Higher Rank Deconfined Quantum Criticality at the Lifshitz Transition

Higher Rank Deconﬁned Quantum Criticality at the Lifshitz Transition and the Exciton Bose Condensate

Han Ma and Michael Pretko Department of Physics and Center for Theory of Quantum Matter, University of Colorado, Boulder, CO 80309, USA (Dated: September 7, 2018) Deconfined quantum critical points are characterized by the presence of an emergent gauge field and exotic fractionalized particles, which exist as well-defined excitations only at the critical point. We here demonstrate the existence of quantum critical points described by an emergent tensor gauge theory featuring subdimensional excitations, in close relation to fracton theories. We begin by reexamining a previously studied Lifshitz transition between two valence bond solid (VBS) phases on a bilayer honeycomb lattice. We show that the critical theory maps onto a rank-two tensor gauge theory featuring one-dimensional particles. In a slightly different context, the same tensor gauge theory also describes a Lifshitz quantum critical point between a two-dimensional superfluid and a finite-momentum Bose condensate, both of which are dual to rank-one gauge theories. This represents an entirely new class of deconfined quantum criticality, in which a critical tensor gauge theory arises on top of a stable conventional gauge theory. Furthermore, we propose that this quantum critical point gives rise to a new finite-temperature phase of bosons, behaving as an exciton Bose condensate, in which excitons (boson-hole pairs) are condensed but individual bosons are not. We discuss how small modifications of this theory give rise to the stable quantum “exciton Bose liquid” phase studied in [Phys. Rev. B 66, 054526 (2002)].

CONTENTS References 15

I. Introduction1 I. INTRODUCTION II. Dual Tensor Gauge Theory of the Critical Point3 Phase transitions beyond the Landau-Ginzburg III. Deconfined Quantum Criticality between paradigm have attracted much attention due to the ex- Condensates4 otic phenomena they exhibit at critical points. The most well-known example is the quantum critical point be- IV. Properties of the Critical Point6 tween a Néelordered antiferromagnet and a valence bond A. Zero-Temperature Properties6 solid (VBS) in two spatial dimensions. Since these phases 1. Exciton Condensation6 break different symmetries, Landau-Ginzburg theory pre- 2. Vortex Solutions7 dicts a first order phase transition between them. In B. Finite-Temperature Behavior8 contrast, modern studies have demonstrated a mecha- 1. Phase Diagram8 nism which allows for a continuous transition between 2. Properties of the Exciton Bose these phases.1–18 This critical point is described by a non- Condensate9 compact CP1 theory in terms of emergent fractionalized excitations coupled with a gauge field. The deconfined V. Lattice Model 10 excitations carry non-trivial quantum numbers and their distinct behaviors give rise to different symmetry break- VI. Exciton Bose Liquid 11 ing phases. For example, condensing spinons (spin-1 2 quasiparticles) results in Néelorder. On the other hand, VII. Conclusions 12 VBS order arises when spinons are confined by the prolif-~ eration of monopoles, which carry quantum numbers of arXiv:1803.04980v2 [cond-mat.str-el] 6 Sep 2018 Acknowledgements 12 lattice symmetries. Right at the critical point, spinons are gapless and coupled to a conventional U 1 gauge A. Conventional Boson-Vortex Duality 12 field. A similar deconfined quantum critical point( can) also B. Duality in Reverse 13 occur between two VBS phases breaking different lattice symmetries. In particular, a system of spin-1 2s on C. Finite-Temperature Screening 14 the bilayer honeycomb lattice has been shown to sup- 1. Logarithmically interacting one dimensional port a second order phase transition between two~ VBS particle 14 phases, with an emergent deconfined gauge theory arising 2. Confined interacting one dimensional at the critical point.19 In contrast to the Néel-VBStran- particle 14 sition, this critical point has dynamical exponent z 2,

= 2 featuring low energy modes with quadratic dispersion, the critical Hamiltonian of the VBS-VBS′ transition fea- ω k2. This transition, called Lifshitz transition, can be tures such one-dimensional charge excitations. We note described by a compact 2 1 -dimensional U 1 gauge that a similar relation with tensor gauge theory has theory,∼ with low-energy Hamiltonian given by19: previously been noticed in the context of multicritical ( + ) ( ) Rokhsar-Kivelson (RK) points of certain quantum dimer i ij 2 1 ij 2 20–22 H κE Ei K ∂iEj ∂iAj (1) models , though without noting the existence of sub- 2 dimensional particles. where Ai is= the emergent+ ( gauge) field+ and( Ei is) its corresponding electric field. The deconfined quantum critical Interestingly, the Hamiltonian of Eq. (2) can also de- point occurs at κ 0. (More precisely, this is a fixed line scribe a completely different physical situation, if the parametrized by K, which we will regard as fixed.) Here cos φ term is suppressed by a global U 1 symmetry. In and below, all indices= refer to spatial coordinates, and this free theory, tuning κ across zero realizes the tran- we sum over all the repeated indices in every equation. sition between a conventional superfluid( ) and a finite- The gauge field also couples to deconfined spinon degrees momentum Bose condensate of bosons eiφ. This tran- of freedom, which are gapped at the critical point, un- sition will be studied in Sec.III. In this case, the system like the Néel-VBS transition. Using a standard particle- is invariant under translations of φ, and the dual rank-1 vortex duality mapping (reviewed in AppendixA), this gauge theory of Eq. (1) is non-compact. In this situa- Hamiltonian can also be conveniently written in terms of tion, the phases on the two sides of the transition are a compact scalar field φ as: no longer gapped, but gapless with a linear mode dual 1 to rank-1 U(1) gauge theory by standard boson-vortex H κ ∂ φ 2 K ∂ ∂ φ 2 n2 ... (2) i i j 2 duality. This represents an entirely new type of deconfined quantum criticality, in which a tensor gauge struc- where n is the= canonical( ) + conjugate( ) to+ φ. The+ ellipsis in- ture emerges at a critical point between two conventional cludes interaction terms of φ, such as a γ cos φ term dual gauge theories. We will show that this critical point has to the instantons of the gauge field Ai. At the critical a natural physical interpretation as a condensate of ex- point, κ 0, the instantons are irrelevant (for a certain citons formed by boson-hole pairs, even though isolated range of K of interest), and we obtain a deconfined gauge bosons remain uncondensed as studied in Ref. 69. The theory with= a quadratic photon. Away from the critical one-dimensional particles can then be understood as the point, instantons proliferate and gap the theory resulting vortices of the exciton condensate. We refer to such a in confined phases.19 system as an “exciton Bose condensate” (EBC), in anal- The existence of deconfined spinons at the VBS-VBS′ ogy with the closely-related exciton Bose liquid (EBL) critical point has been well-established. In the present phase.70–75 work, however, we demonstrate that this critical point also features an even more exotic class of fractionalized excitations which have gone unnoticed in previous liter- In Sec.IV, we show that at the critical point, the one- ature. In Sec.II, we will study in detail that this critical dimensional particles carry a logarithmic energy cost, theory is related by a duality transformation to a com- much like conventional vortices in a superfluid, which pact rank-two tensor gauge theory, with Hamiltonian: suggests that the exciton condensate can survive at nonzero temperatures. Like the normal BKT transi- ij 1 ik j` 2 76–78 Hκ=0 KE Eij ∂i∂jAk` (3) tion , where vortices proliferate at a critical temper- 2 ature and destroy the low-energy quasi-long-range order, we argue for the existence of another phase transition at for symmetric tensor= gauge+ field( Aij and its) correspond- which the one-dimensional vortices undergo BKT-like un- ing electric tensor Eij. The first terms in Eq. (1) and Eq. (2) map onto terms of creation and annihilation of binding and destroy the exciton condensate at the critical magnetic fluxes in the tensor gauge theory. Such tensor temperature, resulting in a completely disordered phase. gauge theories have been studied in the context of frac- We present the proposed phase diagram in Fig.4, with a ton phases23–31, a topic of intense recent research32–68, generic parameter regime featuring a finite-temperature where it has been found that the gauge charges subject EBC phase, which shrinks to the quantum critical point severe restrictions on their motion. Specifically, the ten- at zero temperature. We also establish some of the basic sor gauge theory in Eq. (3) studied in this paper has properties of this new finite-temperature region. vector-valued charges via a generalized Gauss’s law:

ij j ∂iE ρ (4) In Sec.V, we provide a concrete lattice boson model which exhibits the physics described above. Additionally, In addition to charge conservation,= these vector parti- in Sec.VI, we show that small modiﬁcations of the critical cles obey an extra conservation law which forces them theory result in the stable quantum EBL phase studied to move only in the direction of their charge vector, re- in Ref. 70, which is also known to exhibit subdimensional sulting in one-dimensional behavior. We will show that particle excitations.70,72,73 3

II. DUAL TENSOR GAUGE THEORY OF THE CRITICAL POINT

We begin by showing that the critical theory of the VBS-VBS′ transition is a tensor gauge theory. In order to obtain the desired mapping, it is simplest to start on the gauge theory side of the duality and map it onto the scalar field Hamiltonian of Eq. (2) at κ 0. (It is also possible to derive the duality in the opposite direction, starting with the critical boson theory and= mapping onto the tensor gauge theory, as we show in AppendixB.) We first review the basic properties of the appropriate tensor gauge theory, known as the “vector charge theory” in the fracton literature.23,24 The theory is formulated in terms of a rank-2 symmetric tensor U 1 gauge field Aij, along with its canonical conjugate variable, which we call the electric tensor Eij. The theory( is) defined in terms of its Gauss’s law:

ij j ∂iE ρ (5) FIG. 1. All excitations and operators in the eﬀective theory of j which is sourced by vector-valued= charges ρ , assumed the VBS-VBS′ transition can be mapped directly onto those to be gapped. Notably, these charges obey two separate of a tensor gauge theory with one-dimensional vector charges. conservation laws:

i 2 i 2 ij Q d x ρ const. L d x xiρj const. for scalar field φ. Since Eij is canonically conjugate to (6) ik jl representing= S conservation= of= charge,S Q(i, and also) = the an- Aij, it follows that φ is conjugate to B ∂i∂jAkl, gular charge moment, . (Note that is analogous to, which we now relabel as n, for reasons which will be- L L = but distinct from, kinetic angular momentum.) This ex- come clear in the next section. Making the appropriate tra conservation law forces the fundamental charges to replacements in Eq. (8), we obtain the dual Hamiltonian: move only in the direction of their charge vector, giving 2 1 2 rise to one-dimensional behavior. The theory also admits H K ∂i∂jφ n (10) i 2 stable bound states with Q 0, but L 0. These bound states, which we refer to as L-particles, are fully mobile which is precisely the= critical( ) point+ of the Hamiltonian = ≠ 2 and correspond to ordinary spinons in the description of in Eq. (2). The relevant κ ∂iφ perturbation cannot rank-1 gauge theory in Eq. (1). be written as a simple local term in terms of the tensor The Gauss’s law of the theory implies that the low- fields, but rather corresponds( to) a non-local flux created ij energy charge-neutral sector (obeying ∂iE 0) is in- by an instanton event of the gauge field, as we will see variant under the gauge transformation: shortly. = First, however, we determine the correspondence be- Aij Aij ∂iαj ∂jαi (7) tween the charges of the tensor gauge theory and gapped topological defects of the critical scalar field theory. To where αi is an arbitrary→ function+ + of spatial coordinates. do this, we consider the total charge enclosed in a planar Gauge invariance then dictates the form of the low-energy region P with boundary C: Hamiltonian at critical point: j 2 j ij k ji ij 1 2 Q d x ρ dniE ds ∂k ∂iφ H KEijE B . (8) P C C (11) 2 ji = S = c ∆ ∂iφ= c ( ) where the magnetic= field is a scalar+ quantity given by ik j` k B ∂i∂jAk`. Since the magnetic field contains two where the line element= ds ( is related) to the normal vec- k ik derivatives, the equations of motion yield a gapless gauge tor via ds dni, and ∆ ∂iφ represents the change in 2 mode= with quadratic dispersion, ω k . In order to find ∂iφ upon going around the closed curve C. Accordingly, the dual description of this gauge theory, we begin in the the fundamental= charges of( the) gauge theory correspond charge-neutral sector, in which the∼ source-free Gauss’s to singular points around which ∂iφ has nontrivial wind- ij law, ∂iE 0, has the general solution: ing. This type of singularity is much less familiar than a conventional winding of a compact scalar field φ. Nev- ij ik j` = E ∂k∂`φ (9) ertheless, on a lattice, compactness of φ automatically

= 4 implies compactness of ∂iφ. If φ is an angular variable, The path integral will then allow sudden changes in Φ defined modulo 2π, then ∂iφ is only defined modulo 2π a, by 2π, just as in a normal compact gauge theory. In the where a is the lattice spacing. Whether or not such de- dual language, flux insertion corresponds to a symmetry- fects can still be sensibly discussed in a true continuum~ is breaking perturbation to the Hamiltonian: unclear. But for a system with an underlying lattice, such as the bilayer honeycomb system under consideration19, HΦ γ cos φ, insertion of flux (16) the compact field φ naturally hosts this type of defect. 19 We will explicitly construct a configuration of φ which ex- This term is irrelevant= at the( critical point , but) gaps the hibits such a singularity (see Eq. (34)). The mobility of gauge field on either side of the transition. In addition those one-dimensional gauge charges pick particular di- to this conventional flux slip event, the path integral will rections on the lattice according to the lattice derivative also admit an additional type of instanton, in which Φ is left unchanged but Πi changes by 2πa in some lattice di- ∂i. At the critical point, these defects have a logarithmic energy cost, analogous to conventional vortices of a rection. In other words, a dipole moment of flux is added superfluid, as we will see later. to the system. Such a dipolar flux insertion corresponds Of course, we expect that our system also contains the to the following perturbation in the dual language: usual windings of φ, vortices which should be normal H κ ∂ φ 2, insertion of flux dipole (17) mobile particles. These conventional vortices map onto Π i i the L-particles of the gauge theory, with Q 0 but L 0, which is= the( only) relevant( operator at the critical) point. which have no constraints on their motion. To see this, All other perturbations can be shown to either be irrel- consider the total angular charge moment= contained≠ in evant or ruled out by symmetries of the bilayer system, the region P bounded by curve C: leading to a generic second order phase transition.19 We therefore see that, in the tensor gauge theory language, 2 jk 2 jk l L d x xjρk d x xj∂ Elk the transition away from the critical point is driven by P P (12) the proliferation of “dipolar” instantons. On the two m j i j = Sds x(∂m∂jφ ) = Sds ∂i (x ∂jφ φ) sides of the transition, single fluxes are mobile due to the C C background of the dipolar flux, then the “monopolar” When= therec are no net= one-dimensionalc (( particles) − ) in the instantons proliferate and lead to conventional confined region P , such that ∂iφ is single-valued on the boundary, gapped phases. L can be written in terms of the winding of φ as: This completes the duality mapping between the bosonic critical theory and a tensor gauge theory, the i L ds ∂iφ ∆φ (13) details of which are summarized in Fig.1. C

Therefore, the L-particles= − c of the gauge= − theory correspond to the ordinary vortices of φ, which have been studied in III. DECONFINED QUANTUM CRITICALITY BETWEEN CONDENSATES previous treatments of the VBS-VBS′ transition.19 These particles exist as gapped deconfined excitations with a ′ 1 r2 interaction at the κ 0 critical point, but become The physics of the VBS-VBS transition is well- confined on either side of the transition. captured by a Hamiltonian in terms of a scalar field φ, ~Finally, we discuss the= role of instantons, arising as: from the compactness of the gauge field. For a 2 2 1 2 noncompact theory, the definition of the magnetic field, H κ ∂iφ K ∂i∂jφ n (18) ik j` 2 B ∂i∂jAk`, would lead to two independent con- served quantities: with the critical= point( ) at+κ ( 0. Importantly,) + the cos φ = perturbation is irrelevant at the critical point, such that Φ d2x B const. Πi d2x Bxi const. (14) the critical theory can be written= entirely in terms of derivatives of φ. This motivates us to use this Hamilto- corresponding= S = to the conservation= S of( flux,) = and also the nian to describe a completely different physical situation, “dipole moment” of flux. In other words, magnetic flux if the monopolar instantons are forbidden by symmetry would behave like a fracton. Since B is the canonical φ φ α. Then we can interpret Eq. (18) as a system of † iφ conjugate to φ, these conservation laws would map onto bosons b e with total number conservation. The vari- the following symmetries in the dual language: able→ n+then corresponds to the boson number operator, † n b b and= the boson current is proportional to ∂iφ. The 2 φ φ α, ∂iφ ∂iφ λi (15) K ∂i∂jφ operator of the critical theory describes two- boson= hopping process (to be discussed in more detail 2 for constants α→and+λi. For a compact→ + theory, however, later( in Fig.) 3), while the relevant κ ∂iφ perturbation these conservation laws and their associated symmetries corresponds to single-boson hopping operators. ′ will be broken. The gauge field Aij is only defined up Unlike in the VBS-VBS transition,( the) γ cos φ term to some compactification radius, which we take to be 2π. is not allowed with the underlying U 1 symmetry. But

( ) 5

be marginally irrelevant for u 0.19,21 For a system starting with positive u, this perturbation will be unimportant at low energies. We can also> rule out all terms having odd powers of φ by imposing φ φ symmetry on the system, corresponding to particle-hole symmetry of the underlying bosons, which naturally→ − arises at half-integer filling factors.70 All other rotationally invariant terms are irrelevant by power-counting. However, we must also worry about non-rotationally- FIG. 2. In the VBS-VBS′ transition, the critical tensor gauge invariant terms arising from the underlying lattice of the theory separates two gapped confined phases. In contrast, the theory. (Recall that we focus on lattice systems, instead superfluid to finite-momentum condensate transition features of a true continuum, in order to sensibly discuss wind- a critical tensor gauge theory separating two stable noncom- ings of ∂iφ.) On the square lattice, there are relevant pact vector gauge theories. anisotropy terms which trigger the proliferation of instantons at the critical point and drive the transition first order.19 If we consider a honeycomb lattice, how- the critical Hamiltonian still possesses a relevant pertur- ever, then to fourth order in derivatives, we only need to 2 bation by the κ ∂iφ term. (We will verify later that consider the rotationally invariant terms, which we have this remains the only relevant perturbation for a system already discussed.19,21,79 Putting it all together, we see with an underlying( honeycomb) lattice.) Equivalently, the that the critical point on a honeycomb lattice has only a dual vector gauge fields in the formulation: 2 single relevant direction, namely the κ ∂iφ term. Thus, up to marginally irrelevant corrections, the Hamiltonian i ij 2 1 ij 2 H κE Ei K ∂iEj ∂iAj (19) of Eq. (18) describes a direct second( order) quantum 2 phase transition between a superfluid and a finite mo- should be regarded= + as (noncompact.) + ( The end) result is mentum condensate. ′ that the two sides of the phase transition are no longer Similar to the VBS-VBS case, this transition can also gapped phases. Rather, both sides are dual to noncom- be understood in the language of tensor gauge theory. pact vector gauge theories, while the critical point re- At the κ 0 critical point, the system is described by an mains a deconfined tensor gauge theory. This provides emergent tensor gauge structure, with Hamiltonian given an example of an entirely new type of deconfined quan- by: = tum criticality, in which a critical tensor gauge theory ij 1 ik j` 2 separates two stable vector gauge theories. The differ- Hκ=0 KE Eij ∂i∂jAk` (20) ences between the two types of deconfined quantum crit- 2 ical points are sketched in Fig.2. in which the one-dimensional= + ( vector charges,) defined by The phases on the two sides of this transition corre- ij j ∂iE ρ , correspond to exotic vortices around which spond to different types of Bose condensates. When the ∂iφ has nontrivial winding. The conventional mobile vor- coefficient κ is positive, we obtain a conventional super- tices (windings= of φ) correspond to the L-particles dis- fluid phase. On the other hand, when κ 0, it becomes i cussed earlier, with Q 0 but L 0. At the critical energetically favorable for ∂iφ to pick up an expectation point, both types of vortices exist as well-defined exci- value, ∂iφ λi, such that the field φ becomes< “tilted,” tations of the system, with= logarithmic≠ interaction be- behaving as φ λ x. In terms of the microscopic boson tween the one-dimensional vortices. Away from κ 0, the ⟨ ⟩ = 2 field b, we then have b b0 exp iλ x , corresponding to κ ∂ φ term, corresponding to proliferation of the dipo- ⃗ i a condensate⟨ of⟩ = the⋅ bosons⃗ at finite momentum. lar instantons, will result in a linearly confining potential= In order to verify that⟨ ⟩ = the Hamiltonian( ⃗ ⋅ ⃗) of Eq. (18) de- between( ) the one-dimensional particles, leaving the con- scribes a direct second order phase transition, we must ventional vortices as gapped logarithimically interacting 2 check that the κ ∂iφ operator is the only relevant per- particles, as expected. (Note that the monopolar instan- turbation at the κ 0 critical point. The argument pro- tons correspond to a γ cos φ perturbation to the Hamil- ceeds largely along( the) same lines as the analysis of the tonian, which is ruled out by the global U 1 symmetry VBS-VBS′ transition,= but with the added advantage of of the boson system.) a global U 1 symmetry. This symmetry rules out all To study the behavior of the conventional( ) vortices terms which involve bare φ operator (i.e. without deriva- across the transition, and to recover a more familiar for- tives), which( ) significantly decreases the number of terms mulation of the superfluid phase, it is useful to rewrite we need to consider. First, we focus on the rotationally- the tensor gauge Hamiltonian in terms of the effective invariant terms, which are insensitive to the underlying gauge field seen by the L-particles. Since these particles lattice of the system. One worrisome operator of this are bound states of the fundamental vector charges, their 4 24 sort is a quartic term, u ∂iφ , which is marginal at the effective gauge field takes the form : power-counting level. This term was analyzed in the con- ′ ij text of the VBS-VBS transition,( ) where it was shown to Ak ∂iAjk (21)

= 6

The corresponding effective electric field Ei seen by the A. Zero-Temperature Properties k 24 L-particles satisfies Eij ik∂ Ej. In terms of these variables, we can rewrite the low-energy Hamiltonian as: 1. Exciton Condensation =

1 We now focus on the critical Hamiltonian taking the H κEiE K ij∂ E 2 ij∂ A 2 (22) i i j 2 i j form: = + ( ) + ( ) 2 1 2 i H K ∂i∂jφ n (23) where κE Ei represents the creation/annihilation of 2 dipolar fluxes of the tensor gauge theory. This is precisely = ( ) + the vector gauge formulation of Eq. (19), in which the iφ electric field is related to the boson field by Ei ij∂ φ. where φ is the phase of the microscopic bosons, b e , j and n is the boson number, b†b. This Hamiltonian looks When combined with Eij ik∂ Ej, we recover the ex- k very similar to that of the superfluid phase, except∼ that pected relationship between the electric tensor= and the the first term features only second derivatives. This leads boson field, Eij ikj`∂ ∂= φ. When κ 0, the K term k ` to a quadratic dispersion of the gapless mode, ω k2, is unimportant, and we recover the conventional gauge as opposed to the linearly dispersing Goldstone mode dual of a superfluid= at κ 0, with photons≠ of the gauge of the superfluid. In order to gain an intuitive under-∼ theory mapping onto the Goldstone modes. In the su- standing of this critical point, it is instructive to con- perfluid phase, separation> of vortices have a logarithmic sider the microscopic origin of the derivative operators. energy cost, as expected. Right at the critical point, how- The first derivative, ∂ φ, arises from single-boson hop- ever, the κE2 term of the Hamiltonian vanishes, leading i ping processes, since b† xi i b xi exp ii∂ φ . At to a finite energy for conventional vortices at large dis- i the critical point, such first derivative terms are absent tance. When κ 0, the logarithmic energy cost is re- from the Hamiltonian, indicating( + ) ( zero) hopping∼ ( matrix) el- stored. On this side of the transition, it is favorable ements for single bosons. In this sense, the fundamental for the vector electric< field to pick up an expectation bosons behave like fractons at the critical point. When value, Ei ij ∂ φ ijλ , corresponding to a finite- j j the hopping matrix elements are turned back on, the momentum condensate of the microscopic bosons. The system flows away from the critical point into a Bose- behavior⟨ of⟩ = vortices⟨ in⟩ = the finite-momentum condensate condensed phase. is then equivalent to logarithmically interacting charges moving in a background electric field. Despite the absence of single-particle hopping in the Hamiltonian, the bosons are not completely nondynam- ical at the critical point. The second derivative operator, ∂i∂jφ, corresponds to two-boson hopping processes. More specifically, second derivatives correspond to processes in which two bosons move in opposite directions by the same amount of distance, thereby conserving center IV. PROPERTIES OF THE CRITICAL POINT of mass. Importantly, we can also regard such a process as the motion of a particle-hole pair (“exciton”), as depicted in Fig.3. We can therefore understand the critical In the previous sections, we identified a quantum crit- point as a system in which excitons are the fundamental ical point described by a tensor gauge theory featuring mobile particles, while single bosons are locked in place. subdimensional particles. In this section, we character- At zero temperature, we then expect that the excitons ize some of the properties of this critical point, including will form a condensate, while single bosons remain un- the consequences of the critical tensor gauge structure for condensed, leading us to call the system an exciton Bose the surrounding parameter space. Most notably, we find condensate. a finite-temperature phase of matter which shrinks to the quantum critical point at zero temperature. This phase We can verify these expectations explicitly by check- is distinct from both the infinite-temperature disordered ing for off-diagonal long-range order in the correlation phase and the zero-temperature ordered phases. For con- functions of both bosons and excitons. Because the time creteness, we will phrase our discussion for the super- correlations of both bosons and excitons have the same fluid transition (though similar logic carries over to the power law behavior even at finite temperature69, below VBS-VBS′ transition). In the bosonic system, the new we focus on the spatial correlations which can distinguish finite-temperature phase represents an exciton Bose con- the exciton and boson condensate. For single bosons, the densate (EBC), in which excitons have condensed while appropriate correlator takes the form: single bosons have not. We will see in a later section how a small modification of the quantum critical point x b† x b 0 ei[φ( )−φ(0)] can also give rise to the quantum “exciton Bose liquid” 1 2 (24) − 2 ⟨[φ(x)−φ(0)] ⟩ ⟨φ(x)φ(0)⟩ (EBL) phase studied in Ref. 70. ⟨ ( ) ( )⟩ = ⟨e ⟩ e = ∼ 7

correspond to charges in the dual tensor gauge theory. Since all charged excitations remain gapped at the critical point, charges will typically have much slower velocity than the gapless gauge mode. As such, the dominant interactions between charges will be electrostatic in origin. We here focus on this electrostatic limit, leaving retar- dation eﬀects to future study. To this end, we ﬁrst introduce a potential formulation, analogous to Ei ∂iϕ in conventional electromagnetism. Similar potential for- mulations for tensor gauge theories have been studied= − in three dimensions in Ref. 24. We begin by noting the form of the Faraday’s equation for this tensor gauge theory:

ik j` FIG. 3. The ∂iφ operators correspond to single-boson hopping ∂tB ∂i∂jEk` 0 (29) processes. Similarly, the ∂i∂j φ operators correspond to two- boson hopping processes conserving center of mass, which can following from the role+ of Eij as a function= of the conju- equivalently be regarded as exciton hopping processes. gate momentum to Aij. In the static limit (∂tB 0), the ik j` electric tensor must obey ∂i∂jEk` 0. The general solution to this constraint takes the form: = The correlation function of the phase field is given by: = E ∂ ξ ∂ ξ (30) eik⋅x ij i j j i φ x φ 0 d2k dω ω2 Kk4 where ξi is a potential function= −( representing+ ) the potential ik⋅x (25) 24 ⟨ ( ) ( )⟩ ∼ S1 2 e 1 energy per unit vector charge. We now consider the d k + log r K k2 K potential arising from a point particle with vector charge qi, which must satisfy the Gauss’s law: where r x and∼ √k Sk . Note that,∼ −√ here and below, ij 2 j j i j (2) inside logarithms, r should be taken in units of the lattice ∂iE ∂ ξ ∂ ∂iξ q δ r (31) spacing, a=.S WeS then obtain= S S the boson correlator as: Note that qj has= − units( of+ length( ))−1=, so Eij(has) units of 1 −2 b† x b 0 (26) length . It can readily be checked that the following rζ potential provides the appropriate( ) solution: ( ) where ζ K−1~2. This⟨ ( indicates) ( )⟩ ∼ that, even at zero tem- 1 q r ri perature, there is no true long-range order (rather only i i ξ 3 log r q 2 (32) ∼ 8π r quasi-long-range order) of the fundamental bosons. To ( ⋅ ) find an operator exhibiting long-range order, we must = ( ) − leading to a logarithmic interaction energy between the consider the corresponding correlator for excitons: one-dimensional particles. The corresponding electric i∂ φ(x) −i∂ φ(0) ⟨∂ φ(x)∂iφ(0)⟩ − √ 1 tensor is given by: e i e i e i e Kr2 (27) 1 q r δij q r rirj q r q r where⟨ we used ⟩ ∼ ∼ Eij 2 i j j i (33) 4π r2 r4 r2 i 2 i ik⋅x ( ⋅ ) ( ⋅ ) ( + ) ∂iφ x ∂ φ 0 d kdωkik φ k φ 0 e = − − Using this form, and the relation Eij ikj`∂ ∂ φ, we 1 (28) k ` ⟨ ( ) ( )⟩ = S . ⟨ ( ) ( )⟩ can then determine the configuration of the φ field as: Kr2 = ∼ −√ 1 ij The above correlation function approaches a nonzero con- φ 2 qirj θ q r log r (34) stant as r , indicating that, unlike the fundamental 4π bosons, the excitons form a condensate with true off- = − ( ) + ( ⋅ ) which provides an explicit example of a singularity with diagonal long-range→ ∞ order at zero temperature. As such, winding of ∂ φ around r 0. We can check that it is appropriate to regard the quantum critical point as i an exciton condensate, separating two conventional Bose- = ijrj klqkrl condensed phases. 4π∂iφ 2ijqjθ qi 1 log r (35) r2 ( ) satisfying= that the+ winding( + of) +∂iφ is ijqj, where θ 2. Vortex Solutions ArcTan ry and r2 r2 r2. rx x y We can also consider a situation where two opposite= Having established the properties of the condensate, vector charges qj are= created+ with a displacement d per- we now investigate the properties of its vortices, which pendicular to the direction of vectors, i.e. an L -particle 8

jk with L qjdk. The potential generated by this charge the free energy per particle becomes negative, signaling configuration satisfies the creation of a particle is energetically favored. Addi- = tion to this single particle argument, a detailed study can ij 2 j j i ∂iE ∂ ξL ∂ ∂iξL be done for the many-body Hamiltonian: (36) j (2) j (2) ˆ jk (2) q δ r q δ r dk L∂kδ r = −( + ( )) i K ′ Hvor ξ qi 3qr qr′ log r r leading to= the solution:( ) − ( + ) = ( ) 8π r,r′ ′ ′ (42) ik = qr = r rQ qr′ ⋅ r r S − S i L rk q 2 ξ (37) ′ 2 y r (L) 4π r2 r r r [ ⋅ ( − )] [ ⋅ ( − )] − + Q S S This is the potential at distance= r away from the source L where y is the fugacityS − ofS the vortex of the exciton con- -particle whose scale is much smaller than r. Notice this densate. This Hamiltonian takes a similar form as that is also a vector potential acting on single vector charge. of the normal vortex in the superfluid, but with addi- i The potential between two L -particles is given by ∂iξ(L). tional vector structure, also named vector Coulomb gas. Then, the corresponding electric tensor takes the form: A similar finite-temperature phase transition as the dis-

ik j jk i location mediated melting transition, corresponding to ij L rkr rkr E (38) an unbinding transition of the one-dimensional particles, (L) 2π r4 r4 is indicated80. A group of similar scaling equations can = + be obtained on the hexagonal lattice81,82: which scales as 1 r2 leading to a finite energy cost for cre- ating an isolated L-particle, unlike the logarithmic energy −1 dKR 9 2 KR 1 KR cost for the one-dimensional~ particles. The configuration πy I0 I1 d` 2 8π 2 8π of the phase field φ for an -particle as source charge is (43) L dy 3K K given by: = 2 R( y )2−πy2I ( R ) d` 8π 0 8π L φ θ (39) = ( − ) + ( ) (L) 2π where I0 and I1 are modified Bessel functions, KR K T and T is the temperature. We can obtain a fixed point which is the expected winding= − of φ for a normal vortex 16π = ~ where KR Tc 3 . According to the exciton correla- of a superfluid. tion function at finite temperature r−η (see Eq. 48), we can get, at(T ), η= 1 0.004749 which is smaller than c2 4πKR the exponent for the KT transition∼η 0.25, noticing B. Finite-Temperature Behavior KT in the conventional= KT= transition, the ηKT is defined for boson correlation function. We can also= evaluate the −ν 1. Phase Diagram other critical exponent ν satisfying ξ eST −Tc2S where ξ is the correlation length. Using the method in Ref. We just found that the one-dimensional particles of 82, we can get ν 0.418099, which is between∼ the value the critical tensor gauge theory have a logarithmic in- for the KT transition νKT 0.5 and the value for the teraction energy. By the usual logic of the BKT dislocation mediated= melting whereν ˜ 0.36963477.80–82 transition76–78, we therefore expect a finite-temperature Above the critical temperature,= one-dimensional vor- phase transition at which these particles proliferate. A tices proliferate, the exciton condensate= is destroyed, similar argument applied for a single particle can be and the system enters the completely disordered normal established. An isolated one-dimensional particle with phase. A priori, this argument could be affected by L- fundamental charge qi has an energy of order Kq2 log `, particles, which cost finite energy and proliferate at any where ` is the system size. Similarly, the entropy per non-zero temperature. In AppendixC, we verify that the particles behaves as T log ` (working in units such that L-particles do not significantly affect the transition prop- kB 1). The resulting free energy per particle takes the erties of the one-dimensional particles. We note that the schematic form: one-dimensional particles will acquire some limited mo- = bility in their transverse direction at finite temperature, F TS Kq2 T log ` (40) due to absorption of thermally excited L-particles. Nev- ertheless, the one-dimensional particles still have strongly At low temperatures,= E − the energy∼ ( term− ) dominates and the free energy per particle is positive, indicating that it is anisotropic motion since they can only freely move along unfavorable to form isolated one-dimensional particles, one direction. Motion along the other direction is a sta- which serve as vortices of the exciton condensate. As tistical process analogous to a random walk, occurring such, the exciton condensate remains intact in this low- only upon the absorption of L-particles. This justifies temperature regime. On the other hand, above a certain our continued use of the term “one-dimensional particle” critical temperature: at finite temperature. At the κ 0 critical point, we have found that the 2 Tc2 Kq (41) system undergoes a finite-temperature phase transition = ∼ 9

2. Properties of the Exciton Bose Condensate

Having established the existence of a new finite- temperature phase of bosons, we now describe some of its properties. This phase is characterized by unprolif- erated one-dimensional vortices, indicating that exciton condensation is still present at finite temperature. To see this explicitly, we repeat our calculation of correlation functions at finite temperature, where thermal fluctua- tions dominate quantum effects. As such, we calculate correlation functions based on the classical free energy:

FIG. 4. The EBC quantum critical point between two con- 2 2 ventional Bose condensates gives rise to a ﬁnite temperature F β d x K ∂i∂jφ (45) EBC phase. For small nonzero SκS, the EBC exists as an intermediate phase between the superﬂuid and disordered phases. The phase correlator= isS then given( by:)

eik⋅x T φ x φ 0 d2k r2 log r (46) β βKk4 K ⟨ ( ) ( )⟩ ∼ ∼ − corresponding to unbinding of one-dimensional vortices. and the boson correlationS function is: Away from κ 0, however, we also expect a BKT unbind- iφ(x) −iφ(0) − T r2 log r ing transition of the conventional superfluid vortices at e e β e K (47) some other critical= temperature Tc1, and the interplay of these two transitions is not immediately obvious. In or- which decays exponentially,⟨ ⟩ indicating∼ the destruction of der to build the picture of the overall phase diagram, to- the boson condensate, as expected. In contrast, the corresponding exciton correlation function behaves as: gether with Tc2, we also estimate the unbinding temperature Tc1 for conventional vortices. The same argument i∂iφ(x) i∂ φ(0) − T log r 1 gives the free energy per vortex as F κ L2 T log `, i K e e β e η (48) where the energy cost for single vortex is κ L2 log `, r where L is fundamental charge of an∼L(S-particle.S − ) And where η T⟨ 4πK . We see⟩ that,∼ at any∼ finite temper- the critical temperature behaves as: S S ature, the exciton condensate still exhibits quasi-long- range order= ~( contributed) by the non-singular part of the field φ. By including the effect of vortices, this power- 2 Tc1 κ L (44) law correlation only persists until Tc2, at which point the one-dimensional vortices unbind and the condensate will ∼ S S be completely destroyed, resulting in exponential decay leading to a sharp suppression of Tc1 in the vicinity of the of all correlation functions. κ 0 critical point. In contrast, the finite-temperature In addition to correlations functions, we can also char- EBC phase has Tc2 almost independent of κ, remaining acterize the finite-temperature EBC phase by an unusual 2 Kq= . In AppendixC2, we also show that as long as thermodynamic property. The low-temperature thermo- the L-particles proliferate, the confined one-dimensional dynamics will be dominated by the quadratically dispers- particles become logarithmically interacting. Therefore, ing gapless mode. Generically, the specific heat contri- their proliferation at Tc2 is unaffected by the non-zero bution from a gapless mode scales as C T d~z, where z Tc1. This leads to the phase diagram depicted in Fig.4. is the dynamical critical exponent, ω kz, and d is the For large κ we recover the expected direct transition spatial dimension. In the present case, d∼ 2 and z 2, between the Bose condensate and the normal phase. In allowing us to conclude: ∼ the vicinityS S of the critical point, however, (specifically = = for κ K q L 2) the system will undergo two phase C T (49) transitions as the temperature is raised from zero, pass- ing throughS S < ( a~ new) intermediate finite-temperature phase. in the EBC phase. Such a T∼-linear specific heat is more At Tc1, the conventional vortices proliferate, and the con- commonly associated with a Fermi (or Bose) surface, and densate of bosons is destroyed. However, even in the provides a clear distinction from conventional superfluid absence of condensation of the fundamental bosons, the phases, where C T 2. excitons can remain condensed, leaving the system in a Finally, we note that the exciton condensate should not finite-temperature EBC phase. It is only at the higher lead to dissipationless∼ transport of any nontrivial quan- temperature Tc2 that the one-dimensional vortices pro- tum numbers besides energy. Motion of an exciton cor- liferate and the exciton condensate is destroyed, giving responds to motion of a particle-hole pair, which does way to the true disordered phase. not carry typical quantum numbers of the fundamental 10

+ x Exx − ∂x∂xφ ∂ ∂ φ ∂+∂+φ − − E+x

eiφ E+ − (∂ + ∂ )∂ φ (∂ + ∂ )∂ φ E (∂+ + ∂ )∂xφ x + x + Exx E ++ − − − −− E+x Ex+ E x E x Ex − iφ − − FIG. 5. Terms in the critical√ Hamiltonian of boson√ e on the 1 3 1 3 E+ E + +ˆ = + −ˆ = − + − − honeycomb lattice. 2 xˆ 2 yˆ and 2 xˆ 2 yˆ bosons, such as charge. Nevertheless, a particle-hole pair does carry energy, so we expect that the exciton con- FIG. 6. The boson field φ lives on the sites of the honeycomb densate will lead to dissipationless heat transport in the lattice. The three diagonal components of the tensor field Eij live on the sites of the triangular lattice (center of hexagons system, as proposed in the context of electronic exciton 83 of the honeycomb lattice), while the off-diagonal components condensates. live on the links of the triangular lattice (links of the honeycomb lattice).

V. LATTICE MODEL ables around a direct lattice link where a vector charge In the previous sections, we have always assumed that lives, involving four off-diagonal variables and two diag- the critical theory arises from an underlying lattice sys- onal variables whose repeating subscript is the same as the direction of the link. The operation Eij Eij 1 tem, in order to have a well-defined vortex of ∂iφ. In this section, we show how to put the critical theory and E → + its dual tensor gauge theory on the honeycomb lattice, −− which can host a continuous phase transition. On the honeycomb lattice, the bosons eiφ live on the ρ − sites. The boson current√ lives on the√ links along three 1 3 1 3 directions: ˆ 2 xˆ 2 yˆ, ˆ 2 xˆ 2 yˆ andx ˆ. There ρx are six distinct terms in the critical Hamiltonian listed in Exx Exx E Fig.5. Based+ = on this+ bosonic− = model,− + we can define the −− gauge variables on the dual triangular lattice. Notice that E++ jk ∂kφ r is the rank-1 dual current perpendicular to the boson current along k direction, which lives on the dual il jk ρ+ links. Then( ) Eij ∂l ∂kφ r is the difference of this rank-1 dual current along l direction and thus it is defined on the site of the= dual lattice( if)i j or at the center of the rhombus made up from two triangular plaquettes. E++ dual For example, Exx ∂yxj∂jφ ∆y=Jy is defined as the difference of two dual currents living on the successive y-links and it lives= at the site= of the triangular lattice. 1 dual dual Meanwhile, E+x Ex+ 2 ∆+˜Jy ∆yJ+˜ where the first term is the y-directed dual current difference FIG. 7. Three rhombus terms represent the Gauss law in the along the direction= perpendicular= ( to (denoted+ by) ˜ˆ) and vector charge theory. the second terms is the y-directed difference of current along the link perpendicular to . Therefore,+ on the+ dual for off-diagonal components creates four vector charges triangular lattice, there are three diagonal components of at once. The same operation for diagonal components the tensor Eii living on the sites+ while three off-diagonal creates two vector charges. These charge configurations components living on the three types of links as shown are listed in Fig. (8). in Fig. (6). Based on the charge pattern created by adding one to The Gauss’s law ∂iEij ρj in the vector charge the- a single Eij, we can immediately write down the gauge ory now corresponds to rhombus terms as shown in transformation for its conjugate Aij. Accordingly, we can Fig. (7). Each rhombus term= is a summation of six vari- write down the gauge invariant B ijkl∂i∂kAjl which

= 11

Exx Axx

A++ - A −− - -2A - −− -2Axx - -2A++ -

A - A++ −−

Axx

FIG. 9. The gauge invariant is a summation of twelve Aij variables including six oﬀ-diagonal components and six diagonal components. The minus sign in front of the variable indicates its sign in the summation for the gauge invariant.

FIG. 8. Possible charge configurations created by Eij → Eij + 1. Two vector charges are created by a diagonal element while four charges are created by an off-diagonal element. nian can be written in the form: 1 H K η ∂2φ 2 η ∂2φ 2 2 ∂ ∂ φ 2 n2 (50) x y x y 2 involves 21 variables within the orange hexagon as shown = ( ( ) + ( ) + ( ) ) + in Fig. (9). Previously, the boson model at η 0 on the square lattice was studied in the context of “exciton Bose liquid” (EBL) phases.70 The simplest EBL= phase is obtained with Hamiltonian: VI. EXCITON BOSE LIQUID 1 H K ∂ ∂ φ 2 n2 (51) EBL x y 2 Throughout this work, we have discussed the rank-two tensor gauge theory of Eq.3, and its dual scalar formula- This theory is similar= to the( isotropic) + critical theory, in tion in Eq.2, as a quantum critical point, either between that the gapless gauge mode has a quadratic dispersion. two different VBS phases or between a superfluid and Notably, however, this theory has two lines (kx 0 and a finite-momentum condensate. However, since there is ky 0) along which the dispersion vanishes exactly, i.e. only a single relevant direction at the critical point, it a “Bose surface.” Note that the Hamiltonian is invariant= seems plausible that some small modification of the the- under= the transformation: ory could eliminate the instability, resulting in a stable two-dimensional quantum phase of matter described by φ φ f x g y (52) a tensor gauge theory. In this section, we will describe a mechanism which can promote the critical tensor gauge where f x and g y →are+ functions( ) + ( of) only a single coor- theory to a stable quantum phase protected by a sub- dinate. This symmetry on φ implies the following con- system symmetry. Below, by stable phase, we mean the servation( law) on the( ) conjugate variable n: phase is stable under perturbations preserving the subsystem symmetry. (This phase was originally proposed dx n x, y constant dy n x, y constant to be stable against symmetry breaking perturbations as (53) well70, but this claim remains controversial.) representingS ( the) = conservation ofS boson( number) = on each The isotropic critical theory can become a stable phase row and column of the lattice. Previous studies on this on the square lattice through a slight modification in- model have shown that, unlike the isotropic theory, due troducing anisotropy. Accounting for square lattice to this subsystem symmetry61, single-derivative pertur- anisotropy, our previously encountered critical Hamilto- bations to the Hamiltonian are irrelevant, along with all 12 other perturbations, within a certain parameter regime.70 new type of deconfined quantum criticality. Such a crit- As such, the EBL describes a stable phase of matter, not ical theory naturally describes the transition between a critical point, as long as the subsystem symmetry is a superfluid and a finite-momentum condensate. Fur- preserved. thermore, this critical point gives rise to a new finite- Just like the EBC quantum critical point, we can also temperature phase of bosons, corresponding to an exci- capture the stable EBL phase with a “tensor” gauge dual, ton Bose condensate. Our work opens a new door in which is a simple repackaging of the previously studied the field of deconfined quantum criticality, allowing for self-duality transformation of this model.70 We obtained future study of exotic quantum critical points featuring the EBL Hamiltonian by dropping diagonal derivatives deconfined tensor gauge theories. from the isotropic theory. Similarly, we can obtain an appropriate gauge dual for the EBL by dropping diagonal elements of the tensor gauge field from the isotropic ACKNOWLEDGEMENTS theory. The resulting tensor will only have a single component, the off-diagonal element Axy, with its conjugate The authors acknowledge useful conversations with Exy, the resulting Hamiltonian takes the form: Mike Hermele, Albert Schmitz, Abhinav Prem, Sheng- Jie Huang, Yang-Zhi Chou, Rahul Nandkishore, Chong 2 1 2 Wang, T. Senthil, Olexei Motrunich, and Leo Radz- H KE ∂y∂xAxy (54) xy 2 ihovsky. MP is supported partially by NSF Grant 1734006 and partially by a Simons Investigator Award to This theory is invariant= under+ ( the pseudo-gauge) trans- Leo Radzihovsky from the Simons Foundation. HM was formation: supported by M. Hermele’s grant from the U.S. Depart- ′ ′ ment of Energy, Office of Science, Basic Energy Sciences Axy Axy f x g y (55) (BES) under Award No. DE-SC0014415. where f ′ x and g′→y are+ functions( ) + ( of) a single coordi- nate, as before. Note that this is not strictly a true gauge transformation,( ) ( ) since the gauge parameter cannot Appendix A: Conventional Boson-Vortex Duality be varied independently at all points in space. Corre- spondingly, the “Gauss’s law” of the theory no longer We here review the standard boson-vortex duality in has a local expression. Instead, we only have the integral 2 1 dimensions, which relates a superfluid of neutral equations: bosons to a non-compact U(1) gauge theory describing an (insulator+ ) of charged particles. These descriptions provide useful complementary ways of understanding not only the dx Exy constant, dy Exy constant (56) superfluid phase, but also the transition to a Mott insulator. In the superfluid, the primary dynamical variable overS each row= and column ofS the lattice,= closely mirror- is the phase field φ of the microscopic boson field, i.e. ing Eq. (53). A generic E configuration obeying these xy b b eiφ. This phase field represents the gapless Gold- conditions takes the form: 0 stone mode of the theory, while all other excitations are gapped.⟨ ⟩ = The low-energy Hamiltonian describing the dy- Exy ∂x∂yφ (57) namics of this field takes the schematic form: where φ is conjugate to n= ∂ ∂ A . Making these 1 x y xy H K ∂ φ 2 n2 (A1) replacements in the tensor gauge theory of Eq. 54, we i 2 obtain precisely the EBL Hamiltonian≡ of Eq. 51. Note where n is the boson= number( ) canonical+ conjugate to the that this duality transformation simply exchanges the angle φ. The system also supports topological excitations two terms of the Hamiltonian, swapping the scalar field where φ winds by 2π around a point, corresponding to φ for a “pseudoscalar” E . We can then regard the EBL xy vortices of the superfluid. Such vortex excitations will phase as being effectively self-dual. interact with each other through a logarithmic potential. In parallel, let us consider the properties of a noncompact U(1) gauge theory coupled to gapped charges, VII. CONCLUSIONS which mirror those the superfluid. This theory features a gapless mode (the photon), and gapped charges interact- In this work, we have initiated the study of quantum ing through a logarithmic potential. The Hamiltonian critical points described by tensor gauge theories featur- describing the gapless gauge sector takes the standard ing subdimensional particles. We first showed that a pre- form: viously studied quantum critical point between two va- i 1 2 lence bond solids maps exactly onto such a tensor gauge H KE Ei B (A2) structure. We further demonstrated that a deconfined 2 tensor gauge theory can exist at a critical point between where Ei is the two-dimensional= + electric vector field and ij two conventional gauge theories, representing an entirely B ∂iAj is the one-component magnetic flux through

= 13 the system. This Hamiltonian gives the gapless gauge decompose the field φ into its smooth and singular pieces mode a linear dispersion, matching with the properties as φ φ˜ φ(s), where φ˜ is a smooth single-valued func- of the Goldstone mode of the superfluid. The gapped tion, and φ(s) is the singular contribution from vortices. charges act as sources for the electric field through For a= system+ of normal superfluid vortices, the singular Gauss’s law: piece obeys:

i ij (s) ∂iE ρ (A3) ∂i∂jφ ρ (B2) with vortex density ρ. For a system featuring the uncon- In two dimensions, this equation= tells us that a point = ventional one-dimensional vortices discussed in the text charge has an electric field scaling as 1 r, leading to a (see Eq. 34), φ will obey a modified source equation: logarithmic interaction potential between charges. The above discussion indicates that the~ two theories, ik j` (s) j ∂i∂k∂lφ ρ (B3) the superfluid and the U 1 gauge theory, have the same j excitation spectrum. We can also directly map the two where ρ is the vector charge= density of the one- theories onto each other( and) match all physical observ- dimensional vortices. When these one-dimensional vor- ables. To begin, focus on the low-energy sector, where tices are confined to bound states, such that only con- there are no charges, so that the electric field obeys the ventional vortices are present in the system, this source i source-free Gauss’s law, ∂iE 0. The general solution equation will reduce to Eq. B2. We now introduce two to this equation takes the form: Hubbard-Stratonovich fields, a scalar B and a symmetric = tensor χij, in terms of which we write the action as: i ij E ∂jφ (A4) 2 1 ij 1 2 ij S d xdt χ χij B χ ∂i∂jφ B∂tφ (B4) for scalar field φ. The fields= Ei and Ai obey canonical 2K 2 commutation relations: = − + − The actionS is now linear in the smooth function φ˜, which Ei x ,Aj y ihδijδ x y (A5) can be integrated out, yielding the constraint: ij It then follows[ that( ) φ (is)] canonically= − ̵ ( conjugate− ) to B ∂tB ∂i∂jχ 0 (B5) ij∂ A , which we now relabel as n B. Plugging these ij i j It is now useful to introduce+ the= rotated field E expressions into the gauge theory Hamiltonian in Equa-= ik j` χk`, in terms of which the constraint: tion A2, we obtain precisely the superfluid= Hamiltonian ik j` = of Equation A1. We can also directly derive the corre- ∂tB ∂i∂jEk` 0 (B6) spondence between gauge charges and superfluid vortices. takes the form of the generalized Faraday’s equation of Consider the total charge enclosed within some curve C: + = the two-dimensional vector charge theory.24 The general 2 i i i solution to this equation can be written in terms of two Q d x ∂iE dn Ei ds ∂iφ ∆φ (A6) C C potential functions, a symmetric tensor Aij and a vector ξi: where= S ∆φ is the change= c in φ =going− c around= the− curve C. ik j` This indicates that a unit of gauge charge is equivalent B ∂i∂jAk` (B7) to a winding of φ, which is the definition of a vortex of which is invariant under transformation Ak` Ak` the superfluid. = ∂kλ` ∂`λk and ij ij i j j i → + + E ∂tA ∂ ξ ∂ ξ (B8) Appendix B: Duality in Reverse We can then write= the− action− ( in the+ form:) In the main text, we showed how to map from the rank- 1 1 S d2xdt EijE B2 ρiξ J ijA (B9) two tensor gauge theory onto the critical theory of the 2K ij 2 i ij VBS-VBS′ transition. For completeness, we here show = S ij ik j` − − − how to obtain the duality in the opposite direction, start- where J ∂i∂j∂t ∂t∂i∂j φ is a tensor current ing from the critical theory in terms of the φ variable. For of the one-dimensional vortices. The action is now in this purpose, it will be most convenient to work in the precisely the= form( of the Lagrangian− ) formulation of the 23,24,28 Lagrangian formalism. The action for the critical theory two-dimensional vector charge theory , with the takes the form: one-dimensional vortices playing the role of the vector charges. This action leads to one gapless gauge mode 1 with quadratic dispersion, ω k2, as expected from our S d2xdt ∂ φ 2 K ∂ ∂ φ 2 (B1) 2 t i j original model. This completes the derivation of the du- ∼ ′ = ( ) − ( ) ality between the critical theory of the VBS-VBS transi- (Note that K hereS differs by a factor of 2 from the definition and the two-dimensional vector charge tensor gauge tion in the main text, chosen for convenience.) We now theory. 14

Appendix C: Finite-Temperature Screening thermal background density n0 of the L-particles. These ij 24 particles see an effective potential given by L ∂iξj . i In the main text, we established the electrostatic prop- Therefore, in the presence of a potential ξ , the density erties of isolated particles at the critical point. In particu- will shift to: ( ) lar, we found a logarithmic interaction potential between ij −βL( ∂iξj ) ij the one-dimensional particles. This hints that the sys- nL n0e n0 1 βL ∂iξj (C3) tem should undergo a finite-temperature phase transition i at which the one-dimensional particles unbind. Unlike where β =1 T , and we have≈ assumed( − that( the)) potential ξ a system of normal logarithmically interacting particles, is small. (This assumption breaks down very close to the = ~ however, the tensor gauge theory also exhibits nontriv- point charge, but will capture the correct long-distance physics.) Using this form of the density, we obtain: ial bound states, namely the L-particles. These bound states have only a finite energy cost and therefore prolif- 2 erate at arbitrarily low temperatures. Previous studies of i i L 2 ′ `j ′ k ′ ξ r ξbare r n0β d r ∂`ξj r ξL r r three-dimensional fracton models27 have indicated that 4π (C4) screening by a thermal bath of nontrivial bound states [ ] = [ ] −ik ′ ( [ ]) [ − ] k Rk S can often significantly modify the interactions between where ξL R R′2 . Taking a Fourier transform and fundamental particles. We must therefore check care- solving for ξi, we obtain: fully whether or not the logarithmic energy cost survives [ ] = 2 j` ik screening by the thermal bath of L-particles. The calcu- j ij L n0β k` kk bare ξ δ 2 2 2 ξi (C5) lation will proceed in AppendixC1 as a straightforward 8π L n0β k extension of the screening analysis of Ref. 27 to the crit- = − ical two-dimensional tensor gauge theory. In the limit of strong+ screening, n0 , the trans- When κ 0 in Eq. 18, the one dimensional particles are verse component of the potential becomes completely confined as we studied in Sec.II. Meanwhile the two di- projected out, leaving us with: → ∞ mensional≠ particles have logarithmical interaction. This case corresponds to the condensates at two sides of the kikj ξj ξbare (C6) critical point. Additionally, at finite temperature, pre- k2 i vious study69 shows that the irrelevant perturbations, = together with the temperature, would also generate cor- With rection to relevant terms, i.e. making the effective κ finite q k q k although we tune it to zero to reach the critical point. In ξbare i i , (C7) these cases, the one-dimensional particles are subject to i k2 2k4 2 ( ⋅ ) interaction proportional to r where r is the separation j j (k⋅q)k = − between two of them. We are able to show in Appendix we get ξ 2k4 . In real space, the potential corre- C 2 that this strong confinement would be reduced to sponds to: logarithmical interaction as long as the two dimensional = particles condensate at any finite temperature. 1 q r rj ξj log r qj (C8) 8π r2 ( ⋅ ) = ( ) + 1. Logarithmically interacting one dimensional From this equation, we can see the survival of the particle logarithmic behavior of the bare potential. We conclude that, even after accounting for screening by ther- We showed earlier that the bare potential of an isolated mal L-particles, the one-dimensional particles still have vector charge qi takes the form: a logarithmic interaction potential, leaving the finite- temperature unbinding transition intact. Furthermore, 1 q r ri the composite object made up of the one-dimensional ξi 3 log r qi (C1) bare 8π r2 particle plus its screening cloud is still carrying a nonzero ( ⋅ ) = ( ) − vector charge, indicating that the screened particle re- In the presence of a screening cloud of L particles, how- mains one-dimensional. ever, the total potential surrounding a single vector charge will be modified to: 2. Confined interacting one dimensional particle i i 2 ′ i ′ i ′ ξ r ξbare r d r nL ξ r ,T ξ(L) r r (C2) When the one-dimensional particles are confined. [ ] = i [ ] + ( [ ] ) [ − ] where nL ξ ,T isS the local density of L-particles at They are subjected to an interaction εi proportional to temperature T and potential ξi, to be determined self- r2. (Note that the interaction energy is no longer equiv- i consistently,( and) ξ(L) is the potential of an L-particle, alent to ξi when κ 0.) In the momentum space, it is bare 4 from Eq. 37. We now assume that there is some finite proportional to εi κqi k . Via a similar analysis to ≠ ∼ ~ 15

Eq. C4 with the L -particles now logarithmically inter- particles are no longer strongly confined and instead they i acting, i.e. ∂iεL log r, it is straightforward to get the interact logarithmically. Therefore, they can proliferate screened interaction of d 1 particles to be: at finite temperature using Kosterlitz-Thouless criterion. ∼ When the proliferation temperature of one-dimensional q j 2 bare = i particles is larger than that of the two-dimensional par- ε k εi κ 2 κqi log r a (C9) k ticles, which is true for small κ, the system will host an From this calculation,∼ we∼ find∼ that due( ~ to) the prolifer- exciton Bose condensate phase. ation of two-dimensional particles, the one-dimensional

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