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Quantum indistinguishability in chemical reactions

Matthew P. A. Fishera,1 and Leo Radzihovskyb,c,1

aDepartment of Physics, University of California, Santa Barbara, CA 93106; bCenter for Theory of Quantum Matter, Department of Physics, University ofColorado, Boulder, CO 80309; and cKavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106

Contributed by Matthew P. A. Fisher, March 26, 2018 (sent for review October 23, 2017; reviewed by Eduardo Fradkin and Ashvin Vishwanath)

Quantum indistinguishability plays a crucial role in many low-energy physical phenomena, from quantum fluids to molecularspectroscopy. It is, however, typically ignored in most high-temperature processes, particularly for ionic coordinates, implic-itly assumed to be distinguishable, incoherent, and thus wellapproximated classically. We explore enzymatic chemical reac-tions involving small symmetric molecules and argue that in manysituations a full quantum treatment of collective nuclear degreesof freedom is essential. Supported by several physical arguments,we conjecture a “quantum dynamical selection” (QDS) rule forsmall symmetric molecules that precludes chemical processes thatinvolve direct transitions from orbitally nonsymmetric molecularstates. As we propose and discuss, the implications of the QDSrule include (i) a differential chemical reactivity of para- and ortho-hydrogen, (ii) a mechanism for inducing intermolecular quantumentanglement of nuclear spins, (iii) a mass-independent isotopefractionation mechanism, (iv) an explanation of the enhancedchemical activity of “reactive oxygen species”, (v) illuminating theimportance of ortho-water molecules in modulating the quantumdynamics of liquid water, and (vi) providing the critical quantum-to-biochemical linkage in the nuclear spin model of the (putative)quantum brain, among others.

quantum chemical reactions | quantum indistinguishability | nuclear spincoherence | Berry phases in chemical reactions

The far-reaching impact of quantum indistinguishability infew-particle collisions, in molecular spectroscopy (1), and in

the low-temperature behavior of macroscopic many-body sys-tems (e.g., superfluidity) is well appreciated and extensively stud-ied (2). However, the role of indistinguishability for the dynamicsof macroscopic systems at high temperature remains virtuallyunexplored, typically neglected due to the presumed absenceof necessary quantum coherence. For cohesion of both solidsand molecules, while electrons are treated quantum mechan-ically, the much heavier ions are treated as distinguishableand classical. Moreover, in chemical reactions of moleculesin solution, nuclear spins are generally believed to play littlerole, despite their macroscopic quantum coherence times [espe-cially for spin-1/2 nuclei (3, 4)]. However, for small symmetricmolecules the Pauli principle can inextricably entangle the coher-ent nuclear spin dynamics with the molecular rotational prop-erties. The latter must modulate chemical reaction rates, evenif weakly, thereby coupling nuclear spin dynamics to quantumchemistry.

Molecular hydrogen offers the simplest setting for discussingthe interplay of indistinguishability and chemical reactivity.While the two electrons are tightly bound in a symmetric molec-ular orbital, the proton nuclear spins are weakly coupled, sothat molecular hydrogen comes in two isomers, parahydrogen(nuclear spin singlet) and orthohydrogen (nuclear spin triplet).Treating the motion of the nuclei quantum mechanically, thePauli principle dictates that molecular para- and orthohydrogenrotate with even and odd angular momentum, respectively.

A natural question, that to the best of our knowledge has notbeen asked, is whether the para- and orthospin isomers of molec-ular hydrogen exhibit different chemical reaction rates in solvent.If yes, as might be expected from the different rotational proper-ties of the two spin isomers, what is the magnitude and “sign” of

the effect? Intuitively, one might expect such effects to be small,especially at temperatures well above the rotational constant.

In this paper we explore this and related questions in anumber of systems, focusing on enzymatic reactions with the sub-strate consisting of a small symmetric molecule, characterizedby a “quasi-angular momentum,” Lquasi (defined in Theoretical

Framework). As we elaborate and motivate in the next section,and in contrast to the aforementioned “conventional wisdom,”such bond-breaking chemical reactions can be very sensitive tonuclear spin states, via Pauli transduction through the allowedmolecular rotations. Our central conjecture is that symmetricmolecules can have only a direct bond-breaking chemical reac-tion from a state with an orbitally symmetric wavefunction, i.e.,with zero quasi-angular momentum Lquasi =0 (e.g., the symmet-ric parahydrogen). On the other hand, a molecule constrainedby Fermi/Bose indistinguishability to have a nonzero odd orbitalangular momentum is precluded from breaking its bond by a“quantum dynamical selection” (QDS) rule. Physically, this isdue to a destructive interference between the multiple possiblebond-breaking processes—one for each of the symmetry-relatedmolecular orbital configurations. We emphasize that QDS isnot of energetic origin and is predicted to be operational evenif the molecule’s rotational constant is much smaller than thetemperature.

It should be noted that the “radical-pair mechanism” (5) hasbeen proposed as a process for nuclear spins modulating chemi-cal reaction rates. However, the radical-pair mechanism is quitedistinct from the QDS mechanism, with no role for the quantummechanics of the nuclear coordinates in the former and no rolefor electron spins (free radicals) in the latter.

Significance

Counter to conventional approaches that treat nuclear coor-dinates classically, we explore quantum indistinguishabilityof nuclei in enzymatic chemical reactions of small symmetricmolecules. Supported by several physical arguments, we con-jecture a far-reaching “quantum dynamical selection” (QDS)rule that precludes enzymatic chemical bond-breaking reac-tions from orbitally nonsymmetric molecular states. We pro-pose and discuss experimental implications of QDS, such as (i)differential chemical reactivity in ortho- and parahydrogen, (ii)a mass-independent mechanism for isotope fractionation, (iii)an explanation of the enhanced chemical activity of “reactiveoxygen species”, (iv) a route to parahydrogen-induced hyper-polarization important for zero-field NMR spectroscopy, and(v) critical quantum-to-biochemical linkage in the nuclear spinmodel of the (putative) quantum brain, among others.

Author contributions: M.P.A.F. and L.R. designed research; M.P.A.F. and L.R. performedresearch; M.P.A.F. analyzed data; and M.P.A.F. and L.R. wrote the paper.

Reviewers: E.F., University of Illinois at Urbana–Champaign; and A.V., Harvard University.

The authors declare no conflict of interest.

Published under the PNAS license.1 To whom correspondence may be addressed. Email: mpaf@kitp.ucsb.edu or radzi-hov@colorado.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1718402115/-/DCSupplemental.

Published online April 30, 2018.

www.pnas.org/cgi/doi/10.1073/pnas.1718402115 PNAS | vol. 115 | no. 20 | E4551–E4558

Before exploring the role of quantum indistinguishability inchemical processes, we briefly comment on the important issueof quantum decoherence, the common prejudice being that rapiddecoherence in a wet solution will render all quantum phenom-ena inoperative. Indeed, elevated temperatures will generallymove a system toward classical behavior. For example, when aphysical process oscillating with a characteristic frequency, !,is immersed in a thermal environment, quantum effects willtypically wash out for T � ~!/kB . At body/room temperatureit is thus only phenomena oscillating at very high frequencies(1013 Hz, say, such as molecular vibrational modes) where quan-tum mechanics can modify the dynamics. But this argumentimplicitly presumes thermal equilibrium.

Nuclei with spin-1/2 in molecules or ions tumbling in waterare so weakly coupled to the solvent that macroscopic coher-ence times of seconds or minutes are possible and regularlymeasured in liquid-state NMR (6). But weak coupling is a two-way street; if the solvent disturbs only weakly the nuclear spins,the nuclear spin dynamics will only weakly disturb the dynam-ics of the molecule and the surrounding solvent. However, smallsymmetric molecules, where quantum indistinguishability canentangle nuclear spin states with molecular rotations, provide anexception.

As we detail in Theoretical Framework and Planar Symmet-

ric Molecules, the symmetry of the nuclear spin wavefunctionin such symmetric molecules will dictate a characteristic quasi-angular momentum, Lquasi—equal to a small integer in units of~—that is symmetry protected even in the nonrotationally invari-ant solvent environment. And, provided the molecule’s thermalangular momentum is much larger than ~ the environment can-not readily measure Lquasi, so that the different nuclear-spinsymmetry sectors will remain coherent with one another forexponentially long times. Remarkably, even though the solventis ineffective at measuring Lquasi, we argue that enzymes (whichcatalyze irreversible bond-breaking chemical reactions) can, ineffect, measure Lquasi—implementing a projective measurementonto Lquasi =0.

The rest of this paper is organized as follows. In Theoreti-

cal Framework, focusing on small symmetric molecules with Cn

symmetry, we formulate the general problem and then for con-creteness specialize to the case of n =2 and n =3. With thisformulation, in Quantum Dynamical Selection Rule we then stateour QDS conjecture, discussing both physical and mathematicalplausibility arguments for it in Arguments for QDS Conjecture.We conclude in Conceptual and Experimental Implications withexperimental implications of the QDS rule, proposing a numberof experiments to test the conjecture.

Theoretical FrameworkBeyond Born–Oppenheimer Approximation. In this section wepresent the basic framework for our subsequent discussion ofthe role of symmetry and quantum indistinguishability in chem-ical processes involving catalytically assisted bond breaking insymmetric molecules. In molecular processes the electrons, asfast degrees of freedom, are appropriately treated as fullyquantum-mechanical indistinguishable fermions. In contrast,the constraints of quantum indistinguishability on the nuclearorbital degrees of freedom when treated within the Born–Oppenheimer approximation are invariably neglected, especiallyin solution chemistry [although not always in molecular spec-troscopy (1)]. The molecular dynamics and chemical reactionsare thus assumed to be fully controlled by classical motion ofthe molecular collective coordinates on the Born–Oppenheimeradiabatic energy surface. While this may be adequate for somesystems, we argue that it can be wholly insufficient for chem-ical reactions in small symmetric molecules. As we discuss, insuch systems, the nuclear and electron spin degrees of freedomcan induce Berry phases that constrain the molecular orbital

dynamics on the adiabatic energy surface, which must thenbe treated quantum mechanically since these Berry phasesdo not enter the classical equations of motion. As we argue,the presence of Berry phases can have strong and previouslyunappreciated order-one effects in chemical bond-breakingprocesses.

Planar Symmetric Molecules. For simplicity of presentation wefocus primarily on molecules which possess only a single n-fold symmetry axis that under a 2⇡/n planar rigid-body rotation(implemented by the operator ˆCn) cyclically permutes n indistin-guishable fermionic nuclei. A water molecule provides a familiarexample for n =2 and ammonia for n =3, wherein the protonsare cyclically permuted.

For such molecules the nuclear spin states can be conve-niently chosen to be eigenstates of ˆCn , acquiring a phase fac-tor (eigenvalue) !�⌧

n with !n = e i2⇡/n and the “pseudospin”⌧ taking on values ⌧ =0, 1, 2, ...,n � 1. Due to Fermi statisticswhen the molecule is physically rotated by 2⇡/n radians aroundthe Cn symmetry axis, the total molecular wavefunction—which consists of a product of nuclear rotations, nuclear spins,and electronic molecular states—must acquire a sign (�1)

n�1

due to a cyclic permutation of fermionic nuclei. Providedthe electron wavefunctions transform trivially under the Cn

rotation, this constraint from Fermi statistics of the nuclei,e i(�2⇡⌧/n+2⇡L/n�⇡(n�1))

=1, implies that the collective orbitalangular momentum of the molecule is constrained by ⌧ , tak-ing on values L=Lquasi +nZ with the quasi-angular momentumgiven by

Lquasi =

⇢⌧ , n odd,⌧ +n/2, n even. [1]

Molecular trimer of identical fermionic nuclei. For illustrativeclarity we first formulate the problem for the case of n =3, spe-cializing to three identical fermionic nuclei with nuclear spin 1/2and electrical charge +1. We focus on a singly ionized trimermolecule T⌘A+

3 undergoing a chemical reaction,

A+3 !A2 +A+

, [2]

into a singly ionized atom, A+, and a neutral dimer molecule,D⌘A2. The process is schematically displayed in Fig. 1, wherethe molecular trimer is composed of the three fermionic nuclei

Fig. 1. Schematic of a bond-breaking chemical reaction. (Left) Initial stateof a C3 symmetric molecule with two molecular electrons (blue) bondingthe three nuclei (red) together. (Center) An intermediate state where thereaction is catalyzed by an enzyme that “grabs” two of the nuclei, weaken-ing their bonds to the third by depletion of electronic charge. (Right) Finalproduct state composed of a molecular dimer and an isolated atom.

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with creation operators, ˆF †↵, in spin state ↵= ", #, that form a C3

symmetric molecular configuration characterized by a collectivecoordinate �. The nuclei part of such a molecular state is createdby the three-nuclei operator,

ˆT †⌧ (�)=

X

↵��

�⌧↵��

ˆF †↵(�) ˆF

†�(�+2⇡/3)ˆF †

� (�+4⇡/3). [3]

The three-nuclei spin wavefunction �⌧↵�� is chosen as a ⌧

representation of cyclic permutations, an eigenstate of ˆC3,

ˆC3�⌧↵�� =�⌧

�↵� =!⌧3�

⌧↵�� , [4]

with !3 = e i2⇡/3, required by (

ˆC3)3=1. Thus, for this trimer

molecule, ⌧ takes one of three values, ⌧ =0, 1, 2 (or equivalently,⌧ =0,±1). By construction ˆT⌧ (�) then also forms an irreduciblerepresentation of ˆC3, satisfying

ˆT⌧ (�+2⇡/3)=!⌧3ˆT⌧ (�). [5]

We will at times refer to ⌧ as a pseudospin that encodes both thenuclear spin and the orbital qubits, entangled through the Pauliprinciple of identical nuclei.

Because a discrete 2⇡/3 rotation executes a fermionic cyclicinterchange, the ⌧ representation of the nuclear spin wavefunc-tion imprints a nontrivial Berry phase !⌧ onto the orbital degreeof freedom, �, which we discuss below. For simplicity we havesuppressed the position coordinate describing the center of massof the trimer molecule as well as the orientation of the normal tothis planar trimer molecule.

In addition to the nuclei, a correct description of a moleculemust also consist of bonding electrons that, within the Born–Oppenheimer approximation, occupy the molecular orbitals.For concreteness we consider a (singly ionized) molecule withonly two electrons that form a spin singlet in the ground-statemolecular orbital T (r;�), where the subscript denotes thetrimer nuclear configuration. This wavefunction transforms sym-metrically under ˆC3, satisfying T (r;�+2⇡/3)= T (r;�). Wedenote the electron creation operator in this orbital as

c†�(�)=

Z

r T (r;�)c†�(r). [6]

The trimer molecular state we thus consider can be written as

|T i=X

⌧

Z

�

⌧ (�)c†"(�)c

†#(�)

ˆT †⌧ (�)|vaci⌦ |Ei� , [7]

characterized by an orbital wavefunction in the ⌧ representation,

⌧ (�+2⇡/3)=!⌧3 ⌧ (�). [8]

Were the general orbital wavefunction ⌧ (�) expanded inangular-momentum eigenstates, e iL�, with L2Z, this constraintimplies that L=Lquasi +3Z with a quasi-angular momentumLquasi = ⌧ , consistent with Eq. 1 for n =3.

In Eq. 7 the ket |Ei� denotes the initial quantum state of theenvironment—i.e., the solvent and enzyme—that is entangledwith the molecular rotations through the angle �, as genericallythe environment can “measure” the molecular orientation. Notethat we have implicitly assumed that the initial state of the envi-ronment does not depend on ⌧ , so that |Ei�+2⇡/3 = |Ei�. For amolecule with thermal angular momentum LT (defined through~2L2

T/I = kBT , with moment of inertia I) that is much greaterthan one, LT � 1, the solvent is ineffective in “measuring” themolecular quasi-angular momentum, which is a small fractionof LT . Indeed, the quasi-angular momentum decoherence time

should be exponentially long for large thermal angular momen-tum, varying as t⌧coh ⇠ t0 exp(cL

2T ) with an order one constant c

and t0 a microscopic time of order 1 ps.Molecular dimer and atom products state. As illustrated in Fig.1, the bond-breaking reaction proceeds through an intermediateenzymatic stage that is challenging to describe microscopically.Through its interaction with the electronic orbital degrees offreedom the enzyme temporarily binds and “holds” two of thenuclei, separating them from the third nucleus. This causesmolecular rotations to cease and also weakens the molecularbonds. We assume that the final product state consists of a neu-tral molecular dimer, D ⌘A2, held together by the two electronsand a singly ionized atom A+. The orientation of the dimeris characterized by a single angle coordinate, which we againdenote as � (Fig. 1).

This final product state |Pi can then be expressed as

|Pi=X

⌧

a⌧

Z

�

X

µ,↵��

µ(�)�⌧↵�� |A↵i|D�� ,�i⌦ |E⇤i� , [9]

where |E⇤i� describes the state of the environment (solvent plusenzyme) after the chemical reaction, |A↵i= ˆF †

rA,↵|vaci denotesthe state of the atom (located at position rA), and the state of thedimer molecule (located at position rD ) is given by

|D�� ,�i= ˆF †rD ,�(�)

ˆF †rD ,�(�+⇡)c†"(�)c

†#(�)|vaci. [10]

The electron creation operators on the dimer molecule aregiven by

c†�(�)=

Z

r D(r;�)c†�(r), [11]

with the ground-state molecular orbital for the dimer molecule(when oriented at angle �) D(r;�) assumed to transformsymmetrically under the C2 symmetry of the dimer molecule, D(r;�+⇡)= D(r;�). The dimer orbital wavefunction µ(�)transforms as µ(�+⇡)= e iµ⇡

µ(�) with µ=0, 1.Because we do not expect the nuclear spins state in each ⌧

sector to change appreciably through the chemical reaction, theatomic and dimer states remain entangled through the originalnuclear spin wavefunction, �⌧

↵�� . We have introduced an overallamplitude, a⌧ , which we discuss further in Quantum Dynamical

Selection Rule.Since an enzymatic chemical reaction will typically be strongly

exothermic (releasing, for example, a fraction of electron volts inenergy), the quantum state of the environment after the reaction,|E⇤i�, will be very different from before the chemical reaction,|Ei�—that is, �hE⇤|Ei� =0.Generalization to arbitrary n-mer. Here we briefly generalizefrom n =3 to a planar molecule with n-fold symmetry consistingof n fermionic spin-1/2 nuclei. The nuclear spin wavefunc-tion �⌧

↵1↵2...↵n can be chosen as an eigenstate of the cyclicpermutation symmetry, ˆCn ,

ˆCn�⌧↵1↵2...↵n =�⌧

↵n↵1↵2...↵n�1=!⌧

n�⌧↵1↵2...↵n , [12]

with !n = e i2⇡/n required by (

ˆCn)n=1. The pseudospin now

takes on one of n values, ⌧ =0, 1, 2, ...n � 1. Due to Fermistatistics of the nuclei the total molecular wavefunction mustacquire a factor of (�1)

n�1 under a molecular rotation by 2⇡/n .Assuming that the bonding electrons transform trivially underCn , the molecular orbital wavefunction (as in Eq. 8) must satisfy ⌧ (�+2⇡/n)= (�1)

(n�1)!⌧n ⌧ (�). Equivalently, the allowed

orbital angular momenta are given by L=Lquasi +nZ with thequasi-angular momentum Lquasi given in Eq. 1.

Fisher and Radzihovsky PNAS | vol. 115 | no. 20 | E4553

Quantum Dynamical Selection RuleWe can now state our conjecture, which we refer to as a QDSrule: A bond-breaking enzymatic chemical reaction on a symmet-ric planar molecule implements a projective measurement ontozero quasi-angular momentum, Lquasi =0.

More generally, including for molecules with 3D rotationalsymmetries such as H2 and CH4, our QDS rule implies thefollowing: Enzymatic chemical reactions that (directly) breakthe bonds of a symmetric molecule are strictly forbidden fromorbitally nonsymmetric molecular states.

Here, “direct” implies that the transition proceeds withoutthe molecule first undergoing a nuclear spin flip. For example,the orthostate of molecular hydrogen (which has odd angularmomentum) cannot undergo a direct bond-breaking transitionwithout passing through the parastate. For the n =3 planarmolecule described in Planar Symmetric Molecules our QDS ruleimplies that the amplitude in Eq. 9 vanishes unless ⌧ =0; that is,a⌧ = �⌧0. In the following section we present an argument thatoffers support for the QDS rule.

Arguments for QDS ConjectureFor conceptual reasons it is helpful to divide the enzyme-mediated chemical reaction into three stages, each a distinctquantum state of the molecule and enzyme: (i) a state a , inwhich the molecule is free to rotate and dynamical processesthat exchange the atoms are allowed; (ii) a state b , in whichthe molecule’s rotations are stopped by the enzyme and dynam-ical processes that exchange the atoms are forbidden; and (iii) astate c , in which the chemical bond is broken and the moleculeis fragmented into its constituents. For each of these three quan-tum states a careful discussion of the properties of the accessibleHilbert space is necessary and is taken up in the first threesubsections below.

The full enzymatic chemical reaction corresponds to a processthat takes the system from state a to b and then to c . InIndistinguishable-to-Distinguishable Projective Measurement, weexplore the Born and tunneling amplitudes between these quan-tum states. Since these transitions are either microscopically ormacroscopically irreversible, the full enzymatic reaction shouldbe viewed as implementing “projective measurements” on themolecule. As we demonstrate, the Born amplitude for the pro-jective measurement vanishes unless Lquasi =0, which offers anargument for the validity of the conjectured QDS rule.

A Rotating Symmetric Molecule. We begin with a precise definitionof the initial quantum mechanical state of the rotating symmetricmolecule, offering three representations of the accessible Hilbert

A B C

Fig. 2. Three representations of a rotating diatomic molecule composedof identical nuclei. (A) An explicit physical representation. (B) An effectivemodel for the dynamics of the angular coordinate, '= 2�2 [0, 2⇡), for amolecule with C2 symmetry, represented as a quantum bead on a ring withBerry flux, �0 determined by the nuclear spin wavefunction. The bead wave-function, a('), is discontinuous across an nth root branch cut (red squigglyline), + = ei�0 �. (C) Placing the bead on an n-fold cover of the ringwith '2 [0, 2⇡n)—a Mobius strip for the n = 2 double cover—resolves anyambiguity in the placement of the branch cut.

space. For the case of a diatomic (n =2) molecule, these areillustrated in Fig. 2.Mapping to quantum bead on a ring. To proceed requires a care-ful discussion of the Hilbert space for the “angle” kets that areeigenkets of the angle operator, ˆ�|�i=�|�i. They are defined(for n =3) as |�i= ˆT †

⌧ (�)|vaci, where ˆT †⌧ (�) was introduced

in Eq. 3, with a natural generalization for all n . For the n-foldplanar molecule we have

|�+2⇡/ni= e i�0 |�i, [13]

with�0

2⇡=

Lquasi

n, [14]

so that the angle kets are redundant on the full interval [0, 2⇡).As such, it is convenient to restrict the angle �2 [0, 2⇡/n). Then,we can define a new angle operator '2 [0, 2⇡) and its canonicallyconjugate angular-momentum operator ˆ`2Z with [', ˆ`] = i via

ˆ�= '/n; ˆL=nˆ`+Lquasi. [15]

Consider the simplest Hamiltonian for a rotating molecule,

ˆHn =

ˆL2

2In+Vn(

ˆ�), [16]

with Vn(�+2⇡/n)=Vn(�) an environmental potential (e.g.,the enzyme, solution, etc.), with its periodicity encoding the Cn

symmetry of the molecule. Reexpressing this Hamiltonian interms of the “reduced” variables ˆHn ! ˆH gives

ˆH=

(

ˆ`+�0/2⇡)2

2I +U ('), [17]

with a 2⇡-periodic potential, U ('+2⇡)=U (')⌘Vn('/n),and a rescaled moment of inertia, I =n2In . This Hamiltoniancan be viewed as describing a “fictitious” quantum bead on a ring,with a fictitious magnetic flux, �0, piercing the ring, as shown inFig. 2.

To expose the physics of this flux it is useful to consider theLagrangian of the quantum bead on the ring,

L=

1

2

I(@t')2 �U (')+LB , [18]

with Berry phase term LB =(�0/2⇡)(@t'). In a real (or imag-inary) time path integral this Berry phase term contributes anoverall multiplicative phase factor,

e iSB= e i�0W

, [19]

with W 2Z a winding number defined via 2⇡W='(tf )�'(ti),where ti , tf are the initial and final times, respectively.

It must be emphasized that the wavefunction for the beadon the ring, which we denote as a('), is not single valued forLquasi 6=0, since a('+2⇡)= e i�0 a('), so that a branch cut isrequired. For the planar molecule with Cn symmetry this willbe an nth root of unity branch cut, while for the ortho-dimermolecule it is a square-root cut. In either case, the wavefunctionacross the branch cut is discontinuous, with the value on eitherside of the branch cut denoted +, �, being related by a phasefactor + = e i�0 � (Fig. 2B). For an isolated molecule thisbranch cut can be placed anywhere (gauge invariance) but duringthe enzymatic bond-breaking process a natural gauge invariantformulation is not readily apparent.

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Mobius strip and umbilic torus. To resolve any ambiguity in theplacement of the branch cut with bond breaking present, it ishelpful to view the bead as living on an n-fold cover of thering. Mathematically, we simply extend the range of ' to liein the interval '2 [0, 2⇡n), so that the wavefunction is peri-odic in this enlarged domain, a('+2⇡n)= a('). For n =2

with Lquasi =1 this corresponds to a quantum bead living onthe (single) “edge” of a Mobius strip, as depicted in Fig. 2C.At a given angle on the Mobius strip the wavefunction on theopposite edges of the strip must have a sign change, since inthis case a('+2⇡)=� a('). For n =3 with Lquasi =±1 thequantum bead lives on a (three-sided) umbilic torus, whichmust be circumnavigated three times before returning to thesame edge.

A “Not-Rotating” Molecule. A first step in the bond-breaking pro-cess requires stopping the molecule’s rotation as it binds to theenzyme. What does it mean for a small symmetric molecule tobe not rotating and perhaps having zero angular velocity? For1D translational motion the linear (group) velocity of a quan-tum particle is vg = @Ep/@p, suggesting that angular (group)velocity should be likewise defined, ⌦= @EL/@L. But angu-lar momentum is quantized in units of ~, so this definition isproblematic.

One plausible definition of a molecule to be not rotating isfor its orbital motion to be described by a real wavefunction.For odd n this is equivalent to requiring zero quasi-angularmomentum, Lquasi =0, while for n even it is not since for Lquasi =

n/2 6=0 a real wavefunction can still be constructed. But in eithercase, once we allow for quantum entanglement between themolecule and the solvent/enzyme the notion of the “molecule’swavefunction” becomes problematic.

We believe the best way to impose a not-rotating restric-tion of the molecule is to impose a constraint that disallows adynamical rotation which implements an exchange of the con-stituent atoms. This can be achieved by inserting impenetrablepotential wedges at angles centered around �=0 and �=⇡, asillustrated in Fig. 3A for the diatomic molecule. These wedgesrestrict the molecular rotation angle so that one of the nucleiis restricted to the upper half-plane, �/2<�⇡� �/2, whilethe other resides in the lower half-plane, ⇡+ �/2<�< 2⇡� �/2.Rotations that exchange the nuclei are thereby strictly forbidden.And the two nuclei, while still identical, have effectively become“distinguishable”—even if their spins are aligned.

Upon mapping to the effective coordinate, �!'/n , thewavefunction of the bead on the ring in this not-rotating configu-ration, which we denote as b('), is restricted to the line segment'2 (�, 2⇡� �). As illustrated in Fig. 3B, the “ring” has in effect

A B

Fig. 3. Two representations of a nonrotating diatomic molecule composedof identical nuclei. (A) An explicit physical representation with impenetrablepotential wedges inserted around �= 0 and ⇡ that restrict the molecularrotation angle and constrain one nucleus to the upper half-plane and theother to the lower half-plane. These wedges strictly forbid all rotationalmotions that dynamically exchange the two nuclei. The identical nuclei thusbecome effectively distinguishable. (B) An effective quantum bead model,where the ring, now restricted to the angular range, '2 [�, 2⇡� �), hasbeen cut open—and the bead is constrained to move on a line segment.

been “cut open.” As such, there is no meaning to be ascribedto the effective flux through the ring, present in Fig. 2 for therotating molecule. In contrast to the n-fold cover of the rotatingwavefunction, a('+2⇡n)= a('), the nonrotating wavefunc-tion b(') is defined on a single cover of the cut-open ring with'2 (�, 2⇡� �).

A Bond-Broken Molecule. Once the molecule is not rotating andthe identical fermions are effectively distinguishable, a breakingof the chemical bond is greatly simplified. As illustrated in Fig.4A, with the impenetrable barriers present the molecular bondcan break and the constituent atoms are free to move off intothe upper and lower half-planes, respectively. Once the atomsare physically well separated, their distinguishability no longerrests on the presence of the impenetrable barriers.

In the quantum-bead representation, illustrated in Fig. 4B, thebond-breaking process corresponds to the bead tunneling off theline segment. As such, the location of the bead must now be spec-ified by both the angle ' and a radial coordinate, r , the beadwavefunction taking the form c(r ,').

Since this bond-breaking process will typically be macroscop-ically irreversible, we assume that once the bead tunnels off theline segment it will not return. The tunneling rate for this pro-cess can be then expressed in terms of a Fermi’s golden rule,�b!c ⇠ |Ab!c |2, with tunneling amplitude

Ab!c =

Z 2⇡��

�

d'Hbc(') ⇤c (R,') b('), [20]

with Hbc a tunneling Hamiltonian and R the radius of themolecule.

Indistinguishable-to-Distinguishable Projective Measurement. Wenow turn to the more subtle process where the “rotating”molecule transitions into the not-rotating state, which occurswhen the enzyme “catches” the molecule. When the moleculeis rotating, in state a , dynamical processes that interchange theidentical fermions are allowed and will generally be present. Assuch, these identical fermions are truly “indistinguishable” inthe rotating state. But when the enzyme catches and holds themolecule in place, in the state b , a projective measurement ofthe atomic positions has been implemented (the enzyme is the“observer”) and the identical fermions are now “distinguished.”

The transition rate for this process, �a!b , can be expressedas a product of a microscopic attempt frequency, !ab , and aBorn measurement probability, Pab , that is �a!b =!abPab . TheBorn probability can in turn be expressed as the squared over-lap of the projection of the distinguishable state, b , onto theindistinguishable state, a , that is Pab = |h b | ai|2.

Since the rotating molecule wavefunction, a('), is definedon the n-fold cover of the ring (Mobius strip for n = 2) with'2 (0, 2⇡n), while the not-rotating wavefunction b(') lives ona single cover of the line segment, '2 (�, 2⇡� �), the two statesare seemingly defined in a different Hilbert space. This ambigu-ity can be resolved by noting that the Born amplitude projectingfrom the Mobius strip onto the open line segment can occur fromany of the n edges of the Mobius strip. We thus conjecture thatthese n processes must be summed over,

h b | ai=n�1X

m=0

Z 2⇡��

�

d' ⇤b (') a('+2⇡m). [21]

Upon using the 2⇡ periodicity condition for the Mobius stripwavefunction, a('+2⇡m)= e im�0 a(') with �0 =2⇡Lquasi/n , this can be reexpressed as

h b | ai= Cn

Z 2⇡��

�

d' ⇤b (') a('), [22]

Fisher and Radzihovsky PNAS | vol. 115 | no. 20 | E4555

A B

Fig. 4. Two representations of a bond-broken diatomic molecule composedof identical but distinguishable nuclei. (A) An explicit physical representa-tion with impenetrable potential wedges inserted at �= 0 and ⇡, allowingfor a bond-breaking process to the distinguishable-nuclei state. (B) An effec-tive quantum bead on a ring model, restricted to the angular coordinaterange '2 [�, 2⇡� �), with bond breaking modeled by tunneling the beadoff the ring.

with the amplitude

Cn =

n�1X

m=0

e im�0=

n�1X

m=0

e i2⇡mLquasi/n=n�Lquasi,0. [23]

The Born probability Pab = |h b | ai|2 thus vanishes unlessLquasi =0, as does the rate,�a!b , that the enzyme grabs and holdsthe molecule in a nonrotating distinguishable configuration.

Physically, for Lquasi 6=0, there is a destructive interferencebetween the n-parallel paths, each projecting from an edge ofthe Mobius strip/torus onto the line segment. For the diatomicmolecule this is illustrated in Fig. 5, where the two contributionswith opposite signs will destructively interfere. The rate for thisprocess which stops the molecule’s rotation, �a!b , will vanishunless Lquasi =0—the molecule simply cannot get caught by theenzyme for nonzero Lquasi.

Since the enzymatic reaction requires both stopping themolecules rotation, with rate �a!b , and subsequently breakingthe chemical bond, with rate �b!c 6=0, the full chemical reactionrate is

�a!c =�a!b�b!c

�a!b +�b!c⇠ �Lquasi,0 [24]

and vanishes unless Lquasi =0. Mathematically, the chemicalbond-breaking process is “blocked” by the presence of theBerry phase, operative whenever the orbital molecular state isnonsymmetric.

More physically, this blocking is due to the destructive inter-ference between the n-possible bond-breaking processes—onefor each of the symmetry-related molecular orbital configura-tions. These considerations thus provide support for our con-jectured QDS rule that states the impossibility of (directly)breaking a chemical bond of a symmetric molecule rotatingnonsymmetrically.

Conceptual and Experimental ImplicationsThere are numerous experimental implications of the QDS rule.Since this rule should indeed be viewed as a conjecture, it willhave to be validated or falsified by comparison of theoretical pre-dictions with experiments. Below, we discuss some implicationsof QDS.

Differential Reactivity of Para-/Orthohydrogen. Hydrogen providesthe most familiar example of molecular spin isomers, parahy-drogen with singlet entangled proton spins and rotating with

even angular momentum, and orthohydrogen with triplet spinentanglement and odd rotational angular momentum. Since theallowed rotational angular momentum of such homonucleardimer molecules with S = 1/2 fermionic ions (protons) is givenby L=Lquasi +2Z, the quasi-angular momentum, while zero forparahydrogen, is equal to one for orthohydrogen. The presenceof the Berry phase in the rotation of orthohydrogen will suppressthe bond-breaking chemical reactivity.

Many microbes in biology (7) use H2 as a metabolite, and theenzyme hydrogenase catalyzes the bond-breaking chemical reac-tion, H2 ! 2H+

+2e�. Based on the QDS rule we would expecta differential reactivity between para- and orthohydrogen, withthe reaction rate suppressed for orthohydrogen. Indeed, if thisreaction were to proceed “directly”—without an ionized inter-mediary or a flipping of nuclear spin—QDS would predict acomplete blocking of orthohydrogen reacting. At body temper-ature in a thermal distribution the ortho:para ratio approaches3:1 (set by triplet degeneracy), while it is possible to preparepurified parahydrogen where this ratio is strongly inverted (say,1:10). One might then hope to observe different enzymatic activ-ity for hydrogenase catalysis in these two situations, with purifiedparahydrogen being significantly more reactive.

Possible differential combustion of para- and orthohydrogenwith, say, oxygen might also be interesting to explore, eventhough this reaction is not enzymatic.

Intermolecular Entanglement of Nuclear Spins. The ability to pre-pare purified parahydrogen molecules in solvent and drive abond-breaking chemical reaction enables the preparation of twoprotons with nuclear spins entangled in a singlet. If–when thesetwo protons bind onto a large molecule with different chemicalenvironments, it is sometimes possible to perform a ⇡ rotation onone of the two nuclei to create alignment of the two spins, termedhyperpolarization. These hyperpolarized proton spins can thenbe used to transfer spin polarization to the nuclei of atoms onthe molecules to which they are bonded.

There is, of course, a long precedent for liquid-state NMR,exploiting the fact of very long decoherence times in the rapidlyfluctuating liquid environment (3). Indeed, soon after Peter Shordeveloped his prime factoring quantum algorithm, liquid-stateNMR quantum computing efforts were the first out of the gate(8). In NMR quantum computing one uses a solvent hosting aconcentration of identical molecules with multiple nuclear spins(say, protons). Ideally, the chemical environments of the differ-ent nuclei are different, so that they each have a different NMRchemical shift, and can thereby be addressed independently byvarying the radio frequency. In principle it is then possible toperform qubit operations on these spins. However, there are twomajor drawbacks to NMR quantum computing—the difficulty inscalability and the challenge of preparing sufficiently entangled

Fig. 5. The enzymatic projective measurement that induces a transitionfrom a rotating to a not-rotating molecular state can be described as a pro-jective overlap between the rotating state with the quantum bead on theMobius strip, a, and a not-rotating state where the bead is on the cut-open outer ring, b. This projective measurement implements a transitionfrom an initial state in which the identical atoms are indistinguishable to afinal distinguishable state—the enzyme acting as an observer.

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initial states. As we now suggest, it is possible that both of thesecan be circumvented by using small symmetric molecules whichare the substrate for bond-breaking enzymes.

By way of illustration, we consider the symmetric biochemicalion pyrophosphate, P2O4�

7 (usually abbreviated as PPi), whichis important in metabolic activity. Pyrophosphate is a phosphatedimer, which consists of two phosphate ions, PO3�

4 , that sharea central oxygen. (The inorganic-phosphate ion, abbreviated asPi, consists of a phosphorus atom tetrahedrally bonded to fouroxygens.) Since the phosphorus nucleus is an S=1/2 fermionand the oxygens are S =0 bosons, the twofold symmetry of PPi,which interchanges the two 31P nuclei and the three end oxygens,will, like molecular hydrogen, have two isomers, para-PPi andortho-PPi. Moreover, para- and ortho-PPi will rotate with evenand odd angular momentum, respectively. Thus, ortho-PPi, withLquasi =1, rotates with a nontrivial Berry phase.

In biochemistry there is an enzyme (called pyrophosphatase)which catalyzes the bond-breaking reaction, PPi ! Pi + Pi. Dueto the Berry phase term in ortho-PPi, we expect that this reac-tion will be strongly suppressed, if not blocked entirely. Then,provided only para-PPi reacts, the two liberated Pi ions will havenuclear spins which are entangled in a singlet. Such intermolecu-lar entanglement of nuclear spins could, in principle, jump startliquid-state NMR quantum computing efforts, allowing for bothscalability and highly entangled initial-state preparation.

QDS Mass-Independent Mechanism for Isotope Fractionation. Iso-tope fractionation refers to processes that affect the relativeabundance of (usually) stable isotopes, often used in isotopegeochemistry and biochemistry. There are several known mech-anisms. Kinetic isotope fractionation is a mass-dependent mech-anism in which the diffusion constant of a molecule varies withthe mass of the isotope. This process is relevant to oxygen evapo-ration from water, where an oxygen molecule, which has one (ortwo) of the heavier oxygen isotopes (17O and 18O), is less likelyto evaporate. This leads to a slight depletion in the isotope ratiosof 17O/16O and 18O/16O in the vapor relative to that in the liquidwater.

Another mass-dependent isotope fractionation phenomenonoccurs in some chemical reactions, where the isotope abun-dances in the products of the reaction are (very) slightly differentfrom those in the reactants. In biochemistry this effect is usuallyascribed to an isotopic mass-induced change in the frequency ofthe molecular quantum zero-point vibrational fluctuations whenbonded in the pocket of an enzyme. This modifies slightly theenergy of the activation barrier which must be crossed for thebond-breaking reaction to proceed.

However, there are known isotopic fractionation processeswhich are “mass independent,” a classic example being theincreased abundance of the heavier oxygen isotopes in the for-mation of ozone from two oxygen molecules (9). In ozone isotopefractionation the relative increased abundance of 17O and 18Ois largely the same. While there have been theoretical propos-als to explain this ozone isotope anomaly, these are not withoutcontroversy (10).

Here, as we briefly describe, our conjectured QDS rule forchemical reactions involving small symmetric molecules leadsnaturally to the prediction of a mass-independent mechanism forisotope fractionation, driven by the quantum distinguishabilityof the two different isotopes. In the presence of isotopes thatdestroy the molecular rotational symmetry, the QDS rule is nolonger operative and one would expect the chemical reaction toproceed more rapidly.

By way of illustration we again consider the enzymatic hydrol-ysis reaction, PPi ! Pi + Pi. As we now detail, in this experimentone would predict a large heavy oxygen isotope fractionationeffect. Indeed, if one of the six “end” oxygens in PPi is aheavy oxygen isotope, the symmetry of PPi under a rotation is

broken and the reaction becomes “unblocked” (independent ofthe nuclear spin state).

If correct, we would then predict a very large mass-indepen-dent oxygen isotope fractionation which concentrates the heavyoxygen isotopes in the products (Pi + Pi). For the early stages ofthis reaction, before the isotopically modified PPi are depleted,one would, in fact, predict a factor of 4 increase in the ratios of17O/16O and 18O/16O in the enzymatic reaction PPi ! Pi + Pi.

To be more quantitative we introduce a dimensionless func-tion, R(f ), where R denotes the ratio of the heavy isotope ofoxygen in the products, relative to the reactants,

R(f )=[

18O/16O]prod

[

18O/16O]react

, [25]

and f 2 [0, 1] is the “extent” of the reaction. In a conventionalisotope fractionation framework, one would expect a very smalleffect; that is, R(f )⇡ 1. But within our QDS conjecture, ifcorrect, one would have

R(f )=1� (1� f )�

f, [26]

with �=4. Experiments to look for this effect are presentlyunderway.

Activity of Reactive Oxygen Species. In biochemistry it is wellknown that during ATP synthesis the oxygen molecule picksup an electron and becomes a negatively charged “superoxide”ion, O�

2 (11). Having an odd number of electrons (with electronspin-1/2) superoxide is a “free radical.” Together with hydrogenperoxide (H2O2) and the hydroxyl radical (the electrically neu-tral form of the hydroxide ion) the superoxide ion is known asa reactive oxygen species (ROS). ROS ions can cause oxidativedamage due in part to their reactivity. Indeed, in the free-radicaltheory, oxidative damage initiated by ROS is a major contributorto aging. In biology there are specific enzymes to break down theROS to produce benign molecules (e.g., water) (12).

In contrast to the ROS, the stable state of molecular oxygen(“triplet oxygen”) is less reactive in biology. As we detail below,we propose that this difference from triplet molecular oxygen canbe understood in terms of our conjectured QDS rule.

First, we note that standard analysis of electronic molecularstates (that we relegate to Supporting Information) shows thatunder C2 rotation the electronic states in the triplet molecu-lar (neutral) oxygen exhibit an overall sign change. Because 16Onuclei are spinless bosons, there is no nuclear contribution to theBerry phase.

Thus, the triplet neutral oxygen molecule O2 exhibits a purelyelectronic ⇡ Berry phase, despite a spinless bosonic character ofthe 16O nuclei. It therefore rotates with odd angular momen-tum, L=Lquasi +2Z, with Lquasi =1, identical to orthohydrogen.Our QDS conjecture then implies that a direct bond-breakingchemical reaction of triplet oxygen is strictly forbidden.

In contrast to triplet oxygen, the superoxide ion O�2 is not

blocked by the QDS rule and can thus undergo a direct chemicalbond-breaking transition. Indeed, as detailed in Supporting Infor-

mation, due to the electronic nonzero orbital and spin angularmomenta aligned along the body axis, the two ends of the super-oxide ion are distinguishable. Thus, in contrast to triplet oxygen,superoxide does not have any symmetry under a 180

�rotation

that interchanges the two oxygen nuclei. The superoxide ion canthus rotate with any integer value of the angular momentum,L=0, 1, 2, 3, ....

As a result, the QDS is not operative and thus there is no selec-tion rule precluding a direct bond-breaking chemical reaction ofthe superoxide ion. We propose that it is this feature of superox-

Fisher and Radzihovsky PNAS | vol. 115 | no. 20 | E4557

ide, relative to triplet oxygen, which accounts, at least in part, forthe high reactivity of superoxide and explains why it is a “ROS.”

Ortho-Water as a Quantum Disentangled Liquid. Molecular waterhas a C2 symmetry axis which exchanges the two protons. Thus,as for molecular hydrogen, water comes in two variants, para-water and ortho-water which rotate with even and odd angularmomentum, respectively. QDS then predicts that the ortho-water molecule (with Lquasi =1) cannot undergo a direct chem-ical reaction that splinters the molecule into a proton and ahydroxide ion, OH�.

Since the difference between the rotational kinetic energy ofa parawater and an ortho-water molecule is roughly 30 K, liquidwater consists of 75% ortho-water molecules and 25% parawa-ter molecules. In one remarkable paper (13) it was reportedthat gaseous water vapor can be substantially enriched in eitherortho- or para-water molecules and then condensed to createortho- and para-liquid water—although attempts to reproducethis work have been unsuccessful (14). If QDS is operative,ortho-liquid water would be quite remarkable, having zero con-centration of either “free” protons or free hydroxide ions, despitethese being energetically accessible at finite temperature. The-oretically, ortho-liquid water would then be an example of a“quantum disentangled liquid” in which the protons are enslavedto the oxygen ions and do not contribute independently to theentropy density (15).

Experimentally, one would predict that ortho-liquid waterwould have vanishingly small electrical conductivity, nonzeroonly due to ortho-to-para conversion. Data on “shocked” super-critical water indicate that above a critical pressure the electricalconductivity increases by nine orders of magnitude (16). Per-haps this is due to a transition from a quantum-disentangledto a thermal state, where most (if not all) of the ortho-watermolecules are broken into a proton and a hydroxide ion, leadingto a significant electrical conductivity?

The properties of ortho-solid ice might also be quite interest-ing, provided QDS is operative. While an extensive equilibriumentropy would still be expected (consistent with the ice rules), thequantum dynamics would be quite different. Rather than protonshopping between neighboring oxygen ions, in ortho-ice theseprocesses would actually correspond to collective rotations of thewater molecules. The nature of the quantum dynamical quench-ing of the entropy when ice is cooled to very low temperatures isworthy of future investigation.

Summary and ConclusionsIn this paper we have explored the role of quantum indistin-guishability of nuclear degrees of freedom in enzymatic chemicalreactions. Focusing on chemical bond breaking in small sym-metric molecules, we argued that the symmetry properties ofthe nuclear spins, which are entangled with—and dictate—theallowed angular momentum of the molecules’ orbital dynamics,

can have an order one effect on the chemical reaction rate.Our central thesis is a QDS rule which posits that direct bond-breaking reactions from orbitally asymmetric molecular statesare blocked, and only orbitally symmetric molecular states canundergo a bond-breaking reaction. This selection rule, which isnot of energetic origin, arises due to a destructive interferencein the Born amplitude for the enzymatically mediated projectivemeasurement.

The QDS rule is intimately linked to the importance ofFermi/Bose indistinguishability of the nuclei during the enzy-matic process which implements a projective measurement ontoorbitally symmetric molecular states. Mathematically, a Berryphase term, which encodes the Fermi/Bose indistinguishability,leads to an interference between the multiple bond-breakingprocesses—one for each of the symmetry-related molecularorbital configurations. For an orbitally nonsymmetric molecu-lar state this interference is destructive, thereby closing off thebond-breaking reaction—offering a mathematical description ofthe QDS rule.

In much of this paper we focused on simple molecules with aplanar Cn rotational symmetry about a particular molecular axis.In this case the Berry phase is determined by a quasi-angularmomentum, Lquasi, set by the symmetry of the nuclear spinwavefunction. For this planar case our QDS rule predicts thatan enzymatic bond-breaking transition implements a projectivemeasurement onto Lquasi =0.

Our QDS rule leads to a number of experimental implicationsthat we explored in Conceptual and Experimental Implications,including (i) a differential chemical reactivity of para- and ortho-hydrogen, (ii) a mechanism for inducing intermolecular quantumentanglement of nuclear spins, (iii) a mass-independent isotopefractionation mechanism, (iv) an explanation of the enhancedchemical activity of ROS, and (v) illuminating the importanceof ortho-water molecules in modulating the quantum dynamicsof liquid water.

ACKNOWLEDGMENTS. We are deeply indebted and most grateful to Stu-art Licht for general discussions on this topic and especially for emphasizingthe importance of oxygen isotope fractionation experiments to access the(putative) role of nuclear spins in the enzymatic hydrolysis of pyrophos-phate, which we discussed in Conceptual and Experimental Implications,QDS Mass-Independent Mechanism for Isotope Fractionation. We also thankJason Alicea, Leon Balents, Maissam Barkeshli, Steve Girvin, Victor Gurarie,Andreas Ludwig, Lesik Motrunich, Michael Mulligan, David Nesbitt, NickRead, T. Senthil, Michael Swift, Ashvin Vishwanath, and Mike Zaletel fortheir patience and input on our work. M.P.A.F.’s research was supportedin part by the National Science Foundation (NSF) under Grant DMR-14-04230 and by the Caltech Institute of Quantum Information and Matter,an NSF Physics Frontiers Center with support from the Gordon and BettyMoore Foundation. M.P.A.F. is grateful to the Heising-Simons Foundationfor support. L.R. was supported by the Simons Investigator Award fromthe Simons Foundation, by the NSF under Grant DMR-1001240, by the NSFMaterial Research Science and Engineering Center Grant DMR-1420736, andby the Kavli Institute for Theoretical Physics (KITP) under Grant NSF PHY-1125915. L.R. thanks the KITP for its hospitality as part of the SyntheticMatter workshop and sabbatical program.

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