Mathieu Taillefumier

Owen Benton

Han Yan

Ludovic Jaubert

Nic Shannon

Frustrating quantum spin ice : a tale of three spin liquids, and hidden order

Frustration in magnetic interactions often gives rise to disordered ground states with subtle and beautiful properties. The spin ices Ho2Ti2O7 and Dy2Ti2O7 exemplify this phenomenon, displaying a classical spin liquid state, with fractionalized magnetic-monopole excitations. Recently there has been great interest in closely-related "quantum spin ice" materials, following the realization that anisotropic exchange interactions could convert spin ice into a massively-entangled, quantum, spin liquid, where magnetic monopoles become the charges of an emergent quantum electrodynamics. Here we show that by allowing for frustration in the transverse interactions, on top of the inherent frustration of spin ice, an even richer scenario can be realized. Using large-scale numerical simulations of the simplest model of a quantum spin ice, the XXZ model on the pyrochlore lattice, we find that frustrated transverse exchange opens the door to not one, but three distinct types of spin liquid, as well as a phase with hidden, spin-nematic, order. We explore the experimental signatures of each of these new phases, making explicit predictions for inelastic neutron scattering. These results show an intriguing similarity to experiments on a range of pyrochlore oxides.

I. INTRODUCTION

The search for spin liquids -disordered phases of magnets which support entirely new forms of magnetic excitation -has become one of the defining themes of modern condensed-matter physics [START_REF] Lee | An end to the drought of quantum spin liquids[END_REF][START_REF] Balents | Spin liquids in frustrated magnets[END_REF]. In this context, the pyrochlore lattice, a corner-sharing network of tetrahedra found in a wide range of naturally-occurring minerals, has proved an amazing gift to science. Pyrochlore magnets play host to a variety of unconventional forms of magnetic order, and include systems which have not been observed to order at any temperature [START_REF] Gardner | Magnetic pyrochlore oxides[END_REF]. Perhaps the most celebrated of these is the "spin ice" found in the Ising magnets Ho 2 Ti 2 O 7 and Dy 2 Ti 2 O 7 [START_REF] Bramwell | Spin Ice State in Frustrated Magnetic Pyrochlore Materials[END_REF]; a classical spin liquid, described by an emergent U(1) lattice gauge theory with magnetic monopole excitations [START_REF] Castelnovo | Spin Ice, Fractionalization, and Topological Order[END_REF].

As the understanding of spin ice has grown, so more attention has been given to the role of quantum effects. These are of particular relevance where a spin-ice arises through anisotropic exchange interactions in a pyrochlore magnet [START_REF] Curnoe | Quantum spin configurations in Tb2Ti2O7[END_REF][START_REF] Hamid | Dynamically Induced Frustration as a Route to a Quantum Spin Ice State in Tb2Ti2O7 via Virtual Crys-tal Field Excitations and Quantum Many-Body Effects[END_REF][START_REF] Onoda | Quantum melting of spin ice: Emergent cooperative quadrupole and chirality[END_REF][START_REF] Onoda | Quantum fluctuations in the effective pseudospin-1 2 model for magnetic pyrochlore oxides[END_REF][START_REF] Ross | Quantum excitations in quantum spin ice[END_REF][START_REF] Gingras | Quantum spin ice: a search for gapless quantum spin liquids in pyrochlore magnets[END_REF], and have the potential to convert classical spin ice into a massively-entangled, quantum, spin liquid, described by an emergent U(1) quantum electrodynamics [START_REF] Gingras | Quantum spin ice: a search for gapless quantum spin liquids in pyrochlore magnets[END_REF][START_REF] Hermele | Pyrochlore photons: The U(1) spin liquid in a S= 1 2 three-dimensional frustrated magnet[END_REF][START_REF] Banerjee | Unusual liquid state of hard-core bosons on the pyrochlore lattice[END_REF][START_REF] Shannon | Quantum ice: A quantum monte carlo study[END_REF][START_REF] Benton | Seeing the light: Experimental signatures of emergent electromagnetism in a quantum spin ice[END_REF][START_REF] Savary | Coulombic quantum liquids in spin-1/2 pyrochlores[END_REF][START_REF] Lee | Generic quantum spin ice[END_REF][START_REF] Hao | Bosonic many-body theory of quantum spin ice[END_REF][START_REF] Mcclarty | Chain-based order and quantum spin liquids in dipolar spin ice[END_REF][START_REF] Kato | Numerical evidence of quantum melting of spin ice: Quantum-to-classical crossover[END_REF][START_REF] Shannon | Spin ice[END_REF]. At the same time, great progress has been made in synthesizing and characterizing magnetic pyrochlore oxides. As well as revealing a number of candidates for quantum spin-ice behaviour [START_REF] Zhou | Dynamic Spin Ice: Pr2Sn2O7[END_REF][START_REF] Chang | Higgs transition from a magnetic Coulomb liquid to a ferromagnet in Yb2Ti2O7[END_REF][START_REF] Fennell | Power-Law Spin Correlations in the Pyrochlore Antiferromagnet Tb2Ti2O7[END_REF][START_REF] Kimura | Quantum fluctuations in spin-ice-like Pr2Zr2O7[END_REF][START_REF] Sibille | Candidate Quantum Spin Liquid in the Ce 3+ Pyrochlore Stannate Ce2Sn2O7[END_REF][START_REF] Sibille | Candidate quantum spin ice in the pyrochlore Pr2Hf2O7[END_REF][START_REF] Anand | Physical properties of the candidate quantum spin-ice system Pr2Hf2O7[END_REF][START_REF] Wen | Disordered Route to the Coulomb Quantum Spin Liquid: Random Transverse Fields on Spin Ice in Pr2Zr2O7[END_REF], these experiments have turned up many unusual and unexpected magnetic states in systems with strongly anisotropic exchange [START_REF] Dalmas De Réotier | Spin Dynamics and Magnetic Order in Magnetically Frustrated Tb2Sn2O7[END_REF][START_REF] Dun | Yb2Sn2O7: A magnetic Coulomb liquid at a quantum critical point[END_REF][START_REF] Yaouanc | Dynamical Splayed Ferromagnetic Ground State in the Quantum Spin Ice Yb2Sn2O7[END_REF][START_REF] Taniguchi | Long-range order and spin-liquid states of polycrystalline Tb2+xTi2-xO7+y[END_REF][START_REF] Hallas | Incipient Ferromagnetism in Tb2Ge2O7: Application of Chemical Pressure to the Enigmatic Spin-Liquid Compound Tb2Ti2O7[END_REF][START_REF] Hallas | Magnetic frustration in lead pyrochlores[END_REF][START_REF] Hallas | Universal dynamic magnetism in Yb pyrochlores with disparate ground states[END_REF][START_REF] Petit | Observation of magnetic fragmentation in spin ice[END_REF][START_REF] Petit | Antiferroquadrupolar correlations in the quantum spin ice candidate Pr2Zr2O7[END_REF][START_REF] Takatsu | Quadrupole Order in the Frustrated Pyrochlore Tb2+xTi2-xO7+y[END_REF].

The main message of this Article is that even the simplest model of a quantum spin ice -the XXZ model on a pyrochlore lattice -has far more to offer than spin ice alone. Concentrating on frustrated interactions, in the classical limit, accessible to large-scale simulation, we find not one, but three, distinct types of spin liquid, each of which is described by a different lattice gauge theory [Fig. 1]. As an added bonus, one of these spin liquids undergoes a phase transition at low temperatures into a state with hidden, spin-nematic, order. Explicit predictions are made for the experimental signatures of each of these different spin liquids [Fig. 2], and molecular dynamics simulations are used to characterise the gapless collective excitations of the spin-nematic phase [Fig. 5]. We find that the portrait which emerges has striking similarities with the behavior of a number of pyrochlore materials.

The simplest model able to capture quantum effects in a spin ice is the XXZ model on the pyrochlore lattice

H QSI = ij J zz S z i S z j -J ± S + i S - j + S - i S + j ( 1
)
where S i = (S x i , S y i , S z i ) is a (pseudo)spin-half operator describing the two states of the lowest energy doublet of a magnetic ion subject to a strong crystal electric field (CEF). The symmetry of the lattice requires that the quantization axis of each spin (here, S z i ) lies on a local [111] axis, as defined in Appendix A.

Ising interactions, J zz > 0, favor states obeying the "ice rules" in which two spins point into, and two spins point out of, each tetrahedron on the lattice. The transverse term, J ± , introduces dynamics about these spin-ice configurations and, for larger values of J ± /J zz , can drive the system into a state with easy-plane order [START_REF] Savary | Coulombic quantum liquids in spin-1/2 pyrochlores[END_REF][START_REF] Lee | Generic quantum spin ice[END_REF][START_REF] Onoda | Effective quantum pseudospin-1/2 model for Yb pyrochlore oxides[END_REF][START_REF] Wong | Ground state phase diagram of generic xy pyrochlore magnets with quantum fluctuations[END_REF][START_REF] Yan | Theory of multiple-phase competition in pyrochlore magnets with anisotropic exchange with application to Yb2Ti2O7, Er2Ti2O7, and Er2Sn2O7[END_REF][START_REF] Benton | Classical and quantum spin liquids on the pyrochlore lattice[END_REF]. The physical meaning of this easy-plane order depends on the nature of the magnetic ion. For Kramers ions like Yb 3+ and Er 3+ all components of S relate to a magnetic dipole moment [START_REF] Ross | Quantum excitations in quantum spin ice[END_REF], and the ordered phase is an easyplane antiferromagnet. However for non-Kramers ions such as Pr 3+ and Tb 3+ [START_REF] Onoda | Quantum melting of spin ice: Emergent cooperative quadrupole and chirality[END_REF][START_REF] Petit | Antiferroquadrupolar correlations in the quantum spin ice candidate Pr2Zr2O7[END_REF], or "dipolar-octupolar" Kramers ions like Nd 3+ or Ce 3+ [START_REF] Huang | Quantum spin ices and topological phases from dipolaroctupolar doublets on the pyrochlore lattice[END_REF], the easy-plane order may have quadrupolar (octupolar) character. In what follows, we consider explicitly the case of Kramers ions.

For J ± > 0, H QSI [Eq. ( 1)] is unfrustrated, in the sense that it is free of sign problems in Quantum Monte Carlo (QMC) simulation. In this case, the phase diagram is already well-established [START_REF] Banerjee | Unusual liquid state of hard-core bosons on the pyrochlore lattice[END_REF][START_REF] Kato | Numerical evidence of quantum melting of spin ice: Quantum-to-classical crossover[END_REF][START_REF] Shannon | Spin ice[END_REF]. For J ± /J zz 0.05, QMC simulations find a crossover from a conventional paramagnet into a classical spin-liquid (spin ice) at a temperature T * /J zz ∼ 0.2, and a second crossover into a quantum spin liquid (QSL) at a much lower temperature T * QSL /J zz ∼ (J ± /J zz ) 3 . In the low temperature quantum spin liquid regime, the magnetic monopoles of classical spin ice become dynamic, fractional, spin excitations (spinons), while the spectrum of the model also includes gapless photons [START_REF] Hermele | Pyrochlore photons: The U(1) spin liquid in a S= 1 2 three-dimensional frustrated magnet[END_REF][START_REF] Benton | Seeing the light: Experimental signatures of emergent electromagnetism in a quantum spin ice[END_REF]. For J ± /J zz 0.05, the U(1) QSL gives way to easy-plane antiferromagnetic order (AF ⊥ ), in which spins lie in the plane perpendicular to the local S z -axis [START_REF] Banerjee | Unusual liquid state of hard-core bosons on the pyrochlore lattice[END_REF][START_REF] Kato | Numerical evidence of quantum melting of spin ice: Quantum-to-classical crossover[END_REF][START_REF] Shannon | Spin ice[END_REF].

Very little is known about the properties of H QSI for J ± < 0 [START_REF] Onoda | Quantum fluctuations in the effective pseudospin-1 2 model for magnetic pyrochlore oxides[END_REF][START_REF] Lee | Generic quantum spin ice[END_REF][START_REF] Petit | Antiferroquadrupolar correlations in the quantum spin ice candidate Pr2Zr2O7[END_REF]. On perturbative grounds, it is expected that the ground state for |J ± |/J zz 1 will also be a U(1) QSL [START_REF] Hermele | Pyrochlore photons: The U(1) spin liquid in a S= 1 2 three-dimensional frustrated magnet[END_REF], albeit one with a modified spinon dis-persion [START_REF] Lee | Generic quantum spin ice[END_REF][START_REF] Chen | The Spectral Periodicty of Spinon Continuum in Quantum Spin Ice[END_REF]. Gauge Mean-field calculations suggest that this QSL persists over a broad range of parameters, -4.13 J ± /J zz < 0 [START_REF] Lee | Generic quantum spin ice[END_REF]. But the nature of competing ordered -or disordered -phases for J ± < 0 remains an open question.

There are many reasons to believe that the properties of the quantum spin ice model, H QSI [Eq. ( 1)] for frustrated coupling J ± < 0, could be even richer than for J ± > 0. In particular, for J ± /J zz = -1 2 , H QSI [Eq. ( 1)] is equivalent (up to a site-dependent spin-rotation), to the Heisenberg antiferromagnet (HAF) on a pyrochlore lattice. Like spin ice, the HAF is known to support a classical spin liquid [START_REF] Moessner | Properties of a classical spin liquid: The heisenberg pyrochlore antiferromagnet[END_REF][START_REF] Moessner | Low-temperature properties of classical geometrically frustrated antiferromagnets[END_REF][START_REF] Isakov | Dipolar spin correlations in classical pyrochlore magnets[END_REF][START_REF] Henley | Power-law spin correlations in pyrochlore antiferromagnets[END_REF], and it has also been argued to support a QSL ground state [START_REF] Canals | Pyrochlore antiferromagnet: A three-dimensional quantum spin liquid[END_REF][START_REF] Canals | Quantum spin liquid: The heisenberg antiferromagnet on the threedimensional pyrochlore lattice[END_REF][START_REF] Burnell | Monopole flux state on the pyrochlore lattice[END_REF][START_REF] Huang | Spin-ice state of the quantum heisenberg antiferromagnet on the pyrochlore lattice[END_REF]. And, crucially, both the classical and quantum spin liquids in the HAF have a qualitatively different character from those found in spin ice. This sets up a competition between two different kinds of spin liquid, namely spin ice for J ± /J zz ≈ 0, and a state homologous to the HAF for J ± /J zz ≈ - 1 2 . It also opens the door for yet more novel magnetic phases for J ± /J zz < -1 2 .

Since the quantum spin ice model, H QSI [Eq. ( 1)], is inaccessible to QMC for J ± < 0, we instead study its finite-temperature properties using classical Monte Carlo (MC) simulation -the results are summarised in the phase diagram Fig. 1. For J ± > 0, this phase diagram is very similar to that previously found in QMC simulations [START_REF] Banerjee | Unusual liquid state of hard-core bosons on the pyrochlore lattice[END_REF][START_REF] Kato | Numerical evidence of quantum melting of spin ice: Quantum-to-classical crossover[END_REF][START_REF] Shannon | Spin ice[END_REF] -at a qualitative level, the only significant difference is the absence of a QSL below T * QSL /J zz ∼ (J ± /J zz ) 3 0.005. At a quantitative level, we find changes in numerical values of the crossover temperature associated with the spin ice regime, T * 1 , and the parameters for zero-temperature boundary between SL I and AF ⊥ . These changes can be ascribed to the fact that the magnetic monopoles (spinons) are not quantized in classical simulations and do not develop phase coherence [55]. Further details of classical MC simulations for J ± > 0 will be presented elsewhere [56].

We now turn to the frustrated case, J ± < 0. At low temperatures, spin-ice correlations persist up to J ± /J zz = -1 2 [START_REF] Onoda | Quantum fluctuations in the effective pseudospin-1 2 model for magnetic pyrochlore oxides[END_REF][START_REF] Petit | Antiferroquadrupolar correlations in the quantum spin ice candidate Pr2Zr2O7[END_REF], as illustrated in Fig. 2a. Upon reaching J ± /J zz = -1 2 the system becomes thermodynamically equivalent to a HAF. This high-symmetry point gives rise to a new form of spin liquid at finite temperature, labelled SL III in Fig. 1. Once again, this spin liquid has algebraic correlations, as shown in Fig. 2c, but with qualitatively different character from spin ice [Fig. 2a]. These correlations persist up to a crossover temperature T * 3 associated with the Curie-law crossover (CLC) in the magnetic susceptibility [START_REF] Jaubert | Topologicalsector fluctuations and curie-law crossover in spin ice[END_REF].

While the correlations measured in the equal-time structure factor S(q) are also different from those found in the HAF [START_REF] Moessner | Low-temperature properties of classical geometrically frustrated antiferromagnets[END_REF][START_REF] Isakov | Dipolar spin correlations in classical pyrochlore magnets[END_REF][START_REF] Conlon | Spin dynamics in pyrochlore heisenberg antiferromagnets[END_REF], the two models are equivalent up to a local coordinate transformation. And, by analogy with earlier work on the HAF [START_REF] Isakov | Dipolar spin correlations in classical pyrochlore magnets[END_REF][START_REF] Henley | Power-law spin correlations in pyrochlore antiferromagnets[END_REF][START_REF] Henley | The "Coulomb Phase" in Frustrated Systems[END_REF], the spin liquid SL III can be described by a U(1)×U(1)×U(1) gauge theory. We refer to this state below as the pseudo-Heisenberg antiferromagnet (pHAF).

The situation for J ± /J zz < -1 2 is even more interesting. Below a second crossover scale, T * 2 < T * 3 , identifiable by a reduction in the fluctuations of the z-components of the spins [see Appendix B], the spin liquid SL III (pHAF), gives way to a new spin liquid, labelled SL II in Fig. 1. Spin correlations in this phase have algebraic character, with pinch-points in S(q) [Fig. 2b]. However these correlations are qualitatively different from those in either spin ice [Fig. 2a], or the pHAF [Fig. 2c]. At a still lower temperature, T Q < T * 2 , the system undergoes thermodynamic phase transition, marked by a clear anomaly in the specific heat. None the less, this phase transition does not give rise to any magnetic Bragg peaks in S(q) and, at least as far as dipolar spin correlations are concerned, the system remains disordered.

While the new phase for T < T Q -labelled Q ⊥ in Fig. 1 -does not exhibit any conventional magnetic order, it does have a hidden, spin-nematic, order. The associated wavevector is q = 0, and the relevant order parameter is,

Q ⊥ = S x2 -S y 2 2S x S y , (2)
where S i = (S x i , S y i , S z i ) is written in the local frame of site i, defined in Appendix A. This order parameter quantifies the spontaneous breaking of the global U(1) symmetry by the singling out of one axis within the local xy-plane. Q ⊥ transforms with the E-irrep of the tetrahedral pointgroup T d , and is formally identical to the order parameter for the spin-nematic phases found in some frustrated magnets in applied magnetic field [START_REF] Shannon | Nematic order in square lattice frustrated ferromagnets[END_REF][START_REF] Shannon | Nematic, vector-multipole, and plateau-liquid states in the classical O(3) pyrochlore antiferromagnet with biquadratic interactions in applied magnetic field[END_REF][START_REF] Smerald | Theory of inelastic neutron scattering in a field-induced spin-nematic state[END_REF]. In common with these systems, the associated Landau theory

F Q ⊥ = a 2 (T ) Q 2 ⊥ + a 4 Q 4 ⊥ + . . . , (3)
lacks a cubic term, and therefore permits a continuous phase transition. Simulations suggest that the phase transition at T = T Q is indeed continuous for J ± /J zz -1 2 , becoming first-order approaching the high-symmetry point J ± /J zz → - 1 2 . Further details of the thermodynamics of this transition are given in Appendix B.

III. THEORY OF THE SPIN LIQUID SL II

Spin correlations in spin ice (SL I) can be described using a U(1) lattice gauge theory [START_REF] Castelnovo | Spin Ice, Fractionalization, and Topological Order[END_REF][START_REF] Henley | Power-law spin correlations in pyrochlore antiferromagnets[END_REF][START_REF] Isakov | Why spin ice obeys the ice rules[END_REF], which gives rise to characteristic "pinch-points" in the spin structure factor S(q) [Fig. 2a]. Meanwhile spin correlations in the HAF -and by extension SL III (pHAF) -can be described using a U(1)×U( 1)×U(1) gauge theory [START_REF] Moessner | Properties of a classical spin liquid: The heisenberg pyrochlore antiferromagnet[END_REF][START_REF] Moessner | Low-temperature properties of classical geometrically frustrated antiferromagnets[END_REF][START_REF] Isakov | Dipolar spin correlations in classical pyrochlore magnets[END_REF][START_REF] Henley | Power-law spin correlations in pyrochlore antiferromagnets[END_REF]. This has qualitatively different pinch-points, as illustrated in Fig. 2c. It is clear that the correlations of SL II [Fig. 2b] are very different from either SL I or SL III. None the less, the presence of pinch points suggests that SL II, too, may be described by some form of gauge theory.

We can develop a field-theory for the spin liquid SL II by applying the methods developed in [START_REF] Benton | Classical and quantum spin liquids on the pyrochlore lattice[END_REF][START_REF] Owen Benton | A spin-liquid with pinch-line singularities on the pyrochlore lattice[END_REF] to treat a different kind of spin liquid. The starting point of this approach is to recast the spins S i in H QSI [Eq. [START_REF] Lee | An end to the drought of quantum spin liquids[END_REF]] in terms of five order-parameter fields

{m λ } = {m A2 , m E , m T1ice , m T1planar , m T2 } (4)
defined on each tetrahedron r. These objects m λ (r) describe the different kinds of four-sublattice magnetic order consistent with the point group symmetry of the pyrochlore lattice. Definitions of each field m λ in terms of the spins S i are given in Appendix D.

The most general exchange Hamiltonian on the pyrochlore lattice can be transcribed exactly in terms of m λ (a) SL I (J ± = 0, T = 0.05Jzz) [42]. This greatly simplifies the determination of classical ground states and, where classical ground states form an extensive manifold, one can use this approach to determine the local constraints which control the resulting spin-liquid [START_REF] Benton | Classical and quantum spin liquids on the pyrochlore lattice[END_REF][START_REF] Owen Benton | A spin-liquid with pinch-line singularities on the pyrochlore lattice[END_REF]. In the case of SL II, for T → 0, we have

(h,h,0) -6 -4 -2 0 2 4
m A2 (r) = 0, m E (r) = 0, m T1ice (r) = 0. ∀r (5)
The spin fluctuations at low temperature are thus dominated by the fluctuations of the remaining fields m T2 (r) and m T1planar (r). These fields have significance as the order-parameters of the competing four-sublattice magnetic orders which would be induced by the symmetryallowed perturbation

δH ±± = ij J ±± γ ij S + i S + j + γ * ij S - i S - j , (6)
where γ ij are complex phase factors arising from the change in coordinate frame between different lattice sites [6, 8-10, 40, 65]. For this reason, the spin-liquid SL II falls very naturally into the "multiple-phase competition" scenario for pyrochlore magnets [START_REF] Yan | Theory of multiple-phase competition in pyrochlore magnets with anisotropic exchange with application to Yb2Ti2O7, Er2Ti2O7, and Er2Sn2O7[END_REF][START_REF] Benton | Classical and quantum spin liquids on the pyrochlore lattice[END_REF][START_REF] Yan | Living on the edge: ground-state selection in quantum spin-ice pyrochlores[END_REF][START_REF] Jaubert | Are Multiphase Competition and Order by Disorder the Keys to Understanding Yb2Ti2O7?[END_REF]. In Fig. 3, we show the classical ground-state phase diagram of anisotropic exchange model

H ex = H QSI + δH ±± . (7)
This contains three distinct regions of 4-sublattice order : the easy-plane ordered phases described by the fields m E (denoted AF ⊥ in Fig. 1), m T1planar , and m T2 (Palmer-Chalker state [START_REF] Palmer | Order induced by dipolar interactions in a geometrically frustrated antiferromagnet[END_REF]). These border a region of spin ice (denoted SL I in Fig. 1), dominated by fluctuations of m T1ice . We note that a closely-related phase diagram has been derived for non-Kramers ions [START_REF] Onoda | Quantum fluctuations in the effective pseudospin-1 2 model for magnetic pyrochlore oxides[END_REF][START_REF] Petit | Antiferroquadrupolar correlations in the quantum spin ice candidate Pr2Zr2O7[END_REF]; in this case easy-plane order must be interpreted in terms of the quadrupole moment of the magnetic ion.

The non-trivial correlations in the spin-liquid SL II arise from the fact that neighbouring tetrahedra share a spin, so that the fields m λ (r) on neighbouring tetrahedra are not independent of one another. This point, combined with Eq. ( 5), imposes spatial constraints on the fluctuations of m T2 (r) and m T1planar (r). After coarse graining to extract the long wavelength physics these constraints may be written in terms of two, independent, vector fluxes

B 1 = 1 2 (2m x T1planar , - √ 3m y T2 -m y T1planar , √ 3m z T2 -m z T1planar ) B 2 = 1 2 (2m x T1planar , -m y T2 + √ 3m y T1planar , -m z T2 - √ 3m z T1planar ) , (8)
which each separately obey their own Gauss' law

∇ • B 1 = 0 , ∇ • B 2 = 0 . (9)
We can therefore write

B 1 = ∇ × A 1 , B 2 = ∇ × A 2 (10)
and the theory has two, independent, U(1) gauge degrees of freedom.

The free energy associated with the fluctuations of these fields is of entropic origin [START_REF] Henley | The "Coulomb Phase" in Frustrated Systems[END_REF]. The only choice of Gaussian free-energy consistent with both the point group symmetry and the U (1) symmetry of H QSI is 7)], for Jzz > 0. Different phases are labelled in terms of the irreps of the tetrahedral symmetry group, T d [Eq. ( 4)], as described in [START_REF] Yan | Theory of multiple-phase competition in pyrochlore magnets with anisotropic exchange with application to Yb2Ti2O7, Er2Ti2O7, and Er2Sn2O7[END_REF]. The minimal model of a quantum spin ice H QSI [Eq. ( 1)] exists on the line J±± = 0 -for J± < -1 2 (white line), two phases with 4-sublattice easy-plane order meet, and the resulting enlarged ground-state manifold gives rise to the spin liquid SL II, and spin-nematic phase Q ⊥ . A closely-related mean-field phase diagram for non-Kramers ions is given in [START_REF] Onoda | Quantum fluctuations in the effective pseudospin-1 2 model for magnetic pyrochlore oxides[END_REF][START_REF] Petit | Antiferroquadrupolar correlations in the quantum spin ice candidate Pr2Zr2O7[END_REF].

F SL II = T V d 3 r λ(B 2 1 + B 2 2 ) = T V d 3 r λ (∇ × A 1 ) 2 + (∇ × A 2 ) 2 (11) T 1ice T 1planar T 2 E (AF ⊥ ) -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 J ± /J zz J ±± /J zz
where the coefficient λ can be determined through fits to simulation, or a large-N expansion [START_REF] Isakov | Dipolar spin correlations in classical pyrochlore magnets[END_REF][START_REF] Owen Benton | A spin-liquid with pinch-line singularities on the pyrochlore lattice[END_REF].

It follows from the existence of the conserved fluxes B 1 and B 2 and the free-energy Eq. ( 11) that SL II is a Coulomb phase with algebraic correlations [START_REF] Henley | The "Coulomb Phase" in Frustrated Systems[END_REF]. The validity of this description is demonstrated in Fig. 4 where we compare analytic calculations of the flux structure factor At finite temperature, we anticipate that the spin liquid SL II will be perturbatively stable against terms such as δH ±± [Eq. ( 6)], which retain the point-group symmetry of the lattice but lift the U (1) symmetry of the spins. Bµ (q) [Eq. ( 12)]. Left half of panel: structure factor S xx B 1 (q) as calculated from the theory Eq. ( 11), with λ = 1. Right half of panel: structure factor S xx B 1 (q) as calculated in classical Monte Carlo simulations of H QSI [Eq. ( 1)]. The pinch point centered on q = (0, 0, 0), follows from the zero-divergence conditions on the fields Bµ [Eq. ( 9)]. Simulations were carried out for a cubic cluster of N = 8192 spins, with J ± /Jzz = -1, T = 0.01Jzz, as described in Appendix E.

S αβ Bµ (q) = B α µ (-q)B β µ (q) (12
In this case the free energy will be modified :

F SLII → F SLII + δF SLII ( 13
)
δF SLII = T V d 3 r λ (B x 1 ) 2 - 1 2 (B y 1 ) 2 + (B z 1 ) 2 -(B x 2 ) 2 + 1 2 (B y 2 ) 2 + (B z 2 ) 2 - √ 3 [B z 1 B z 2 -B y 1 B y 2 ] (14)
This form of free energy will still lead to pinch points in S αβ Bµ (q) and S(q), but these will take on a more anisotropic character.

IV. SPIN DYNAMICS IN THE SPIN-NEMATIC PHASE

For temperatures, T < T Q the spin-liquid SL II gives way to hidden spin-nematic order, labelled Q ⊥ in Fig. 1. As far as the dipole moments of spins are concerned, the spin-nematic phase is disordered, and neutron scattering experiments would reveal exactly the same algebraic correlations as in the spin liquid SL II. However the pinch points in S(q) [cf Fig. 2b] hide a great wealth of interesting spin excitations.

To better understand the dynamics of the spinnematic phase, we have calculated the dynamical structure factor S(q, ω), within a semi-classical moleculardynamics (MD) simulation, using the methods described in [START_REF] Taillefumier | Semiclassical spin dynamics of the antiferromagnetic heisenberg model on the kagome lattice[END_REF]. Relevant definitions are given in Appendix C. For ω/J zz 0.2, S(q, ω) presents a featureless, nondispersing continuum [Fig. 5a]. Relics of dispersing ex-(a) Dynamical structure factor for spins, S(q, ω)

q Γ (1,1,1) (2,2,2) (2,2,0) (1,1,0) (1,1 ,1) ω/
χ Q ⊥ (q, ω) q Γ (1,1,1) (2,2,2) (2,2,0) (1,1,0) (1,1,1) 0 1 2 3 ω/J zz 10 -4 10 -3 10 -2 10 -1 1 10 q ( 3 2 , 3 2 , 3 2 ) (2,2,2) (2,2,1) 0 0.5 ω/Jzz Figure 5.
Spin dynamics in the phase with hidden spin-nematic order (Q ⊥ ). (a) Dynamical structure factor for spin degrees of freedom, S(q, ω), showing gapless continuum of excitations at low energies. (b) Dynamical susceptibility for fluctuations of quadrupole moments, χ Q ⊥ (q, ω). Inset : details of correlations at low energies near the zone center q rl = (2, 2, 2), showing a linearly-dispersing Goldstone mode at low energies. Results are taken from molecular-dynamics simulations of H QSI [Eq. ( 1)] for a cluster of N = 65536 spins, with J±/Jzz = -1.0, T /Jzz = 0.002.

. citations are visible in S(q, ω) at higher energies, but these are explicitly not Goldstone modes, and have nothing to do with the hidden spin-nematic order. Results for S(q, ω) in the spin-nematic phase are very similar to those found in the spin liquid SL II for T > T Q .

Incoherent, non-dispersing structure of the type shown in Fig. 5a is reminiscent of theoretical predictions [START_REF] Knolle | Dynamics of a two-dimensional quantum spin liquid: Signatures of emergent majorana fermions and fluxes[END_REF][START_REF] Punk | Topological excitations and the dynamic structure factor of spin liquids on the kagome lattice[END_REF][START_REF] Bieri | Gapless chiral spin liquid in a kagome heisenberg model[END_REF] and experimental measurements [START_REF] Petit | Antiferroquadrupolar correlations in the quantum spin ice candidate Pr2Zr2O7[END_REF][START_REF] Han | Fractionalized excitations in the spinliquid state of a kagome-lattice antiferromagnet[END_REF][START_REF] Joseph | Continuous excitations of the triangular-lattice quantum spin liquid YbMgGaO4[END_REF], for a wide range of different spin liquids. The presence of a nondispersing continuum reflects the fact that, unlike conventional spin waves (magnons), single elementary excitations of a spin liquid cannot be created by local processes. It follows that, when a neutron scatters from a spin liquid, the energy, momentum and angular momentum (spin) transferred are not absorbed by a single excitation with a well-defined energy and momentum, but rather shared between multiple excitations [START_REF] Savary | Quantum spin liquids: a review[END_REF]. This scenario also holds for the spin liquid, SL II. However the fact that S(q, ω) only records dipolar spin correlations obscures another important fact -the spin-nematic order which is present for T < T Q which breaks a continuous, U(1), symmetry of the Hamiltonian. And, by Goldstone's theorem, it must, therefore, also support gapless Goldstone modes.

In order to resolve this conundrum, it is necessary to examine the dynamical correlations of the quadrupole moments of spin. In Fig. 5b we present MD simulation results for the dynamical susceptibility χ Q ⊥ (q, ω), which measures fluctuations of the order parameter Q ⊥ [Eq. ( 2)]. A sharp excitation, with dispersion

ω ≈ v Q |q -q rl | (15)
can now be resolved near to the zone centers with q rl = (0, 0, 0), (1, 1, 1), (2, 2, 2). These are the same zone centers for which the Bragg peaks associated with the hidden spin-nematic order Q ⊥ would occur in a quadrupolar structure factor, which might, in principle, be measured in resonant X-ray experiments [START_REF] Savary | Probing Hidden Orders with Resonant Inelastic X-Ray Scattering[END_REF].

It is interesting to compare these results with the behavior expected of a quantum spin nematic. Fieldtheoretic analysis [START_REF] Smerald | Theory of inelastic neutron scattering in a field-induced spin-nematic state[END_REF][START_REF] Smerald | Theory of spin excitations in a quantum spin-nematic state[END_REF][START_REF] Starykh | Excitations and quasi-one-dimensionality in field-induced nematic and spin density wave states[END_REF][START_REF] Smerald | Theory of NMR 1/T1 relaxation in a quantum spin nematic in an applied magnetic field[END_REF], based on the symmetry of the order parameter, predicts that spin-nematics support gapless Goldstone modes, visible in χ Q ⊥ (q, ω). This Goldstone mode has dispersion ω ∝ |q| [cf. Eq. ( 15)], and at zero temperature the associated intensity diverges as ∼ 1/ω for ω → 0 [START_REF] Smerald | Theory of inelastic neutron scattering in a field-induced spin-nematic state[END_REF]. In the present case, we find a linearly-dispersing Goldstone mode [Eq. ( 15)], with intensity which diverges for ω → 0 [cf inset to Fig. 5b].

However, the form of divergence observed in MD simulations is ∼ 1/ω 2 . This follows from the fact that simulations are performed at finite temperature and probe thermal rather than quantum fluctuations. Further details of the spin dynamics in the spin-nematic phase, and specifically the characterization of the Goldstone mode are given in Appendix F.

V. DISCUSSION

Spin-liquids and spin-nematics are both unconventional phases expected to arise in certain frustrated magnets, for which there has been a long experimental search. This makes the possibility of realizing both of these in the same system rather an enticing one. We therefore turn to discuss the possibility of observing the SLII and Q ⊥ phases in rare-earth pyrochlore magnets.

It is worth noting at this point that spin-nematic order of the Q ⊥ phase is distinct from the multipolar orders proposed as possible ground states of non-Kramers ions (Pr 3+ [START_REF] Onoda | Quantum melting of spin ice: Emergent cooperative quadrupole and chirality[END_REF][START_REF] Lee | Generic quantum spin ice[END_REF][START_REF] Petit | Antiferroquadrupolar correlations in the quantum spin ice candidate Pr2Zr2O7[END_REF], Tb 3+ [START_REF] Onoda | Quantum melting of spin ice: Emergent cooperative quadrupole and chirality[END_REF][START_REF] Taniguchi | Long-range order and spin-liquid states of polycrystalline Tb2+xTi2-xO7+y[END_REF][START_REF] Takatsu | Quadrupole Order in the Frustrated Pyrochlore Tb2+xTi2-xO7+y[END_REF][START_REF] Guitteny | Anisotropic Propagating Excitations and Quadrupolar Effects in Tb2Ti2O7[END_REF]) and of dipoleoctupole Kramers doublets (Nd 3+ [START_REF] Huang | Quantum spin ices and topological phases from dipolaroctupolar doublets on the pyrochlore lattice[END_REF], Ce 3+ [START_REF] Li | Symmetry enriched U(1) topological orders for dipole-octupole doublets on a pyrochlore lattice[END_REF]). In those cases the multipolar order is actually a "classical" ordering of the transverse part of the pseudospins S i . This ordering must then be interprated as a multipolar order due to the symmetry properties of the crystal field doublet which S i describes. Where multipole order of this kind occurs, experiments which probe the dynamics of dipoles will see a gapped response and a sharp excitation spectrum. By contrast, in the case developed in this Article, the dipole moments remain in an essentially liquid like state with strong fluctuations at low temperature and a broad, gapless response coexisting with the hidden nematic order [Fig. 5a].

Where then might we observe this unusual magnetic state? Further experimental work will be necessary to definitively answer this question, but there are already a few trails to follow. In particular, the Pr-based pyrochlores have the recommended single-ion and interaction anisotropies [START_REF] Onoda | Quantum melting of spin ice: Emergent cooperative quadrupole and chirality[END_REF][START_REF] Lee | Generic quantum spin ice[END_REF][START_REF] Petit | Antiferroquadrupolar correlations in the quantum spin ice candidate Pr2Zr2O7[END_REF]. Coupling parameters of Pr 2 Zr 2 O 7 for example have been suggested to sit in the AF ⊥ phase of Fig. 1 [START_REF] Petit | Antiferroquadrupolar correlations in the quantum spin ice candidate Pr2Zr2O7[END_REF], although it seems that the coupling of structural disorder to the non-Kramers doublets plays a significant role [START_REF] Wen | Disordered Route to the Coulomb Quantum Spin Liquid: Random Transverse Fields on Spin Ice in Pr2Zr2O7[END_REF]. Since chemical pressure has already proven to be a useful tool to move a family of compounds across a phase diagram [START_REF] Hallas | Universal dynamic magnetism in Yb pyrochlores with disparate ground states[END_REF][START_REF] Yan | Theory of multiple-phase competition in pyrochlore magnets with anisotropic exchange with application to Yb2Ti2O7, Er2Ti2O7, and Er2Sn2O7[END_REF][START_REF] Jaubert | Are Multiphase Competition and Order by Disorder the Keys to Understanding Yb2Ti2O7?[END_REF][START_REF] Dun | Chemical pressure effects on magnetism in the quantum spin liquid candidates Yb2X2O7 (X = Sn, Ti, Ge)[END_REF][START_REF] Wiebe | Frustration under pressure: Exotic magnetism in new pyrochlore oxides[END_REF], Pr 2 X 2 O 7 (X=Sn,Hf,Pb) are promising candidates to investigate, with ferromagnetic correlations consistent with positive J zz and no dipole order yet observed [START_REF] Zhou | Dynamic Spin Ice: Pr2Sn2O7[END_REF][START_REF] Sibille | Candidate quantum spin ice in the pyrochlore Pr2Hf2O7[END_REF][START_REF] Anand | Physical properties of the candidate quantum spin-ice system Pr2Hf2O7[END_REF][START_REF] Hallas | Magnetic frustration in lead pyrochlores[END_REF][START_REF] Matsuhira | Low-temperature magnetic properties of the geometrically frustrated pyrochlore Pr2Sn2O7[END_REF].

The notion of hidden order also resonates with the elusive physics of Yb-based pyrochlores. As far as we know, Yb pyrochlores lie in a different regime of magnetic interactions than the H QSI model of Eq. ( 1) [START_REF] Ross | Quantum excitations in quantum spin ice[END_REF][START_REF] Robert | Spin dynamics in the presence of competing ferromagnetic and antiferromagnetic correlations in Yb2Ti2O7[END_REF][START_REF] Thompson | Quasiparticle Breakdown and Spin Hamiltonian of the Frustrated Quantum Pyrochlore Yb2Ti2O7 in Magnetic Field[END_REF]. The properties of Yb 2 Ti 2 O 7 in particular seem to be connected with a different phase boundary than the one associated with SLII [START_REF] Yan | Theory of multiple-phase competition in pyrochlore magnets with anisotropic exchange with application to Yb2Ti2O7, Er2Ti2O7, and Er2Sn2O7[END_REF][START_REF] Jaubert | Are Multiphase Competition and Order by Disorder the Keys to Understanding Yb2Ti2O7?[END_REF]. That being said, the sim-ilarities between our results and the Yb-pyrochlores are striking: a gapless continuum of spin excitations, oblivious to the transition [START_REF] Hallas | Universal dynamic magnetism in Yb pyrochlores with disparate ground states[END_REF][START_REF] Ross | Two-Dimensional Kagome Correlations and Field Induced Order in the Ferromagnetic XY Pyrochlore Yb2Ti2O7[END_REF][START_REF] Maisuradze | Anomalously slow spin dynamics and short-range correlations in the quantum spin ice systems Yb2Ti2O7 and Yb2Sn2O7[END_REF] [Fig. 5(a)], and robust in temperature up to a broad feature in specific heat [START_REF] Hallas | Universal dynamic magnetism in Yb pyrochlores with disparate ground states[END_REF] (here between SL III and SL II). And while the magnetic order in Yb-pyrochlores is, at least partially, an order of dipolar moments [START_REF] Chang | Higgs transition from a magnetic Coulomb liquid to a ferromagnet in Yb2Ti2O7[END_REF][START_REF] Dun | Yb2Sn2O7: A magnetic Coulomb liquid at a quantum critical point[END_REF][START_REF] Yaouanc | Dynamical Splayed Ferromagnetic Ground State in the Quantum Spin Ice Yb2Sn2O7[END_REF][START_REF] Yasui | Ferromagnetic Transition of Pyrochlore Compound Yb2Ti2O7[END_REF][START_REF] Lhotel | First-order magnetic transition in Yb2Ti2O7[END_REF][START_REF] Hallas | XY antiferromagnetic ground state in the effective S = 1 2 pyrochlore Yb2Ge2O7[END_REF], recent experiments have indicated that the primary order parameter may be "hidden", and distinct from a standard dipole order [START_REF] Hallas | Universal dynamic magnetism in Yb pyrochlores with disparate ground states[END_REF]. Thus, although the specific case developed in this manuscript probably does not apply to the Ybpyrochlores, related physics may be at play. Furthermore, since nematic order does not prevent dipole disorder, our work provides a model Hamiltonian for the co-existence between emergent gauge fields and long-range order. Hence, the Q ⊥ phase offers an unconventional counterpart to the Coulombic ferromagnet [START_REF] Savary | Coulombic quantum liquids in spin-1/2 pyrochlores[END_REF][START_REF] Powell | Ferromagnetic coulomb phase in classical spin ice[END_REF], and spin-ice fragmentation [START_REF] Brooks-Bartlett | Magnetic-Moment Fragmentation and Monopole Crystallization[END_REF] that has recently been observed in Nd 2 Zr 2 O 7 [START_REF] Petit | Observation of magnetic fragmentation in spin ice[END_REF][START_REF] Benton | Quantum origins of moment fragmentation in Nd2Zr2O7[END_REF] and Ho 2 Ir 2 O 7 [START_REF] Lefrançois | Magnetic charge injection in spin ice: a new way to fragmentation[END_REF] materials.

We also note that many other magnetic systems outside the rare earth oxides R 2 X 2 O 7 feature moments located on a pyrochlore lattice. Of particular interest are materials such as NaCaCo 2 F 7 and NaSrCo 2 F 7 [START_REF] Ross | Static and dynamic xy-like short-range order in a frustrated magnet with exchange disorder[END_REF][START_REF] Ross | Single-ion properties of the S eff = 1 2 XY antiferromagnetic pyrochlores NaA Co2F7 (A = Ca 2+ , Sr 2+ )[END_REF] which boast XY like interactions with much higher energy scales than observed in the rare-earth oxides. If such a case could be found with frustrated transverse coupling J ± < 0 then it would render the physics discussed here accessible at a much more amenable temperature range.

VI. SUMMARY AND CONCLUSIONS

"Quantum spin ice", in which magnetic ions on a pyrochlore lattice interact through highly-anisotropic exchange interactions, have become an important paradigm in the search for quantum spin liquids. In this Article we have used large-scale Monte-Carlo simulation to explore the physics of the minimal model of a quantum spin ice H QSI [START_REF] Lee | An end to the drought of quantum spin liquids[END_REF]. We find that this model has far more to offer than spin ice alone, supporting three distinct types of spin liquid, each with a different emergent gauge symmetry. These include a completely new form of spin liquid, described by a U(1)×U(1) gauge theory, which exhibits a hidden spin-nematic order at low temperatures [Fig. 1]. So far as experiment is concerned, the main lesson of these results is that "quantum spin-ice" materials can play host to a great many different spin-liquid and (hidden-)order phases -crudely-put : pinch-points do not imply spin ice [Fig. 2]. The existence of a sharp Goldstone mode in the nematic phase Q ⊥ also serves as a salutary reminder that broad, non-dispersing continua of excitations can hide a multitude of secrets [Fig. 5].

From the theoretical point of view, this work opens an interesting new perspective on quantum spin liquids on the pyrochlore lattice. The effect of quantum fluctuations on the novel ordered and spin-liquid phases shown in Fig. 1 for J ± < 0 remains an open question. Experi-ence with QMC simulation of H QSI [Eq. ( 1)] for J ± > 0 suggests that quantitative values of the crossover temperature T * 2 and T * 3 may be substantially renormalized, but that the qualitative structure of the phase diagram should remain the same down to very low temperatures [START_REF] Banerjee | Unusual liquid state of hard-core bosons on the pyrochlore lattice[END_REF][START_REF] Kato | Numerical evidence of quantum melting of spin ice: Quantum-to-classical crossover[END_REF][START_REF] Shannon | Spin ice[END_REF]. The high-symmetry point, J ± /J zz = -1/2 is also a high-symmetry point for quantum spins, and so remains the anchor for the spin liquid SL III (pHAF) None the less, the fate of this U(1)×U( 1)×U(1) spin liquid for quantum spins at T = 0 remains an open question [START_REF] Canals | Pyrochlore antiferromagnet: A three-dimensional quantum spin liquid[END_REF][START_REF] Canals | Quantum spin liquid: The heisenberg antiferromagnet on the threedimensional pyrochlore lattice[END_REF][START_REF] Huang | Spin-ice state of the quantum heisenberg antiferromagnet on the pyrochlore lattice[END_REF][START_REF] Tsunetsugu | Antiferromagnetic Quantum Spins on the Pyrochlore Lattice[END_REF]. And, to the best of our knowledge, quantum analogues of the new spin liquid, SL II, which has a U(1)×U(1) gauge structure, remain unexplored. However it seems reasonable to speculate that quantum effects will enhance, rather than suppress, the fluctuations which drive SL II and SL III, and that the phase Q ⊥ will survive as hidden quantum spin-nematic order, within a quantum spin liquid. All of these questions open exciting avenues for future research.

The phase diagram shown in Fig. 1 was extracted from classical Monte Carlo (MC) simulations of the quantum spin ice model, H QSI [Eq. ( 1)]. Spins were treated as classical vectors with fixed length |S i | = 1 2 . These simulations were carried out for a cubic cluster of 8192 spins, using a single spin flip algorithm combined with simulated annealing, parallel tempering and over-relaxation. The phase diagram is obtained using 75000 simulated annealing steps using a stepwise decrease of temperature starting from T = 10J zz down to the target temperature. Each annealing step consists of 10 Monte Carlo steps (a Monte Carlo step consists of a full sweep of the lattice combined with over-relaxation). The simulated annealing is followed by 1000 parallel tempering steps with 500 Monte Carlo steps in between, and then by 200000 Monte Carlo steps for thermalization at fixed temperature. Measurements consist of 200000 samples separated by 10 Monte Carlo steps and combined to parallel tempering every 50 measures. We use 256 different replicas with temperature set in linear scale for J ± /J zz > - 1 2 and 256 temperatures in logarithmic scale for J ± /J zz ≤ - 1 2 . The phase boundary of the antiferromagnetically ordered (AF ⊥ ) phase, T N , was extracted from the susceptibility of the relevant order parameter, m E , as defined in Appendix D.

The crossover scale for the spin-ice regime (SL I), T * 1 , was estimated from the Schottky-like peak in the heat capacity.

The crossover scale T * 3 for the spin-liquid SLIII was estimated from the Curie-Law crossover shown in Fig. 6.

For J ± < -1 2 , the crossover scale T * 2 is associated with a weakening of the correlations of the local z-components of the spins. This can be observed by measuring the susceptibility, χ T1Ice (T ), of the field m T1Ice , defined in Appendix D]. Decreasing the temperature for -1 < J± Jzz < -0.5 the quantity T χ T1Ice (T ) first increases during the crossover from the paramagnet to SLIII and then drops as the system enters SLII. We define the crossover temperature T * 2 as the point at which the quantity T χ T1ice (T ) drops below its infinite temperature value

T * 2 χ T1ice (T * 2 ) = lim T →∞ T χ T1ice (T ) (B1)
This is illustrated in Fig. 7b. The spin-nematic ordering temperature T Q is estimated from the peak in the order-parameter susceptibility ) [cf. Fig. 7d], where the relevant order parameter Q ⊥ [cf. Fig. 7c], is defined through Eq. (2). Fig. 7 is obtained using 300 temperatures in logarithmic scale coverring 3 orders of magnitude, parallel tempering every 100 Monte Carlo steps, simulated annealing and thermalization at temperature T for 100000 Monte Carlo steps each. Measurements consist of 100000 different samples with 10 Monte Carlo steps between each sample. Error bars were estimated by comparing the results of three independent runs of the simulation.

χ Q ⊥ = N T Q 2 ⊥ -|Q ⊥ | 2 . ( B2
where the dynamical structure factor S(q, ω) is defined through

S(q, ω) = αβ δ αβ - q α q β q 2 m α (-q, ω)m β (q , ω) m α (q, ω) = iβγ R αβ i g βγ i S γ i (t)e iωt dt e iq•ri , (C2)
and the g βγ i is the g-tensor written in the local coordinate frame [START_REF] Yan | Theory of multiple-phase competition in pyrochlore magnets with anisotropic exchange with application to Yb2Ti2O7, Er2Ti2O7, and Er2Sn2O7[END_REF]. For simplicity, we have here taken g βγ i = 2δ βγ for all of the calculations in this paper. R αβ i is a rotation matrix which rotates from the local coordinate frame on site i, to the global, crystal coordinate frame. The definition of the local coordinate frame is given in Appendix A. Results for S(q) are shown in the left half-panels of Fig. 2. These results were taken from classical MC simulations of H QSI at a given temperature, with further averaging provided by numerically integrating the semi-classical equations of motion for the spins. This secondary molecular-dynamics (MD) simulation was carried out using methods described in Ref. [START_REF] Taillefumier | Semiclassical spin dynamics of the antiferromagnetic heisenberg model on the kagome lattice[END_REF].

It is also useful to decompose the structure factor into the spin-flip (SF) and non spin-flip (NSF) channels measured in polarised neutron-scattering experiments.

S SF (q) = 1 q 2 dω |m(q, ω) • (n × q)| 2 S NSF (q) = dω |m(q, ω)

• n| 2 , ( C3
)
where n is the direction of polarization of the neutron magnetic moment. Following Fennell et al. [START_REF] Fennell | Magnetic Coulomb Phase in the Spin Ice Ho2Ti2O7[END_REF], we take n = (1, -1, 0)/ √ 2. Simulation results for S SF (q) and S NSF (q) are shown in the right half-panels of Fig. 2.

We have also used MD simulation to calculate the dynamical structure factor S(q, ω). Results for S(q, ω) within the spin-nematic phase of the quantum spin ice model are shown in Fig. 5a. Further details of the calculation of dynamical properties can be found in Appendix F. showing the expected behaviour (q) = q at small q. (b) Intensity I(q) of the peak as a function of momentum q. The dashed line shows the expected behaviour at finite temperature, I(q) ∝ 1/q 2 . Results are taken from moleculardynamics simulations of a cluster of N = 65536 spins, for J±/Jzz = -1.0, T = 0.002Jzz. Momentum q is measured relative to q = (0, 0, 0).

where fluctuations of spin-nematic order are given by δQ ⊥ (q, ω) = i dt Q ⊥ (r i , t) -Q⊥ (t) e iωt e iq•ri

Q⊥ (t) = 1 N i Q ⊥ (r i , t) , (F2)
and the order parameter Q ⊥ (r i , t) is defined through Eq. ( 2). χ Q ⊥ (q, ω) is calculated numerically from 200 sample configurations extracted from Monte Carlo simulations on a system of linear size L = 16. We used 20000 steps for the simulated annealing spaced by 10 Monte Carlo steps between each simulated annealing step. The other parameters for the thermalization and parallel tempering are identical to the parameters used to calculate the phase diagram [Appendix B].

The ensemble of configurations obtained from Monte Carlo is then evolved in time according to the equation of motion,

dS i dt = H i (t) × S i (t) (F3)
where

H i (t) = j∈nn i J ij • S j (t) (F4)
is the effective exchange field acting on site i, J ij is the anisotropic exchange interaction tensor and the sum in Eq. (F4) runs over the neighbors of i. The numerical integration of this nonlinear equation of motion proceeds as described in Ref. [START_REF] Taillefumier | Semiclassical spin dynamics of the antiferromagnetic heisenberg model on the kagome lattice[END_REF].

In Fig 8 we plot the dispersion of the Goldstone mode found in molecular dynamics (MD) simulations of H QSI within the spin-nematic phase for J ± /J zz = -1. The dispersion was extracted from the position of low-energy dispersing peak in χ Q ⊥ (q, ω), as shown in the inset to Fig. 5b.

Figure 1 .

 1 Figure 1. (a) Finite-temperature phase diagram of the XXZ model on a pyrochlore lattice, H QSI [Eq. (1)]. The model posses three distinct spin-liquid phases (SL), as well as ground states with easy-plane antiferromagnetic (AF ⊥ ) and spin-nematic (Q ⊥ ) order. Associated crossover temperatures (phase transitions) are indicated with dashed (solid) lines. For J±/Jzz = -1/2 (dash-dotted line), the model is thermodynamically equivalent to the Heisenberg antiferromagnet on a pyrochlore lattice. Results are taken from classical Monte Carlo simulation of a cubic cluster of N = 8192 spins. (b) Representative configuration of quadrupoles in the Q ⊥ phase with nematic order (c) "Two in, two out" configuration of spins in the spin ice regime (SLI). (d) Representative configuration of spin dipoles in the ordered AF ⊥ phase.

6 (Figure 2 .

 62 Figure 2. Comparison of correlations in spin-liquid (SL) and spin-nematic (Q ⊥ ) phases. a) Spin ice (SL I), showing "pinch points" indicative of algebraic spin correlations. Left half-panel : equal-time structure factor S(q), as measured in unpolarised neutron-scattering experiments. Right half-panel : S(q) resolved into spin-flip (SF, top) and non spin-flip (NSF, bottom) components, as measured in polarised neutron-scattering experiments (cf. [54]). Definitions of each structure factor are given in Appendix C. b) Phase with spin-nematic order (Q ⊥ ), showing the absence of magnetic Bragg peaks, and persistence of algebraic spin correlations, which is also characteristic of the spin liquid SL II. c) Spin liquid SL III, for parameters thermodynamically equivalent to a Heisenberg antiferromagnet, showing algebraic spin correlations distinct from those in spin ice (SL I) or SL II. Results are taken from Monte Carlo simulations of H QSI [Eq. (1)], for a cubic cluster of N = 8192 spins.

Figure 3 .

 3 Figure 3. (Color online). Classical ground-state phase diagram of the anisotropic exchange model Hex [Eq. (7)], for Jzz > 0. Different phases are labelled in terms of the irreps of the tetrahedral symmetry group, T d [Eq. (4)], as described in[START_REF] Yan | Theory of multiple-phase competition in pyrochlore magnets with anisotropic exchange with application to Yb2Ti2O7, Er2Ti2O7, and Er2Sn2O7[END_REF]. The minimal model of a quantum spin ice H QSI [Eq. (1)] exists on the line J±± = 0 -for J± < -1 2 (white line), two phases with 4-sublattice easy-plane order meet, and the resulting enlarged ground-state manifold gives rise to the spin liquid SL II, and spin-nematic phase Q ⊥ . A closely-related mean-field phase diagram for non-Kramers ions is given in[START_REF] Onoda | Quantum fluctuations in the effective pseudospin-1 2 model for magnetic pyrochlore oxides[END_REF][START_REF] Petit | Antiferroquadrupolar correlations in the quantum spin ice candidate Pr2Zr2O7[END_REF].

  ) based on Eq.[START_REF] Gingras | Quantum spin ice: a search for gapless quantum spin liquids in pyrochlore magnets[END_REF] with the results of Monte Carlo simulation. Pinch point singularities are clearly seen in both analytic and numerical calculations. It is the same fluctuations of B 1 and B 2 which are responsible for the characteristic pinch-point structures in the (spin) structure factor measured by neutron scattering [cf. Fig.2b].

Figure 4 . U( 1 )

 41 Figure 4.U(1)×U(1) gauge structure of the spin liquid SL II, as demonstrated by pinch-points in equal-time structure factors S αβ Bµ (q) [Eq.[START_REF] Hermele | Pyrochlore photons: The U(1) spin liquid in a S= 1 2 three-dimensional frustrated magnet[END_REF]]. Left half of panel: structure factor S xx B 1 (q) as calculated from the theory Eq. (11), with λ = 1. Right half of panel: structure factor S xx B 1 (q) as calculated in classical Monte Carlo simulations of H QSI [Eq. (1)]. The pinch point centered on q = (0, 0, 0), follows from the zero-divergence conditions on the fields Bµ [Eq. (9)]. Simulations were carried out for a cubic cluster of N = 8192 spins, with J ± /Jzz = -1, T = 0.01Jzz, as described in Appendix E.

  Dynamical susceptibility for quadrupoles,

3 Figure 6 .

 36 Figure 6. (Color online). Crossover in the magnetic susceptibility χ(T ), from a traditional Curie law in the hightemperature paramagnet, to a low-temperature Curie law in the spin liquids, as seen by different plateaux in the function T χ, plotted as a function of log(T ). The crossover temperature T * 3 /Jzz ≈ 0.3 (red dot) is estimated from the point of inflection of T χ. The extraction of the crossover temperature T * 2 is explained in Fig. 7. Results are taken from classical Monte Carlo simulations of H QSI [Eq (1)], for a cubic cluster of N = 8192 spins, with J±/Jzz = -1.

Figure 7 .

 7 Figure 7. Thermodynamics of the QSI in the region of spinnematic order. (a) Specific heat cV (T ), showing an upturn followed by a shallow maximum in the region of the crossover into the spin liquid SL II at T * 2 /Jzz ≈ 10 -1 , and small peak associated with the onset of spin-nematic order at TQ/Jzz ≈ 10 -2 . (b) Correlation function T χ T 1 ice (T ) used to determine the crossover temperature into the spin-liquid SLII. χ T 1 ice is the susceptibility of the field m T 1 ice defined in Appendix D. The crossover temperature T * 2 is defined by the point at which the quantity T χ T 1 ice (T ) drops below its infinite temperature limit [Eq. B1] (b) Norm of the order parameter Q ⊥ (T ) [cf. Eq. (2)], showing continuous phase transition into phase with hidden spin-nematic order at TQ/Jzz ≈ 10 -2 . (c) Orderparameter susceptibility χ Q ⊥ (T ) [Eq. (B2)], used to estimate the transition temperature TQ. The crossover temperature T * 3 has been extracted as explained in Fig.6. Results are taken from classical Monte Carlo simulation of H QSI [Eq (1)], for cubic clusters of N = 1024, N = 8192 and N = 27648 spins, with J±/Jzz = -1.

Figure 8 .

 8 Figure 8. Dispersion and intensity of the Goldstone mode in the phase with hidden spin-nematic order, as shown in Fig.5b. (a) Dispersion (q) of low-energy peak in χQ ⊥ (q, ω) showing the expected behaviour (q) = q at small q. (b) Intensity I(q) of the peak as a function of momentum q. The dashed line shows the expected behaviour at finite temperature, I(q) ∝ 1/q 2 . Results are taken from moleculardynamics simulations of a cluster of N = 65536 spins, for J±/Jzz = -1.0, T = 0.002Jzz. Momentum q is measured relative to q = (0, 0, 0).

ACKNOWLEDGEMENTS

We are pleased to acknowledge stimulating discussions with Karim Essafi. This work was supported by the Theory of Quantum Matter Unit of the Okinawa Institute of Science and Technology Graduate University (OIST). Numerical calculations were carried out using HPC facilities provided by OIST.

Appendix A: Definition of local coordinate frame

We describe the local-coordinate frame which is defined for four spins on a pyrochlore tetrahedron S 0 , S 1 , S 2 , S 3 occupying positions

where a is the length of a cubic, 16-site unit cell of the pyrochlore lattice.

The pseudospins in the global, crystal, coordinate frame S i relate to the pseudospins in the local frame S i [Eq. ( 1)] as

Where

and

Appendix C: Definitions of dynamical structure factors

In Fig. 2 we show predictions for neutron scattering experiments, based on the equal-time (i.e. energyintegrated) structure factor S(q) = dω S(q, ω) , (C1)

Appendix D: Definitions of local order parameter fields

The definitions of the local order parameter fields m λ which appear in the theory of the spin liquid SLII [Section III] are given in Table I.

Here we give the definitions in terms of the spins written in the local coordinate frame S i (defined in Appendix A), cf. Ref. [START_REF] Yan | Theory of multiple-phase competition in pyrochlore magnets with anisotropic exchange with application to Yb2Ti2O7, Er2Ti2O7, and Er2Sn2O7[END_REF] where definitions are given in the global, crystal basis.

Definition in terms of spins within tetrahedron

Table I. Order-parameter fields m λ , derived from irreducible representations (irreps) of the tetrahedral point-group T d . Spin components Si = (S x i , S y i , S z i ) are written in the local frame of the magnetic ions, see Appendix A for a definition of this coordinate frame. The convention for the labelling of the spins with an tetrahedron is given in Appendix A.

Appendix E: Numerical simulation of the correlations of the flux

Values of the flux field B µ (r) are calculated for each tetrahedron r according to Eq. ( 8) and the definitions of m λ given in Table I.

The tetrahedra of the pyrochlore lattice may be divided into two sets A and B. The centres of each set of tetrahedra each form an FCC lattice.

To calculate S αβ Bµ we use Eq. ( 12) where B µ (q) is defined as the lattice Fourier transform of B µ (r) over only the A sublattice of tetrahedra.

where N u.c. is the number of unit cells in the system. Simulations were carried out using local spin updates, augmented by over-relaxation, within a parallel tempering scheme with 300 temperatures distributed on a log scale between T = 0.003 J zz and T = 0.1 J zz . Thermalisation was accomplished through a process of simulated annealing, with 10 4 Monte Carlo steps (MCs) of annealing from high temperature to temperature T , followed by 10 4 MCs of thermalization at temperature T , and 10 5 MCs of measurements at temperature T . Spin configurations were sampled every 100 MCs during the measurements, giving an ensemble of 1000 samples. To study the Goldstone mode associated to the development of spin-nematic order, we calculate the dynamical correlation function.


Aboubakry Ly

Alois Würger

Hydrodynamic interactions in DNA thermophoresis

Keywords:

We theoretically study the molecular-weight dependence of DNA thermophoresis, which arises from mutual advection of the n repeat units of the molecular chain. As a main result we find that the dominant driving forces, i.e., the thermally induced permittivity gradient and the electrolyte Seebeck effect, result in characteristic hydrodynamic screening. In comparison with recent experimental data on single-stranded DNA (2 ≤ n ≤ 80), our theory quantitatively describes the increase of the drift velocity up to n = 30; the slowing-down of longer molecules is well accounted for by a simple model for counterion condensation. It turns out that thermophoresis may change sign as a function of n: For an appropriate choice of the salt-specific Seebeck coefficient, short molecules move to the cold and long ones to the hot; this could be used for separating DNA by molecular weight.

When applying a temperature gradient on a colloidal dispersion, one observes thermally driven transport towards the hot or the cold [1,2]. In recent years, thermophoresis has been shown to provide a versatile means for manipulating DNA, including translocation through plasmonic nanopores [3], stretching in nanochannels [4,5], separation by molecular weight [6], sequencespecific detection with functionalized nanoparticles [7], and force-free trapping of single molecules [START_REF] Braun | Proc. SPIE 9922, Optical Trapping and Optical Micromanipulation[END_REF]. Protein thermophoresis has become a standard technology in biomedical analysis [START_REF] Jerabek-Willemsen | [END_REF], and the accumulation of RNA in hydrothermal pores is discussed as a scenario for biomolecular synthesis in the early evolution of life [10].

In the last decade, much progress has been made concerning the physical mechanisms of thermophoresis of charged colloids. It has been shown that, in addition to thermo-osmosis [11,12], the electrolyte Seebeck field [13][14][15][16][17][18] and concentration gradients of salt [16] or nonionic polymers [19,20], play an important role. These companion fields arise from specific solvation enthalpies of salt ions or nonionic solutes, and are at the origin of the "inverse" Soret effect, where the colloids accumulate in hot regions [16,19]. Regarding the size dependence, there is conclusive evidence that the mobility of colloidal beads does not vary with the radius [21,22].

In spite of the many experimental studies mentioned above, little is known on the molecular-weight dependence of DNA thermophoresis. If the hydrodynamic slowing-down of Brownian motion is well understood in terms of mutual advection of the repeat units [START_REF] De Gennes | Scaling concepts in Polymer Physics[END_REF], a more complex picture arises for phoretic motion where external forces are absent and which is driven by non-equilibrium surface properties. For short-ranged dispersion forces, hydrodynamic interactions are irrelevant and the thermophoretic velocity is constant [START_REF] Brochard | [END_REF]25]; deviations observed for very short polymers in organic solvents [26], arise probably from chemically different end groups. For DNA in a weak electrolyte, however, the electrostatic interaction length may attain tens of nanometers, which suggests an incomplete screening of hydrodynamic coupling.

In this paper we study hydrodynamic effects on DNA

r dV i j λ a) b)
FIG. 1: Schematic view of a charged polymer in an electrolyte solution. a) The drag on the molecular unit j consists of two contributions: First, the stokeslet of bead i, driven by the force F, drags the neighbor j. Second, due to the force f (r) exerted by the bead i, the fluid element dV moves and in turn exerts a drag on bead j. These contributions cancel each other at distances well beyond the Debye length λ. b) Since the persistence length is of the order of the Debye length, the molecule may be treated as a rigid rod within the reach of electrostatic interactions.

thermophoresis. We consider the two dominant chargerelated surface forces, i.e., the thermally induced permittivity gradient and the electrolyte Seebeck effect, and derive the respective hydrodynamic correction factors in the rigid-rod limit. With a simple model for counterion condensation, we compare our theory to recent Soret data for single-stranded DNA [17]. Hydrodynamic interactions. -Consider a polyelectrolyte chain of n building blocks, as illustrated in Fig. 1. Unit i creates a flow field v(rr i ) in the surrounding fluid and thus drags its neighbor j. Then the overall velocity u of the chain is given by the sum of the monomer contribution u 1 and the mutual advection,

u = u 1 + 1 n i,j =i v(r ij ) , (1)
where the angular brakets • • • indicates the configurational average with respect to r ij = r j -r i .

Thermophoresis arises from the solute-solvent interactions. The force density exerted on the counterion cloud surrounding a charged monomer reads as [27]

f = - E 2 2 ∇ + ρE T , (2)
where the first term is proportional to the thermally induced permittivity gradient ∇ = (d /dT )∇T , with the charged monomer's electric field E. Since the permittivity decreases with rising temperature, d /dT < 0, the surrounding water moves to the hot, as recently confirmed experimentally for thermoosmosis in a capillary [12]. By reaction, the molecule migrates toward the cold. The second term in (2) describes the force exerted by the macroscopic thermoelectric field E T = S∇T on the monomer's counterion density ρ [14]. The electrolyte Seebeck coefficient S is a salt-specific quantity that may take either sign, resulting in motion along the temperature gradient or opposite to it [16]. Eq. ( 2) gives the dominant thermal forces to leading order in the ratio a/λ of the monomer radius and the Debye length. Additional companion fields, such as the salinity gradient, arise in the colloid limit where a is comparable to or larger than λ [16].

The force density f acts on the surrounding water and, by reaction, the molecular unit is subject to the opposite force F = -dV f [27]. Thus the velocity field induced by the moving bead i at the position of its neighbor j, consists of two contributions,

v(r ij ) = G(r ij ) • F + G(r ij -r) • f (r)dV, (3)
where G(r) = (1 + r r)/8πηr is the Oseen tensor with the viscosity η and r = r/r [START_REF] Kim | Microhydrodynamic: Principles and Selected Applications[END_REF]. The first term describes the long-range velocity field v ∼ 1/r ij or "stokeslet" of particle i at the position j, due to the force F; it gives rise to strong hydrodynamic effects on diffusion and sedimentation [START_REF] De Gennes | Scaling concepts in Polymer Physics[END_REF]. The second term is characteristic for phoretic motion; it may be viewed as the sum of stokeslet flows of strength f dV and centered at a distance r from particle i, as illustrated in Fig. 1a. Since both E and ρ vanish well beyond the Debye length, the second term cancels the first one at large distances, r ij λ, whereas it is small for nearby beads.

As a consequence of this hydrodynamic screening, the advection velocity (3) varies as 1/r ij within the Debye length but vanishes at larger distances. When performing the configurational average in (1) with the (isotropic) equilibrium distribution function, the only finite component of the mean drag velocity is along the force density f , that is, along the temperature gradient. Then the tensor equation simplifies to a scalar one, and Eq. ( 1) becomes

u = u 1 + 1 n i,j =i G(|r ij -r|) -G(r ij ) f (r)dV, (4)
with the Oseen tensor replaced by its diagonal part G(r) = 1/6πηr. This form shows that mutual advec- tion vanishes for distant pairs with r ij λ, thus nicely displaying hydrodynamic screening.

(1+χ ϵ )z  2 (1+χ S )z  1+χ ϵ 1+χ S 1
In order to evaluate (4) we need to explicit the force density f (r). The electrostatic potential of a single bead of valency ẑ is well described by the Debye-Hückel expression

ψ = - ẑe 4π r e -r/λ = ζ 1 a r e -r/λ , (5)
where the second equality defines the single-bead surface potential ζ 1 = -ẑe/4π a, which we assume to be negative. One readily obtains the radial electric field E = -dψ/dr and the counterion charge density ρ = -ψ/λ 2 which determine the force density (2). Then the volume integrals in (4) can be performed in closed form [START_REF]Details for the derivation of the thermophoretic mobility, the effective charge, and the experimental data and graphs are given in Supplementary Material[END_REF], resulting in the factors e -2rij /λ /r 2 ij and e -rij /λ /r ij . Since the main contribution to Eq. ( 4) stems from within the screening length λ, which in turn is comparable to the molecular persistence length [START_REF] Tinland | [END_REF], the chain may be treated as rigid such that the distance of beads i, j simplifies to r ij = |i -j| d. Replacing moreover the double sum by integrals over i and j, we obtain

u = ζ 2 1 3η (1 + χ ) ∇ + 2 ζ 1 3η (1 + χ S )E T , (6)
where the quantities χ and χ S account for hydrodynamic interactions (see Fig. 2.) With χ = 0 = χ S one has the explicit expression for the monomer velocity u 1 defined in (1).

The hydrodynamic correction factor for motion driven by the permittivity gradient reads

χ = a 2 d 2 (1 + 2n d) E 2 d -E 2n d n + e -2 d - e -2n d n , (7)
with the shorthand notation E x = Ei(-x) for the exponential integral function, and d = d/λ for the ratio of the monomer length and the Debye length. For the Seebeck term we find

χ S = 2a d E n d -E d + e -n d -e - d n d , (8)
The factor 2 in the exponential and Ei functions in χ arises from the screening factor of the force density, E 2 ∝ e -2r/λ , whereas the factors in χ S are related to the decay of the screening cloud, ρ ∝ e -r/λ . Fig. 2 shows χ and χ S as a function of the molecular weight. Both vanish for monomers, n = 1, whereas for long molecules they tend toward the constants

χ ∞ = (a/d) 2 (2 dE 2 d + e -2 d) and χ ∞ S = -2(a/d)E d.
Note that χ S is identical to the hydrodynamic correction of electrophoresis [31]. Flexible molecules with a power law r 2 ij ∝ |i -j| 2ν , would result in cumbersome expressions with the incomplete Gamma function Γ(ν -1 , nd/λ) instead of the exponential integral function, without changing the qualitative features.

Counterion condensation.-A polyelectrolyte carries a line charge e/d. If the bead spacing d is larger than the Bjerrum length l B ≈ 7 Å, Debye-Hückel approximation is valid even for long chains, and the electrostatic potential reads i ψ i (r -r i ). Yet in the opposite case d < l B , which is relevant for DNA, this linear superposition ceases to be valid as n increases. Because of the strong Coulomb interaction, the counterions partly condense onto the polymer until its linear charge density is reduced to the critical value e/l B [32]. The remaining free counterions are well described by Debye-Hückel theory.

In a mean-field model, counterion condensation is described by an effective valency

ẑ = ξ -1 + 1 -ξ -1 1 + β n , (9)
where ξ = l B /d > 1 is the Manning parameter. For a monomer the quantity β 1 vanishes, and one has ẑ = 1. For long chains, β n tends to infinity, thus resulting in ẑ = ξ -1 and reducing the charge density to its critical value ẑe/d = e/l B [32]. The progressive condensation of the counterions on the chain, and the dependence of β n on n and λ, constitute an intricate problem which is beyond the scope of the present paper [33]. Here we use the simple form β n = (n 2 -1)n -2 0 which, with n 0 = 80, fits rather well the experimental data. Note that this model does not depend on the electrolyte strength.

Phoretic coefficients.-The thermophoretic mobility is defined through the drift velocity u = -D T ∇T in a temperature gradient. From (6) we find [17]. The theory curves are calculated from ST = DT /D, where DT is given by Eq. ( 10) with Ŝ = 0, a = 4.25 Å, and d = 3 Å. The values of the diffusion coeffcient D are shown in the inset; those for n = 5, ..., 50 are measured [17], that for n = 2 is extrapolated from the previous, and the one at n = 80 corresponds to the known power law for long molecules [35].

D T = k B 12πηa l B a ẑ2 (1 + χ )τ + 2ẑ(1 + χ S ) Ŝ , (10)
○ ○ ○ ○ ○ ○ ○ ○ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ○ ○ ○ ○ ○ ○ ○ ○ 2 4 8 10
with the parameter τ = -d ln /d ln T ≈ 1.4 which arises from the permittivity gradient, and the dimensionless Seebeck coefficient Ŝ = S(e/k B ). For monomers the mobility is independent of the Debye length, whereas for longer chains, the correction factors give rise to complex dependencies on λ and n. In Fig. 2 we plot the two contributions to D T as a function of n. The initial increase results from hydrodynamic interactions (dashed lines), whereas the decrease at larger n is due to counterion condensation (solid lines). Both factors reach a finite value at large n; for typical parameters of DNA in a weak electrolyte, the permittivity term shows an overall decrease, ẑ2 (1 + χ ∞ ) < 1, whereas the Seebeck term is enhanced, ẑ(1 + χ ∞ S ) > 1. The stationary DNA concentration c is achieved when thermophoretic drift and gradient diffusion with coefficient D cancel each other, cu -D∇c = 0. This "Soret equilibrium" is usually written in the form ∇c+cS T ∇T = 0, since experiments probe the Soret coefficient S T = D T /D rather than the mobility D T . In Fig. 3 we compare our theory with Soret data for single-stranded DNA as a function of the Debye length λ, taken from Ref. [17]. The theoretical curves are calculated with (10) and a simple model for the measured diffusion coefficient D, as described in [START_REF]Details for the derivation of the thermophoretic mobility, the effective charge, and the experimental data and graphs are given in Supplementary Material[END_REF]. The best agreement with the data is obtained when retaining in (10) the permittivity-gradient term only, that is, for zero Seebeck coefficient, Ŝ = 0. The increase of S T with the Debye length arises mainly from the hydrodynamic correction χ . For short chains, n < 30, the variation with n is of purlely hydrodynamic The full curves are calculated from Eq. ( 10) with Ŝ = 0, a = 4.25 Å and d = 3 Å. The data points give DT = DST , with ST and the hydrodynamic radius from Ref. [17]; for details see [START_REF]Details for the derivation of the thermophoretic mobility, the effective charge, and the experimental data and graphs are given in Supplementary Material[END_REF].

○ ○ ○ ○ ○ ○ ■ ■ ■ ■ ■ ■ ▲ ▲ ▲ ▲ ▲ ▲ ▼ ▼ ▼ ▼ ▼ ▼ ◆ ◆ ◆ ◆ ◆ ◆ ▮ ▮ ▮ ▮ ▮ ▮ • • • • • • ○ ○ ○ ○ ○ ○ λ = 10 nm ■ λ = 8 nm ▲ λ = 7 nm ▼ λ = 6 nm ◆ λ = 5 nm ▮ λ = 4 nm • λ = 3 nm ○ λ = 2 nm
origin, whereas for larger chains counterion condensation plays an important role, as is clear from Fig. 2.

In order to clearly display the effect of hydrodynamic interactions, we plot in Fig. 4 the thermophoretic mobility (10) as a function of the molecular weight n. The experimental points are obtained from D T = DS T , with measured S T and D [17] as described in Fig. [START_REF]Details for the derivation of the thermophoretic mobility, the effective charge, and the experimental data and graphs are given in Supplementary Material[END_REF]. The theoretical curves are calculated with the permittivitygradient only ( Ŝ = 0). The initial increase of the data up to n = 22 agree quantitatively with the relation (7), thus providing strong evidence for the role of hydrodynamic interactions. The maximum and the subsequent decrease are well described by counterion condensation according to [START_REF] Jerabek-Willemsen | [END_REF]. Adding a significant thermoelectric contribution would not improve the quality of the fit, quite on the contrary. This suggests that the Seebeck field in NaCl solution is small, confirming a previous analysis of Soret data for polystyrene beads [16].

The electrolyte Seebeck effect was discarded in the above analysis of Soret data in NaCl solution. In Fig. 5 we plot the complete mobility D T as a function of n, for several values of the dimensionless Seebeck coefficient Ŝ. As the most striking feature, for negative Ŝ the superposition of the two contributions in (10) may result in a change of sign of the D T . From Fig. 2 it is clear that for short chains, the permittivity gradient term prevails, whereas for longer moleculese the Seebeck term dominates because of its much larger hydrodynamic factor χ S .

The resulting velocity difference could be used for specific accumulation of one component at a heated spot, or for separating DNA by molecular size. For example, in an electrolyte with Ŝ = -0.3, the permittivity-gradient 6) drives the molecules toward the hot, whereas the permittivity gradient points toward the cold. Since the latter dominates for short molecules and the latter for long ones, DT changes sign as the n increases.

term dominates for short molecules (n < 50) which move to the cold accordingly, whereas longer chains (n > 50) are driven to the hot by the thermoelectric field E T . The stagnation molecular length n c , where D T = 0, is easily adapted by chosing an appropriate salt mixture. The change of sign has been observed for nano-size micelles [15] and micron-size polystyrene beads [16] in mixed electrolytes NaOH x Cl 1-x ; the values of Ŝ used in Fig. 5 are realized by with 0.1 < x < 0.4.

Conclusion. -We briefly summarize our main results on DNA thermophoresis. First, D T does not vanish in the limit of high salinity or small Debye length, contrary to what is known for micron-size colloidal particles [16] and what was assumed to hold true for DNA: A finite value for small λ was observed in a recent experiment [17], and interpreted as a non-ionic contribution due to dispersion forces. In our Eq. (10) this limit is obtained by letting χ → χ ∞ and ẑ → ξ -1 , revealing the existence of a large residual double-layer contribution.

Second,we find that DNA thermophoresis is rather sensitive to hydrodynamic interactions. With increasing chain length n, the mobility is enhanced due to mutual advection of the repeat units, then passes through a maxium, and finally decreases below the monomer value due to counterion condensation. Comparison with measured data in Fig. 4, provides strong evidence that the molecular-weight dependence arises from the interplay of hydrodynamic interactions and non-linear charge effects.

Fourth, the interplay between the dominant driving forces, that is, the permittivity gradient and a thermoellectric field with negative Seebeck coefficient, results in a change of sign as a funciton of n: Short molecules move to the cold, and long ones to the hot.

FIG. 2 :

 2 FIG. 2: Molecular-weight dependence of the two contributions to DT . Both hydrodynamic factors χ and χS increase with n, albeit with differnt amplitudes, as shown by the dashed lines; parameters are the monomer distance d = 3 Å and the Debye length λ = 5 nm. Counterion condensation results in the factors ẑ2 and ẑ, which significantly reduce the mobility (solid lines), according to (9) with βn = (n 2 -1)n -2 0 and n0 = 80.

1 )FIG. 3 :

 13 FIG.3: The Soret coefficient ST as a function of the Debye length λ for DNA of different length n. The data points, measured at 15 • C, are taken from Ref.[17]. The theory curves are calculated from ST = DT /D, where DT is given by Eq. (10) with Ŝ = 0, a = 4.25 Å, and d = 3 Å. The values of the diffusion coeffcient D are shown in the inset; those for n = 5, ..., 50 are measured[17], that for n = 2 is extrapolated from the previous, and the one at n = 80 corresponds to the known power law for long molecules[35].

1 FIG. 4 :

 14 FIG.4: Thermophoretic mobility DT as a function of the molecular length n for three values of the Debye length λ. The full curves are calculated from Eq. (10) with Ŝ = 0, a = 4.25 Å and d = 3 Å. The data points give DT = DST , with ST and the hydrodynamic radius from Ref.[17]; for details see[START_REF]Details for the derivation of the thermophoretic mobility, the effective charge, and the experimental data and graphs are given in Supplementary Material[END_REF].

1 )FIG. 5 :

 15 FIG.5: Thermophoretic mobility DT as a function of the chain length n, for different values of the dimensionless Seebeck coefficient Ŝ. For negative Ŝ the thermoelectric field in (6) drives the molecules toward the hot, whereas the permittivity gradient points toward the cold. Since the latter dominates for short molecules and the latter for long ones, DT changes sign as the n increases.

The authors acknowledge funding by Agence Nationale de la Recherche through contract ANR-13-IS04-0003.