How magnetic field can transform a superconductor into a Bose metal

We discuss whether a simple theory of superconducting stripes coupled by Josephson tunneling can describe a metallic transport once the coherent tunneling of pairs is suppressed by the magnetic field. For a clean system, the conclusion we reach is negative: the excitation spectrum of preformed pairs consists of Landau levels; once the magnetic field exceeds a critical value, the transport becomes insulating. Nevertheless, we suggest that it is still possible to realize a Bose metal by introducing strong disorder. Such disorder may localize some pairs; the coupling between propagating and localized pairs broadens the Landau levels, resulting in a metallic conductivity. Our model respects the particle-hole symmetry, which leads to a zero Hall response. It has also been demonstrated that the resulting anomalous metallic state has no Drude peak and the spectral weight of the cyclotron resonance must vanish at low temperatures.


I. INTRODUCTION
The discovery of anomalous superconducting state in the stripe-ordered LBCO [1,2] has gradually arouse interest in a possibility of Pair Density Wave (PDW) state -a superconducting state where pairs carry finite momentum (see [3] for the most recent review of the field). At zero field this 3D layered material exhibits an unusual 2D superconductivity with an in-plain Berezinskii-Kosterlitz-Thouless (BKT) transition coexisting with a finite resistivity along the c-axis. This resistivity vanishes at much smaller temperature marking an onset of 3D superconductivity with Meissner effect.
The recent experiments on several stripe-ordered superconductors have revealed that once the superconductivity is destroyed by the applied magnetic field, a peculiar resistive metallic state with zero Hall response emerges, which persists down to lowest temperatures [4][5][6]. The Hall response vanishes at the BKT temperature, and it remains zero throughout the entire low temperature region even when the superconductivity is destroyed. The critical field is relatively small; the electrical resistance gradually increases with the field, and at fields ∼ 25 − 30T the sheet resistance reaches a plateau at R ≈ 2π /2e 2 .
Similar anomalous metallic states have been observed in disordered thin films (see, for instance, [7] and references therein) and Josephson junction arrays [8]. The situation with Josephson junction arrays bears the closest resemblance to the one which takes place in the stripeordered LBCO. It has been theoretically analyzed under assumption that the space between the superconducting granules is filled with an ordinary metal [9][10][11]. Since the fermionic quasiparticles are necessary there, one cannot * Electronic address: tren@bnl.gov † Electronic address: atsvelik@bnl.gov call such state a Bose metal. The purpose of this paper is to find out whether one can describe the anomalous metal without invoking quasiparticles. Experimentally, the stripe-ordered LBCO appears to be a good candidate for this. Its simplest description [12] does not invoke quasiparticles and this is the the prism through which we will consider this system. We augment the model by the long range Coulomb interaction and call it Wire Theory (WT for short). WT may be considered as a minimalistic model of PDW since it does not introduce any other entities besides preformed pairs. It describes the stripe-ordered state as an assembly of one dimensional superconducting wires (stripes) separated by insulating regions and coupled by Josephson tunneling. Each wire constitutes a Luther-Emery liquid, where the interactions generate a spin gap responsible for the superconducting pairing. The bulk superconductivity emerges when the pair tunneling establishes a global coherence. The distinguishing feature of WT is the suggestion that the Josephson coupling has the wrong sign so that the sign of the superconducting order parameter alternates between the neighboring stripes. Since the stripe orientation changes along the c-axis, the sign alternation frustrates the pair tunneling in this direction. It is also assumed that the low energy sector of the system is occupied by bosonic excitations -charged pairs and spin fluctuations from the undoped copper oxide chains between the stripes. If WT is correct and there are no fermionic quasiparticles, then the low temperature resistive state is some kind of Bose metal where the transport is carried by incoherent pairs.
Below we will study WT both in zero and finite magnetic field. We demonstrate that in its simplest form the theory cannot explain the experiments and hence requires certain modifications. It is shown that the excitation spectrum of pairs experiences Landau quantization. Although the nonlinear effects lead to a certain broadening of the Landau levels, in the clean system this effect is not sufficiently strong to lead to metallic transport. As a consequence, once magnetic field destroys the coherence of pairs, the system becomes an insulator. We have found a loophole which allows existence of resistive state in the purely bosonic theory of PDW, but does not explain the universal features of the transport. A sufficient broadening of the Landau levels is achieved if we assume the presence of small localized pairs. Due to their small size, these states do not experience Landau quantization and serve as a reservoir of low energy states for the transport. An alternative possibility was described in Section III. A. 3 of [13] where one of the authors suggested that the resistive state in the stripe-ordered LBCO owes its existence to the presence of fermionic quasiparticles. This brings us to theories described in [9][10][11].
Our model respects the particle-hole symmetry, which leads to a zero Hall response that is consistent with the experiments [4,6,14]. The resulting anomalous metallic state possesses no Drude peak. As for AC conductivity, the spectral weight of the cyclotron resonance vanishes at low temperatures, which is also consistent with the experiment [7].

II. THE MODEL
As we have stated in Introduction, the model of the stripe-ordered state we consider is of a 3D array of coupled Luther-Emery liquids augmented with the long range Coulomb interaction. One can consider two versions of it: one where the stripes consist of single doped chains and another where they consist of double chains (see Fig. 1). In the first (second) case the superconductivity competes with 2k F (4k F ) charge density wave, or CDW for short, since at weak interactions both susceptibilities are singular. If the paring susceptibility is singular in the second case, the charge density correlations at 2k F are short ranged (see, for example [15]). To simplify matters, we will consider the case when the CDW matrix elements are zero. In this case, the Lagrangian can be written as (2.1) We find this form suggested in [16] where dual bosonic fields θ, φ are both present more convenient.
If we adopt the gauge where ∇ · A = 0, then the A 0 component decouples and can be integrated out. The resulting contribution to the Lagrangian density is where a 0 is the distance between the stripes. Since the backscattering term is absent in the current case, we can also integrate over φ-fields and obtain a closed expression for the Lagrangian: where the matrix C ij is defined through This completes our construction of WT. Before we consider the transport properties of WT, we first discuss two relevant issues: the plasmon mode in zero magnetic field and the pairing susceptibility of a single stripe.
A. The plasmon mode in zero magnetic field In the superconducting phase where θ fields are pinned either at 0 (J > 0) or at 0 and π on alternating stripes (J < 0). The later case can be reduced to the former by the substitution θ j = jπ +θ j whereθ is a slow function of coordinates. Then we can expand the cosine term and obtain the spectrum where a 0 is the distance between the stripes and C(q) is the Fourier component of the matrix C ij . In the arrangement relevant to stripe-ordered cuprates, the stripes comprise a three dimensional structure and the Fourier transform of C ij depends on all three components of the wave vector. At small wave vectors C(q) ∼ |q| −2 , therefore the spectrum in 3D is gapped, as is customary for 3D plasmons. In literature related to the stripe-ordered states, the Goldstone excitations in Fulde-Ferrel-Larkin-Ovhinnikov (FFLO) superconductor have been considered [17], but the issue of the Coulomb interaction were not addressed. As for experimental data, the value of the in-plane plasma frequency for x = 1/8 LBCO extracted from the optical measurements [18] is ∼ 1600 cm −1 (200 meV). The same experiments give the spectral gap the estimate ∼ 20 meV so that the plasma frequency is well above this cutoff, and we can ignore the effect of the plasmon mode in following sections.

B. Pairing susceptibility for a single stripe
As far as pairing susceptibility is concerned, we will show that the long range nature of the Coulomb interaction does not change matters qualitatively. For this, we calculate the pairing susceptibility for a single stripe. We assume that the system is 3D and integrate over the transverse momentum Q over the 2D slice of the Brillouin zone. In what follows we neglect the q z -dependence of C(Q, q z ). Then the pairing susceptibility for a single stripe is obtained as where v(Q) = v[C(Q)/π] 1/2 and N is a normalization factor such that we get back to the noninteracting result when we set e = 0. This function can be approximately replaced by the standard power law with the scaling dimension where α = 16e 2 /π, d 0 = 1/4. As a result, we can imitate the effect of the long range Coulomb interaction as equivalent to the renormalization of the Luttinger parameter.
In what follows, we focus on the case where d 1 when the Josephson tunneling is relevant.

III. FINITE MAGNETIC FIELD. RPA APPROACH
In this section we study the low temperature regime in strong magnetic field when the superconductivity is suppressed. We treat the magnetic field inside the sample as uniform. We expect this approximation to be valid in strong magnetic fields. Below we consider the case with J < 0. In our analysis, we use the Random Phase Approximation (RPA), which replaces the original action for the order parameter field Φ = exp(iθ) by the Gaussian one: where χ P is the pairing susceptibility for a single stripe calculated in Section II B. The formal expansion parameter of RPA is the inverse number of nearest neighbors, so we expect our results to be valid only qualitatively. In the momentum representation things become even more convenient: where h = 2eHa 0 /c. To be compared with experimental data, it is convenient to express it as where µ B is the Bohr magneton and m e is the electron mass. To calculate the Green's function G ≡ −i ΦΦ † we have to solve the equation The formal solution is expressed via normalizable eigenfunctions satisfying 5) such that the Green's function can be expressed as Taking into account that a 0 ≈ 1.5 nm we have 2 /(4m e a 2 0 ) ≈ 10 2 K. For H = 30 T, we obtain ha 0 ≈ 1/3, which means that for the entire experimental range of [4][5][6] it is reasonable to adopt ha 0 1. Then we can reformulate the eigenvalue problem (3.5) as the differential equation: It is more convenient to perform the calculations in imaginary time, since then the Green's function is real, and to do analytic continuation afterwards. For T = 0 we have where Λ is the high energy cutoff, and we will set Λ = 1 and restore it when necessary. Let us perform the rescaling such that Eq. (3.7) assumes the dimensionless form: where the eigenfunctions Ψ k (x) can be chosen to be real. Then the Green's function after analytic continuation iω n → ω + i0 becomes Since the eigenvalue for Ω n = 0 is on the order of one, the critical value of h corresponding to Ω n = 0 is , (3.12) above which the superconductivity is lost. It should be noted that the present calculation is valid only for h > h c , when the cyclotron radius for pairs v/ 0 is finite.
To simplify the expressions, below we will set 0 = v = 1, and recover them when necessary. For real Ω n , Schrödinger equation (3.10) has a discrete spectrum with eigenvalues depending on Ω n . The wave functions for even k are even, for odd k are odd and vanish at x = 0. It is interesting to express the Green's function in real space: It is obvious that for d = 0, functions f k (iΩ n ) have branch cuts so that under analytic continuation iΩ n → ω + i0 they become complex. This means that the discrete levels of the Schrödinger equation (3.10) acquire finite width. Moreover, since CDW is suppressed since we are considering stripes made of double chains, the CDW operator will not couple strongly to disorder.
We can calculate δf k (ω) = f k (ω) − f k (0) for small ω using the first order perturbation theory: 14) followed by the analytic continuation iΩ n → ω +i0. Then the imaginary and real parts of δf k (ω) are The calculation of the imaginary part depends on k. For even k, we have (3.17) To summarize, we use the following short notations: where α k , β k are dimensionless constants depending on the scaling dimension d. Close to the critical field at frequencies ω 0 we can write the Green's function as

19) where the first term in the summation is
20) and the other terms with nonzero k are similar. It can seen from the poles of G R that the excitation spectrum of the pairs correspond to damped Landau levels. Another important feature of the Green's function is that it possesses the particle-hole symmetry such that G R (ω) = [G R (−ω)] * = G A (−ω). This will lead to a zero Hall response, which we will discuss in Sec. III A.

A. Conductivities
The current operators for currents along and transverse to the stripe direction are given by the following expressions: The Kubo formula for the conductivity is We now focus on the DC conductivity at h > h c , which can be calculated by setting q = 0 in Eq. (3.22) and taking the limit ω → 0. The conductivity along the stripe direction is obtained as (3.24) Performing the analytic continuation iΩ n → ω + i0 and we obtain F k1,k2 (ω) = 1 4J 2 π dy coth(y/2T ) − coth[(y + ω)/2T ] ω mχ k1 (y) mχ k2 (y + ω). (3.25) If we perform a shift of the integration variable y → y −ω followed by a reflection y → −y, using the particle-hole symmetry that mχ k (ω) = − mχ k (−ω), we then obtain F k1,k2 (ω) = F k2,k1 (ω). This ensures that the summation over k 1 and k 2 won't make the longitudinal conductivity σ xx (ω) vanish. For magnetic fields above the transition point in the limit of zero frequency we obtain (3.26) The DC conductivity σ xx decreases with temperature, resulting in an insulator, though without a sharp gap. The conductivity transverse to the stripe direction is obtained as This expression is even in h as it should be, and it has a similar structure as σ xx (ω), where the matrix elements are nonzero only when the eigenfunctions Ψ k have different parity. Consequently, σ yy (ω → 0) has the same behavior as σ xx (ω → 0) at low temperatures. The Hall conductivity is obtained as (

3.28)
This expression is odd in h as it should be. Since F k1,k2 (ω) = F k2,k1 (ω) as shown above, the summand in this expression is odd under the exchange k 1 ↔ k 2 . As a result, the summation over k 1 and k 2 makes σ xy (ω) vanish identically. This shows that the particle-hole symmetry ensures a zero Hall conductivity, which is consistent with the experiments [4,6,14]. To summarize, the DC transport calculation in this section shows that both longitudinal conductivities, along or transverse to the stripe direction, vanishes as powers of T at low temperatures, while the Hall conductivity vanishes identically. As a result, when superconductivity is suppressed, WT gives out an insulator with a soft gap on the level of RPA, excluding the possibility of a Bose metal in a clean system.

IV. A POSSIBLE RESCUE
So it appears that WT in a clean system is not capable to describe the metallic state. One possible loophole is to imagine that due to a strong disorder the system supports localized pairs.
A particular fine structure of the stripe configuration is that above some critical magnetic field ∼ h c , there is an intermediate region between the superconducting stripes, termed as charged insulator [19] . This region can accommodate charges in the form of pairs localized by quenched disorder (see Fig. 2). Such localized pairs are likely to be present between the superconducting stripes. Due to their small size, they are not subject to Landau quantization and can act as a reservoir of low energy state necessary for broadening of the Landau levels. Localized pairs are coupled to the order parameter field Φ j (x) through the Andreev reflection mechanism: where the Pauli matrix operator σ + (x) creates a localized pair at point x within the charged insulator, and h(x) suffers from quenched disorder. In the second order of perturbation theory in g, the contributions of the localized pairs to the RPA self energy is After averaging over randomh we obtain for the imaginary part where c is the concentration of localized pairs, provided the distribution of disorder is wide enough to encompass the frequency ω. As a result, to leading order in g 2 , Eq. (3.4) is modified as where we ignored the effect of eΣ R , which can be incorporated into renormalization of the Josephson coupling constant. The resulting disorder-averaged Green's function for h > h c still has the form in Eq. (3.19), but the function χ k (ω) is modified by the extra imaginary term in Eq. (4.4). For example, the function χ 0 (ω) in Eq.
The extra imaginary term in Eq. (4.4) introduces another contribution to the longitudinal conductivity σ xx along the stripe direction in Eq. (3.26): where ω 0 corresponds to the first Landau level and σ 0 ∼ γ 2 is a constant. As a result, the leading term for σ xx (ω → 0) is finite even at T → 0, resulting in a Bose metal. As we can also see, this anomalous metal does not have a Drude peak. The same analysis also applied to calculation of the longitudinal conductivity σ yy (ω → 0) transverse to the stripe direction, while the Hall conductivity σ xy (ω → 0) still vanishes due to the symmetry argument stated below Eq. (3.28). It should be noted, however, that in contrast to the experiment which shows that at high fields the sheet conductance approaches the universal value ∼ 2e 2 /2π , the mechanism we discuss does not. As far as AC conductivity is concerned, we will be interested in the cyclotron resonance peak. Assuming that the transition frequency ω k − ω 0 >> T , from (3.25) we obtain for ω near the cyclotron resonance that (4.7) Since the integrand contains no divergence in the limits of integration, this integral is nonsingular. Therefore, there is no cyclotron resonance at T → 0. It is easy to see that the absence of the cyclotron resonance at T → 0 is related to the absence of the Fermi sea for particles with Bose statistics. Hence for temperatures much smaller than the energy of the first peak, the spectral weight of the cyclotron resonance is small ∼ T . This is consistent with the experiment [7].

V. DISCUSSION AND CONCLUSIONS
In this paper, we discussed a simple theory of superconducting stripes separated by insulating region and coupled by Josephson tunneling. Our results suggest that in a sufficiently strong magnetic field the excitation spectrum of the preformed bosonic pairs consists of Landau levels. The corresponding excitations have a finite life time due to the interactions, disorder and temperature effects. In a clean system, the broadening is modest leading to power law temperature dependence of the longitudinal conductivity, which is insufficient to keep the conductivity metallic above the critical magnetic field when superconductivity is lost. So in magnetic field the clean system undergoes a transition from a 2D superconductor to a weak insulator.
One way to explain the observations of the anomalous metal in the recent experiments in the stripe-ordered LBCO [4] is to invoke quasiparticles as was suggested in Tsvelik's earlier paper [12]. This scenario has been discussed in the context of think films and Josephson junction arrays [9][10][11]. There are theories where PDW coexists with fermionic quasiparticles [20][21][22]. Some of them [20,21] respect the particle-hole symmetry, the others [22] do not.
In this paper we considered a different possibility, namely when all low energy excitations remain bosonic. The essence of our proposal is that strong disorder leads to creation of localized pairs or, perhaps, small superconducting grains. Due to their small size, their spectrum does not experience Landau quantization and hence provides a reservoir of low energy states for the metallic transport. This alternative also provides a mechanism for Bose metal other than the exotic proposal of quantum phase glass [23,24], where the Josephson coupling constant is random in sign in the latter proposal. The anomalous metallic state described in the present paper possesses some of the features observed in the stripeordered LBCO [4] and the indium oxide films [7]. It has zero Hall response, no Drude peak and no cyclotron res-onance at low temperatures. It would be interesting to measure terahertz conductivity for LBCO.

VI. ACKNOWLEDGEMENTS
We are grateful to Andrey Chubukov and John Tranquada for very valuable discussions. We thank Peter Armitage for attracting our attention to Ref. [7]. This work was supported by the Office of Basic Energy Sciences, Material Sciences and Engineering Division, U.S. Department of Energy (DOE) under Contract No. DE-SC0012704.

Appendix A: Effect of Disorder in Josephson Coupling
In this appendix, we consider the effect of disordered Josephson coupling on the simple WT. The quenched disorder of the Josephson coupling between superconducting stripes is assumed to be δ-correlated in space: where we also assume that the disorder is weak ∆ J J 2 such that it won't change the sign of J. We use the replica trick to calculate the disorder-averaged Green's function: where the replicated action is It is more convenient to express it in momentum space: We calculate the one-loop correction to the diagonal Green's function, assuming that the replica symmetry is unbroken. By taking the limit n → 0, we obtain the equation for the diorder-averaged Green's function: where G(ω 2 , q 2 , q 2 ; p 2 ) is the disorder-free Green's function in Eq. (3.6). By performing the summation, we arrive at essentially the same equation as in Eq. (3.4), only the Josephson coupling constant J is replaced by an effective one: For h < h c , we have χ k (ω) < 0, so the weak disorder weakens the Josephson coupling. For weak disorder ∆ J J 2 , the Josephson coupling remains its sign with a smaller magnitude, so we have a stable superconducting phase, only the effective critical magnetic field is reduced.
We now consider the situation where the disorder is weak to maintain the sign of the Josephson coupling, but yet strong enough to reduce the critical magnetic field greatly. Then we can make the approximation by setting h = 0 in Eq. (A.6), resulting in an effective Josephson coupling constant as which determines the effective critical magnetic fieldh c through Eq. (3.12) by replacing J withJ. Then the disorder-averaged Green's function G can be obtained from Eq. (3.19) by replacing h c withh c , and the analysis of conductivities aboveh c showed in Sec. III A can be carried through without change. As a result, the weak disorder in Josephson coupling between superconducting stripes only shifts the critical point without changing the nature of the phase transition, so it still results in an insulator above the transition point.
The above conclusion can be understood using a coarse-graining picture. In presence of disorder, we will have regions where the Josephson coupling between stripes is strong as well as regions where it is weak. The effective critical magnetic field is determined by the Josephson coupling constant via Eq. (3.12). For two stripes where the Josephson coupling between them is strong such that the effective critical magnetic field exceeds the applied magnetic field h c > h, we have superconducting correlation across them and we can fuse them into a single stripe. After one step of such coarsegraining, we end up with exactly the same WT defined in Eq. (2.1), only with a smaller Josephson coupling constant between the effective stripes. If this coarse-graining procedure can be carried on until we have a single stripe, we are in the superconducting phase. If otherwise, we ob-tain a WT with effective critical magnetic field h c < h, then the calculation of the conductivities in Sec. III A tells us that we are in the insulating phase. As a result, a Bose metal cannot arise.