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 descrip- tion. 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 ˆij, 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.