Vison-Generated Photon Mass in Quantum Spin Ice: a Theoretical Framework

PHYSICAL REVIEW B 102, 125113 (2020)

Vison-generated photon mass in quantum spin ice: A theoretical framework

M. P. Kwasigroch London Centre for Nanotechnology, University College London, 17-19 Gordon Street, London WC1H 0AH, United Kingdom and T.C.M. Group, Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom

(Received 31 October 2019; accepted 17 June 2020; published 10 September 2020)

Describing the experimental signatures of quantum spin ice has been the focus of many theoretical efforts, as definitive experimental verification of this candidate quantum spin liquid is yet to be achieved. Gapped excitations known as visons have largely eluded those efforts. We provide a theoretical framework, which captures their dynamics and predicts new signatures in the magnetic response. We achieve this by studying the ring-exchange Hamiltonian of quantum spin ice in the large-s approximation, taking into account the compact nature of the emergent U(1) gauge theory. We find the stationary solutions of the action—the instantons—which correspond to visons tunneling between lattice sites. By integrating out the instantons, we calculate the effective vison Hamiltonian, including their mass. We show that in the ground state virtual vison pairs simply renormalize the speed of light and give rise to an inelastic continuum of excitations. At low temperatures, however, thermally activated visons form a Debye plasma and introduce a mass gap in the photon spectrum, equal to the plasma frequency, which we calculate as a function of temperature. We demonstrate that this dynamical mass gap should be visible in energy-resolved neutron scattering spectra but not in the energy-integrated ones. We also show that it leads to the vanishing of the susceptibility of an isolated system, through a mechanism analogous to the Meissner effect, but that it does not lead to confinement of static spinons.

DOI: 10.1103/PhysRevB.102.125113

I. INTRODUCTION potential condensation of electric charges can lead to the confinement of magnetic charges. Quantum spin ice (QSI) is a candidate quantum spin liquid It is believed that at zero temperatures visons simply lead [1], where an emergent U(1) gauge symmetry prevents mag- to the renormalization of the speed of light, but at nonzero netic order down to zero temperature and gives rise to gauge temperatures, it is anticipated that their effects are less be- and fractionalized excitations: a gapless photon, magnetic nign. Reference [1] already drew comparisons with (2 + 1)- monopoles (spinons) and emergent electric charges (visons). dimensional compact lattice gauge theory, which is always in In spite of this long list of predicted exotic excitations, defini- the confined phase and has a gapped photon spectrum [11]. tive experimental verification is still missing, and this situation At increasing temperatures, the (3 + 1)-dimensional QSI be- is shared by QSI with many other proposed spin liquids [2]. comes more like a (2 + 1)-dimensional compact lattice gauge Hence, it is important for theoretical efforts to characterize as theory and perhaps a condensation of visons leads to a photon many signatures of these excitations as possible. In particular, mass gap and confinement of magnetic monopoles. the theoretical description of the inelastic response, through Despite their importance, so far, few experimental sig- the dynamical structure factor, has recently been successful in natures of visons have been predicted [12,13]. They are α providing evidence of a potential spin liquid in -RuCl3 [3–8]. sources of a fictitious electric field and hence cannot easily be In our previous work [9], we analytically characterized the probed directly in experiments. They have also largely eluded photon excitation of QSI by including quantum fluctuations theoretical efforts of quantum Monte Carlo [14,16–20], due around the classical limit via a large-s expansion. We were to the limited resolution of obtained excitation spectra, as also able to calculate the energy of the gapped vison excita- well as mean-field treatments [21–25]. In this context, the tion, but were not able to study its dynamics. In this paper, we work of Ref. [26], made recent progress by looking at visons extend the semiclassical description of our previous work to through classical Monte Carlo, characterising their effect on capture the vison dynamics and its contribution to the inelastic emergent field correlators and the heat capacity. This work magnetic response of QSI. analysed vison signatures in the classical s =∞limit. Here, The vison is an emergent excitation of QSI, a consequence we will show that including nonperturbative corrections in of the compact U(1) gauge symmetry [10]. (The gauge group s, makes the vison inertia finite and allows us to study their is compact because of the quantization of the spins.) Visons quantum dynamics at low temperatures. We demonstrate that are sources of flux of the emergent electric field. Even though these dynamics couple to the magnetic response of QSI via the field is divergenceless, a Dirac string carrying flux that is a electromagnetic induction, which can be probed in neutron multiple of 2π has zero energy in a compact theory, allowing scattering experiments. the charges at its ends to behave as free excitations. The state In our theoretical analysis, we are motivated by the suc- of these electric charges has profound consequences on the cess of our previous work [9], where we used a large-s force between the magnetic charges (spinons). In particular, semiclassical description to successfully capture the emergent

2469-9950/2020/102(12)/125113(20) 125113-1 ©2020 American Physical Society M. P. KWASIGROCH PHYSICAL REVIEW B 102, 125113 (2020) electrodynamics of the ring-exchange Hamiltonian [10] and where the sum is taken over the four pyrochlore lattice sites obtained a photon spectrum that was in good quantitative ij that belong to the diamond lattice site i, i.e., over the four agreement with quantum Monte Carlo calculations for s = corners of the tetrahedron centered on i (note that a positive 1/2[14]. As in our previous work, we map the ring-exchange sign is taken for “up” tetrahedra and negative for “down” Hamiltonian in the large-s limit onto compact U(1) lattice tetrahedra). Qi ∈ Z are static magnetic charges (magnetic gauge theory using the Villain spin representation. However, monopoles) introduced into the system. The charges are static this time we do not neglect spin quantization, or equivalently, because the constraint commutes with the Hamiltonian and Qi the periodic nature of the emergent dual field, which allows us are therefore constants of motion. to include the dynamics of visons in the theoretical description. We show that, in the ground state the effects of these A. Electric charge representation charges are rather innocuous; they exist as virtual, tightly bound pairs and simply renormalize the speed of light. At The visons are made more manifest in the dual, electric nonzero temperatures, however, they enter into existence as charge representation of the Hamiltonian. Rather than work- ˆz physical, thermally activated excitations, that form a plasma. ing with the conjugate magnetic field Sij and the electric φ We show that propagation of the photon through the vison vector potential îj, we shall be working with the conjugate plasma will appear gapped, which can be probed in neutron electric field Eαβ and the magnetic vector potential Aαβ.We scattering experiments. We calculate typical scattering inten- introduce the new conjugate operators via sities that might be observed. We note that a similar effect had Eˆαβ = curlαβφ,ˆ (3) been anticipated in the context of hot QED by Ref. [27]. ˆz = ˆ + ψ, The paper is structured as follows. In Sec. II, we present the Sij curlijA ij (4) ring-exchange Hamiltonian in the Villain spin representation, where ψ is a scalar field (not an operator) defined on diamond which was introduced in our previous work. We then move i lattice sites with ψ ≡ ψ − ψ . Note that, because the to the dual electric charge representation, where the visons ij j i electric field Eαβ is defined as the lattice curl it necessarily are made more manifest. We conclude the section by writing has zero divergence. Equation (4) is just the lattice Helmholtz down the imaginary-time action of our model and finding its decomposition for the magnetic field and because its diver- normal modes. In Sec. III, we analyze the stationary solutions genceful part is a constant of motion, it can be expressed as of the action—the instantons—which correspond to the quan- ψ with ψ a scalar field. The commutation relations for the tum tunneling of visons between neighboring lattice sites. In ij i ˆz ˆ Sec. IV, we look at the ground state properties of our model. magnetic Sij and electric field Eαβ operators (which follow φˆ , z = Sec. V studies the effects of visons at nonzero temperatures from [ ij Sij] i) imply that the new conjugate operators and presents the main results of this paper. Finally, in Sec. VI, have the canonical commutator [Eˆαβ, Aˆαβ] =−i. we summarize all our main findings. Because the Hamiltonian is periodic in Eˆαβ, the noninteger 0 part of Aˆαβ, Aαβ is a constant of motion and is analogous to crystal momentum. It is therefore useful to make the following II. THE MODEL replacement: Following on from our previous work [9], we study the 0 Aˆαβ → Aˆαβ + A , (5) ring-exchange Hamiltonian of QSI [10] in the Villain spin αβ representation [28] where the new operators Aˆαβ have strictly integer eigenvalues. The energy eigenstates, which are analogous to Bloch states, φ/ˆ 1 φ/ˆ Hˆ = g eicurlαβ 2 s˜2 − Sˆz 2 2 eicurlαβ 2 + H.c. , are now given by ij αβ ij∈αβ |= | ⊗ 0 cAαβ Aαβ Aαβ (1) Aαβ 0 where the spin azimuthal angle φ ∈ (−π,π], its projection = dEαβ (Eαβ )|Eαβ⊗ Aαβ , (6) on the z-axis Sz is an integer or half-integer with |Sz| s, 1 φ,ˆ ˆz = = + αβ αβ [ S ] i,˜s s 2 , and the product is over all six py- where (E ) is a periodic function of E and the kets rochlore lattice sites ij belonging to the plaquette centered on are eigenstates of the respective operators. After the above site αβ of the dual pyrochlore lattice. The curl is taken around replacement, the magnetic field becomes this plaquette. Just in like our previous work, Latin letters {i} z 0 Sˆ = curl Aˆαβ + B , (7) index the sites of the diamond lattice (bond midpoints {ij} ij ij ij correspond to pyrochlore lattice sites on which the spins live) 0 where Bij is a static background field and the Greek letters index the sites of the dual diamond lattice {αβ} 0 = 0 + ψ. (bond midpoints correspond to the dual pyrochlore Bij curlijAαβ ij (8) lattice sites on which the plaquettes are centered). The spins Different constraints on the allowed values of the magnetic satisfy the constraint ˆz field Sij can be implemented as constraints on the allowed (i) eigenstates of Aˆαβ and values of the static background field z ≡ z = , 0 diviS Sij Qi (2) Bij. For instance, to realize the constraint that the magnetic ij field is half-integer valued we simply choose any background

125113-2 VISON-GENERATED PHOTON MASS IN QUANTUM SPIN … PHYSICAL REVIEW B 102, 125113 (2020)

0 β field Aαβ that is half-integer valued. The field Aˆαβ then be- given by = ). | z | Nτ comes unconstrained. The kinematic constraint Sij s is π +∞ +∞ 0 the hardest to implement, since for any background field Aαβ, Z = dEαβ (τ ) dAαβ (τ ) dϕγ (τ ) −π −∞ −∞ there are forbidden eigenstates of Aˆαβ. As discussed in our τ,αβ,γ previous√ work [9], in the large-s limit, the typical fluctuations −S z ∼ × e , Sij s and hence this constraint is largely irrelevant. jαβ (τ ) Any magnetic charges Qi introduced into the system [see Eq. (2)] uniquely determine ψi via the lattice Laplace equation 2 S = i Aαβ (τ )τ Eαβ (τ ) + g˜ Eαβ (τ ) 0 ≡∇2ψ = , τ αβ αβ diviB i Qi (9) gz˜ 2 where ∇2 is the lattice Laplacian. To realize the constraint that + curl A(τ ) + B0 i s2 ij ij the magnetic field is integer (half-integer) valued, we must ij 0 0 also choose a configuration Aαβ that makes Bij integer (half- integer)—this is important for confinement and is analysed in + iϕγ (τ )[divγ E(τ ) − 2πqγ (τ )] detail in Appendix F. γ Because energy eigenstates are periodic in Eαβ, without + π τ τ , loss of generality, we will restrict Eαβ ∈ (−π,π], i.e., mea- 2 ijαβ ( )Aαβ ( ) (13) sure the field modulo 2π. The electric field now acquires a τ,αβ π nonzero divergence quantized in units of 2 where τ Eαβ ≡ Eαβ (τ + ) − Eαβ (τ ), the sum over integers jαβ (τ ) ensures that Aαβ (τ ) are integer-valued, and ϕγ (τ )are (α) Lagrange multipliers that ensure the divergence of the electric ˆ ≡ ˆ = π , divαE Eαβ 2 qα (10) field at each dual diamond lattice site γ is an integer multiple αβ qγ of 2π for all τ. The Lagrange multipliers can be interpreted as the scalar electric potential. where qα ∈ Z are the emergent electric charges ((α) identifies that the sum is taken over the four corners of the tetrahedron α Note that the zero modes, arising from the U(1) gauge symmetry of the action, Aαβ (τ ) → Aαβ (τ ) + χβ (τ ) − χα (τ ), of the dual diamond lattice and again positive sign is taken for ϕ τ → ϕ τ + χ τ + − χ τ “up” tetrahedra and negative for “down” tetrahedra). A single γ ( ) γ ( ) γ ( ) γ ( ) enforce vison charge conservation electric charge at the tetrahedron α (qα =±1) gives rise to a smooth, long-wavelength modulation of the electric field τ qγ + divγ j = 0, (14) ∝ 1 Eαβ 2 , where R is the distance from the electric charge. R for all tetrahedra γ . We can see that jαβ (τ ) can be interpreted This long-wavelength modulation is a gapped topological as the vison current between the tetrahedra touching at αβ. excitation known as the vison. To summarize this subsection, we write down the ring- exchange Hamiltonian in the electric charge representation C. Normal modes of the action We first conveniently parametrize the sites of the dual py- 1 iEˆαβ /2 2 0 2 2 rochlore lattice. The dual pyrochlore lattice is a superposition Hˆ =−g e s˜ − curlijAˆ + B ij of four fcc lattices indexed by μ = 1, 2, 3, and 4. Taking a αβ ij∈αβ single up tetrahedron from the dual diamond lattice, with its

αβ / center located at the position vector r, the position vectors of × eiE 2 + H.c. . (11) its four corners (μ = 1, 2, 3, 4) are (r + eμ/2), where the four basis vectors eμ are given by We will be working in the s 1 limit and expand the Hamil- a0 a0 tonian accordingly e = (1, 1, 1), e = (1, −1, −1), 1 4 2 4 a a gz˜ 2 − = 0 − , , − , = 0 − , − , . Hˆ = g˜ Eˆ 2 + curl Aˆ + B0 + O(s 2 ), (12) e3 ( 1 1 1) e4 ( 1 1 1) (15) αβ s2 ij ij 4 4 αβ ij The centres of all up tetrahedra map out an fcc lattice. Each μ / whereg ˜ = gs6 and z = 6. We have also replaceds ˜ with s,as fcc lattice is then a translation of this lattice by eμ 2 and cor- we will be working to the ﬁrst nonvanishing order in s for all responds to the set of all those up tetrahedra corners that are / physical quantities that we calculate. displaced by eμ 2 from their centres. We thus identify each dual pyrochlore lattice site αβ by an index μ, corresponding to the fcc lattice to which this site belongs, and its position B. Partition function vector on that fcc lattice. This is reﬂected in the following To study the nonperturbative effects of dynamical electric change of notation for the variables charges, we begin with the partition function for the above Eαβ (τ ) → Eμ(rα + eμ/2,τ), Hamiltonian, obtained in the usual way by inserting two Aαβ (τ ) → Aμ(rα + eμ/2,τ), (16) resolutions of the identity in the Eˆαβ and Aˆαβ bases into each of the Nτ Suzuki-Trotter time slices (the time slice width is jαβ (τ ) → jμ(rα + eμ/2,τ),

125113-3 M. P. KWASIGROCH PHYSICAL REVIEW B 102, 125113 (2020) where rα + eμ/2 is the position vector of the site αβ and μ original pyrochlore lattice identifies the fcc lattice to which it belongs. rα and rβ are the Aμ(r + eμ/2,τ) position vectors of the up and down tetrahedra, respectively, which touch at the site αβ, and eμ = rβ − rα. 1 † ik· r+eμ/2 −iωτ = √ U (k)Aλ(k,ω)e ( ) , We also take the continuum limit → 0ofS, where N β μλ s k∈BZ,ω,λ β/2 → τ, d Eμ(r + eμ/2,τ) τ −β/2 1 † ik·(r+eμ/2)−iωτ τ Eαβ = √ Uμλ(k)Eλ(k,ω)e , (22) → E˙αβ, N β s k∈BZ,ω,λ λ = , , , jαβ (τ )Aαβ (τ ) where 1 2 3 and 4 index the normal modes of the τ action, the wave vectors k are summed over the first Brillouin zone of the fcc lattice, ω are Matsubara frequencies, Ns is the β/2 † number of sites of the fcc lattice, and Uμλ(k)are4× 4 unitary → dτAαβ (τ ) jαβ (τ0 )δ(τ − τ0 ) , (17) −β/2 τ matrices. In this basis, the curl operator can be written as 0 1 † 0 curlij(A) = √ Zμν (k)Uνλ(k) and we will set the background field Bij to zero for now and N β return to analyzing its effects later. s k∈BZ,ω ν,λ τ ik·(rα +eμ/2)−iωτ Because Aαβ ( ) is a continuous variable, provided we × e Aλ(k,ω) enforce charge conservation in Eq. (14), we can fix its gauge 1 † via the usual Faddeev-Popov procedure [29]. We choose to = √ ξλ(k)U (k) N β μλ work in the Coulomb gauge s λ,k∈BZ,ω

ik·(r +eμ/2)−iωτ (α) × e i Aλ(k,ω), (23) divαA ≡ Aαβ = 0, (18) † where the columns of the matrix Uνλ(k) are eigenvectors of αβ the matrix Zμν (k) = 2i sin (k · μν ) with eigenvalues where the longitudinal and transverse parts of the electric √ ξ =± 2 · , field Eαβ (τ ) decouple. (The longitudinal part of Eαβ satisfies λ=1,2(k) 2 sin (k μν ) μν curlijE(τ ) = 0 everywhere, whereas the transverse part satisfies divαE(τ ) = 0 everywhere.) In the Coulomb gauge, the ξλ=3,4(k) = 0. (24) action can be decomposed as follows: This identifies λ = 3 and 4 as the longitudinal modes since S = Slong. + Stran., (19) they vanish under the action of the curl operator. It follows that the transverse part of the action becomes diagonal in the where Slong. is a functional of the longitudinal part of Eαβ (τ ) new basis only and Stran. of the transverse part only. Stran. = ωAλ(−k, −ω)Eλ(k,ω) 1. Transverse modes ω k∈BZ,λ=1,2

S zg˜ 2 The transverse part of the action tran. is diagonal in the + gE˜ λ(−k, −ω)Eλ(k,ω) + ξ (k)Aλ(−k, −ω) 2 λ eigenbasis of the curl operator, which is defined as s × Aλ(k,ω) + 2πijλ(k,ω)Aλ(−k, −ω) , (25) curlijA ≡ ±Aν (ri + eμ/2 ± μν ,τ), (20) ν =μ,± where in the Coulomb gauge Aλ=3,4(k,ω) = 0 and where ,ω = √1 + / ,τ × jλ(k ) Uλμ(k) jμ(r eμ 2 ) a0 eμ eν Nsβ μν ≡ √ , (21) μ,τ,r∈fcc 8 |eμ × eν | × eiωτ−ik·(r+eμ/2). (26) and ri + eμ/2 is the position vector of the plaquette center ij on the original pyrochlore lattice. The decomposition for 2. Longitudinal modes the original pyrochlore lattice works in the same way as that We proceed to finding the normal modes of the longitu- for the dual pyrochlore lattice: r gives the position vector of i dinal part of the action. The constraint in Eq. (10) uniquely the center of the i up tetrahedron, of the original diamond determines the longitudinal (λ = 3 and 4) modes of the elec- lattice, and centres of all up tetrahedra map out an fcc lattice, tric field Eαβ (τ ) via the lattice Laplace equation. To form which can be translated by eμ/2 to give one of the four the equation, we write the longitudinal field as the lattice μ = 1, 2, 3, and 4 fcc lattices the plaquette center ijbelongs derivative to. We transform to the eigenbasis of the curl operator in the same way that we have done in our previous work [9]forthe E(r + eμ/2,τ) = d(r + eμ,τ) − u(r,τ), (27)

125113-4 VISON-GENERATED PHOTON MASS IN QUANTUM SPIN … PHYSICAL REVIEW B 102, 125113 (2020) where the variables u/d(r) are defined on the centres of The above vison self-energy and Coulomb interaction agree up/down tetrahedra that make up two fcc lattices. Making with the results of Ref. [26] in the quadratic approximation. the above substitution, the constraint in Eq. (10), for the up (Note that our definition of g and the definition given in tetrahedra, can be written as Ref. [26] differ by a factor of 2.) 2πqu(r,τ) = [d(r + eμ,τ) − u(r,τ)] (28) μ III. INSTANTONS and for the down tetrahedra as A. Stationary solutions of the action Stationary solutions of the Euclidean action make an im- 2πq (r,τ) =− [ (r,τ) − (r − eμ,τ)], (29) d d u ∝ −s μ portant nonperturbative ( e ) contribution to the partition function, that restores the periodic symmetry with respect to ,τ ,τ where qu(r ) and qd(r ) are the vison occupation numbers the electric field Eαβ (τ ). The instanton is precisely such a sta- at position r of an up or down tetrahedron respectively. Fourier tionary solution. For illustration, let us first consider a single transforming the two equations we obtain a unique solution instanton at r = 0 and imaginary time τ = 0 in the direction ,τ (up to a constant) for u/d(r ) eσ , i.e., jμ=σ (0, 0) = 1 with all other jαβ vanishing. To satisfy π the continuity equation in Eq. (14), we will consider a vison u(k,τ) 1 γ (k) qu(k,τ) = ∗ , of charge 2π tunneling from the up tetrahedron at r =−eσ /2 d(k,τ) 2(|γ (k)|2 − 1) γ (k)1 qd(k,τ) to the down tetrahedron at r = eσ /2. Correspondingly, the vi- (30) son occupation number qu(−eσ /2,τ)[qd(eσ /2,τ)] decreases τ = γ = 1 ik·eμ = √1 −ik·r (increases) at 0 by one. (Alternatively, we could have where (k) μ e , u/d(k) r∈u/d e u/d(r) 4 Ns considered a vison of charge −2π tunneling in the opposite √1 −ik·r and qu/d(k) = ∈ / e qu/d(r)(r is summed over the Ns r u d direction or a creation of a vison-antivison pair.) positions of up/down tetrahedra, respectively). Substituting When deriving the stationary solution of the action, it this back into S in Eq. (13), we obtain the unique longitudinal proves convenient to briefly reinstate the longitudinal part of part of the action Aαβ (τ )[A(k,ω)λ= , normal modes] in the action so that the 3 4 β/2 continuity expressed in Eq. (14) is automatically taken care of. Slong. = dτ V (r − r )qu(r,τ)qu(r ,τ) (This, of course, has no effect on the physical, gauge-invariant −β/ 2 rr∈u observables and is the only place where we are not working in the Coulomb gauge.) We can see how continuity is enforced + V (r − r )q (r,τ)q (r,τ) d d as follows. Substituting the longitudinal part of Aαβ (τ ), which rr∈d can be written as Aαβ (τ ) = χβ (τ ) − χα (τ ), into the action S, + − ,τ ,τ , and collecting all terms where it enters, we obtain Vud(r r )qu(r )qd(r ) (31) r∈u r∈d −i χα (τ )[divατ E(τ ) + 2πdivα j(τ )]. (36) where α,τ 2 g˜ π · − χ τ V (r − r ) = eik (r r ), The longitudinal part α ( ) can thus be interpreted as a N 1 −|γ (k)|2 Lagrange multiplier that enforces the continuity expressed in s k∈BZ Eq. (14). The instanton solution can now be derived from the π 2γ g˜ 2 (k) ik·(r−r ) transverse part of the action in Eq. (25) by extending the sum Vud (r − r ) = e . (32) N 1 −|γ (k)|2 to include the longitudinal modes λ = 3, 4. Minimising S in s k∈BZ Eq. (25) with respect to variations in the variables Eλ(k,ω) Assuming the visons are far apart, it is useful to decompose and Aλ(k,ω), we obtain the following equations of motion the sum into its diagonal and off-diagonal parts β/2 β/2 ωAλ(k,ω) = 2˜gEλ(k,ω), 2 S . = dτμ qσ (r,τ) + dτ π long V 2zg˜ 2 2 i −β/2 −β/2 ωEλ(k,ω) + ξ (k)Aλ(k,ω) + √ Uλσ (k) = 0. σ =u/d,r∈fcc 2 λ β s Ns × V˜ (r − r )qσ (r,τ)qσ (r ,τ), (33) (37) σ,σ,r,r∈fcc Solving for the electric field, we obtain the instanton solution where V˜ (r) is the asymptotic Coulomb part of the vison π ω interaction energy with the additional constraint that V˜ (0) = 0 inst. const. 1 i Uλσ (k) E (k,ω) = E (k)δω, − √ , (38) λ λ 0 2zg˜2 ω2 and the vison chemical potential (self-energy) is given by Nsβ ξ 2 + s2 λ (k) 2 π 2 g˜ const. μ = = C g˜, (34) where Eλ (k) is constant in time and is a purely divergence- V 2 1 Ns 1 −|γ (k)| const. = k∈BZ ful field [i.e., Eλ=1,2(k) 0] due to a pair of visons of charge π ± / = . = at positions eμ 2. This is to ensure that the longitudinal where C1 17 7. For r 0, inst. ,τ part of Eμ (r ) is the electric field due to a vison of charge 3 2 d k 8˜ga π 2π,atr =−eσ /2forτ<0, and at r = eσ /2forτ>0. (See V˜ (r) = 0 eik·r. (35) (2π )3 |k|2 Appendix A for further details.)

125113-5 M. P. KWASIGROCH PHYSICAL REVIEW B 102, 125113 (2020)

It is important to check that our solution lies within the are governed by the following action: domain |Eαβ (τ )| π. By symmetry, the solution will reach β 2 2|δ ˙ ,τ |2 its maximum at τ = 0, r = 0 s Eλ(k ) δStran[δEλ(k,τ)] = dτ β 2 − 4zg˜ξλ (k) ∞ ω −π ω † 2 k∈BZ,λ=1,2 inst. d i Uσλ(k)Uλσ (k) E 0, = β σ ( 0) 2 ω2 −∞ 2πN 2zg˜ ξ 2 + 2 1 . s λ, ∈ 2 λ (k) sm 2 k BZ s 2 + g˜ dτ V Eαβ (τ ) δEαβ (τ ), − β 2 =±π, (39) 2 αβ (43) where the field jumps from −π to π at τ = 0 and we have const. sm. ,τ =− πδ sm. ,τ − π + =− πδ used the fact that Eμ (r = 0) = 0. We have thus verified where V (Eμ (r )) 4 (Eμ (r ) ) 2 4 sm. that |Eαβ (τ )| π everywhere. Note that we are working (τ )δr,0δμσ /E˙σ (0, 0) + 2. The normal modes of this action, β/τ sm. in the low-temperature limit QF 1, where finite-size which include the zero translational mode E˙ (r,τ) (see ≈ effects in imaginary time can be neglected and ω∈2πn/β Appendix B for an explicit proof) determine the instanton ∞ β dω . τ = √s is the characteristic timescale of quan- measure −∞ 2π QF g˜ z − 1 tum fluctuations. Notice that β/τQF 1 implies that the ˙ sm.| ˙ sm. ˜ 2 ξ˜ ξ E E det K = gC˜ 3d 0 , temperatures are low by comparison with the photon d ˜0 √ √ (44) π det K s bandwidth [9]. 2 where C3 = 5.13, ξ˜0 specifies the position of the instanton core, detK˜ is the determinant of the kernel of the fluctuation B. The instanton measure action in Eq. (43), excluding the zero eigenvalue, and det K is To compute the measure associated with a single instanton, the determinant of the kernel of the original action in Eq. (41). we first remove the discontinuity in the instanton solution See Appendix C for a detailed derivation of the measure. inst. Eμ (r,τ)atτ = 0[seeEq.(38)] and let Eσ (0,τ)windby π 2 instead C. Instanton interactions

Eμ(r,τ) → Eμ(r,τ) − δr,0δμσ [πsgn(τ ) + π]. (40) The power-law decay (see Appendix A) of the single instanton solution gives rise to long-range interactions between Note that this transformation removes the jump in the instantons. This infrared effect has consequences for the low- inst. longitudinal part of Eμ (r,τ) that occurs at τ = 0(see energy properties of the system. Just like the visons interact Appendix B). The longitudinal part is now the electric field via the Coulomb potential, the instantons, which correspond due to a vison of charge 2π at r =−eσ /2forτ>0as to vison currents, interact in Euclidean space-time via forces well as τ<0. The longitudinal part is thus a constant and that obey the inverse square law. the dynamical part is purely transverse: (λ = 1, 2) are the If the instantons are far apart the interactions will be taking only components we need to consider, when calculating the place in a region of space with small electric field. To derive instanton measure. Considering the action S in Eq. (13), the long-range part of instanton interactions, we can therefore π , , the above transformation removes the 2 ijσ (0 0)Aσ (0 0) relax the constraint on the electric field Eαβ ∈ (−π,π] and 2 ,τ > π term, translates the quadratic term Eσ (0 0) by 2 ,al- integrate it out to obtain an effective action Sinst. that describes ters the constraint enforced by ϕα (τ ) so that divE = 2π at instanton-instanton interactions, i.e., interactions between vi- r = eμ/2 for all τ, and changes the range of integration son currents, over Eσ (0,τ) from |Eσ (0,τ)| π to πsgn(τ ) < Eσ (0,τ) S = − ,τ − τ ,τ ,τ , 2π + πsgn(τ ) (the range of integration over the other Eαβ (τ ) inst. Vμν (r r ) jμ(r ) jν (r ) (45) variables remains unaltered). Integrating out the variables τ,τ r,r∈fcc Aλ=1,2(k,ω) in the transverse part of the action Stran.,weare where left with the following action for the variables Eλ= , (k,ω) 1 2 − ,τ − τ together with a smooth instanton solution E sm.(r,τ) Vμν (r r ) μ π 2 ∞ ω iω(τ −τ )+ik·(r−r ) β/2 2 2 d e s |E˙λ(k,τ)| = S λ ,τ = τ π ω2 zg˜ 2 tran[E (k )] d 2 Ns −∞ 2 + ξ (k) −β/ 4zg˜ξ (k) λ=1,2,k∈BZ 4˜g s2 λ 2 k∈BZ,λ=1,2 λ † ×Uμλ(k)Uλν (k). (46) + g˜ V (Eαβ (τ )) , (41) In the limit where instantons are far apart, it is useful to αβ decompose the sum into its diagonal and off-diagonal parts sm. ,τ = inst. ,τ + δ δ π τ + π , Eμ (r ) Eμ (r ) r,0 μσ [ sgn( ) ] (42) Sinst. = V˜μν (r − r ,τ − τ ) jμ(r,τ) jν (r ,τ ) 2 where V (Eαβ (τ )) = min (Eαβ (τ ) − 2πn) is a continued τ,τ r,r ∈fcc n parabolic potential and the minimum is taken with respect to + μ j2 (r,τ), (47) integer n. See Appendix B for verification that E sm.(r,τ)is I μ μ τ, ∈ indeed a stationary solution of the above action. r fcc Fluctuations around the smooth instanton solution where V˜μν (r − r ,τ − τ ) is given by the asymptotic in- sm. δEμ(r,τ) = Eμ(r,τ) − Eμ (r,τ) are also smooth and verse square law of Vμν (r − r ,τ − τ ) with the additional

125113-6 VISON-GENERATED PHOTON MASS IN QUANTUM SPIN … PHYSICAL REVIEW B 102, 125113 (2020) ⎛ ⎞ n constraint that V˜μμ(0, 0) = 0, and the instanton chemical po- β 2 C gd˜ τ tential (its self-energy) is given by × ⎝ 3√ e2πiAαβ +iαβ θ + e−2πiAαβ −iαβ θ ⎠ β αβ − s ∞ † 2 π 2 dω U (k)Uλ (k) μ = V (0, 0) = 1λ 1 I 11 ω2 zg˜ −S Ns −∞ 2π + ξ 2 ≡ θ τ ϕ τ τ O(2) , λ=1,2,k∈BZ 4˜g s2 λ (k) d γ ( ) d γ ( ) dAαβ ( ) e qγ (τ ) π 2s Z2 (k) = √ 11 = C s, (48) |ξ |3 2 where the O(2) vison action is given by zNs ∈ 1(k) k BZ β 2 2 2 where Z (k) is given by the matrix product Z μ(k)Zμ (k) SO(2) = S0 + dτ (μV qγ (τ ) + 2πiqγ (τ )ϕγ (τ ) 11 μ 1 1 β γ − and C2 = 0.624. (Note that the self-energy is exact, i.e., 2 unaffected by the above relaxation of the constraint on the β −μI 2 C3ge˜ dτ size of the electric ﬁeld and can be obtained by substituting − iqγ (τ )θ˙γ (τ )) − √ − β s the single instanton solution in Eq. (38) into the action.) The 2 above decomposition separates the UV (self-energy) and IR × θ τ + π τ . (long-range interaction) contributions to the instanton action cos( αβ ( ) 2 Aαβ ( )) (51) and can equivalently be obtained by modifying the original αβ action in Eq. (13) as follows. One imposes a small cutoff Motivated by the success of our previous work [9], where a <

125113-7 M. P. KWASIGROCH PHYSICAL REVIEW B 102, 125113 (2020)

by β. Under RG, βμV grows linearly from its initial bare value of ∼βg˜, whereas K decays exponentially from its initial bare value of ∼βg˜. Eventually, once K has decayed sufficiently, βμV begins to fall linearly. Hence, the visons dissociate before they begin to proliferate. Even though initially the vison self-energy and interaction energy are comparable, at long length scales, the interaction can be neglected by comparison with the self-energy and the visons can be treated as free particles. Classical Monte Carlo calculations of Ref. [26] found that the energy of a nearest-neighbor vison pair is actually lower than that of a single vison, so that at low temperatures, vison pairs are the dominant species and the visons form a weak electrolyte. Here, we argue that the model can be mapped, by integrating out tightly bound vison pairs, to a coarse-grained 3D Coulomb gas where free visons are the dominant species. Note that a finite vison inertia (finite s) should only increase their ability to screen each other, reducing interactions even further. Therefore, if the visons are in the plasma phase for infinite s, we expect them to also be in the plasma phase for finite s.

B. Semiclassical limit at low temperatures We now consider the dynamics of the vison plasma at small nonzero temperatures. At low temperatures the typical vison spacing will be large and their hardcore interactions FIG. 1. The unpolarized scattering intensity at T = 0fors = 1 , negligible. We can therefore treat visons as free bosons of 2 two flavors coupled to the electromagnetic field. Introducing as might be seen in a typical neutron diffraction experiment along + − π τ τ the direction k = 2 (h, h, h). Generation of vison-antivison pairs two complex bosonic fields γ ( ) and γ ( ) corresponding a0 to visons and antivisons respectively, the vison plasma action gives rise to a continuum of excitations above 2μV .Below,we can see the linearly dispersing photon excitation. To simulate the can be written as follows: finite resolution of the measuring apparatus, we have convoluted σ the photon intensity with a Gaussian of width 0.02g. (Note that SVP γ (τ ) the photon and vison spectra have separate intensity scales and β the relative intensity will be very sensitive to the resolution of the 2 σ σ = S0 + dτ ¯ γ (∂τ + 2πiσϕγ + μV )γ measuring apparatus.) − β σ =± 2 γ −μI C3ge˜ σ σ −2πiσ Aαβ vison continuum had been expected in previous works, e.g., − √ ¯ α β e + H.c , (53) Ref. [13]. In this work, our focus will instead be on the effect s αβ of thermally excited visons on the photon part of the scattering spectrum. which is the action of lattice bosons coupled to a gauge field. Notice that the above action is particle conserving, i.e., vison- V. NONZERO TEMPERATURES antivison pair creation and annihilation has been neglected. As demonstrated in Appendix D, these can be treated perturba- A. 3D plasma of visons tively in the large-s limit and give rise to a continuum of exci- μ At nonzero temperatures the visons are not so innocuous. tations above 2 V and a small renormalization of the speed of They become real thermal excitations and form a 3D Coulomb light. Our focus here will be on a more dramatic reshaping of the photon√ spectrum. At sufficiently low temperatures, where gas. The gas is always in the plasma phase [30], however large −μ β I / the vison self-energy μV . To see this, we consider the RG flow ge˜ s 1, [31] the visons move at the bottom of the equations of the gas parameters. In the s →∞limit, where band and a continuum (low k) approximation for the bosonic the visons are static, and at low temperatures, where βμV 1, fields can be made the RG equations are given by [30] β 3 2 d r − σ σ 2 1 SVP[ (r,τ)] = S0 + dτ μV | | dK −1 2 β 3/ = K + y , σ =± − a0 8 dl 2 2 −μI dy 8˜gC3a0e σ σ 2 = 3y − Ky, (52) + √ |∇ − 2πiσ A | dl s −βμ where the fugacity y ≡ e V and K is the coefficient of the σ σ + ¯ (∂τ + 2πiσϕ) , (54) vison Coulomb interaction energy V˜ (r)inEq.(35) multiplied

125113-8 VISON-GENERATED PHOTON MASS IN QUANTUM SPIN … PHYSICAL REVIEW B 102, 125113 (2020)

μ where A(r,τ) [defined by A (r,τ) = A(r,τ) · eμ] and potentials, the single-particle vison action becomes ϕ ,τ (r ) are coarse-grained vector and scalar potentials. Note that we have neglected nonperturbative corrections to the m∗ω2 − = | ω |2 ± πω , −ω · ω μ C2s Ssingle x( ) 2 A( x ) x( ) vison energy V of order e . This is for self-consistency h2 ω =0 2¯ as we have already neglected perturbative corrections which come from higher order terms in the expansion of the Hamil- ± 2πi βϕ(x,ω = 0). (60) tonian in Eq. (12). The vison plasma action describes bosons μ with a thermal√ activation energy of V , effective mass of μ We note here that visons do not couple to the static component ∗ = h¯2 se I π m 2 and an effective charge of 2 h¯. of the gauge field A(x,ω = 0), and hence also the static 16C3a g˜ 0 0 The bosonic excitations can be treated semiclassically at background field Aαβ, be it from the half-integer constraint βμV low temperatures because their typical separation ∼a0e on the magnetic field or from magnetic monopoles introduced 2πh¯2 into the system. This is because we have neglected the Lorentz λ , where λ = ∗ is the thermal de Broglie wavelength B B kTm force experienced by visons. The visons are only coupled of the bosons. The vison excitations are thus not quantum to the electric field in our description which can only be degenerate and can be treated through a single-particle for- induced by a time-varying magnetic field. Of course, visons malism. Considering the Feynman path integral of a single also respond to static magnetic fields. Loops of vison currents vison excitation, we obtain will align in the direction of an applied static magnetic field β ∗ 2 through the action of the Lorentz force. This effect will m 2 = τ ∂τ ± π ,τ · ∂τ ∇× 2 Ssingle d 2 ( x) 2 iA(x ) x generate a ( A) term in the action and only renormalize − β 2¯h 2 the speed of light slightly. The effect we are describing here is far more dramatic. The response of visons to the electromotive ± 2πiϕ(x,τ) , (55) force induced by a time varying magnetic field, as we shall see shortly, generates a mass gap in the photon spectrum. where x(τ ) is the vison’s position vector and ± depends on Integrating over the fluctuations in the particle’s position whether the vison is positively or negatively charged. The x(ω) with ω = 0, we obtain typical quantum fluctuations in the position of the particle 2 will be 2π 2h¯ S = |A(x,ω)|2 ± 2πi βϕ(x,ω = 0). single m∗ √ 2πh¯2β ω =0 |x| ∼ 2πβ|∂τ x| = ≡ λ . (56) m∗ B (61) ,τ At low temperatures, fractional variations in A(x ), or Summing over many-particle configurations, we obtain the ϕ ,τ (x ), due to the fluctuating position of the particle x will partition function of the system be negligible −nβμV −S e |∇A| λB 1 Z = dϕγ (τ ) dAαβ (τ ) e 0 λ ∼ ∼ 1, (57) B ∗ 2 γ,αβ n! |A| λA 2πβm c n n 3 2π2h¯2 2 where λA = 2πhc¯ β is the typical wavelength of the gauge d xi − ∗ |A(x ,ω)| √ × e m ω =0 i = 2˜g za0 λ3 field and c sh¯ is the speed of the photon excitation. The i=1 B √ √ temperatures here are small relative to the vison rest energy π βϕ , − π βϕ , ∗ μ × 2 i (xi 0) + 2 i (xi 0) m c2 ∼ ge˜ I . At these temperatures, the visons are moving e e ∗ − 1 = β 2 3 nonrelativistically with a typical speed v ( m ) c.We d x −βμ = dϕγ (τ ) dAαβ (τ )exp − S + e V can therefore approximate 0 λ3 γ,αβ B A(x,τ) ≈ A(x,τ), − 2π2h¯2 | ,ω |2 × m∗ ω =0 A(x ) π βϕ , . ϕ(x,τ) ≈ ϕ(x,τ), (58) e 2 cos(2 (x 0)) (62) where x is the average position of the vison. With regards −βμ V to vison dynamics, we have neglected the Lorentz force At low temperatures, we are in the Debye limit e 1, experienced by visons by making the above approximation, where the action can be expanded to quadratic order. One can only leaving the electric field force. This is of course a see that the gauge fields are effectively small in this limit by good approximation at nonrelativistic speeds. The particle’s performing the following rescaling: position can be decomposed as βμ / → V 3 , x e x 1 ωτ x(τ ) =x+√ x(ω)ei , (59) −βμ / β ω → e V 3ω, ω =0 −βμ / ,ω → V 6 ,ω , ω = 2πn A(x ) e A(x ) where the Matsubara frequency β and n is an integer. −βμ / Using the above approximation for the vector and scalar ϕ(x,ω) → e V 6ϕ(x,ω). (63)

125113-9 M. P. KWASIGROCH PHYSICAL REVIEW B 102, 125113 (2020)

C. Inelastic magnetic response We can now extract several correlators from the above quadratic gauge action. In particular, relevant for neutron scattering experiments are magnetic field correlators: z 2 χλ(k, iω) ≡ Sλ(k,ω) 2 2˜gξλ (k) = + δω, Cλ(k), (67) − ω 2 + 2 0 (i ) Ek √ ξ 2 = 2 ξ 2 / 2 + ω2 = 2˜g λ (k) − where Ek 4˜g z 1 (k) s p and Cλ(k) 2−ω2 Ek p 2 2˜gξλ (k) 2 . (Notice that we have replaced the low-k expansion Ek with ξλ(k) to restore periodicity across the Brillouin zone.) FIG. 2. Plasma frequency as a function of temperature. At higher temperatures the plasma frequency will level off as the vison density saturates. 1. Equal-time structure factor A particularly useful quantity to measure in neutron To quadratic order, the gauge part of the action is given by diffraction experiments is the equal-time (energy integrated) structure factor. We calculate the structure factor in the spin- 2| |2 ω2 ω2 flip channel for a polarized neutron-scattering experiment gza˜ 0 k p S = + + (1 − δω, ) considered by Ref. [33], gauge 2 0 ω,| |<,λ= , s 4˜g 4˜g k 1 2 yy z z 3 S (k) ≡ S (k,τ = 0)S (−k,τ = 0) (eˆμ · eˆ )(eˆν · eˆ ) 2 d k 1 μ ν y y × |Aλ(k,ω)| + π 3 μ,ν ω |k|< (2 ) 4˜ga0 − 1 † × | |2 + λ 2δ |ϕ ,ω |2, = χλ(k, iω)(eˆμ · eˆy )(eˆν · eˆy )Uμλ(k) k D ω,0 (k ) (64) β μ,ν ω,λ=1,2 where the Debye length is given by † ×Uνλ(−k) − −βμ − 1 2 λ = π 2 λ 3β V 2 , = g˜ Zμν (k)(eˆμ · eˆ )(eˆν · eˆ ) D 16 a0 B ge˜ (65) y y μν βμ and in the Debye limit e V 1, there are many visons 1 + 2n (E ) TC (k) −βμ βμ / B k 1 V λ /λ 3 ∼ V 2 × + , (68) e ( D B ) e 1 in the Debye volume. The E g˜ξ 2(k) plasma frequency (multiplied byh ¯) is given by k 1 = k×(1,−1,0) = 1 2 where eˆ , n (E ) βE , and Zμν (k)isgiven 2 2 −βμV y |k×(1,−1,0)| B k e k −1 16˜gπ h¯ a0e ω = , (66) by the matrix product Zμσ (k)Zσν(k). p λ3 m∗ σ B Because the mass gap ωp does not couple to the zero Matsubara frequency component, it can only have an effect and agrees with the continuum derivation presented in on the equal-time structure factor at low temperatures, but this Appendix G. The vison dynamics has generated a dynamical is precisely where it is exponentially damped. The effects of mass gap for the gauge field, equal to the plasma frequency. the mass gap on the equal-time structure factor are therefore Figure 2 shows the behavior of the plasma frequency with small across the temperature range and cannot be easily seen temperature. The mass gap is dynamical because it does not as shown by Figs. 3 and 4. Motivated by the success of our couple to the static ω = 0 component of the gauge field. We previous work [9], we extrapolated the large-s results to s = 1 . have derived the mass gap√ in the large-s and low-temperature 2 β / βμ Both figures are in good agreement with the results of Monte limits, where√s 1, g˜ z s 1 (which implies V 1), −μ Carlo simulations of Ref. [14]. and βge˜ I / s 1, i.e., the temperature is small by comparison with the photon and vison bandwidths respectively. However, we expect the above picture to extend to mod- 2. The dynamical structure factor erate temperatures too, although the instanton energy, and We should instead look for experimental signatures of the hence the mass gap, would acquire significant renormalization mass gap in the unintegrated neutron scattering spectra. We due to finite-size effects along imaginary time. Note that we do this by calculating the dynamical structure factor have also not included tightly bound vison pairs in our semiclassical analysis of vison dynamics since their oscillations do z z iωt not contribute to the generation of the mass gap [32] but only Sλ(k,ω) = dt Sˆλ(k, t )Sˆλ(−k, 0) e . (69) serve to renormalize the speed of light (see Appendix D for a detailed calculation of this effect).

125113-10 VISON-GENERATED PHOTON MASS IN QUANTUM SPIN … PHYSICAL REVIEW B 102, 125113 (2020)

FIG. 3. Equal-time structure factor in the spin-ﬂip channel FIG. 4. Equal-time structure factor in the spin-ﬂip channel along π 2π 1 = 2 , , = . = 1 along k = (h, h, l)atT = 0.15g,fors = and with the vison k a (h h l)atT 0 15g,fors 2 and without the vison con- a0 2 0 ω contribution. tribution (i.e., with p set to zero). Figures 3 and 4 look identical to the eye and there are no clear signatures of visons.

We can analytically continue the analytic part of χ(k, iω)to μν ,ω = ,ω † † − obtain the spectral density where S (k ) λ=1,2 Sλ(k )Uμλ(k)Uνλ( k). The results are plotted in Figs. 5 and 6 and show a clear development ρλ(k,ω) = Im lim χλ(k,ω+ i ) →0 of a mass gap at nonzero temperatures, that are relatively small with respect to the photon bandwidth for s = 1 . πξ2 2 g˜ λ (k) We compare our dynamical structure factor plots with the = (δ(ω − Ek ) − δ(ω + Ek )). (70) Ek quantum Monte Carlo calculations of Ref. [20]. The reported . The structure factor is then given by bandwidth (4 28g) is significantly higher than the one we calculate (0.6g). However, we believe this discrepancy is 2ρλ(k,ω) mostly due to higher order corrections in the 1/s expansion. Sλ(k,ω) = 2πTCλ(k)δ(ω) + 1 − e−βω In fact, our previous work [9] shows that the next order . . 2 already renormalizes the bandwidth from 0 6g to 1 6g. Fur- 2˜gπξλ (k) = (nB(Ek )δ(ω + Ek ) + (1 + nB(Ek )) ther, the plasma frequency and the speed of light should not Ek be compared numerically in our calculation. This is because × δ(ω − Ek )). + 2πTCλ(k)δ(ω). (71) the calculation of the former is nonperturbative and includes all orders in 1/s, whereas the speed of light has only been See Ref. [15] for further details of the relations between calculated to finite order in 1/s. This does not impact the main χ ,ω , ,ω ρ ,ω λ(k ) Sλ(k ), and λ(k ). The presence of the delta observation though, which is that the photon acquires a mass function in the dynamical structure factor signals that corre- gap equal to the plasma frequency. Our dynamical structure lations of the magnetic field are infinitely long-lived in time. plots are in rough qualitative agreement with the work of This is because of persistent vison currents that are excited Ref. [20]. However, the quantum Monte Carlo calculations (through electromagnetic induction) with any fluctuation of do not have the required resolution to ascertain whether the the magnetic field. These currents and the magnetic field they photon dispersion is linear, let alone whether there is a small give rise to do not decay with time. energy gap. The total unpolarized scattering intensity is proportional to [14] D. Susceptibility (k · eˆμ)(k · eˆν ) μν I(k,ω) = eˆμ · eˆν − S (k,ω), (72) As discussed previously in this section, the mass gap is 2 μν k a consequence of the induced electric field coupling to the

125113-11 M. P. KWASIGROCH PHYSICAL REVIEW B 102, 125113 (2020)

= = 1 FIG. 6. The unpolarized scattering intensity at T = 0.15g for FIG. 5. The unpolarized scattering intensity at T 0fors 2 with the vison contribution, as might be seen in a typical neu- s = 1 with the vison contribution, as might be seen in a typical 2 π tron diffraction experiment along the direction k = 2π (h, h, h). To neutron diffraction experiment along the direction k = 2 (h, h, h). a0 a0 simulate the finite resolution of the measuring apparatus, we have Notice the development of a sizable gap equal to ωp at temperatures convoluted the intensity with a Gaussian of width 0.01g. that are relatively small by comparison with the photon bandwidth. To simulate the finite resolution of the measuring apparatus, we have convoluted the intensity with a Gaussian of width 0.01g. visons. Hence, it only shows up in the presence of time- varying magnetic fields. To demonstrate this, we calculate ization of the speed of the photon excitation and only change here the static isothermal and isolated susceptibilities using the value of the above constant.) the formulas derived in Ref. [15]. For the isothermal sus- For the isolated susceptibility, we assume the sys- ceptibility, we assume the system to be in thermal equilib- tem to be isolated at all times and in thermal equilibrium at temperature T and in the presence of the follow- rium to begin with i.e., at t →−∞.Atime-dependent δ ˆ = ing time-independent perturbation in the Hamiltonian H perturbation is then slowly turned on δHˆ (t ) =−h (1 − − h (Sˆz (k) + Sˆz (−k)). The isothermal susceptibility is then 2 2 λ λ (t ))e t (Sˆz (k) + Sˆz (−k)), where (t ) is the Heaviside step defined as λ λ function and → 0+. The isolated susceptibility is then de- z z fined as 1 δ Sˆλ(k) + Sˆλ(−k) χ T ≡ T λ (k) δ ˆz + ˆz − ω ρ ,ω 2 δh I 1 Sλ(k) Sλ( k) I d λ(k ) χλ(k) ≡ = ω ρλ ,ω β λ 2 δh 2π ω = d (k ) + C (k) 2π ω 2 ξ 2 = g˜ λ (k) = 1 1 λ = , ω for 1 and 2 gξ 2 k E 2 4gs4 + pa0 2 ˜ λ ( ) 1 k 1 ξ = = for λ = 1 and 2, (73) hc¯ 1 (k) E 2 − ω2 4gs4z k p (74) where the expectation value is taken with respect to the where the expectation value is evaluated at t = 0forthe thermal ensemble. We can see that the isothermal transverse isolated system. We can see that the isolated transverse (λ = (λ = 1 and 2) susceptibility is a constant, and as expected, is 1 and 2) susceptibility is no longer a constant but begins to unaffected by visons because the applied magnetic field is fall to zero at small |k|∼ωp/(¯hc) (see Fig. 7). This is due to time-independent. We would obtain the same expression for the screening contribution of persistent vison currents that are a massless photon. (Note that including in our description set up as the external field is switched on, and is analogous the effects of the Lorentz force on visons, from the time- to the Meissner effect in superconductors. Following on from independent magnetic field, would simply lead to a renormal- Eq. (G3), where E = 0 once equilibrium is reached, we have

125113-12 VISON-GENERATED PHOTON MASS IN QUANTUM SPIN … PHYSICAL REVIEW B 102, 125113 (2020)

2 photon heat capacity with visons photon heat capacity without visons vison heat capacity 1.5 ) 3

γΤ 1 C/(

0.5

0 0 0.1 0.2 T/g FIG. 7. The transverse isothermal (χT ) and isolated (χI ) susceptibilities along k = 2π (h, h, h)atT = 0.15g. The isolated suscepti- a0 FIG. 8. The photonic contribution to the heat capacity per unit bility deviates from the isothermal one and decays to zero on length volume with and without visons. Note that because the vison- ∼ω / scales p (¯hc). This is because of persistent vison currents that are generated photon mass is suppressed exponentially at low temper- set up as the external magnetic field is switched on, whose magnetic atures, it has no effect on the γ T 3 behavior of the heat capacity fields oppose the applied field. γ = 4π2 in this region [ 15(ch¯)3 ]. By equipartition theorem, the photon −3 heat capacity tends to 8a0 at high temperatures regardless of the the following screening equation presence of the mass gap. Hence, the effects of the mass gap are ω2 most pronounced at intermediate temperatures. ∇2A = p A, (75) c2h¯2 of the equal-time structure factor, the dynamical mass gap where ch¯/ω is the characteristic screening length. Note p has a small effect on the heat capacity. Because the visons that, in contrast to superconductors, this corresponds to the are thermally activated, they are exponentially suppressed at screening of the ’magnetic field’ defined as B ≡ Sz , not the ij ij small temperatures and do not affect the low-temperature T 3 physical applied field, which is not eliminated. behavior of the photonic heat capacity. At high temperatures, Measuring the isolated susceptibility thus provides a pos- the mass gap does not affect the equipartition limit of the sible experimental verification of the presence of dynamical photon heat capacity, i.e., k per mode. There is only a visons in the system. In particular, if a uniform field h along B small difference between the photonic heat capacity with the global x direction is applied, the perturbation takes the and without visons at intermediate temperatures. Figure 8 form demonstrates this (see Appendix E for the derivation of the z δHˆ =−hη Sˆμ(r + eμ/2)(eˆμ · eˆx ) heat capacity). The contribution of the vison gas to the heat r,μ capacity is also included for comparison. √ η 2h Ns z z = Sλ= (k = 0) − Sλ= (k = 0) , (76) F. Confinement of spinons 3 1 2 where η is the individual spin’s magnetic moment. The result- We note that the dynamical mass gap generated by ing magnetization per lattice site, along the global x direction, the vison plasma does not cause confinement of mag- is given by netic monopoles (spinons). Introducing a pair of magnetic η monopoles into the system corresponds to choosing an appro- z 0 M ≡ Sˆμ(r + eμ/2) (eˆμ · eˆ ) priate static background field Aλ(k), which only couples to the x 4N x s r,μ zero Matsubara frequency component of the dynamical gauge ⎧ field Aλ(k,ω = 0), which is not gapped. Hence, the vison ⎨ 2η2hχ I (0) 1 = 0 isolated, plasma does not alter the Coulomb interaction between static = 9 ⎩ 2η2hχ T (0) η2h magnetic monopoles introduced into the system. We refer the 1 = isothermal. 9 18gs4z reader to Appendix F for a detailed mechanism of how a static Thus visons lead to the vanishing of the uniform isolated sus- mass gap, e.g., one generated by the condensation of visons in ceptibility. In fact, it is a straightforward generalization of the the ground state, causes confinement of magnetic monopoles. above to show that visons cause the uniform ac susceptibility of an isolated system to vanish as long as the frequency of the VI. CONCLUSION applied field is different from the plasma frequency. We have described some of the possible experimental signatures of gapped excitations of QSI, known as visons. E. Heat capacity Because visons are sources of emergent electric field, they We have also investigated the effect of the mass gap on are not easily accessible to experimental probes. Perhaps, the photon contribution to the heat capacity. As in the case magnetostriction or Dzyaloshinskii-Moriya effects can couple

125113-13 M. P. KWASIGROCH PHYSICAL REVIEW B 102, 125113 (2020) real electric fields to the emergent electric field of the visons and the destruction of the Coulombic quantum spin liquid. [26,34–36], in which case visons can leave a signature in Our large-s description predicts this to happen as soon as the electric response measurements. In this paper, we have instead temperature becomes nonzero due to the development of a focused on how the dynamics of the vison’s emergent electric finite, albeit exponentially small, density of visons. There is field couple to the dynamics of the physical magnetic field, thus a smooth cross-over between the Coulombic quantum via electromagnetic induction. This coupling leaves signatures spin liquid at zero temperature and the paramagnetic phase at in neutron scattering experiments, which probe the magnetic higher temperatures, i.e., the above spinon confinement length field dynamics. scale and timescale diverge exponentially as we approach zero We have been able to capture the dynamics of the visons temperature. This is in contrast to the gauge mean-field theory analytically and quantitatively for the first time by study- analysis of Ref. [23], which predicts a first order transition ing the ring-exchange Hamiltonian in the large-s limit. We between these two phases, even in the perturbative Jz J± extended the perturbative analysis of our previous work [9] limit that we have considered in this work. by including nonperturbative corrections that correspond to the tunneling of visons between lattice sites. These quantum ACKNOWLEDGMENTS corrections also go beyond the recent classical description of visons in Ref. [26] and endow them with a finite mass This work was supported in part by UKRI Grant Nos. (exponentially small in s), which we have calculated. EP/K028960/1 and EP/P013449/1, and by the UKRI Net- In agreement with previous works, we have shown that workPlus on “Emergence and Physics far from Equilibrium.” visons give rise to a continuum of excitations above the We gratefully acknowledge discussions with C. Castelnovo, ground state and renormalize the speed of light. Both were A. Szabó, and G. Goldstein. calculated in the large-s limit. What is perhaps more novel, is that in the large-s description, as soon as the temperature APPENDIX A: THE LONGITUDINAL AND TRANSVERSE becomes nonzero, the visons have a dramatic effect on the PARTS OF THE INSTANTON SOLUTION IN THE photon spectrum. The visons form a dilute 3D Coulomb gas, CONTINUUM LIMIT which, as is well known, is always in the plasma phase, i.e., the visons are deconfined. We have investigated how this It is very insightful to consider the continuum limit of plasma interacts with the long-wavelength degrees of freedom the instanton solution in Eq. (38). We first write down the of the gauge field. We have found that in the Debye limit, time-dependent part of the instanton solution as a sum of where a quadratic description is viable, plasma oscillations longitudinal and transverse parts inst. const. tran. long. introduce a dynamical mass gap in the photon spectrum, Eμ (r,τ) − Eμ (r) = Eμ (r,τ) + Eμ (r,τ), (A1) which follows an Arrhenius law at low temperatures. This tran. long. mass gap should be observable in inelastic magnetic response where by definition divαE (τ ) = 0 and curlijE (τ ) = 0 measurements. In particular, it should show up in energy- everywhere. From the solution in Eq. (38), it follows that resolved neutron-scattering experiments and we have calcu- 1 ∞ −iωe−iωτ+ik·r lated the scattering intensity as would be seen in a typical E tran.(r,τ) = dω μ 2zg˜2 ω2 Ns −∞ ξ 2 + experiment. We have compared our scattering intensity results λ=1,2,k∈BZ s2 λ (k) 2 with the recent quantum Monte Carlo (QMC) simulations of × † Ref. [20]. These are in rough agreement, although, because of Uμλ(k)Uλσ (k) the limited resolution of QMC, the linear photon dispersion | | 3 3 ∞ −iωτ+ik·r k a0 1 a d k −iωe cannot be resolved, let alone a potential small gap in the ≈ 0 dω 3 2zg˜2a2 ω2 4(2π ) 0 0 | |2 + spectrum. s2 k 2 We also show that the photon mass gap generated by visons 3 does not couple to the zero Matsubara frequency component × [eˆσ · eˆμ − (kˆ · eˆσ )(kˆ · eˆμ)], (A2) of the gauge field, and hence does not result in confinement 4 of static magnetic monopoles (spinons) introduced into the where we have used the identity system. However, it leads to the vanishing of the uniform † 1 † 2 susceptibility of an isolated system through a mechanism U (k)Uλσ (k) = U (k)ξλ (k)Uλσ (k) μλ ξ 2(k) μλ analogous to the Meissner effect. In fact, visons will lead to λ=1,2 1 λ the vanishing of the uniform ac susceptibility as the long as 1 the system is isolated and the frequency of the applied field is = Zμν (k)Zνσ (k) ξ 2 different from the plasma frequency. 1 (k) ν An interesting question to consider is the effect of the ka01 −4 photon mass gap on dynamical spinons. We leave this to be ≈ (k · μν )(k · νσ ) |k|2a2 fully addressed by future works, but note here that induced 0 ν vison currents are expected to lead to the decay of coherent 3 = ((eˆμ · eˆσ ) − (kˆ · eˆμ)(kˆ · eˆσ )). spinon excitations by Lenz’s law. The induced currents will 4 also cause the Coulombic magnetic field of a spinon, which (A3) is propagating nonrelativistically and at constant speed, to form a flux tube of radius (ch¯)/ωp on a timescale ofh ¯/ωp. The transverse part of the instanton solution is transient in Visons will thus lead to confinement of propagating spinons time with a power-law tail ∝ 1/|τ|3 at long times.

125113-14 VISON-GENERATED PHOTON MASS IN QUANTUM SPIN … PHYSICAL REVIEW B 102, 125113 (2020) √ The longitudinal part of the electric field can also be Taking the low temperature limit βg˜ z/s →∞, we obtain extracted from the solution in Eq. (38) ∞ ω π inst. d d Uλσ (k) −iωτ Eλ (k,τ) = √ e . π · 2zg˜2 ω2 long ik r † dτ −∞ 2π Ns ξ 2 + Eμ (r,τ) =− sgn(τ ) e Uμλ(k)Uλσ (k) s2 λ (k) 2 Ns √ λ=3,4,k∈BZ |ξ | sπUλσ (k) d −|τ| 2˜g z λ(k) = √ e s | | 3 3 k a0 1 3π a d k 2˜g N zξλ(k) dτ ≈ τ 0 eik·r s sgn( ) 3 √ 4 4(2π ) πsgn(τ )Uλσ (k) −|τ| 2˜g z|ξλ(k)| =− √ e s . (B2) 1 Ns × + (kˆ · eˆσ )(kˆ · eˆμ) 3 The smooth instanton solution in Eq. (42) is given by π 3 3 a0 = τ δ + ,τ · μ, √ sgn( ) (r) AE(r ) eˆ (A4) π |ξ | 16 sm. Uλσ (k) −|τ| 2˜g z λ(k) Eλ (k,τ) = √ [−sgn(τ )e s + sgn(τ ) + 1], √ Ns = 3 2 where the factor of A 4 a0 links the lattice electric flux (B3) Eμ(r,τ) with the continuum flux density E(r,τ) and follows from approximating a sum over a large closed surface S by a and we can see that the jump in the field has been removed. → dS ,τ surface integral (r,μ)∈S S A . E(r ) is the continuum Straightforward differentiation gives electric field due to a dipole at r = 0 with a dipole moment of √ π τ π τ 2ξ 2 |ξ | sgn( )eσ sm. sgn( )Uλσ (k) 4zg˜ λ (k) −|τ| 2˜g z λ(k) E¨ (k,τ) =− √ e s . (B4) λ 2 Ns s 3 · d k k eσ ik·r E(r,τ) =−πisgn(τ )∇ e The smooth instanton solution is the stationary solution of (2π )3 |k|2 the action in Eq. (41). Minimising that action with respect 1 to variations in the dynamical, transverse part of the field = πsgn(τ )∇ eσ ·∇ . (A5) |r| Eλ=1,2(−k,τ) we obtain the following saddle point equations

− 2 δ ,τ We have also used the identity 2s Eμ(r ) E¨λ(k,τ) + g˜ V [Eμ(r,τ)] = 0, 4zg˜ξ 2(k) δEλ(−k,τ) λ r,μ † = δ − † Uμλ(k)Uλσ (k) μσ Uμλ(k)Uλσ (k) (B5) λ=3,4 λ=1,2 | | λ = k a0 1 3 1 for all k and 1 and 2, where ≈ + (kˆ · eˆσ )(kˆ · eˆμ) . (A6) 4 3 δ ,τ Eμ(r ) 1 † −ik·r = √ Uμλ(−k)e (B6) δEλ(−k,τ) N The longitudinal part of the electric field is uniquely de- s termined by the charge distribution via Laplace’s equation and long. in Eq. (30). Because Eμ (r,τ) has a lattice divergence −π τ − μ/ π τ μ/ of sgn( )at e 2 and sgn( )ate 2, it necessarily V [Eμ(r,τ)] = 2Eμ(r,τ) − 2πδμσ δr,0[sgn(τ ) + 1]. (B7) corresponds to the electric field due to a pair of charges π and −π at positions −eμ/2 and eμ/2 respectively for τ<0 The first derivative of the continued parabolic potential τ> that switch positions for 0. We therefore obtain a dipole V [Eμ(r,τ)] is given by 2Eμ(r,τ), when |Eμ(r,τ)| π.This field as above in the continuum limit. It also follows that, is true in the case of the above smooth instanton solution π τ = sm. to describe the hopping of a vison of charge 2 at 0, Eμ (r,τ), everywhere except for the single electric field const. ,τ the constant background electric field Eμ (r ) has to be Eμ=σ (r = 0,τ), which lies between π and 3π for positive π the longitudinal field due to a pair of charges at positions imaginary times. In this case V [Eμ(r,τ)] = 2Eμ(r,τ) − 4π, ±eμ/2. thus justifying the above expression. It follows that the smooth sm. instanton solution Eμ (r,τ) satisfies the saddle point equa- tioninEq.(B5) APPENDIX B: THE SMOOTH INSTANTON SOLUTION s2 We begin with the instanton solution in Eq. (38) [in the fol- =− ¨ sm. ,τ + sm. ,τ 0 2Eλ (k ) 2 2˜gEλ (k ) lowing, the instanton solution is given relative to the constant 4zg˜ξλ (k) and divergenceful background field E const.(k)] λ πg − √2 ˜ τ + . Uλσ (k)[sgn( ) 1] (B8) Ns d 1 πUλσ (k) E inst. k,τ = √ e−iωτ . λ ( ) 2 ω2 (B1) sm. dτ β 2zg˜ ξ 2 + We now turn to proving that E˙λ (k,τ) is the zero-energy Ns ω 2 λ (k) s 2 mode of the action in Eq. (43) describing fluctuations around

125113-15 M. P. KWASIGROCH PHYSICAL REVIEW B 102, 125113 (2020) the stationary smooth instanton solution The instanton measure can be obtained by working out β/2 2 2 the contribution of the one-instanton sector to the partition s |δE˙λ(k,τ)| δS [δEλ(k,τ)] = dτ function relative to the zero-instanton sector tran 2 −β/ zgξ k 1 2 2 ∈ ,λ= , 4 ˜ λ ( ) − = λ˜ iξ˜ ˜ i|˜ i k BZ 1 2 dξ˜i ˜ i|˜ ie 2 i 0 i i √ β/2 − 1 λ ξ 2 | g˜ . dξ | e 2 i i i i i + dτ V E sm (τ ) δE 2 (τ ). i i i i αβ αβ −β/2 2 αβ ˜ 0|˜ 0 det K = dξ˜0 , (C5) (B9) 2π detK˜ The zero-mode of the above action satisfies the following δ ,τ = ξ ,τ = where we have written down E(r ) i i i(r ) differential equation for λ = 1 and 2: ξ˜ ˜ ,τ ˜ i i i(r ) in terms of the real eigenvectors of K and K s2 respectively in the (r,τ) basis and the zero eigenvalue is 0 =− δE¨λ(k,τ) 2 excluded from the determinant of K˜ .From 4zg˜ξλ (k) − δ ,τ K˜ = K(1 − λK 1vv†), (C6) Eμ(r ) g˜ sm. + V E τ δEαβ τ . δ − ,τ αβ ( ) ( ) (B10) Eλ( k ) 2 αβ it follows that det K˜ sm. −1 † We see that δEαβ (τ ) = E˙ (τ ) is a solution of the equation, = det(1 − λK vv ). (C7) αβ det(K) because 2 + λ −1 † − s ... . δEμ(r,τ) g˜ The matrix (1 K vv ) has N 1 eigenvectors perpen- − sm + sm. τ ˙ sm. τ − E λ (k) V Eαβ ( ) Eαβ ( ) dicular to v with eigenvalue 1 and an eigenvector K 1v with 4zg˜ξ 2(k) δEλ(−k,τ) 2 λ αβ eigenvalue equal to (1 − λv†K−1v). Therefore (B11) det K˜ = (1 − λv†K−1v). (C8) is proportional to the time derivative of the left-hand side of det(K) Eq. (B5). When λ takes on its physical value given by Eq. (C4) and − −1 APPENDIX C: INSTANTON MEASURE equal to (v†K 1v) , det K˜ and the above ratio vanish. To avoid this, we will perturb λ from its physical value as follows The fluctuation action in Eq. (43) can be written as follows: 1 2ω2 λ = − δλ. (C9) s 2 † −1 δ = + |δ λ ,ω | v K v S g˜ 2 E (k ) 4zg˜ξλ (k) ω k,λ=1,2 The matrix K˜ changes as a result by δK˜ = δλvv†, and stan- 2πg˜ dard perturbation theory gives the shift of the zero eigenvalue − δE 2(0, 0), (C1) sm. σ of K˜ E˙σ (0, 0) −1 |δ ˜ | −1 † −1 2δλ δλ˜ = K v K K v = (v K v) . where √ 0 − − − (C10) π K 1v|K 1v v†K 2v sm. 2 z g˜ † E˙ (0, 0) = U (k)|ξλ(k)|Uλσ (k) σ sN σλ With the perturbed λ, we can now evaluate s k,λ ˜ ˜ † −1 † −2 √ det K det K v K vδλ v K v π σμ μσ = = = , (C11) 2 z g˜ Z (k)Z (k) † −1 = det(K) δλ˜ 0det(K) δλ˜ 0 v K v sNs |ξ1(k)| k∈B.Z.,μ and then take the limit δλ → 0, where ⎛ ⎞ C2g˜ 2 ≡ 3 , (C2) † −2 1 † ⎝ 1 ⎠ s v K v = U (k) Uλσ (k) β σλ + ω2s2 = . Ns ω ,λ g˜ 2 and C3 5 13. It is convenient to express the fluctuation k 4zg˜ξλ (k) action in matrix form 2 C3 δS = E†KE˜ = E†(K − λvv†)E, (C3) = , (C12) 4πgs˜ √ where and we have summed over ω in the βg˜ z/s →∞limit and [E]λωk = δEλ(ω, k), used the identity quoted in Eq. (C2). 1 We also calculate the norm of the zero mode ∞ [v]λωk = √ Uλσ (k), β sm. 2 Ns ˜ 0|˜ 0= |E˙λ (τ,k)| dτ −∞ s2ω2 k,λ [K]λω ,λω = δλω ,λω g˜ + , √ k k k k 2 † 2 4zg˜ξ (k) π 2 U ξ k Uλσ k λ = 2 zg˜ σλ λ ( ) ( ) 2πs 1 Nss |ξ1(k)| λ = = , (C4) k,λ C2 v†K−1v 3 C2πg˜ = 3 , (C13) where the final identity follows from Eq. (C2). s

125113-16 VISON-GENERATED PHOTON MASS IN QUANTUM SPIN … PHYSICAL REVIEW B 102, 125113 (2020) where we have again used the identity quoted in Eq. (C2). 4π 2w2 = ω2| ,ω |2 ω . μ Aλ(k ) for g˜ 1 Inserting everything into Eq. (C5), the instanton measure μ3 e2 I becomes V ω,k,λ=1,2 (D2) C gd˜ ξ˜ 3√ 0 , (C14) We can see that virtual vison pairs lead to a small renormal- s 2 ization of the ω coefficient in the gauge field action S0, and hence a small renormalization of the speed of light. We can ξ where 0 specifies the position of the instanton in imaginary also see that vison pair loops introduce a pole in χλ(k,ω) time. at ω = 2μV corresponding to the energy of creation of a single static vison-antivison pair. To obtain a continuum of APPENDIX D: CONTRIBUTION OF VISON PAIRS excitations above this threshold, we need to consider vison We will look at the contribution of vison pairs to the Sz pair loops accompanied by more than two instantons, i.e., correlator in the ground state. the visons execute hops before annihilating. It will be more convenient to do this using the O(2) rotor description of visons given in Eq. (51). 1. Renormalization of the speed of light In the large-s limit, virtual vison pairs will be far away 2. Continuum of excitations above the ground state from each other because each vison pair loop is accompanied There will be a continuum of excitations at energies above by at least two instantons and will enter the partition function twice the vison energy 2μV . In the large-s limit, the ground −2μ −2C s with a weight at least as small as ∼e I = e 2 . We can state of the O(2) action in Eq. (51) is characterized by therefore neglect the hardcore interactions between virtual qγ (τ ) = 0 everywhere with small virtual vison fluctuations as vison pairs. Following the action in Eq. (49), the partition a perturbation. In the same limit, the multi-particle excited function of a dilute gas of such loops can be written as states are described by the action in Eq. (53), as long as the excitations are dilute. The excited states consist of bosons of two −S 1 flavors (corresponding to visons and antivisons). For small Z = dAαβ (τ ) e 0 n! momenta and gauge fields, these excitations are described by n ⎛ ⎞ the action in Eq. (54). The two-particle excited states are then n 2 τ T T connected to the ground state via the following perturbation, ⎝ w dTd 2πiAαβ (τ− )−2πiAαβ (τ+ ) ⎠ × e 2 2 + c.c. , which creates or annihilates vison pairs μV T 2μI αβ e e δ = π 3 τ − · ∇++ ¯ + · ∇¯ −+ . . , (D1) S± 2 v d r d (i A i A H c ) 3 3 where n is the number of loops considered, T is the length −2πv d k1d k2 + − = √ ( (k ,ω ) (k ,ω ) τ β π 6 1 1 2 2 of the loop in imaginary time, the position of its center and ω ,ω (2 ) = C√3g˜ 1 2 w s the instanton measure. As we are focusing on the ef- τ fect on the Aαβ ( ) gauge field only, we have omitted the scalar × (k1 − k2 ) · A(−k1 − k2, −ω1 − ω2 ) + c.c.)(D3) field ϕγ (τ ) from the above partition function. The contribution = 4¯h2 of vison pairs to the gauge field action, at quadratic order, can where v m∗a3 . The above perturbation gives a correction 0 thus be written as to the gauge field correlator Aλ(k,ω)Aλ (k ,ω ). The zeroth order correlator can be found using the continuum limit of the 2 2 4π w −μ action S0 given in Appendix G δ = τ V T Spairs μ dT d e e2 I 3 3 αβ 2˜g(2π ) δ (k + k )δω,−ω δλλ Aλ(k,ω)Aλ (k ,ω ) = . 0 ω2 + 2 2 2 T T 2 c h¯ k × Aαβ τ + − Aαβ τ − (D4) 2 2 π 2 2 ω2| ,ω |2 Standard perturbation theory then gives the following lowest = 16 w Aλ(k ) μ order correction to the gauge field correlator e2 I 4μ2 + ω2 ω,k,λ=1,2 V

2 c δ ± λ ,ω λ ,ω ( S ) A (k )A (k ) 0 δAλ(k,ω)Aλ (k ,ω )= 2 4πv2 d3k d3k d3q d3q = 1 2 1 2 ¯ +(q , )+(k ,ω ) ¯ −(q , )−(k ,ω ) β (2π )12 1 1 1 1 0 2 2 2 2 0 ω1,ω2,1,2 × ,ω ,ω − · − − , −ω − ω − · + , + c Aλ(k )Aλ (k )(k1 k2 ) A( k1 k2 1 2 )(q1 q2 ) A(q1 q2 1 2 ) 0

125113-17 M. P. KWASIGROCH PHYSICAL REVIEW B 102, 125113 (2020) π 2 3 3 3/ 3/ 4 v d k1d k2 a0 8 a0 8 = Aλ(k,ω)Aλ (k ,ω )(k − k ) 6 1 2 β (2π ) −iω1 + (k1) −iω2 + (k2 ) ω1,ω2 c · A(−k1 − k2, −ω1 − ω2 )(k1 − k2 ) · A(k1 + k2,ω1 + ω2 ) 0 4πv2 d3k d3k a3/8 a3/8 = 1 2 0 0 6 β (2π ) −iω1 + (k1) iω1 − iω2 + (k2 − k1) ω1,ω2 × ,ω ,ω − · − , −ω − · ,ω c , Aλ(k )Aλ (k )(2k1 k2 ) A( k2 2 )(2k1 k2 ) A(k2 2 ) 0 (D5) where as usual only connected diagrams are included, the APPENDIX E: HEAT CAPACITY → − ω → ω − ω transformation k2 k2 k1, 2 2 1 was performed ,ω h¯2k2 Integrating out gauge fields Aλ=1,2(k ) from the action in the final step and (k) = μ + ∗ is the vison en- V 2m in Eq. (64) gives us the photon contribution to the partition · ,ω = ergy. In the Coulomb√ gauge k2 A(k2 2 ) 0, and we function can write A(k ,ω ) = a Aλ(k ,ω )nˆ λ(k ), where √ 2 2 0 λ=1,2 2 2 2 π π/β nˆ λ(k) · nˆ λ (k) = δλλ and nˆ λ(k) · k = 0. This allows us to ZP = (E1) ω2 rewrite the correction to the gauge field correlator as g˜β gz˜ p ω2 ω,k,λ=1,2 |ξλ |2 + − δω, + s2 (k) 4˜g (1 0 ) 4˜g δAλ(k,ω)Aλ (k ,ω ) where the factor of π is a result of previously integrat- 16πv2 d3k d3k g˜β = 1 2 ,ω β π 6 ing out the Eλ=1,2(k ) fields from the action in Eq. (13). ω ,ω (2 ) 1 2 λ1,λ2 Ignoring temperature-independent prefactors in the partition 3 3 function and performing the sum over Matsubara frequencies a 8 (k · nˆ λ (k )) (a /8)(k · nˆ λ (k )) × 0 1 1 2 0 1 2 2 we obtain the photon contribution to the free energy −iω + (k ) iω − iω + (k − k ) 1 1 1 2 2 1 c Ek − / × λ ,ω λ ,ω λ ,ω λ − , −ω = + − Ek T − . a0 A (k )A (k )A 1 (k2 2 )A 2 ( k2 2 ) 0 FP kB T ln(1 e ) T ln Ek (E2) 3 3 ,λ= , 2 = (2π ) δ (k + k )δω,−ω δλλ k 1 2 πv2a7g˜2 d3k k2 − (k · kˆ )2 The photon contribution to the heat capacity per unit volume × 0 1 1 1 , 2 2 2 2 2 3 is then given by (ω + c h¯ k ) (2π ) (k1) + (k − k1) − iω T ∂2F (D6) C =− P P V ∂T 2 where by symmetry the integral over k vanishes when λ = / 1 1 k (TE − E )2eEk T TE λ = B k k − k 2. The correction to the imaginary-time magnetic field corre- / V T 2(eEk T − 1)2 2 lator, defined in Eq. (67), is given by k,λ=1,2 2 2 2 3 2 − · ˆ 2 TEk 2TEk T Ek T (Ek ) d k1 k1 (k1 k) − + + − , (E3) δχλ(k, iω) = E /T 2 3 e k − 1 Ek Ek E (2π ) (k1) + (k − k1) − iω k π 2 7 2ξ 2 where the primes indicate temperature derivatives and the v a0g˜ λ (k) × (D7) N a3 2 2 2 2 2 = s 0 (ω + c h¯ k ) volume V 4 . At low temperatures the vison contribution to the partition and the spectral density by function is given by −nβμ n δρλ(k,ω) = Im lim δχλ(k,ω+ i ) e V V →0 Z = . (E4) V n! λ3 d3k n B = 1 k2 − (k · kˆ )2 (2π )3 1 1 The contribution to the heat capacity per unit volume is given by πv2a7g˜2ξ 2(k) × δ( (k ) + (k − k ) − ω) 0 λ 1 1 ω2 + 2 2 2 2 T 2 ∂2 ln Z ( c h¯ k ) C = V V 2 ∗ 5 3 V ∂T 2 2 7 2ξ 2 2 (m ) v a0g˜ λ (k) (W )W √ = , (D8) 1 −βμ 6πh¯5(ω2 + c2h¯2k2 )2 = λ2 Te V (4μ2 + 12μT + 15T 2 ). (E5) 4 B h¯2k2 where W = ω − 2μ − ∗ and (W ) is the Heaviside step V 4m APPENDIX F: CONFINEMENT OF SPINONS function. The contribution to the total unpolarized scattering intensity I(k,ω) from the generation of vison pairs can now In this section, we demonstrate how a nondynamical mass be found using Eq. (72) and the fact that at zero temperature gap, i.e., one that couples to the zero Matsubara frequency the dynamical structure factor Sλ(k,ω) = 2ρλ(k,ω). component, would generate confinement.

125113-18 VISON-GENERATED PHOTON MASS IN QUANTUM SPIN … PHYSICAL REVIEW B 102, 125113 (2020)

−ik·e2/2 2 We introduce two oppositely charged monopoles situated Uλ2(k)e | = 2atk · aˆ = 0. The above result shows that on up tetrahedra with centres r ± N(e1 − e2 ). This results in a static mass gap generates a constant conﬁning force equal to a line of background ﬂux linking them gz a /ξ 2 ξ √ = ˜ ( √0 ) ln . 0 −ik·(r+eμ/2) F (F5) N B (k) = e Uλμ(k)Bμ(r + eμ/2) s λ 2πa0 r,μ N APPENDIX G: PLASMA OSCILLATIONS −ik·(na+e1/2) = (Uλ1(k)e − Uλ2(k) IN THE CONTINUUM LIMIT n=−N We begin with the action in Eq. (13) and take the contin- − · + / × e ik (na e2 2)) uum limit as follows:

−ik·e1/2 −ik·e2/2 = (Uλ1(k)e − Uλ2(k)e ) Aαβ (τ ) → A(r,τ) · (rβ − rα ), 1 sin k · a(N + ) Eαβ (τ ) → E(r,τ) · (rβ − rα ), × 2 , (F1) sin k · a β 3 2 d r 2 gza˜ 0 2 = − S = dτ iE · A˙ + g˜E + (∇×A) . where a e1 e2.ByEq.(8), the noninteger background a s2 0 0 0 0 vector potential is then given by Aλ(k) = Bλ(k)/ξλ(k)for λ = 1 and 2. In its presence, the static part of the action in (G1) Eq. (64) becomes We can also consider adding a vison current term to the action 2 which couples to the magnetic vector potential gza˜ 0 2 0 2 −2 2 S/β = k |Aλ(k) + Aλ(k)| + ξ |Aλ(k)| , s2 3 k,λ=1,2 δS =− d rJ· A. (G2) (F2) Moving from imaginary to real time (which we measure in where ξ −2 is a static mass gap that could be generated by units ofh ¯), we obtain the saddle point equations of the action, vison condensation in the ground state. Integrating out the i.e., Maxwell’s equations field Aλ(k), we obtain A˙ =−2˜gE, gza˜ 2 k2ξ −2 /β = 0 0 2. 2˜ga2z S − Aλ(k) (F3) ˙ = 0 ∇× ∇× − . s2 k2 + ξ 2 E ( A) Ja0 (G3) k,λ s2 βμ We can see that in the absence of a static mass gap, i.e., in In the Debye limit e V 1, when there are many visons the limit ξ →∞, there is no energy cost arising from the inside the Debye volume, the visons will respond to the gauge 0 transverse part of the background field Bλ(k). There is only field coherently and we can use a hydrodynamic description. a Coulombic energy cost arising from its longitudinal part, The charges accelerate in the presence of the electric field 0 i.e., ijψ in Eq. (8). Substituting in for Aλ(k) and integrating according to k · aˆ ∗ along first, we obtain J˙ =−2(2πh¯)2nA˙ /m , (G4) − 3π 3 ξ 2 | | 2 · −βμ − gza˜ 0 n d k a sin nk a = V λ 3 π S/β = where n e B is the vison number density and (2 h¯) | | π 3 2 + ξ −2 π · 2 a |k|< (2 ) k n(k a) is the vison charge. Differentiating the second equation in ˙ −ik·e1/2 −ik·e2/2 2 Eq. (G3) with respect to time, substituting in for A with the ×|Uλ1(k)e − Uλ2(k)e | help of the first equation and using J˙ =−2(2πh¯)2nA˙ /m∗,we 2πgza˜ 3n d2k ξ −2 = 0 obtain the plasma equation | | π 3 2 + ξ −2 a |k|< (2 ) k 2 2∇2 = ¨ + ω2 , h¯ c E E pE (G5) /ξ 2 ξ gz˜ (a0 ) ln −βμ 4˜g2a2z π 2 2 V ≈ n √ , (F4) 2 = 0 ω2 = 16 h¯ a0e g˜ where c 2 2 , p ∗λ3 , and we have assumed 2π h¯ s m B plasma neutrality so that ∇·E =−∇2ϕ = 0 everywhere. where we have worked in the limit a0 1 and used the fact |a| sin2 nk·a −ik·e /2 This result agrees with the derivation presented in the main that lim →∞ ( ) = δ(k · aˆ ) and |Uλ (k)e 1 − n πn (k·a)2 1 body of this paper.

[1]M.J.P.GingrasandP.A.McClarty,Rep. Prog. Phys. 77, [4] A. Banerjee, C. A. Bridges, J.-Q. Yan, A. A. Aczel, L. Li, 056501 (2014). M. B. Stone, G. E. Granroth, M. D. Lumsden, Y. Yiu, J. [2] J. Knolle and R Moessner, Annu. Rev. Condens. Matter Phys. Knolle, S. Bhattacharjee, D. Kovrizhin, R. Moessner, D. A. 10, 451 (2019). Tennant, D. G. Mandrus, and S. E. Nagler, Nat. Mater. 15, 733 [3] A. Kitaev, Ann. Phys. 321, 2 (2006). (2016).

125113-19 M. P. KWASIGROCH PHYSICAL REVIEW B 102, 125113 (2020)

[5] A. Banerjee, J. Yan, J. Knolle, C. A. Bridges, M. B. Stone, [20] C.-J. Huang, Y. Deng, Y. Wan, and Z. Y. Meng, Phys.Rev.Lett. M. D. Lumsden, D. G. Mandrus, D. A. Tennant, R. Moessner, 120, 167202 (2018). and S. E. Nagler, Science 356, 1055 (2017). [21] L. Savary and L. Balents, Phys.Rev.Lett.108, 037202 [6] S.-H. Do, S.-Y. Park, J. Yoshitake, J. Nasu, Y. Motome, Y. S. (2012). Kwon, D. T. Adroja, D. J. Voneshen, K. Kim, T.-H. Jang, J.-H. [22] S. B. Lee, S. Onoda, and L. Balents, Phys. Rev. B 86, 104412 Park, K.-Y. Choi, and S. Ji, Nat. Phys. 13, 1079 (2017). (2012). [7] A. Banerjee, P. Lampen-Kelley, J. Knolle, C. Balz, A. A. Aczel, [23] L. Savary and L. Balents, Phys. Rev. B 87, 205130 (2013). B. Winn, Y. Liu, D. Pajerowski, J. Yan, C. A. Bridges, A. T. [24] Z. Hao, A. G. R. Day, and M. J. P. Gingras, Phys.Rev.B90, Savici, B. C. Chakoumakos, M. D. Lumsden, D. A. Tennant, R. 214430 (2014). Moessner, D. G. Mandrus, and S. E. Nagler, Quantum Mater. 3, [25] O. Benton, L. D. C. Jaubert, R. R. P. Singh, J. Oitmaa, and N. 8 (2018). Shannon, Phys. Rev. Lett. 121, 067201 (2018). [8] Y. Kasahara, T. Ohnishi, Y. Mizukami, O. Tanaka, S. Ma, K. [26] A. Szabó and C. Castelnovo, Phys. Rev. B 100, 014417 (2019). Sugii, N. Kurita, H. Tanaka, J. Nasu, Y. Motome, T. Shibauchi, [27] A. M. Polyakov, Phys. Lett. B 72, 477 (1978). and Y. Matsuda, Nature (London) 559, 227 (2018). [28] J. Villain, J. Phys. 35, 27 (1974). [9] M. P. Kwasigroch, B. Douçot, and C. Castelnovo, Phys.Rev.B [29] L. D. Faddeev and V. N. Popov, Phys. Lett. B 25, 29 (1967). 95, 134439 (2017). [30] J. M. Kosterlitz, J. Phys. C 10, 3753 (1977). [10] M. Hermele, M. P. A. Fisher, and L. Balents, Phys. Rev. B 69, [31] In this limit, the typical instanton separation along τ will be 064404 (2004). small by comparison with β, so that ﬁnite-size effects associ- [11] A. M. Polyakov, Nucl. Phys. B 120, 429 (1977) ated with ﬁnite temperature are negligible. [12] G. Chen, Phys. Rev. B 94, 205107 (2016). [32] Consider a pair of tightly bound visons of opposite charge, [13] G. Chen, Phys. Rev. B 96, 195127 (2017). where one of them executes two instanton hops returning to [14] O. Benton, O. Sikora, and N. Shannon, Phys.Rev.B86, 075154 its original position. Because, there is a growing energy cost (2012). associated with separating the two instanton hops in imaginary [15] P. C. Kwok and T. D. Schultz, J. Phys. C: Solid State Phys. 2, time, the oscillation will not generate a mass gap but only 1196 (1969). renormalize the speed of light. [16] N. Shannon, O. Sikora, F. Pollmann, K. Penc, and P. Fulde, [33] T. Fennell, P. P. Deen, A. R. Wildes, K. Schmalzl, D. Phys.Rev.Lett.108, 067204 (2012). Prabhakaran, A. T. Boothroyd, R. J. Aldus, D. F. McMorrow, [17] Y. Kato and S. Onoda, Phys. Rev. Lett. 115, 077202 (2015). and S. T. Bramwell, Science 326, 415 (2009). [18] A. Banerjee, S. V. Isakov, K. Damle, and Y. B. Kim, Phys. Rev. [34] D. Khomskii, Nat. Commun. 3, 904 (2012). Lett. 100, 047208 (2008). [35] E. Lantagne-Hurtubise, S. Bhattacharjee, and R. Moessner, [19] J.-P. Lv, G. Chen, Y. Deng, and Z. Y. Meng, Phys. Rev. Lett. Phys. Rev. B 96, 125145 (2017). 115, 037202 (2015). [36] S. Nakosai and S. Onoda, J. Phys. Soc. Jpn. 88, 053701 (2019).

125113-20