixing

text: properties

text: of

text: this

text: θ,T

text: kernel

text: can

text: be

text: very

text: poor

text: due

text: to

text: path

text: degeneracy

text: To

text: address

text: this

text: fundamental

text: problem

text: we

text: now

text: turn

text: to

text: our

text: new

text: procedure,

text: PGAS.

text: The

text: idea

text: is

text: to

text: sample

text: a

text: new

text: value

text: for

text: the

text: index

text: variable

text: aN

text: in

text: an

text: t

text: ancestor

text: sampling

text: step.

text: While

text: this

text: is

text: a

text: small

text: modiﬁcation

text: of

text: the

text: algorithm,

text: the

text: improvement

text: in

text: mixing

text: can

text: be

text: quite

text: considerable;

text: see

text: Section

text: and

text: the

text: numerical

text: evaluation

text: in

text: Section

text: The

text: ancestor

text: sampling

text: step

text: is

text: implemented

text: as

text: follows.

text: At

text: time

text: t

text: we

text: consider

text: the

text: part

text: of

text: the

text: reference

text: trajectory

text: x

text: ranging

text: t:T

text: from

text: the

text: current

text: time

text: t

text: to

text: the

text: ﬁnal

text: time

text: point

text: T.

text: The

text: task

text: is

text: to

text: artiﬁcially

text: assign

text: a

text: history

text: to

text: this

text: partial

text: path.

text: This

text: is

text: done

text: by

text: connecting

text: x

text: to

text: one

text: of

text: t:T

text: the

text: particles

text: xi

text: N

text: Recall

text: that

text: the

text: ancestry

text: of

text: a

text: particle

text: is

text: encoded

text: via

text: the

text: corresponding

text: ancestor

text: index.

text: Hence,

text: we

text: can

text: connect

text: the

text: partial

text: reference

text: path

text: to

text: one

text: of

text: the

text: particles

text: xi

text: N

text: by

text: assigning

text: a

text: value

text: to

text: the

text: variable

text: aN

text: N

text: To

text: do

text: this,

text: ﬁrst

text: we

text: compute

text: the

text: weights

text: t

text: γ

text: ((xi

text: x

text: wi−

text: wi−

text: θ,T

text: t:T

text: t

text: T

text: t

text: γ

text: (xi

text: θ,t

text: N.

text: Here,

text: (xi

text: x

text: refers

text: to

text: the

text: point

text: in

text: XT

text: formed

text: by

text: t:T

text: P

text: concatenating

text: the

text: two

text: partial

text: trajectories.

text: Then,

text: we

text: sample

text: aN

text: with

text: (aN

text: t

text: t

text: i)

text: wi−

text: The

text: expression

text: above

text: can

text: be

text: understood

text: as

text: an

text: application

text: of

text: t

text: T

text: Bayes’

text: theorem,

text: where

text: the

text: importance

text: weight

text: wi−

text: is

text: the

text: prior

text: probability

text: of

text: t

text: the

text: particle

text: xi

text: and

text: the

text: ratio

text: between

text: the

text: target

text: densities

text: in

text: can

text: be

text: seen

text: as

text: the

text: likelihood

text: that

text: x

text: originated

text: from

text: xi

text: A

text: formal

text: argument

text: for

text: why

text: t:T

text: provides

text: the

text: correct

text: ancestor

text: sampling

text: distribution,

text: in

text: order

text: to

text: retain

text: the

text: invariance

text: properties

text: of

text: the

text: kernel,

text: is

text: detailed

text: in

text: the

text: proof

text: of

text: Theorem

text: in

text: Section

text: The

text: sampling

text: procedure

text: outlined

text: above

text: is

text: summarized

text: in

text: Algorithm

text: and

text: the

text: family

text: of

text: PGAS

text: kernels

text: is

text: formally

text: deﬁned

text: below.

text: Note

text: that

text: the

text: only

text: dif-

text: ference

text: between

text: PG

text: and

text: PGAS

text: is

text: on

text: line

text: of

text: Algorithm

text: (where,

text: for

text: PG,

text: we

text: would

text: simply

text: set

text: aN

text: N).

text: However,

text: as

text: we

text: shall

text: see,

text: the

text: eﬀect

text: of

text: this

text: small

text: t

text: modiﬁcation

text: on

text: the

text: mixing

text: of

text: the

text: kernel

text: is

text: quite

text: signiﬁcant.

text: Deﬁnition

text: (PGAS

text: kernels).

text: For

text: any

text: N

text: and

text: any

text: θ

text: Θ,

text: Algorithm

text: maps

text: x

text: stochastically

text: into

text: thus

text: implicitly

text: deﬁning

text: a

text: Markov

text: kernel

text: PN

text: θ

text: on

text: (XT

text: T

text: The

text: family

text: of

text: Markov

text: kernels

text: PN

text: θ

text: Θ

text: indexed

text: by

text: N

text: θ

text: is

text: referred

text: to

text: as

text: the

text: PGAS

text: family

text: of

text: kernels.

text: ##LTLine##


text: (cid:44) xk = xb1:T (xb1, . . . , xbT ), (10) 1:T 1:T 1 T where the indices b are given recursively by the ancestor indices: b = k (cid:44)1:T × { } − T and b = abt+1. Let Ω XNT 1, . . . , N N(T 1)+1 be the space of all random t t+1 variables generated by Algorithm 2. Following [6], we then deﬁne a function (cid:55)→ R φ : Ω as follows: θ (cid:89) − (cid:89)− (cid:89) | φ (x , a , k) = φ (xb1:T , b )φ (x b1:T , a b2:T xb1:T , b ) θ 1:T 2:T θ 1:T 1:T θ 1:T 2:T 1:T 1:T (cid:44) γ¯ (xb1:T ) N T N (cid:124) θ,TN(cid:123)(cid:122)T1:T (cid:125) (cid:124) rθ,1(xi1) (cid:123)(cid:122) Mθ,t(ait, xit(cid:125)), (11) i(cid:54) =1 t=2 i(cid:54)=1 i=b1 i=bt marginal conditional where we have introduced the notation − { − } − { − − } x i = x1, . . . , xi 1, xi+1, . . . , xN , x b1:T = x b1, . . . , x bT t t t t t 1:T 1 T and similarly for the ancestor indices. By construction, φ is nonnegative and θ integrates to one, i.e., φ is a probability density function on Ω. We refer to θ this density as the extended target density. The factorization into a marginal and a conditional density is intended to reveal some of the structure inherent in the extended target density. In partic- { } ular, the marginal density of the variables xb1:T , b is deﬁned to be equal to 1:T 1:T − the original target density γ¯ (xb1:T ), up to a factor N T related to the index θ,T 1:T { } variables b . This has the important implication that if x , a , k are 1:T 1:T 2:T distributed according to φ , then, by construction, the marginal distribution of θ xb1:T is γ¯ . 1:T θ,T By constructing an MCMC kernel with invariant distribution φ , we will θ thus obtain a kernel with invariant distribution γ¯ (the PGAS kernel) as a θ,T byproduct. To prove Theorem 1 we will reinterpret all the steps of the PGAS algorithm as partially collapsed Gibbs steps for φ . The meaning of partial θ collapsing will be made precise in the proof of Lemma 2 below, but basically it refers to the process of marginalizing out some of the variables of the model in the individual steps of the Gibbs sampler. This is done in such a way that it does not violate the invariance property of the Gibbs kernel, i.e., each such Gibbs step will leave the extended target distribution invariant. As a consequence, the invariance property of the PGAS kernel follows. First we show that the PGAS algorithm in fact implements the following sequence of partially collapsed Gibbs steps for φ . θ (cid:48) (cid:48) ∈ Procedure 1 (Instrumental reformulation of PGAS). Given x ,b1:T XT and (cid:48) ∈ { } 1:T b 1, . . . , N T : 1:T 10 ##LTLine##

text: xk

text: xbT

text: T

text: where

text: the

text: indices

text: b

text: are

text: given

text: recursively

text: by

text: the

text: ancestor

text: indices:

text: b

text: k

text: T

text: and

text: b

text: Let

text: Ω

text: XNT

text: N

text: N(T

text: be

text: the

text: space

text: of

text: all

text: random

text: t

text: variables

text: generated

text: by

text: Algorithm

text: Following

text: we

text: then

text: deﬁne

text: a

text: function

text: R

text: φ

text: Ω

text: as

text: follows:

text: θ

text: φ

text: (x

text: a

text: k)

text: φ

text: b

text: )φ

text: (x

text: a

text: b

text: θ

text: θ

text: θ

text: γ¯

text: N

text: T

text: N

text: Mθ,t(ait,

text: i=bt

text: marginal

text: conditional

text: where

text: we

text: have

text: introduced

text: the

text: notation

text: x

text: i

text: xi

text: xN

text: x

text: x

text: x

text: bT

text: t

text: t

text: t

text: t

text: t

text: T

text: and

text: similarly

text: for

text: the

text: ancestor

text: indices.

text: By

text: construction,

text: φ

text: is

text: nonnegative

text: and

text: θ

text: integrates

text: to

text: one,

text: i.e.,

text: φ

text: is

text: a

text: probability

text: density

text: function

text: on

text: Ω.

text: We

text: refer

text: to

text: θ

text: this

text: density

text: as

text: the

text: extended

text: target

text: density.

text: The

text: factorization

text: into

text: a

text: marginal

text: and

text: a

text: conditional

text: density

text: is

text: intended

text: to

text: reveal

text: some

text: of

text: the

text: structure

text: inherent

text: in

text: the

text: extended

text: target

text: density.

text: In

text: partic-

text: ular,

text: the

text: marginal

text: density

text: of

text: the

text: variables

text: b

text: is

text: deﬁned

text: to

text: be

text: equal

text: to

text: the

text: original

text: target

text: density

text: γ¯

text: up

text: to

text: a

text: factor

text: N

text: T

text: related

text: to

text: the

text: index

text: θ,T

text: variables

text: b

text: This

text: has

text: the

text: important

text: implication

text: that

text: if

text: x

text: a

text: k

text: are

text: distributed

text: according

text: to

text: φ

text: then,

text: by

text: construction,

text: the

text: marginal

text: distribution

text: of

text: θ

text: is

text: γ¯

text: θ,T

text: By

text: constructing

text: an

text: MCMC

text: kernel

text: with

text: invariant

text: distribution

text: φ

text: we

text: will

text: θ

text: thus

text: obtain

text: a

text: kernel

text: with

text: invariant

text: distribution

text: γ¯

text: (the

text: PGAS

text: kernel)

text: as

text: a

text: θ,T

text: byproduct.

text: To

text: prove

text: Theorem

text: we

text: will

text: reinterpret

text: all

text: the

text: steps

text: of

text: the

text: PGAS

text: algorithm

text: as

text: partially

text: collapsed

text: Gibbs

text: steps

text: for

text: φ

text: The

text: meaning

text: of

text: partial

text: θ

text: collapsing

text: will

text: be

text: made

text: precise

text: in

text: the

text: proof

text: of

text: Lemma

text: below,

text: but

text: basically

text: it

text: refers

text: to

text: the

text: process

text: of

text: marginalizing

text: out

text: some

text: of

text: the

text: variables

text: of

text: the

text: model

text: in

text: the

text: individual

text: steps

text: of

text: the

text: Gibbs

text: sampler.

text: This

text: is

text: done

text: in

text: such

text: a

text: way

text: that

text: it

text: does

text: not

text: violate

text: the

text: invariance

text: property

text: of

text: the

text: Gibbs

text: kernel,

text: i.e.,

text: each

text: such

text: Gibbs

text: step

text: will

text: leave

text: the

text: extended

text: target

text: distribution

text: invariant.

text: As

text: a

text: consequence,

text: the

text: invariance

text: property

text: of

text: the

text: PGAS

text: kernel

text: follows.

text: First

text: we

text: show

text: that

text: the

text: PGAS

text: algorithm

text: in

text: fact

text: implements

text: the

text: following

text: sequence

text: of

text: partially

text: collapsed

text: Gibbs

text: steps

text: for

text: φ

text: θ

text: Procedure

text: (Instrumental

text: reformulation

text: of

text: PGAS).

text: Given

text: x

text: XT

text: and

text: b

text: N

text: T

text: ##LTLine##


text: r2 r Ρ (cid:72) (cid:76) 0.06 0.05 accurate 0.04 KS SCE (cid:45) 0.03 KS SCE LDA (cid:45) (cid:43) 0.02 KS SCE LVee,d (cid:45) (cid:43) 0.01 HF r 2 4 6 8 10 12 − FIG. 5. r2ρ(r) for H and various approaches. to the total energy, which for these systems is not very tight as can be seen from the underestimation of versus the actual value . This is due to Z ≈ 0.7307 Z ≈ 0.9110289 crit crit the inherent strong correlation nature of the electrons in the SCE formulation that results in underestimating the electron-electron repulsion energy. Self-consistently thus, the KS-SCE densities become quite compact until the kinetic energy starts to dominate in Eq. (19). This − is manifested in Fig. 5 where the density of H is displayed for several methods, the KS-SCE yielding the most compact density. Physically, this is due to the fact that the two electrons, being perfectly correlated, can avoid each other as much as possible and can get much closer to the nucleus to lower the total energy. The local corrections to the SCE functional improve considerably the predicted and Z crit give a more realistic description of the electronic interactions. We observe that the KS- SCE+LDA density is too spread out compared to the accurate data (e.g. Fig. 5). This can be attributed to the self-interaction error that is introduced by the LDA correction which is obvious from Eq. (33) – the energy densities do not vanish for a density integrating to 1. In Fig. 5 we display also the Hartree-Fock density. It is possible to do this because the HOMO eigenvalue is negative even though . (If the electron number were E < E N=1 N=2 treated as a variational parameter, the minimum energy would be attained for .) We N < 2 20 ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: r

text: Ρ

text: accurate

text: KS

text: SCE

text: KS

text: SCE

text: LDA

text: KS

text: SCE

text: LVee,d

text: HF

text: r

text: FIG.

text: for

text: H

text: and

text: various

text: approaches.

text: to

text: the

text: total

text: energy,

text: which

text: for

text: these

text: systems

text: is

text: not

text: very

text: tight

text: as

text: can

text: be

text: seen

text: from

text: the

text: underestimation

text: of

text: versus

text: the

text: actual

text: value

text: This

text: is

text: due

text: to

text: Z

text: Z

text: crit

text: crit

text: the

text: inherent

text: strong

text: correlation

text: nature

text: of

text: the

text: electrons

text: in

text: the

text: SCE

text: formulation

text: that

text: results

text: in

text: underestimating

text: the

text: electron-electron

text: repulsion

text: energy.

text: Self-consistently

text: thus,

text: the

text: KS-SCE

text: densities

text: become

text: quite

text: compact

text: until

text: the

text: kinetic

text: energy

text: starts

text: to

text: dominate

text: in

text: Eq.

text: This

text: is

text: manifested

text: in

text: Fig.

text: where

text: the

text: density

text: of

text: H

text: is

text: displayed

text: for

text: several

text: methods,

text: the

text: KS-SCE

text: yielding

text: the

text: most

text: compact

text: density.

text: Physically,

text: this

text: is

text: due

text: to

text: the

text: fact

text: that

text: the

text: two

text: electrons,

text: being

text: perfectly

text: correlated,

text: can

text: avoid

text: each

text: other

text: as

text: much

text: as

text: possible

text: and

text: can

text: get

text: much

text: closer

text: to

text: the

text: nucleus

text: to

text: lower

text: the

text: total

text: energy.

text: The

text: local

text: corrections

text: to

text: the

text: SCE

text: functional

text: improve

text: considerably

text: the

text: predicted

text: and

text: Z

text: crit

text: give

text: a

text: more

text: realistic

text: description

text: of

text: the

text: electronic

text: interactions.

text: We

text: observe

text: that

text: the

text: KS-

text: SCE+LDA

text: density

text: is

text: too

text: spread

text: out

text: compared

text: to

text: the

text: accurate

text: data

text: (e.g.

text: Fig.

text: This

text: can

text: be

text: attributed

text: to

text: the

text: self-interaction

text: error

text: that

text: is

text: introduced

text: by

text: the

text: LDA

text: correction

text: which

text: is

text: obvious

text: from

text: Eq.

text: the

text: energy

text: densities

text: do

text: not

text: vanish

text: for

text: a

text: density

text: integrating

text: to

text: In

text: Fig.

text: we

text: display

text: also

text: the

text: Hartree-Fock

text: density.

text: It

text: is

text: possible

text: to

text: do

text: this

text: because

text: the

text: HOMO

text: eigenvalue

text: is

text: negative

text: even

text: though

text: (If

text: the

text: electron

text: number

text: were

text: E

text: E

text: treated

text: as

text: a

text: variational

text: parameter,

text: the

text: minimum

text: energy

text: would

text: be

text: attained

text: for

text: We

text: N

text: ##LTFigure##

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