Charge-response of the Majorana toric code

# Charge-response of the Majorana toric code

Ananda Roy JARA Institute for Quantum Information, RWTH Aachen University, 52056 Aachen, Germany    Fabian Hassler JARA Institute for Quantum Information, RWTH Aachen University, 52056 Aachen, Germany
###### Abstract

At zero temperature, a two dimensional lattice of Majorana zero modes on mesoscopic superconducting islands has a topologically ordered toric code phase. Recently, a Landau field theory has been proposed for the system that captures its different phases and the associated phase-transitions. It was shown that with the increase of Josephson tunneling between the islands, a continuous symmetry-breaking 3D-XY transition gets transformed into a discrete symmetry-breaking 3D-Ising transition through a couple of tricritical points and first order transitions. Using the proposed field theory, we analyze the charge-response of the system at the different continuous phase-transitions. We calculate the universal conductivity at the 3D-XY transitions and the change in the superconducting density at the Ising transition using 1/N expansion. Furthermore, by computing a one-loop correction to the field theory, we show that an additional tricritical point is likely to be present in the phase-diagram. Finally, we provide a mean-field calculation that supports the earlier proposed field theory.

## I Introduction

One of the most promising platforms for fault-tolerant quantum computationDiVincenzo (2009) is the toric code proposed by Kitaev.Kitaev (2003, 2006); Fowler et al. (2012) The ground space of the toric code is four-fold degenerate where two qubits can be encoded. The primary advantage of this way of encoding is that the degeneracy of the ground space depends only on the topology of the space on which the toric code is implemented and thus, is robust towards local perturbations. A direct realization of the toric code is by placing qubits on the links of a large square lattice with periodic boundary conditions and performing measurements that project the system to the topologically ordered ground space. Kitaev (2003) Alternately, a Hamiltonian can be designed whose the low-energy sector can be described by the toric code Hamiltonian.Levin and Wen (2006); Kitaev (2006) This is the case for a system of spins located at the edges of a 2D honeycomb lattice, interacting via alternating , and interactions around each plaquette of the lattice. For a suitable choice of interaction strengths, the system is in the toric code phase. Kitaev (2006)

In contrast to the aforementioned schemes of realizing the toric code using spins/qubits, alternative approaches have been proposed using Majorana zero modes.Xu and Fu (2010); Terhal et al. (2012); Nussinov et al. (2012); Vijay et al. (2015); Landau et al. (2016); Karzig et al. (2017); Litinski et al. (2017) In these approaches, a 2D lattice of mesoscopic superconducting islands is considered. On each of these islands, two Kitaev wiresKitaev (2001) are embedded which contain localized Majorana zero modes. Due to the finite size, each island has a finite charging energy, denoted by , where is the capacitance of each island to a ground plane. Due to the Josephson effect, the Cooper-pairs tunnel between different neighboring islands at a rate . Furthermore, the Majorana zero modes enable tunneling of single electrons between two neighboring islands at a rate .Fu (2010) For vanishing Josephson tunneling rate and , the system is in the topologically ordered toric code phase.Xu and Fu (2010); Landau et al. (2016) Upon increasing the single-electron tunneling rate, the system undergoes a topological phase-transition of 3D-XY type into a topologically trivial state.Xu and Fu (2010) Moreover, in the limit of infinite Cooper-pair tunneling rate, the system is also in the toric code phase. In this case, the phase-transition to the topologically trivial state is of 3D-Ising type. A Landau field theory analysis for the system has been done in Ref. Roy et al., 2017. It was shown that upon increasing the Josephson coupling strength from zero, the line of the 3D-XY topological phase-transition terminated at a 3D-XY tricritical point. Subsequently, it turned first order, which then terminated at a 3D-Ising tricritical point. Further increase of the Josephson coupling made the phase-transition a 3D-Ising one. The charge signatures of the different phases are as follows. For , the system is a Mott-insulator. Upon Increasing , the system undergoes an additional 3D-XY phase-transition into a charge- superconductor. Fisher et al. (1989); Fazio and Schön (1998); Fazio and van der Zant (2001); Herbut (2007); Sachdev (2011) Most importantly, the system stays in the toric code phase both as a Mott-insulator and as a charge- superconductor. Upon increasing from either one of these phases, the system makes a topological phase-transition to a charge- superconductorWen (1991); Balents et al. (1999); Senthil and Fisher (2000). Depending on the strength of , it is the nature of this phase-transition that changes from 3D-XY to 3D-Ising.

Given that the phases have distinct charge signatures, we investigate the charge response across the different phase transitions. Measurement of the superconducting densities and correlation lengths reveals the critical exponents.Roy et al. (2017) Most importantly, a measurement of the electrical conductivity provides a unique signature that distinguishes between the different phase-transitions. The main goal of this work is to provide quantitative predictions of the conductivity that are amenable to experimental verification.

In the context of the Bose-Hubbard model at zero temperature, the conductivity at a 3D-XY transition has been shown to be a universal value. This conductivity value has been computed numerically using Monte-Carlo methodsCha et al. (1991) and analytically using 1/N expansion Cha et al. (1991) and expansion Fazio and Zappalà (1996). These results can be directly applied to the two 3D-XY transitions and the 3D-XY tricritical point of our model. However, for the Ising transition and the Ising tricritical point, a fresh computation is necessary. This computation is nontrivial due to the fact that the current is carried by two coupled Ising degrees of freedom, out of which only one undergoes the phase transition. We do this computation to leading order in the interaction using 1/N expansion. We show that as the system undergoes the transition from a charge- to a charge- superconductor, there is a jump in the superconducting density. In contrast to the 3D-XY transitions, there is no dissipative component to the conductivity.

In addition, we compute a one-loop correction to the field theory describing the system’s transition from a Mott-insulator to a charge- superconductor. This calculation indicates that another tricritical point of a 3D-XY type is likely to be present in the phase diagram. Lastly, we also perform a mean-field analysis of the model that provides additional support to the Landau field theory proposed in Ref. Roy et al., 2017. Throughout this work, we restrict ourselves to zero temperature.

The paper is outlined as follows. First, we describe the basic building block of our model, Majorana zero modes on mesoscopic superconducting islands, in Sec. II. Then, we describe the microscopic Hamiltonian of the system in Sec. III. We map the problem to coupled spins and rotors, with nearest-neighbor interactions using Jordan-Wigner transformation in Sec. IV. We compute the one-loop correction to the field theory in Sec. V. We present the conductivity calculations in Sec. VI. We provide a concluding summary in Sec. VII. Finally, Appendix A contains the mean-field calculation.

## Ii Superconducting islands in the topological regime

In this section, we provide a concise summary of how charging effects can be consistently incorporated in the mathematical description of topological superconductors. We consider a toy model for a topological superconductor, the Kitaev wire. Kitaev (2001) It describes the effect of coupling a quantum wire (modeled by a chain of spinless, fermionic modes) to an infinitely large superconductor. As the superconductor screens the charge of the electrons in the quantum wire, the relevant fermionic degrees at low energies are chargeless (Majorana) fermions.Chamon et al. (2010); Beenakker (2014) Even more interesting, in the topological phase of the Kitaev wire, Majorana zero modes, denoted by operators , appear at its ends. These operators are Hermitian and obey the Clifford algebra . Furthermore, the Majorana zero modes lead to a twofold degeneracy of the ground state. In a topological superconductor, these exotic degrees of freedom are in fact non-Abelian particles.Read and Green (2000); Ivanov (2001) As a result, topological superconductors realize a platform for topological quantum computation.Nayak et al. (2008); Beenakker (2013)

All these results essentially rely on the fact that the superconducting phase is a classical variable that has a well-defined value. The problem of extending the results to a superconducting island with a fluctuating phase has been addressed, e.g., in Refs. Fu, 2010; Xu and Fu, 2010; van Heck et al., 2011. It has been shown that the connection between the neutral Majorana zero mode operators and the charged electronic annihilation operator is given by .Alicea (2012) Note that the charge on the superconducting island is carried by the Cooper-pairs in the condensate and thus, is measured by where is the elementary charge. Thus, the process of removing an electron from the superconducting island (described by ) both flips the fermion parity (because of the action of ) and removes a charge from the island (because of the action of ). After all, fundamentally, the particles on the topological superconductor are electrons. So any physical state of the system has to be reachable by applying the electronic operators , to the vacuum state without any particles present.

These arguments lead to the conclusion that the physical Hilbert space is not simply given by the tensor product of the condensate Hilbert space (on which and act) and the Majorana Hilbert space (on which and act). Instead, the states are physical only if they are connected to the vacuum state by addition or removal of electrons. Counting the fermion parity, denoted by the operator , with the charge on the island, denoted by the operator , yields the constraint

 (−1)Q/eψ=Pψ. (1)

All states in the physical Hilbert space have to fulfill this constraint. Therefore, in the case of a pair of Majorana zero modes on an island, even though the zero modes are chargeless, the two ground states (that differ in fermion parity with ) are not degenerate in energy. This is because they necessarily correspond to a different charging of the mesoscopic superconducting island: for the case of and for the case of .

On first sight, it might appear that the finite charging energy of a mesoscopic superconducting island prevents any degeneracy due to the neutral Majorana zero modes. However, it can be easily seen that this conclusion is incorrect for the situation where there are two Kitaev wires and thus four Majorana zero modes on the island. In this case, the fermion parity constraint still is given by (1) but with a fermion parity operator that involves all of the zero modes. In particular, we imagine to pair () with the ground state degeneracy given by the two eigenvalues of the partial parity (). In the Majorana sector, the ground state is four fold degenerate with the four states given by , with . As before, the finite charging energy will split states that correspond to different eigenvalues of the charge operator . However, in this case the constraint (1) only implies that the states with different total fermion parity are not degenerate. This leaves the degeneracy of the set of states and of the set of states . So even at finite charging energy, there is a finite topological degeneracy of the ground state remaining.111Note that for case of Majorana zero modes, the remaining topological degeneracy will be .

Physically, the degeneracy of the states can be understood due to the fact that a single unpaired electron can be moved from the fermionic mode to (or the other way round). As a result, the fermion parity on each of the two pairs changes without changing the charging energy. More interestingly, the degeneracy of the state with arises from the fact that starting with the state can be reached by breaking a Cooper pair and filling both the modes and . This process does not involve taking any charge to or from the island and thus, does not change the charge on the superconducting island. Note that due to the presence of the zero modes, one does not pay the usual superconducting energy gap to break the Cooper-pair.

A remaining question is how to describe the tunneling of single charges between two superconducting islands . Note that in a multi-island situation the parity constraint (1) has to be fulfilled on each of the islands separately. For concreteness, let us assume that the zero modes on island is close to the mode on island . The resulting tunneling of single electrons leads to a term

 He↔e∝iγidγjacos(ϕi−ϕj2), (2)

also dubbed -periodic Josephson effect.Kitaev (2001) It is easy to see that under the action of this Hamiltonian [Eq. (2)], the state still satisfies the constraint [Eq. (1)] both on island and . The reason is that while the -term adjusts the charge (by increasing it by on one of the islands and decreasing it on the other island correspondingly), the factor flips the fermion parity on each side such that physical states are mapped onto physical states in this process.

## Iii Model Hamiltonian of the Majorana toric code

With this introduction, it can be seen that the Hamiltonian of the system depicted in Fig. 1 is given by , where

 HC =4EC∑in2i,HJ=−EJ∑⟨i,j⟩cos(ϕi−ϕj), HM =−EM∑⟨i,j⟩Vi,jcos(ϕi−ϕj2). (3)

Here, the superconducting phase and the excess charge (in units of Cooper pairs) on the -th island are canonically conjugate. We treat the idealized case of zero offset charges in the absence of disorder. The Majorana tunneling operator between the two neighboring islands is given by (see Fig. 1) Terhal et al. (2012)

 Vμ+^z,μ−^x =iγμ+^zcγμ−^xb, Vμ−^z,μ−^x =iγμ−^zaγμ−^xd, Vμ−^z,μ+^x =iγμ+^xcγμ−^zb, Vμ+^z,μ+^x =iγμ+^xaγμ+^zd, (4)

where the are Hermitian operators. The fermion parity on the -th island is given by the operator . As the charge is constraint by the fermion parity, the (physical) Hilbert-space for the Hamiltonian is spanned by the wavefunctions satisfying [cf. Eq. (1)].Fu (2010) At finite charging energy, the ground state is in the even parity sector on each island (). In this sector, the four Majorana zero modes on each island encode one qubit Bravyi (2006) and, neglecting , a perturbation calculation in yields the toric code Hamiltonian.Xu and Fu (2010); Landau et al. (2016)

## Iv Mapping to a coupled spin-rotor Hamiltonian

In this section, we map the Hamiltonian to that of spins coupled to rotors using Jordon-Wigner transformation. Note that the Jordan-Wigner mapping presented in this section is different from the one presented in Ref. Terhal et al., 2012. Unlike the mapping of Ref. Terhal et al., 2012, the current one keeps the size of the Hilbert space invariant and thus, captures the degeneracies in the spectrum of the Hamiltonian.

First, we perform a gauge transformation in order to simplify the Hilbert-space to -periodic functions.van Heck et al. (2012) This is done by applying a unitary transformation

 H→Ω†HΩ, ψ→Ω†ψ (5)

where

 Ω=∏ieiqiϕi/2, qi=1−Pi2. (6)

As a result, now only -periodic wavefunctions correspond to the physical states of the system. After the transformation, stays invariant, while the gets transformed as

 HC =4EC∑i(ni+qi2)2, HM =−EM2∑⟨i,j⟩(e−iqiϕiVi,jeiqjϕj+H.c.). (7)

Next, we map the Majorana zero modes into spins using a Jordan-Wigner transformation. Consider an lattice with periodic boundary conditions. Starting from any point on the lattice, we enumerate the Majorana zero modes traversing the lattice first in the direction (along the blue arrows in Fig. 2). Once all the lines along the direction have been traversed, we do the same in the direction (along the orange arrows in Fig. 2). Thus, each island is traversed twice, once in the and then in the direction. For the -island, while traversing in the direction, the map maps the Majorana zero modes to spins as follows:

 M(γ2ik−1) =[ik−1∏j=1σzj]σxik, M(γ2ik) =[ik−1∏j=1σzj]σyik, k=1,2. (8)

Under this mapping, the fermion parity operator of the -island is given by

 Pi=−γiaγibγicγid=γ2i1−1γ2i1γ2i2−1γ2i2=σzi1σzi2. (9)

Using Eqs. (III) and (IV), for any link at the interior of the lattice (connecting sites with neither being a multiple of ), the are transformed as

 Vik,jk=σxikσxjk, k=1,2. (10)

The interactions on the links at the boundary, where the lattice wraps around (connecting sites with or being a multiple of ) are nonlocal. For instance, along the first line of enumeration, the interaction coupling the Majorana modes and is

 VL,1=[L∏i=1σzi]σx1σxL. (11)

However, one can check that the product of the fermion parity along each of these lines is a conserved quantity.Nussinov et al. (2012) Thus, the Hilbert space splits into sectors where each of these . In what follows, we will fix all the these quantities to be . Similar analysis can be done for other choices. Naturally, for open boundary conditions, this nonlocal interaction does not arise. Thus, the thermodynamic properties are described by the interactions on-site and those mediated by the internal links.

Next, we reduce the effective Hilbert space size that the interaction Hamiltonian acts on. To that end, we lay down a Bell-basis for the spins on the -island.Terhal et al. (2012) We define a sign qubit () and a target qubit () on each island, whose joined state is given by :

 |s=0,t=0⟩ =1√2(|00⟩+|11⟩), |s=0,t=1⟩ =1√2(|01⟩+|10⟩), |s=1,t=0⟩ =1√2(|00⟩−|11⟩), |s=1,t=1⟩ =1√2(|01⟩−|10⟩). (12)

Thus, is the sign bit and the is the two-qubit parity bit of information in the superposition. In this basis, the operators , get mapped to:

 σxi1|s,t⟩=σxi,tσzi,s|s,t⟩, σxi2|s,t⟩=σxi,t|s,t⟩, σzi1|s,t⟩=σxi,s|s,t⟩, σzi2|s,t⟩=σzi,tσxi,s|s,t⟩, (13)

where are the Pauli operators for the sign (target) qubits. Thus, the interaction Hamiltonian along direction gets mapped to an effective interaction between the target qubits where the sign of the interaction is determined by the eigenvalue of the sign qubits. We can define a classical bit on each link connecting the and the island in the direction that is the product of the -eigenvalues of the corresponding sign qubits (cf. Fig. 3).

The resultant interaction Hamiltonian, after a trivial rotation on the target qubits, is then written by (we drop the -index on the spins for clarity)

 HC =4EC∑i(ni+1+σzi4)2, (14) HM =−EM2∑⟨i,j⟩si,j{σ−iσ−j(eiϕi+eiϕj) +σ−iσ+j[1+ei(ϕi−ϕj)]+H.c.},

while remains invariant.Coo () Here, the sign of the interaction is determined by gauge bits . The state-space is subjected to the following constraints:

 ∏iσzi=1, ∏w∏□sij=1, (15)

where denote the product over all the white plaquettes (see Fig. 3).222Note that is trivially true. The above constraints arise from the fact that the total fermion number parity of all the islands is conserved. The spectrum of the Hamiltonian in Eq. (III) is given by the union of the spectra of Eq. (14) for different gauge bit configurations. Here, gauge bit configurations are nonequivalent if their product around any plaquette differs.Fradkin (2013) The Hamiltonian is invariant under the simultaneous transformations :

 eiϕi↦eiϕieiθ, σ+i↦σ+ieiθ/2. (16)

Physically, this global symmetry originates from the fact that the spins correspond to single-electrons that carry half of the charge of the Cooper pairs.

The different phases of the system arise depending on how or break the symmetry.Roy et al. (2017) In the phase when the Josephson tunneling is the strongest, the symmetry is spontaneously broken, the rotors align and the ground state of the system is invariant under rotation by multiples of . Then, the system is a conventional superconductor of Cooper pairs (of charge ). We denote this phase by . When the Majorana-assisted single electron tunneling is the strongest, the symmetry is again broken. However, now the spins are aligned and the ground state is invariant only under rotation by multiples of . Moreover, the alignment of the spins in turn aligns the rotors. As a result, the system is an ‘exotic’ superconductor Wen (1991); Balents et al. (1999); Senthil and Fisher (2000) of charge- bosons. We denote this phase by . Finally, when the charging energy is the strongest, the complete is restored, the rotors and the spins are disordered and the system is a Mott insulator. We denote this phase by .

The relevant field theory capturing the phase-diagram of this system was proposed in Ref. Roy et al., 2017. In the next section, we compute the one-loop correction to the field theory describing the transition between the phases and . Our calculation shows that an additional tricritical point is likely to be present in the phase-diagram which was not captured by the mean-field analysis of Ref. Roy et al., 2017.

## V One loop correction to the field theory

Consider the coarse-grained expectation values of and in imaginary time , denoted by complex fields and . The relevant degrees of freedom are given by the low-frequency, long-wavelength behavior of these complex fields. The microscopic symmetry is elevated to the symmetry on the coarse-grained variable that has to be respected in the effective field theory. Since close to the phase transitions, the fields are small, the action can be expanded in a Taylor and gradient expansion in . The partition function at zero temperature is given by with the Euclidean action

 S= ∫d2rdτ[|∂τψs|2+|∂τψr|2+Ks|∇ψs|2+Kr|∇ψr|2 +rM|ψs|2+rJ|ψr|2+us|ψs|4+ur|ψr|4 +β|ψs|2|ψr|2−α(ψ∗s2ψr+ψ2sψ∗r)]. (17)

The parameter is used to tune through the phase transition and corresponds to , where . From stability considerations, must be positive and we choose without loss of generality.

The phase transition between the phases and is the conventional Mott-insulator to superconductor transition.Fisher et al. (1989); Herbut (2007); Sachdev (2011) Across this transition, stays zero, while turns finite. Integrating over small fluctuations around the saddle point , we get an effective partition function withRoy et al. (2017)

 S(b) =∫d2rdτ(|∂τψr|2+Kr|∇ψr|2+rJ|ψr|2+ur|ψr|4).

Here, we have kept the renormalization of the couplings to zeroth order in . The phase transition line is given by and it becomes metastable in the phase . Next, we perform calculations going beyond the zeroth order in . We show that due to the cubic term proportional to in Eq. (V), this line of phase transition also terminates in a tricritical point. After this point, the transition turns first order. For the ease of computation, we define and rescale the spatial axis so as to set . The rescaled stiffness of the rotor sector is denoted by .

Considering small-fluctuations around the saddle point: and expanding to second order, the effective action can be written as

 S(b)=S[δψs]+S[ψr]+S[δψs,ψr] (18)

where the actions for the spin and rotor fields are given by:

 S[δψs] =∫x{|∂τδψs|2+|∇δψs|2+rM|δψs|2}, S[ψr] =∫x{|∂τψr|2+~Kr|∇ψr|2+rJ|ψr|2+ur|ψr|4},

while the interaction between the two is given by

Here, . Now, the field can be perturbatively integrated out. The diagrams contributing to the renormalization of the action to one loop order are shown in Fig. 4.

Thus, the effective action for the rotor field is given by

 S[ψr] =∫x{(1+δ~Kr)|∂τψr|2+(~Kr+δ~Kr)|∇ψr|2 +(rJ+δrJ)|ψr|2+(ur+δur)|ψr|4},

where

 δ~Kr =4α2∫p1(rM+p2)3 δrJ =−4α2∫p1(rM+p2)2+β∫p1(rM+p2) δur =−β22∫p1(rM+p2)2+4α2β∫p1(rM+p2)3 −144α4∫p1(rM+p2)4, (19)

where . Note that there are no infrared singularities in these diagrams since as we are far from the phase-transition in the spin-sector. The renormalization of the spin-wave stiffness can be evaluated to be:

 δ~Kr=18πα2r3/2M (20)

The renormalization of the gap-parameter has the usual cut-off-dependent shift as is common in one-loop renormalization group analysis:

 δrJ=−α22πr1/2M−βr1/2M4π+βr1/2M2π2Λ, (21)

where is the ultraviolet cut-off. Finally, the renormalization of is given by:

 δur =−116πβ2r1/2M+18πα2βr3/2M−94πα4r5/2M. (22)

For large , . Also, for both large and small values of , . Thus, without fine-tuning, for generic parameter choices, . This implies that the stabilizing quartic interaction is depressed. Thus, it is likely that this line of 3D-XY phase-transition also terminates in a tricritical point. Then, a sextic term generated from integrating out the field stabilizes the theory. After the tricritical point, the transition turns first order. The phase-diagramRoy et al. (2017), including the additional tricritical point , is shown in Fig. 5.

## Vi Conductivities at the different phase-transitions

In this section, we investigate the charge-response signatures of the different continuous phase-transitions of our model. As discussed in the previous section, the transition between the phases and is of 3D-XY type, which likely terminates in a 3D-XY tricritical point. The same was shown to be true for the transition between the phases and .Roy et al. (2017) The universal value of conductivity across the 3D-XY phase-transition lines has been calculated analytically using 1/N expansion Cha et al. (1991) and expansionFazio and Zappalà (1996). These results can be directly applied for the 3D-XY transitions of our model and the 3D-XY tricritical points. On the other hand, a 3D-Ising line terminating at a 3D-Ising tricritical point separates the the phases and . The charge response across this transition has not been analyzed before. We do this calculation using 1/N expansion.

In what follows, first, we summarize the results for the 3D-XY transitions. Since we need the 1/N formalism for the subsequent calculation, we sketch the details in the context of the 3D-XY transitions. Then, we perform the relevant analysis for the 3D-Ising transition. Finally, we provide the conductivities across the different tricritical points.

The Kubo formula for the conductivity is given byMahan (2013); Sachdev (2011)

 σxx(ikz)=ℏkz∫d3xeik⋅xδ2FδAx(x)δAx(0)∣∣A=0, (23)

where , and . The coupling to the electromagnetic field is performed by making the previously encountered global U(1) symmetry, denoted by angle [cf. Eq. (16)], a local one, space-time dependent: . Thus, can be obtained from the earlier action given by Eq. (V) after the minimal substitution: . Here, and are the charges of the spin and rotor fields and are given by and respectively. This is because while the rotor sector conducts electric current via Cooper pairs, the spin sector, arising out of the Majorana zero modes, conduct current through single electrons. The DC-conductivity is obtained by first analytically continuing our result to , followed by taking the limit of (see Chap. 3 of Ref. Mahan, 2013). The electric current is carried by both spin and rotor current fields, given by

 Js(r)(x) =12i{ψ∗s(r)(x)∇ψs(r)(x)−ψs(r)(x)∇ψ∗s(r)(x)}. (24)

From Eq. (23), we get:

 σxx(ikz)=e2sℏρs(kz)kz+e2rℏρr(kz)kz−4eserℏρsr(kz)kz, (25)

where

 ρs(r)(kz) =2⟨ψ∗s(r)(0)ψs(r)(0)⟩ −4∫d3x⟨Js(r),x(x)Js(r),x(0)⟩eik⋅x, (26) ρsr(kz) =∫d3x{⟨Js,x(x)Jr,x(0)⟩ +⟨Jr,x(x)Js,x(0)⟩}eik⋅x. (27)

### vi.1 Conductivities at the 3D-XY transitions

Consider the 3D-XY transition between the phases and [line (a) in Fig. 5]. The relevant action is given byRoy et al. (2017)

 S(a) =∫d3x{|∂τψs|2+Ks|∇ψs|2+rM|ψs|2 +uM2|ψs|4+~uM|ψs|6}. (28)

For the , the transition is of the 3D-XY type and the sextic term can be neglected. Chaikin and Lubensky (2000) We will treat this case in this subsection. The tricritical points are treated in Sec. VI.3. We perform perturbation in the inverse of the dimensionality of the order parameter .Ma (1973); Coleman (1988); Cha et al. (1991); Zinn-Justin (2002) For this case, with complex order parameter, . The effective action in terms of the N-component order parameter is given by

 S(a)eff = ∫d3x{∇ψ∗s,α∇ψs,α+rMψ∗s,αψs,α (29) +uM2(ψ∗s,αψs,α)2},

where now denotes the 3D gradient. We have rescaled the spatial axis so as to set . We consider complex fields, , , where .

For this effective field theory, the conductivity is given by

 σxx(ikz)=e2sℏρs(kz)kz, (30)

where is defined in Eq. (VI) after the transformation: . The current operator is defined by Eq. (24) after the same transformation. We define the Fourier transform of :

 ψs,α(x)≡1√Ω∑qeiq⋅xψs,α(q), (31)

where is the 3D volume element. In Fourier domain, the effective action is given by

 S(a)eff =∑q(q2+rM)ψ∗s,α(q)ψs,α(q)+12Ω∑quM ∑p,p′ψ∗s,α(p+q)ψ∗s,β(p′)ψs,α(p)ψs,β(p′+q)}. (32)

The expression for is given by:

 (33)

Since in three dimensions, the quartic coupling is relevant, we treat it perturbatively using the 1/N-expansion. To obtain a non-trivial result, we must choose , where stays constant. Following Ref. Coleman, 1988, we can decouple the interaction by Hubbard-Stratonovich transformation, introducing an auxiliary field , leading to an effective action:

 S(a)eff[ψs,α,ζ] =∫d3x{∇ψ∗s,α∇ψs,α+rMψ∗s,αψs,α +N2gMζ2+iζψ∗s,αψs,α}. (34)

The linear term in the above action can be removed by shifting the field: , where is determined by

 ∂~S(a)eff[ζ]∂ζ∣∣ζ=¯ζ=0, (35)

with

 e−~S(a)eff[ζ]=∫Dψs,αDψ∗s,αe−S(a)eff[ψs,α,ζ]. (36)

The action then becomes

 S(a)eff[ψs,α,ζ] =∫d3x{∇ψ∗s,α∇ψs,α+~rMψ∗s,αψs,α +N2gMζ2+iζψ∗s,αψs,α+NgM¯ζζ}, (37)

where , is the renormalized gap parameter that goes to zero at the phase-transition. The last two terms cancel, thereby eliminating the linear term in . The free propagator for the field is given by

 Gψ0(q)=1q2+~rM (38)

and the resummed two-body interaction mediated by the is given by

 ~uM(k)=uM1+uMN′Πψ(k), (39)

where

 Πψ(k)=∫d3q(2π)3Gψ0(k+q)Gψ0(q). (40)

Note that as , the effective action is free with . The 1/N-expansion can now be performed by replacing by . The details of the calculation can be found in Ref. Cha et al., 1991. Here, we merely state the result:

 (41)

where in the last equality, we have set for the 3D-XY transition.

For the phase-transition between the and the phases [line (b) in Fig. 5], the same calculation can be performed, with the following substitution: . Thus, in that case, the DC-conductivity is given by

 σxx=π2(1−329π2)e2h. (42)

### vi.2 Conductivity at the 3D-Ising transition

For the effective action for the transition between and [line (c) in Fig. 5], we use the parametrizationRoy et al. (2017)

 ψr = (¯ρr+δρr)eiθr/¯ρr, (43) ψs = (σ+iw)eiθr/2¯ρr (44)

where is the saddle point value of and the real fields , and , denote the fluctuations of and . The fluctuations in correspond to the massless Goldstone mode associated with the symmetry breaking in the rotor sector. They decouple from the rest. As described in Ref. Roy et al., 2017, the emergent Ising degree of freedom undergoes the transition, while the fields stay regular with gap parameters . The effective action for the Ising degree of freedom is given by

 S(c) =∫d3x{(∂τσ)2+Ks(∇σ)2+tcσ2+uc2σ4 (45) +~ucσ6}.

Again, in this subsection, we consider the case when (see Sec. VI.3 for ). In this new parametrization, the electric current is carried by [using Eq. (24) and Eqs. (43,44)]

 Js =σ∇w−w∇σ+σ2+w22¯ρr∇θr, (46) Jr =(¯ρr+δρr)2¯ρr∇θr. (47)

Defining Fourier transforms as in Eq. (31) for , , and , we can write the effective action in Fourier domain. From Eq. (45), the action for the -field, undergoing the Ising transition, is given by

 S[σ] =∑q(q2+tc)σ(q)σ(−q)+12Ω∑puc ∑q,q′σ(q)σ(q′)σ(p−q′)σ(−p−q′). (48)

Since the fields remain regular, we only need the action governing them to quadratic order. The action for the -field is given by

 S[w]=∑q(q2+rw)w(q)w(−q), (49)

The mean-field form of is given byRoy et al. (2017)

 rw =rM+α√−2rJur−βrJ2ur (50) =α√−8rJur, (51)

where the last line is valid only on the Ising-transition line. The actions for