Geometric frustration produces long-sought Bose metal phase of quantum matter

Significance

Traditional bosonic lore is dictated by the binary dynamics of perfect flow or no flow. In recent decades, empirical observations support the unexpected phenomenon of bosons exhibiting dissipative transport at low temperature. Several specialized theories attempt to explain this surprising phenomenon, but a universal theory more representative of experimental observations is lacking. We propose a scenario in which frustration confines the low-energy transport to a lower dimension. The result is a failed insulator state where gapless insulating behavior exhibits dissipative transport at finite temperature or through weak disorder. This result modifies our fundamental understanding of bosons by producing a phase of quantum matter intervening between superfluid and insulator.

Abstract

Two of the most prominent phases of bosonic matter are the superfluid with perfect flow and the insulator with no flow. A now decades-old mystery unexpectedly arose when experimental observations indicated that bosons could organize into the formation of an entirely different intervening third phase: the Bose metal with dissipative flow. The most viable theory for such a Bose metal to date invokes the use of the extrinsic property of impurity-based disorder; however, a generic intrinsic quantum Bose metal state is still lacking. We propose a universal homogeneous theory for a Bose metal in which geometric frustration confines the essential quantum coherence to a lower dimension. The result is a gapless insulator characterized by dissipative flow that vanishes in the low-energy limit. This failed insulator exemplifies a frustration-dominated regime that is only enhanced by additional scattering sources at low energy and therefore produces a Bose metal that thrives under realistic experimental conditions.

Superfluidity has fascinated many with the phenomenon of perfect (dissipationless) flow since its discovery (1, 2). One of the few homogeneous phases of matter known to disrupt this low-temperature behavior is the Mott insulator (3, 4) characterized by a complete lack of flow. Bosons underlie both phases of matter and are well known to exhibit other exotic and extreme properties such as Bose–Einstein condensation. For decades it had been understood that bosons can undergo a transition from the perfect flow of superfluidity directly into the Mott insulating state of no flow, exemplifying this extreme nature. In the absence of inhomogeneities this abrupt transition was understood to dominate low-energy bosonic systems in general.

Therefore, an unexpected mystery arose over the last several decades as experiments (5–10) have continued to find evidence that seemingly disparate bosonic systems exhibit more mundane dissipative transport. Surprisingly, the prevailing theories could not account for this behavior whatsoever. In the intervening decades, several theories were developed to fill this unexpected gap, but despite these efforts, there remains no consensus even for a qualitative account of these observations. Identifying a universal mechanism that allows for a stable phase of dissipative bosonic transport at low-temperature remains a holy grail of condensed matter physics and ultracold atom research.

The extreme nature of bosons has guided theoretical developments toward several specialized directions as opposed to a universal qualitative understanding of this metallic phase. Early research focused on superconducting grain models (11–14), but apparent metallic behavior was shown to be unstable at finite temperature (15), and a large class of models were further ruled out by scaling arguments (16). Another approach used exotic interactions on a lattice such as ring exchange models (17, 18), but these are unstable toward insulating phases with the realistic introduction of weak disorder (19). A more recent approach involves the so-called moatband models (20–26). These models are solved in the extreme dilute limit where the particle density scales much slower than the volume, which is not the regime relevant to the experimental observations. Even more recently, a phenomenological field theory has been developed to account for metallic crossover behavior in two-dimensional (2D) systems (27), but such methods would be inapplicable deep inside a stable phase of matter. A top contender (28) applies extrinsic phase frustration to avoid the issues of the above systems but leaves the question of a universal intrinsic mechanism unanswered.

Instead of attempting to tame the extreme tendencies of bosons by hand, we propose a universal picture where both extremes coexist and stabilize metallic behavior. Consider an emergent system of freely flowing bosons composed of lower-dimensional subsystems. If flow between subsystems vanishes at low energy due to perfect phase interference, then each subsystem is effectively disconnected. We find a gapless insulating state at zero temperature protected by complete frustration of the quantum phases that becomes metallic upon introduction of finite temperature or weak disorder. This failed insulator is mediated by a dimensional crossover between freely moving bosons and disconnected subsystems, providing a universal origin for stable metallic behavior, so we expect this type of Bose metal to be ubiquitous in nature.

We consider a relatively simple implementation of this Bose metal by stacking 2D checkerboard lattice xy layers along the z axis to form a 3D lattice. Alternatively, this system can be viewed as two overlapping sets of stacked vertical xz and yz slabs. We study the regime in which the hopping between such vertical slabs is weaker than that within each slab. In the low-energy limit these vertical slabs become disconnected. This model has been suggested (29) to represent the underdoped cuprates beyond the superconducting dome, a class of materials suspected to exhibit Bose metal behavior through the formation of an emergent Bose liquid (29–33). Even if we drive each slab into superfluidity by introducing a small repulsive local interaction, transport between slabs is still suppressed at low energy. Although proving that a superfluid phase exists can be somewhat subtle (see, e.g., ref. 34 for a discussion on this issue), proving that it does not exist is far less stringent. The lack of transport between effectively independent slabs at low energy forbids superflow, and upon introduction of temperature or disorder we find a Bose metal as a failed insulator.

Results and Discussion

The intralayer structure of the 2D checkerboard lattice as well as its inherent two-band nature is shown in Fig. 1. In the xy layer, this lattice contains hopping to four nearest neighbors (NN) $\tau \prime >0$ as well as an alternating set of two next-nearest neighbors (NNN) $\tau ″>0$. We study the case where the z axis has simple NN hopping with ${\tau }_z<0$, which we suppress below for brevity. Finally, we include on-site repulsive interactions U > 0 resulting in an xy-layer Hamiltonian $\mathcal{H}$ given by

$$\mathcal{H}={\displaystyle \sum _i\left\{{\displaystyle \sum _{j\in \text{NN}}\tau }\prime a_i^{†}{a}_j+{\displaystyle \sum _{j\in \text{NNN}}\tau }″a_i^{†}{a}_j+U{a}_i^{†}{a}_i^{†}{a}_i{a}_i\right\}},$$ (1)

where $a_i^{†}$ and a_j are the bosonic creation and annihilation operators at sites i and j, respectively.

Fig. 1.

The checkerboard lattice. (A) There are two sublattices, denoted red and blue for horizontal and vertical site orientation, respectively, with NN hopping $\tau \prime >0$ between sublattices and NNN hopping $\tau ″$ following site orientation. (B) A symptom of the frustrated regime ($\tau ″>\tau \prime$) is the appearance of a line degeneracy in the band minimum denoted in green. (C) The corresponding dispersion with orbital weight given in red and blue as in A with Γ, X, M, and Z representing (0,0,0), ($\pi ,0,0$), ($\pi ,\pi$,0), and ($0,0,\pi$), respectively. Note the clear separation of orbitals along the line degeneracy. (D) The 3D DOS (in states per 100$\tau ″$ per unit cell) converges to a constant at low energy due to the emergent 2D nature of the excitations near the line degeneracy. (E and F) Magnification at low energy shows that the low-lying excitations are quadratic.

We study the frustrated $\tau ″>\tau \prime$ regime of this model in the low-temperature $T\to 0$ limit. The noninteracting (U = 0) dispersion for the two bands is given by

$$\begin{array}{l}{\mathrm{ϵ}}_{\pm ,\mathbf{k}}=2{\tau }_z(\text{cos}{k}_z-1)+\tau ″(\text{cos}{k}_x+\text{cos}{k}_y+2)\\ \hspace{1em}\pm \sqrt{4\tau {\prime }^2(1+\text{cos}{k}_x)(1+\text{cos}{k}_y)+\tau \prime {\prime }^2{(\text{cos}{k}_x-\text{cos}{k}_y)}^2}.\end{array}$$ (2)

This differs from the unfrustrated regime $\tau \prime >\tau ″$, where there is a single lowest-energy momentum state. In that case, the bosons condense into that state at low temperature, and a d-wave superfluid is obtained (30). Here, as we discuss below and prove in SI Appendix, throughout the frustrated regime denoted by $\tau ″>\tau \prime >0$, interslab transport via $\tau \prime$ is completely disabled in the low-energy limit, and superflow does not occur between slabs.

Effectively Independent Slabs due to Frustration

Geometrical frustration in any lattice creates a set of states that are effectively independent from one another. Consider a special one-body state as an example, illustrated in red in Fig. 2, defined in a single slab with alternating phase differences along the slab. This particular state has unusually high symmetry such that at every point where two slabs intersect, this state has odd parity in the x direction and even parity otherwise. Since parity is a good symmetry of the system, a particle in this state cannot propagate to any intersecting slab. However, this is the only path in the Hamiltonian (via $\tau \prime$) that allows the particle to leave the slab, so this particle is effectively localized to a lower dimension. Since the many-body dressing of a particle must respect its underlying symmetry, the confinement of this state’s propagation to a single slab will persist in the interacting system despite its additional quantum fluctuations in the vicinity of the slab.

Fig. 2.

Ground state phase structure of one xz slab (red) and one yz slab (blue) centered at the point where they cross within the primitive cell (X, Y, Z) outlined in gray. The signs at each orbital are the phases relative to their neighbors within the same slab. Each slab hosts alternating phases along $\tau ″>0$. As a result, this illustration clearly shows that the parity of the wave function in a single slab at the point where these two lines cross is odd along a given line and even otherwise.

A superficial consequence of the decoupling between these states is the occurrence of a line degeneracy in momentum space along the direction of the decoupling in the band structure. The fact that such degeneracy is present for any arbitrary choice of parameters in the Hamiltonian (cf. SI Appendix, Fig. S3) reflects the profound symmetry-based origin of this decoupling. This geometrical frustration induced decoupling distinguishes the line degeneracy in this case from those in existing literature (cf. ref. 35) since subdimensional decoupling is a much more stringent condition than just having a degenerate line in momentum space.

Under the right conditions, when the many-body ground state is formed from the dressed version of the above confined states, one would expect novel physics such as the Bose metal studied here. In our checkerboard lattice example, we expect this to be the case when $\tau ″>\tau \prime >0$ and U is not too big. This is clear from Fig. 1, which shows that the line degeneracy, corresponding to these states, has the lowest energy under this condition at U = 0. As long as the interaction is not so overwhelmingly strong that it drives the system into a completely different state (e.g., a Mott insulator), we expect it to simply dress the confined particles within the many-body ground state. In summary, due to geometric frustration, each particle in the many-body ground state is still unable to propagate between slabs. In other words, the slabs are effectively independent at low energy.

Stability

Since the low-energy transport is dominated by the physics of each 2D slab, we can study their characteristics in order to estimate the thermal stability of this system. To facilitate this understanding we recall (see, e.g., ref. 36) that the low-energy physics of a 2D superfluid is dominated by phase-mode gapless excitations with a linear spectrum. Correspondingly, the one-body density of states (DOS) is linear at low energy. A DOS that vanishes linearly at low energy immediately implies that the bosonic system is stable at low temperature. Such a suppressed DOS at low energy indicates a diminishing channel for fluctuations at low temperature and therefore establishes the thermal stability of our system.

Similarly, it is easy to verify the stability of this system against disorder. Interestingly, introduction of a single impurity breaks parity locally and ruins the perfect interference in $\tau \prime$. In the regime where disorder is weaker than the scale of the phase stiffness of each slab defined by U, such disorder will not challenge the rigidity of each slab and can only establish a small coupling between extended states in the slab. Since the effect of an impurity and the lack of coherence due to geometrical frustration are both one-body effects, it is sufficient to analyze the energy scale of the induced coherence by analyzing the scaling in a one-body Hilbert space formed by the 2L special states defined above. As outlined in SI Appendix, the coupling between extended states of different slabs along their lines of intersection then scales as $\mathcal{O}(1/L\sqrt{L})$ assuming that the number of impurities scale with the system size. Therefore, the leading order energy scale of the coherence scales as $\mathcal{O}(1/L^3)$. There are L² intersections between pairs of slabs, so the total relevant energy correction scales as $\mathcal{O}(L^2/L^3)=\mathcal{O}(1/L)$ and thus vanishes in the large system limit. Therefore, this coupling is not thermodynamically meaningful. This analysis does not account for the incoherent nature of disorder between different intersections, so in that sense it is an upper bound estimate of the coherence generated by impurities. In summary, disorder cannot introduce meaningful 3D coherence of the one-body states that form the many-body ground state, and the slabs remain effectively independent.

Our Failed Insulator Is a Bose Metal

Since each particle in the ground state cannot propagate between slabs, our system naturally does not exhibit 3D superfluidity. This is illustrated in Fig. 3, where despite the superflow propagation allowed along each slab, transition between slabs is suppressed. As a result, in the low-energy limit, there is no conducting path from an arbitrary source to sink, separated by distance $r=\sqrt{r_x^2+r_y^2}$ much larger than the lattice spacing a. Correspondingly, there is no superflow along this general path either.

Fig. 3.

Hypothetical experimental probe of the current with two gray contacts separated by a distance $r=\sqrt{r_x^2+r_y^2}$ much larger than the lattice spacing a. Particles emerge from the source and travel in straight lines along a given coherent slab. The probability for the current to flow between sublattices vanishes in the low-energy limit, so no path remains to reach the sink if either r_x or r_y are large enough to place the leads on distinct coherent slabs.

On the other hand, since the ground state is formed from effectively independent slabs, naively, one might expect an insulator. However, each slab hosts gapless excitations common to 2D superfluidity, so there is no energetic protection for this insulating behavior. These excitations will be modified by $\tau \prime$, coupling the slabs above the ground state and defeating the insulation found there. We refer to such a gapless insulator as a failed insulator.

In this failed insulator, one could imagine finite temperature normal conductivity (in the presence of phonons and disorder, for example) to be limited by the weak connection between coherent slabs. However, since there is no gap, we do not have an exponentially small activation at low temperature. Furthermore, as shown in Fig. 3, the path connecting sink to source can be made with as little as one $\tau \prime$ connection, whose leading effect is polynomial in energy. So, the temperature enhancement of the conductivity will scale faster than $e^{-\mathrm{\Delta }/T}$ in contrast to a typical insulator with an activation scale Δ. We have therefore proven that this system is neither a superfluid nor an insulator protected by an energy scale, so by definition it is a metal. However, this metal should be distinguished from a regular metal since it has a large resistivity at low temperature even for a rather clean sample.

Experienced readers might notice that our 2D slab could still host 2D superfluidity since it is effectively a 2D interacting bosonic subsystem. However, it is well known that 2D superfluidity is particularly susceptible to disorder in relatively low-density systems (see, e.g., refs. 3, 5, 36). Therefore, we can expect this residual 2D superfluid response along the slab to be replaced by a low resistivity along the slab directions in the presence of weak disorder such as expected in real materials (as long as the boson density is relatively low). The resulting anisotropic transport can actually account for the puzzling observation of unexpectedly reversed anisotropy in some materials (37, 38).

Demonstration via Controlled Approximation

To further illustrate the stable Bose metal derived above, we implement a controlled approximation in which a reference state is first established as the dominant contribution to the zero temperature many-body ground state and expand the Hamiltonian to bilinear order in fluctuations about this reference state. We construct the many-body eigenstates using the solution to the Hamiltonian in order to compute the current–current response function in the low-frequency limit. From the result we establish the conductivity and therefore the low-temperature phase of matter.

As detailed in SI Appendix, we adopt an effective local density and phase operator formalism to represent our system. We expand the Hamiltonian in small fluctuations about a many-body reference state chosen based on the general properties of the many-body ground state identified above. In particular, we use a reference state with uniform average density and with average local phases as illustrated in Fig. 2, namely, π phase difference across a $\tau ″$ bond and overall phase freedom for each slab.

As a demonstration, Fig. 4 represents the solution corresponding to a coincidentally coherent phase structure

$${\varphi }_{\alpha \mathit{XYZ}}=\pi (X+Y),$$ (3)

where $\alpha \in \{1,2\}$ are the two orbitals in the (X, Y, Z) primitive unit cell. This choice maintains momentum as a good quantum number, and as such it provides an upper bound on the superfluid density of the system. Choosing phases that do not preserve lattice translational symmetry can only introduce additional interference and weaken any physics that relies on long-range coherence.

Fig. 4.

The checkerboard lattice in the presence of local repulsive interactions U > 0. (A) Each shade of orange represents an independent coherent slab with a randomly selected overall phase. (B and C) An example with $(\pi ,\pi ,0)$ periodic phase structure shows that the band structure maintains the line degeneracy and separation of orbitals from Fig. 1. (D) The corresponding DOS goes to zero linearly with energy indicating that the ground state is stable at low temperature. (E and F) The low-lying excitations are now stiffened into a linear spectrum.

All of the key features of our Bose metal that were proven in general above are borne out of this controlled approximation as illustrated in Fig. 4. First, Fig. 4 C, E, and F clearly identify gapless excitations, so our insulating ground state is not protected, and the system is therefore not an insulator. Fig. 4 B, C, E, and F indicate that the decoupling remains as evidenced by its symptom of a large degeneracy, so the system is not a 3D superfluid. Finally, Fig. 4D identifies a linear DOS at low energy, which stabilizes the system at finite temperature and proves that this result is not simply a zero-temperature idealization.

Similarly, this approximate solution also produces zero superfluid density as expected from the general proof above. To that end, we use the following thought experiment to identify the current–current response in the low-frequency, long-wavelength limit. Two conducting leads are attached to a sample in an arbitrary orientation as in Fig. 3. These leads are centered at x and $\mathbf{x}\prime$ such that the number of lattice spacings a between them is large ($r=|\mathbf{x}-\mathbf{x}\prime |\gg a$). The sizes of the leads themselves span many lattice spacings as well but far fewer than the distance between the leads. Long-range coherent transport is achieved when the current–current response from one lead to the other converges to a finite value for arbitrarily large r and arbitrarily small frequency ω. The conductivity can be evaluated via

$$\sigma (\mathbf{x}-\mathbf{x}\prime ,\omega )=\frac{1}{\omega }\text{Im}{\chi }^{ll}(\mathbf{x}-\mathbf{x}\prime ,\omega ),$$ (4)

where ${\chi }^{ll}(\mathbf{x}-\mathbf{x}\prime ,\omega )$ is the longitudinal component of the current–current response function at frequency ω between points x and $\mathbf{x}\prime$. SI Appendix, Eq. 4 is shown to reproduce the correct superfluid contribution in the unfrustrated regime $\tau \prime >\tau ″$ as expected.

In the frustrated regime $\tau ″>\tau \prime$, our choice of overall phases for the slabs given by Eq. 3 once again provides an upper bound on the superfluid density. We find that at zero temperature the superfluid contribution $\sigma (r,\omega =0)=0$.

This result can be readily understood from the disconnected nature of the slabs at low energy. Consistent with our general proof above, the hopping parameter $\tau \prime$ that couples neighbor coherent slabs is suppressed at low energy within this controlled approximation. As a result, any transport that crosses through successive coherent slabs is also suppressed. As demonstrated in Fig. 3, no coherent slabs simultaneously intersect, for example, the source at x and the sink at $\mathbf{x}\prime$. Therefore, the DC conductivity vanishes between x and $\mathbf{x}\prime$, which includes both the superfluid and normal DC response at zero temperature.

At finite temperature the normal contribution will develop a finite value given the fact that the gapless excitations at finite energy have weak but finite coherence. More generally, we have proven above and in SI Appendix that this coherence grows faster than exponential with increasing energy. Therefore, the zero temperature insulating behavior fails at any finite temperature, and a metal is obtained.

Other Considerations and Implications

Although the dressed particle is localized in a single slab due to geometric frustration, this physics is qualitatively unrelated to Anderson localization. In the latter, randomness causes localization and creates an energy scale that must be overcome to reach a channel for transport. If we remove the randomness, then we remove this scale and the localized states. In our case, the localization is caused by perfect geometric frustration, which is unrelated to randomness and should be regarded as a high-energy constraint beyond the scale of the Hamiltonian. Any state other than this particular many-body ground state is unprotected by perfect frustration and is not localized. The perfect geometric frustration is protected by symmetry, so even when we choose overall phases for each slab that appear coherent as in our above demonstration, each particle in the ground state remains confined to the slab, and the superfluid density vanishes.

On the other hand, full coherence can in principle be recovered via disruption of the perfect interference, for example, by introducing anisotropy in $\tau \prime$. In that case, the ground state would host a superfluid with p-wave symmetry. Accepting the claim (30) that this model is representative for the extreme low-doping ($<5\%$) nonsuperconducting regime of the cuprates, this implies that applying enough uniaxial pressure along the (110) crystallographic direction will generate a second superconducting dome in the phase diagram. This dome is separated from the original dome of d-wave superfluidity by a quantum critical point due to the difference in their local symmetry.

Conclusions

By introducing perfect geometrical frustration among many adjacent nearly free flowing subsystems, we have discovered a long-sought universal stable Bose metal phase intervening between superfluid and insulator. Utilizing a gapless failed insulator in the presence of temperature or disorder immediately leads to dissipative transport. We demonstrate this concept via a two-band Bose–Hubbard model in an extended checkerboard lattice with frustrated coupling between vertical slabs and a particle density that scales with the system volume. We find that each dressed particle in the many-body ground state is confined to a single slab such that the low-frequency conductivity and the superfluid response both vanish in a general direction at zero temperature. At finite temperature, the system is stable due to the suppressed DOS at low energy. The universal mechanism of our Bose metal leads to a stable phase of quantum matter that is robust under realistic conditions and should therefore be ubiquitous in nature. Engineering this Bose metal in the laboratory (using ultracold atomic gases, for example) and explaining otherwise mysterious metallic behavior plaguing many prototypical strongly correlated materials are just a few of the exciting possibilities that the discovery of this paradigm entails.

Data Availability

All study data are included in the article and/or SI Appendix.