Bose Metal via Failed Insulator: A Novel Phase of Quantum Matter
BBose Metal via Failed Insulator: A Novel Phase of Quantum Matter
Anthony Hegg, Jinning Hou ( 侯 晋 宁 ), and Wei Ku ( 顧 威 )
2, 3, ∗ Tsung-Dao Lee Institute, Shanghai 200240, China Tsung-Dao Lee Institute & School of Physics and Astronomy,Shanghai Jiao Tong University, Shanghai 200240, China Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), Shanghai 200240, China (Dated: February 23, 2021)Two of the most prominent phases of bosonic matter are the superfluid with perfect flow and the insulatorwith no flow. A now decades-old mystery unexpectedly arose when experimental observations indicated thatbosons could organize otherwise into the formation of an entirely different intervening third phase: the Bosemetal with dissipative flow. The most viable theory for such a Bose metal to-date invokes the use of the extrinsicproperty of impurity-based disorder, however a generic intrinsic quantum Bose metal state is still lacking. Wepropose a universal homogeneous theory for a Bose metal in which phase frustration confines the quantumcoherence to a lower dimension. The result is a gapless insulator characterized by dissipative flow that vanishesin the low-energy limit. This failed insulator exemplifies a frustration-dominated regime that is only enhancedby additional scattering sources at low-energy and therefore produces a Bose metal that thrives under realisticexperimental conditions.
Superfluidity has fascinated many with the phenomenon ofperfect (dissipationless) flow since its discovery[1, 2]. Oneof the few homogeneous phases of matter known to disruptthis low-temperature behavior is the Mott insulator[3, 4] char-acterized by a complete lack of flow. Bosons underlie bothphases of matter and are well-known to exhibit other exoticand extreme properties such as Bose-Einstein condensation.For decades it had been understood that bosons can undergoa transition from the perfect flow of superfluidity directly intothe Mott insulating state of no flow, exemplifying this extremenature. In the absence of inhomogeneities this abrupt transi-tion was understood to dominate low-energy bosonic systemsin general.Therefore, an unexpected mystery arose over the last sev-eral decades as experiments[5–10] have continued to find ev-idence that seemingly disparate bosonic systems exhibit moremundane dissipative transport. Surprisingly, the prevailingtheories could not account for this behavior whatsoever. Inthe intervening decades, several theories were developed tofill this unexpected gap, but despite these efforts there remainsno consensus even for a qualitative account of these observa-tions. Identifying a universal mechanism that allows for a sta-ble phase of dissipative bosonic transport at low-temperatureremains a holy grail of condensed matter physics and ultra-cold atom research.The unavoidable extreme nature of bosons has guided the-oretical developments toward several specialized directions asopposed to a universal qualitative understanding of this metal-lic phase. Early research focused on superconducting grainmodels[11–14], but apparent metallic behavior was shownto be unstable at finite temperature[15] and a large classof models were further ruled out by scaling arguments[16].Another approach used exotic interactions on a lattice suchas ring exchange models[17, 18], but these are unstable to-ward insulating phases with the realistic introduction of weakdisorder[19]. A more recent approach involves the so-calledmoatband models[20–26]. These models are solved in the extreme dilute limit where the particle density scales muchslower than the volume, which is not the regime relevant tothe experimental observations. A top contender[19] appliesextrinsic phase frustration to avoid the issues of the abovesystems but leaves the question of a universal intrinsic mech-anism unanswered.Instead of attempting to tame the extreme tendencies ofbosons by hand, we propose a universal picture where bothextremes coexist and stabilize metallic behavior. Consideran emergent system of freely flowing bosons composed oflower dimensional subsystems. If flow between subsystemsvanishes at low energy due to perfect phase interference, theneach subsystem is effectively disconnected. We find a gaplessinsulating state protected by full frustration of the quantumphases that is highly vulnerable to temperature or disorder.This failed insulator is mediated by a dimensional crossoverbetween freely moving bosons and disconnected subsystems,providing a universal origin for stable metallic behavior, sowe expect this type of Bose metal to be ubiquitous in nature.We consider a relatively simple implementation of thisBose metal by stacking 2D checkerboard lattice xy -layersalong the z -axis to form a 3D lattice. Alternatively, this sys-tem can be viewed as two overlapping sets of stacked vertical xz - and yz - slabs. We study the regime in which the hoppingbetween such vertical slabs is weaker than that within eachslab. In the low-energy limit these vertical slabs become dis-connected. This model has been suggested[27] to representthe underdoped cuprates beyond the superconducting dome,a class of material suspected to exhibit Bose metal behaviorthrough the formation of an emergent Bose liquid (EBL)[27–30]. Even if we drive each slab into superfluidity by intro-ducing a small repulsive local interaction, transport betweenslabs is still suppressed at low energy. Although proving thata superfluid phase exists can be somewhat subtle[31], provingthat it does not exist is far less stringent. The lack of flowbetween slabs at low energy forbids superflow, so the intro- a r X i v : . [ c ond - m a t . s up r- c on ] F e b duction of temperature or disorder results in a Bose metal as afailed insulator. RESULTS AND DISCUSSION
Γ MX E n e r gy ( 𝜏 ′′ ) (e) (f) (a) (c) (b) k x k y y x z ZΓ τ"τ" τ'τ' (d) DOS20
FIG. 1. The checkerboard lattice. (a) There are two sublattices, de-noted red and blue for horizonal and vertical site orientation respec-tively, with nearest neighbor hopping τ (cid:48) > τ (cid:48)(cid:48) following site orientation. (b) Inthe frustrated regime ( τ (cid:48)(cid:48) > τ (cid:48) ) the band structure has a line degen-eracy for its minimum denoted in green. (c) The corresponding dis-persion with orbital weight given in red and blue as in (a) with Γ , X,M and Z representing (0 , , π , , π , π ,0) and (0 , , π ) respec-tively. Note the clear separation of orbitals along the line degeneracy.(d) The 3D density of states (DOS in states per 100 τ (cid:48)(cid:48) per unit cell)converge to a constant at low energy due to the emergent two di-mensional nature of the excitations near the line degeneracy. (e-f)Magnification at low energy shows that the low-lying excitations arequadratic. The intralayer structure of the 2D checkerboard lattice aswell as its inherent two-band nature is shown in Fig.(1). Inthe xy -layer, this lattice contains hopping to four nearest-neighbors (NN) τ (cid:48) > τ (cid:48)(cid:48) >
0. We assume the z -axishas simple nearest neighbor hopping with τ z <
0. Finally, weinclude on-site repulsive interactions U > H = ∑ i (cid:40) ∑ j ∈ NN τ (cid:48) a † i a j + ∑ j ∈ NNN τ (cid:48)(cid:48) a † i a j + Ua † i a † i a i a i (cid:41) (1)where a † i and a j are the bosonic creation and annihilation op-erators at sites i and j respectively.We study the frustrated τ (cid:48)(cid:48) > τ (cid:48) regime of this model in thelow-temperature T → U = ε ± , k = τ z ( cos k z − ) + τ (cid:48)(cid:48) ( cos k x + cos k y + ) (2) ± (cid:113) τ (cid:48) ( + cos k x )( + cos k y ) + τ (cid:48)(cid:48) ( cos k x − cos k y ) . In the unfrustrated regime τ (cid:48) > τ (cid:48)(cid:48) , there is a single lowestenergy momentum state. The bosons condense into this stateat T = d -wave superfluid obtains[32]. However, in thefrustrated regime τ (cid:48)(cid:48) > τ (cid:48) the lowest energy momenta form adegenerate line along the intralayer Brillouin zone boundaryin momentum space as shown in Fig.(1). This line degeneracyis a reflection of the decoupling of the xz - and yz -slabs at low-energy due to perfect interference of the τ (cid:48) hopping.Systems composed of arbitrarily many lower-dimensionalsubsystems are poorly studied to-date. Therefore, we usemany-body methods directly when possible. We use a con-trolled approximation in which a reference state is first es-tablished as the dominant contribution to the zero temperaturemany-body ground state, and we then expand the Hamiltonianto bilinear order in fluctuations about this reference state. Weconstruct the many-body eigenstates using the solution to theHamiltonian in order to compute the current-current responsefunction in the low-frequency limit. From the result we estab-lish the conductivity and therefore the low-temperature phaseof matter. Low-Energy Effective Theory and Stability
The number of sites V and the number of particles N areassumed to scale with the system size, whereas the number oflower-dimensional systems M must necessarily scale slowersuch that M (cid:28) V . In a non-interacting system the N particleswould be placed in the M subsystems over all possible (indis-tinguishable) configurations equally. In our system the repul-sive interactions reduce the number of lowest-energy statesconsiderably by spreading the particles out as uniformly aspossible in each subsystem. We assume the repulsion is cho-sen to be weak enough to avoid the Mott insulator regime,so we do not need to distinguish between commensurate andincommensurate density.We choose to use the physically meaningful density andphase operator formalism to represent the system. Such a rep-resentation has a long and unresolved history[33]. The dif-ficulties of this representation are eliminated if we allow forparticle-hole symmetry and study fluctuations of density andphase about our reference state[34], which is chosen from thesolution to Eq.(1) at U =
0. We first transform to density n andphase φ operator formalism and formally expand each opera-tor in fluctuations η and θ about the local average values n and φ i of our reference state a i = e i φ i √ n i ≈ e i φ i √ n (cid:18) + η i n + i θ i (cid:19) (3) i δ ii (cid:48) = [ η i , θ i (cid:48) ] . (4)To enforce the internal consistency of the theory in terms oflocal density we add a Lagrange multiplier K = H − µ ∑ i ( n i − n ) to constrain the average particle number. We expand tobilinear order in the fluctuation operators and find the eigen-solution K = E + ∑ l E l b † l b l , (5)where b † l creates a single particle eigenstate of energy E l .Many-body eigenstates | l λ (cid:105) close to the reference state arenow | l λ (cid:105) ≈ {| Ω (cid:105)} ∪ { b † l a λ | Ω (cid:105)} (6) E l λ = { E } ∪ { E + E l } , (7)where | Ω (cid:105) is the many-body ground state with energy E and a λ removes a particle from the slab λ .Note that E is independent of the overall phase φ i ∈ λ ofeach slab. That is, the overall phase of each slab remains un-correlated, and therefore the slabs are still disconnected. Thisleaves a large amount of remaining degeneracy in the many-body ground state.As a demonstration, Fig.(2) represents the solution E l cor-responding to a coincidentally coherent phase structure φ α xyz = e i π ( x + y ) , (8)where α ∈ { , } and xzy is the location of a primitive unitcell. This choice maintains momentum as a good quantumnumber ( l → k ). Note that the line degenerate minimum ismaintained, and we find the characteristic linear spectrum atlow-energy in the direction perpendicular to that line indica-tive of stiffened phase within each slab. Correspondingly, thedensity of states vanishes in the low energy limit, proving thatthis system is stable at finite temperature and not simply azero-temperature idealization. Metallicity
We now demonstrate the absence of superfludity in this sys-tem via the computation of the current-current response func-tion in the low-frequency limit χ bb (cid:48) ( x , x (cid:48) , ω ) = ∑ l λ (9) × (cid:40) (cid:104) Ω | J b ( x i ) | l λ (cid:105) (cid:104) l λ | J b (cid:48) ( x (cid:48) i (cid:48) ) | Ω (cid:105) ω − ( E l λ − E ) + i η − (cid:104) Ω | J b (cid:48) ( x (cid:48) i (cid:48) ) | l λ (cid:105) (cid:104) l λ | J b ( x i ) | Ω (cid:105) ω + ( E l λ − E ) + i η (cid:41) , where | l λ (cid:105) , | Ω (cid:105) denote the many-body eigenstates of the sys-tem identified in the previous section, x i is the real space loca-tion x corresponding to site i , b and b (cid:48) are vector componentsof the current operator J .
2Γ MX (d) E n e r gy ( 𝜏 ′′ ) (e) (f) 0 (a) (c) (b) k x k y DOS y x z ZΓ FIG. 2. The checkerboard lattice in the presence of local repulsiveinteractions U >
0. (a) Each shade of orange represents an indepen-dent coherent slab with a randomly selected overall phase. (b-c) Anexample with ( π , π , ) periodic phase structure shows that the bandstructure maintains the line degeneracy and separation of orbitalsfrom Fig.(1). (d) The corresponding DOS goes to zero linearly withenergy indicating that the ground state is stable at low-temperature.(e-f) The low-lying excitations are now stiffened into a linear spec-trum. We use the following thought experiment to identify an ob-servable corresponding to long-range coherent transport. Twoconducting leads are attached to a sample in an arbitrary ori-entation as in Fig.(3). These leads are centered at x and x (cid:48) such that the number of lattice spacings a between them islarge ( R = | x − x (cid:48) | (cid:29) a ). The size of the leads themselves spanmany lattice spacings as well, but far fewer than the distancebetween the leads. Long-range coherent transport is achievedwhen the current-current response from one lead to the otherconverges to a finite value for arbitrarily large R and arbitrar-ily small frequency ω . We represent this using the followingpart of Eq.(9) σ ( x − x (cid:48) , ω ) = ω Im χ ll ( x − x (cid:48) , ω ) , (10)where χ ll ( x − x (cid:48) , ω ) is the longitudinal component of thecurrent-current response function at frequency ω betweenpoints x and x (cid:48) . We verified that Eq.(10) produced the cor-rect superfluid density in the unfrustrated regime τ (cid:48) > τ (cid:48)(cid:48) .In the frustrated regime τ (cid:48)(cid:48) > τ (cid:48) we continue to use the co-incidentally coherent phase structure defined above for con-venience. This provides an upper bound on the coherent re-sponse since any randomness in the phase would only reducecoherence and therefore transport. We find that σ ( R , ω → ) → . (11)This result can be readily understood from the disconnectednature of the slabs in the solution of Eq.(5). The hopping pa-rameter τ (cid:48) that couples neighbor coherent slabs is suppressed Sink Source 𝑅 𝑦 ≫ 𝑎 𝑅 𝑥 ≫ 𝑎 𝑎 FIG. 3. Hypothetical experimental probe of the current with two graycontacts separated by a distance R = (cid:113) R x + R y much larger thanthe lattice spacing a . Particles emerge from the source and travel instraight lines along a given coherent slab. The probability for the cur-rent to flow between sublattices vanishes in the low-energy limit, Sono path remains to reach the sink if either R x or R y are large enoughto place the leads on distinct coherent slabs. at low energy. As a result, any transport that crosses throughsuccessive coherent slabs is also suppressed. As demonstratedin Fig.3, no coherent slabs intersect the source at e.g. x and thesink at e.g. x (cid:48) simultaneously. Therefore, the DC conductiv-ity vanishes[35] identically for all x , x (cid:48) , and λ , which includeboth the superfluid and normal dissipative DC response.In essence the reason for this insulating result is not thatdifferent from the microscopic mechanism of Anderson in-sulators, in which randomness provides enough interferenceto localize states without opening a gap. In our many-bodyground state we have perfect frustration between slabs andas a result we find inter-slab localization. Anderson insula-tors, however, have an energy scale protecting the localizedstates, which must be overcome in order to reach a channelfor transport. Our inter-slab frustration has no such protec-tion as the frustration is imperfect for any state other than theabove many-body ground state.This is precisely why in reality this idealized result shouldfail to remain insulating in the presence of finite temperatureor weak disorder. At finite temperature the response is gener-ated by the presence of dissipative carriers b † l a λ . Introductionof weak local disorder disrupts the perfect frustration couplingneighboring slabs and provides a nearby channel for transport.We refer to such a system as a failed insulator, and we have used it here to obtain a Bose metal. The key physics that leadsto metallic transport through perfect frustration comes froma dimensional crossover between lower-dimension free mov-ing boson systems embedded in a higher-dimension systemsuch that they are isolated at low-energy. In this example ef-fectively independent slabs exhibit long-range coherence in-dividually while avoiding macroscopic coherence.In fact, full coherence can be recovered via disruption of theperfect interference, for example, by introducing anisotropyin τ (cid:48) . In that case, the ground state would host a superfluidwith p -wave symmetry. Accepting the claim[28] that thismodel is representative for the extreme low-doping ( < d -wave superfluidity by a quantum criticalpoint due to the difference in their local symmetry. CONCLUSIONS
By introducing perfect frustration among many adjacentnearly free flowing subsystems, we have discovered a long-sought universal stable Bose metal phase intervening betweensuperfluid and insulator. Utilizing a gapless failed insulatorin the presence of temperature or disorder immediately leadsto dissapative transport. We demonstrate this concept via a 2-band Bose-Hubbard model in an extended checkerboard lat-tice with frustrated coupling between vertical slabs and a par-ticle density that scales with the system volume. We findthe many-body ground state consists of disconnected slabs.We find that the system is stable at finite temperature due tothe long-range phase coherence within each slab. The low-frequency conductivity and the superfluid response both van-ish in a general direction. The universal mechanism of ourBose metal leads to a novel stable phase of quantum matterthat is robust under realistic conditions and should thereforebe ubiquitous in nature. Engineering this Bose metal in the lab(using ultra-cold atomic gasses for example) and explainingotherwise mysterious metallic behavior plaguing many pro-totypical strongly correlated materials are just a few of theexciting possibilities that the discovery of this new paradigmentails.We thank Anthony Leggett for valuable comments concern-ing our idea. We also thank Jianda Wu and Zi-Jian Langfor helpful discussions. This work is supported by NationalNatural Science Foundation of China (NSFC) ∗ corresponding email: [email protected] [1] P. Kapitza, Nature , 74 (1938).[2] J. F. Allen and A. D. Misener, Nature , 75 (1938).[3] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher,Phys. Rev. B , 546 (1989).[4] M. Greiner, O. Mandel, T. Esslinger, T. W. H¨ansch, andI. Bloch, Nature , 39 (2002).[5] H. M. Jaeger, D. B. Haviland, B. G. Orr, and A. M. Goldman,Phys. Rev. B , 182 (1989).[6] A. Yazdani and A. Kapitulnik, Phys. Rev. Lett. , 3037 (1995).[7] C. Christiansen, L. M. Hernandez, and A. M. Goldman, Phys.Rev. Lett. , 037004 (2002).[8] K. A. Parendo, K. H. S. B. Tan, and A. M. Goldman, Phys. Rev.B , 100508 (2007).[9] A. W. Tsen, B. Hunt, Y. D. Kim, Z. J. Yuan, S. Jia, R. J. Cava,J. Hone, P. Kim, C. R. Dean, and A. N. Pasupathy, NaturePhysics , 208 (2015).[10] A. T. Bollinger, G. Dubuis, J. Yoon, D. Pavuna, J. Misewich,and I. Boˇzovi´c, Nature , 458 (2011), this experiment identi-fies a superconducing to non-superconducting phase transitioncomposed of charge 2e bosons, supporting the existence of astable phase of bosonic matter with dissipative transport.[11] M. V. Feˇigel’Man and A. M. Tsvelik, Soviet Journal of Experi-mental and Theoretical Physics , 1222 (1979).[12] B. Spivak, A. Zyuzin, and M. Hruska, Phys. Rev. B , 132502(2001).[13] K.-H. Wagenblast, A. van Otterlo, G. Sch¨on, and G. T. Zim´anyi,Phys. Rev. Lett. , 1779 (1997).[14] D. Das and S. Doniach, Phys. Rev. B , 1261 (1999).[15] M. V. Feigelman, V. B. Geshkenbein, L. B. Ioffe, and A. I.Larkin, Phys. Rev. B , 16641 (1993).[16] P. Phillips and D. Dalidovich, Phys. Rev. B , 081101 (2002).[17] A. Paramekanti, L. Balents, and M. P. A. Fisher, Phys. Rev. B , 054526 (2002).[18] T. Tay and O. I. Motrunich, Phys. Rev. Lett. , 187202(2010). [19] D. Dalidovich and P. Phillips, Phys. Rev. B , 052507 (2001),arXiv:cond-mat/0005119 [cond-mat].[20] L. P. Gor’kov and E. I. Rashba, Phys Rev. Lett. , 037004(2001).[21] E. Cappelluti, C. Grimaldi, and F. Marsiglio, Phys Rev. Lett. , 2 (2007).[22] G. Goldstein, C. Aron, and C. Chamon, Phys Rev. B , 020504(2015).[23] J. P. Vyasanakere and V. B. Shenoy, Phys Rev. B , 094515(2011).[24] A. V. Chaplik and L. I. Magarill, Phys Rev. Lett. , 2 (2006).[25] S. Takei, C.-H. Lin, B. M. Anderson, and V. Galitski, Phys Rev.A , 023626 (2012).[26] S. Sur and K. Yang, Phys. Rev. B , 024519 (2019).[27] Y. Yildirim and W. Ku, Phys. Rev. X , 011011 (2011).[28] Y. Yildirim and W. Ku, Phys. Rev. B , 180501 (2015).[29] S. Jiang, L. Zou, and W. Ku, Phys. Rev. B , 104507 (2019).[30] Z.-J. Lang, F. Yang, and W. Ku, arXiv e-prints ,arXiv:1902.11206 (2019).[31] A. Leggett, Quantum Liquids (Oxford University Press, NewYork, 2006) Chap. 1.[32] Y. Yildirim, W. Ku, et al. , Physical Review B , 180501(2015).[33] P. CARRUTHERS and M. M. NIETO, Rev. Mod. Phys. , 411(1968).[34] R. Lynch, Physics Reports256