## New Research In

### Physical Sciences

### Social Sciences

#### Featured Portals

#### Articles by Topic

### Biological Sciences

#### Featured Portals

#### Articles by Topic

- Agricultural Sciences
- Anthropology
- Applied Biological Sciences
- Biochemistry
- Biophysics and Computational Biology
- Cell Biology
- Developmental Biology
- Ecology
- Environmental Sciences
- Evolution
- Genetics
- Immunology and Inflammation
- Medical Sciences
- Microbiology
- Neuroscience
- Pharmacology
- Physiology
- Plant Biology
- Population Biology
- Psychological and Cognitive Sciences
- Sustainability Science
- Systems Biology

# Nonergodic metallic and insulating phases of Josephson junction chains

Contributed by Boris L. Altshuler, November 20, 2015 (sent for review August 24, 2015; reviewed by Eugene Bogomolny and Leonid I. Glazman)

## Significance

Conventional equilibrium statistical physics that aims to describe dynamical systems with many degrees of freedom relies crucially on the equipartition postulate: After evolving for a sufficiently long time, the probabilities to find the system in states with the same energy are equal. Time averaging is thus assumed to be equivalent to the averaging over the energy shell—the famous ergodic hypothesis. In this study we show that this hypothesis is not correct for a large class of quantum many-body models that can be implemented in the laboratory. These models are predicted to show a novel type of behavior that we name bad metal, which is neither a many-body insulator nor a conventional conductor.

## Abstract

Strictly speaking, the laws of the conventional statistical physics, based on the equipartition postulate [Gibbs J W (1902) *Elementary Principles in Statistical Mechanics, developed with especial reference to the rational foundation of thermodynamics*] and ergodicity hypothesis [Boltzmann L (1964) *Lectures on Gas Theory*], apply only in the presence of a heat bath. Until recently this restriction was believed to be not important for real physical systems because a weak coupling to the bath was assumed to be sufficient. However, this belief was not examined seriously until recently when the progress in both quantum gases and solid-state coherent quantum devices allowed one to study the systems with dramatically reduced coupling to the bath. To describe such systems properly one should revisit the very foundations of statistical mechanics. We examine this general problem for the case of the Josephson junction chain that can be implemented in the laboratory and show that it displays a novel high-temperature nonergodic phase with finite resistance. With further increase of the temperature the system undergoes a transition to the fully localized state characterized by infinite resistance and exponentially long relaxation.

The remarkable feature of the closed quantum systems is the appearance of many-body localization (MBL) (1): Under certain conditions the states of a many-body system are localized in the Hilbert space resembling the celebrated Anderson localization (2) of single particle states in a random potential. MBL implies that macroscopic states of an isolated system depend on the initial conditions (i.e., the time averaging does not result in equipartition distribution and the entropy never reaches its thermodynamic value). Variation of macroscopic parameters (e.g., temperature) can delocalize the many-body state. However, the delocalization does not imply the recovery of the equipartition. Such a nonergodic behavior in isolated physical systems is the subject of this paper.

We argue that regular Josephson junction arrays (JJAs) under the conditions that are feasible to implement and control experimentally demonstrate both MBL and nonergodic behavior. A great advantage of the Josephson circuits is the possibility to disentangle them from the environment, as was demonstrated by the quantum information devices (3). At low temperatures the conductivity *σ* of JJA is finite (below we call such behavior metallic), whereas as

In the bad metal regime the dynamical evolution starting from a particular initial state does not lead to the thermodynamic equilibrium, so that even extensive quantities such as entropy differ from their thermodynamic values. Starting with the seminal paper of Fermi et al. (6) the question of nonergodicity in nonintegrable systems was extensively studied (7). However, the difference between the long time asymptotics of the extensive quantities and their thermodynamic values was not analyzed to the best of our knowledge. We believe that the behavior of JJA that we describe here can be observed in a variety of nonlinear systems.

JJA is characterized by the set of phases *i* are canonically conjugated. The Hamiltonian *H* is the combination of the charging energies of the islands with the Josephson coupling energies. Assuming that the ground capacitance of the islands dominates their mutual capacitances (this assumption is not crucial for the qualitative conclusions) we can write *H* as

The ground state of the model Eq. **1** is determined by the ratio of the Josephson and charging energies, *Supporting Information, section 1* and Fig. S1). The quantum transition at *U* is the energy per superconducting island (

The main qualitative finding of this paper is the appearance of a nonergodic and highly resistive “bad metal” phase at high temperatures, *T* and violation of thermodynamic identities. We support these findings by semiquantitative theoretical arguments. Finally, we present the results of numerical diagonalization and tDMRG [time density matrix renormalization group (9)] of quantum systems that demonstrate both the nonergodic bad metal and the MBL insulator.

It is natural to compare the nonergodic state of JJA with a conventional glass characterized by infinitely many metastable states. The glass entropy does not vanish at

## Qualitative Arguments for MBL Transition

In a highly excited state *Supporting Information, section 2* and Fig. S2. This computation also shows that resistivity is indeed infinite in this phase, contrary to the statement of work (11) that true MBL transition is impossible in translationally invariant systems.

If

Classical regime is realized at *T*. One can express the charge of an island *q* through the dimensionless time

We solve Eq. **2** for the various initial conditions corresponding to a given total energy and compute the energy *τ*.

The ergodicity implies familiar thermodynamic identities; for example

From Eq. **1** it follows that *Supporting Information, section 3*]. One can thus rewrite Eq. **3** as

Results of the numerical solution of Eq. **2** are compared with Eq. **3** in Fig. 2. For any given evolution time, **4**). At a fixed *u*, *u* rather than *T* is the proper control parameter. The effective temperature, defined as *Inset*.

## Qualitative Interpretation

Large dispersion of charges on adjacent islands, *i* satisfies the recursion relation*Supporting Information, section 4*):*ρ*, we thus have **7**

To determine the current caused by voltage *V* across the chain we solved Eq. **2** with modified boundary conditions, **8** (Fig. 3). The range of the resistances set by realistic computation time is too small to detect the logarithmic factor in Eqs. **7** and **8**; however, a relatively large slope, **8** that gives

The qualitative picture of triads separated by log-normally distributed resistances of silent regions allows one to understand the long time relaxation of *l* (Fig. 2). Each resistance can be viewed as a barrier with a tunneling rate *τ* the barriers with

The numerical simulations confirm that the energy variance *w* relaxes in agreement with Eq. **10** as shown in Fig. 4. The best fit to Eq. **10** yields parameters close to the expected, *l*, which is significantly smaller than thermodynamic value

Another test of the ergodicity follows from the fluctuation-dissipation theorem (FDT) that relates conductivity and current fluctuations. In the low frequency limit the noise power spectrum is *Inset*. Note that both the energy and current fluctuations are less than expected in equilibrium.

## Quantum Behavior

In contrast to a classical limit **1** to a finite number of charging states at each site,

Fig. 6, *Inset* shows the time dependence of the entropy at

Fig. 6 presents

Deep in the insulator the time dependence of the entropy is extremely slow, roughly linear in *Supporting Information, section 5*). This resembles the results of the works (16, 17) for the conventional disordered insulators. The extremely long relaxation times can be attributed to rare pairs of almost degenerate states localized within different halves of the system. The exponential decay with distance of the tunneling amplitude that entangles them leads to the exponentially slow relaxation.

To locate the MBL transition we analyzed the time dependence of the charge fluctuations. In a metal the charge fluctuations relaxation rate depends weakly (as a power law) on the sample size, in contrast to the exponential dependence in the insulator. Comparing the dependencies of the rates on the system size for different

The variances of the charge in the RHS model at **1** at finite *T* coincide at **1** at **1**. The transition line shown in Fig. 1 is a natural connection of this point with *T*, in the opposite limit

## Possible Experimental Realization

MBL and the violation of the ergodicity can be observed only at sufficiently low temperatures when one can neglect the effects of thermally excited quasiparticles that form the environment to model Eq. **1**. This limits temperatures to **1** to be relevant. Realistic measurements of such array include resistance

Recently numerical studies (18⇓–20) investigated whether MBL can exist in disorder-free 1D systems. These works studied models different from ours. De Roeck and Huveneers (21) argued that the localization should happen in a similar model at infinite *T*. Very recently the same group (11) claimed to prove the absence of true localization in periodic systems. This proof was based on the assumption of ergodicity, specifically the eigenstate thermalization hypothesis, in the metallic regime. This assumption (which allowed the authors to argue in terms of moving ergodic bubbles) is incorrect for the model studied here because the metallic phase is generally nonergodic. Furthermore, results concerning Kullback–Liebler (KL) divergence (Fig. S5 and *Supporting Information, section 6*) indicate that the metallic phase is not fully described by Gaussian ensemble at any value of parameters. Qualitatively, a bubble is subject to the effective random potential, which originates from its boundaries and is exponentially stronger than the bubble bandwidth, resulting in the bubble localization. Similarly one can ask why Arnold diffusion does not lead to delocalization in the classical limit (22). The reason is the random environment created by the triad neighbors that destroys subtle many-body resonance. We have checked numerically that a short classical chain does not display triad diffusion.

In conclusion, we presented strong numerical evidence for the MBL transition and its semiquantitative description in a regular, disorder-free Josephson chain. Probably the most exciting finding is the intermediate nonergodic conducting phase (bad metal) between the MBL insulator and good ergodic metal. This phase distinguishes Josephson junction chain from the spin

## Methods

### Simulation of the JJA in the Classical Regime.

At large *u* averaging out temporal fluctuations requires exponentially long times. Moreover, at *u* factorially, leading to a strong heating in the computation of resistance unless the measurement current is factorially small. Observation of a small current against the background of a low-frequency noise requires increasingly long times. Accordingly, for the realistic evolution times

### Simulation of the Quantum Problem.

The time dependence of the entropy and the charge fluctuations for system of sizes

## 1. Berezinski–Kosterlitz–Thouless Critical Point at Zero Temperature

In this section we give the details of the numerical methods that allowed us to determine the value of the ratio of Josephson to charging energies, *η*, for the Berezinsky–Kosterlitz–Thouless phase transition of the model Eq. **1** at zero temperature. Finite order derivatives of the energy do not display any discontinuity at this transition. Thus, we have to use more sophisticated numerical methods than the ones used in the case of second-order phase transitions.

The idea of our approach is to compute the fidelity of the ground state defined by the equation*Inset* as a function of

## 2. Charge Propagation in the Quantum Insulating Phase

In the insulating phase the charge transfer by distance *r* appears in the *P*, of

We estimate the transition temperature by demanding

The estimate of the charge transfer Eq. **S1** neglects the charge discreteness that is expected to become irrelevant at *n* (resonances). Such degeneracies are lifted in the next order of the perturbation theory: The energy difference between the states with charge *i* becomes

Close to the transition line we estimate

These resonances occur with the probability **S4** is satisfied. We conclude that the shift of the transition line upwards is small in **S4**:**S3** this renormalization is of the order of

According to Eq. **S2** the charge propagation is controlled by the average logarithm of the charge difference. This suggests that the simulations in which for the charge is evenly distributed in the interval **1** with effective temperature

## 3. Free Energy of the Chain in Thermodynamic Equilibrium

Evaluation of the free energy with Hamiltonian Eq. **1** gives*u* in terms of *T*:**S6** one can check the validity of Eq. **3**, which takes the form

## 4. Noise Distribution

At high temperatures the frequencies of individual phases are typically large, **2** in perturbation theory. Furthermore, because the effect of noise decreases exponentially with distance one can use the forward propagation approximation in which the evolution of the next phase is determined exclusively by the previous one:

Looking for the solution in the form *i* we get**S8** the noise recursion**S9** in the limit **5** in the main text. The exponential decrease of the noise away from its source in the classical regime is in agreement with the numerical computation in the quantum regime.

The recursion Eq. **5** of the main text implies that the noise generated by a single triad at distance **5** and **S10**) implies the charge localization. The dc charge transport in a macroscopically large array requires interaction of different triads through the quiet regions. A quiet interval **S10** with the Poisson distribution for the sizes of the quiet regions, *r*, yields the log-normal distribution for the resistances

## 5. Charge and Entropy Relaxation in the Quantum Regime

Here we give the details of the numerical results for charge and entropy evolution in the insulating and metallic regimes that are both qualitatively different from the nonergodic bad metal phase.

We begin with the insulating phase. The behavior of the entropy in a wide time range (

At shortest times

The striking feature of the insulating state in this model is a very slow (logarithmic) growth of entropy at long times (

In contrast to the entropy, the charge fluctuations display simple monotonic behavior at all times. The characteristic charge relaxation time increases extremely rapidly with the system size. We conclude that this phase is a genuine insulator whose phase space is separated into a thermodynamically large number of independent compartments.

In a good metal there is only one regime for charge and entropy time evolution. The charge relaxes quickly and this relaxation does not show any sign of getting slower at larger system sizes (Fig. S4). Accordingly, the entropy increases rapidly and saturates at the values that approach

## 6. KL Divergence

We present results concerning KL divergence in RHS of Eq. **1**. For vectors *ψ* and *φ* KL divergence is defined by*N* is Hilbert space dimension. The value of KL in the Gaussian orthogonal ensemble (GOE) is:

Feigel’man et al. (28) compute KL between nearest energy eigenvectors of a Hamiltonian that has an MBL transition. It finds that KL diverges in MBL phase whereas it goes to GOE value in the metallic regime, just after the MBL transition. An important point is that the value of

Here, we are interested in the average of KL for many eigenstates. Notice that we have to use many vectors to average KL, because our Hamiltonian does not contain disorder. Specifically, we use 0.4 of the total states in the middle of the band. We do not include KL between nearest and second-nearest eigenstates in energy because they can be degenerate due to inversion or other symmetries.

In Fig. S5, KL is plotted for different size *Inset*. This is a signature of MBL transition. We notice that the crossing point moves toward smaller values of

Remarkably, KL does not reach the value of GOE for any of the sizes studied, in contrast with the results in ref. 28. This is also the case for the largest sizes

The large value of KL significantly above MBL transition and the fact that it differs significantly from

## Acknowledgments

We thank I. L. Aleiner, M. Feigelman, S. Flach, V. E. Kravtsov, and A. M. Polyakov for useful discussions. This work was supported in part by Templeton Foundation Grant 40381, Army Research Office Grant W911NF-13-1-0431, and Agence Nationale de la Recherche QuDec.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: bla{at}phys.columbia.edu.

Author contributions: M.P., L.B.I., and B.L.A. performed research, analyzed data, and wrote the paper.

Reviewers: E.B., Université Paris-Sud, CNRS, Laboratoire de Physique Théorique et Modèles; and L.I.G., Yale University.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1520033113/-/DCSupplemental.

## References

- ↵
- ↵
- ↵.
- Devoret MH,
- Schoelkopf RJ

- ↵.
- Efetov KB

- ↵
- ↵.
- Fermi E,
- Pasta J,
- Ulam S

- ↵.
- Galavotti G

- ↵.
- Kosterlitz JM,
- Thouless DJ

- ↵
- ↵.
- Kauzmann W

- ↵
- ↵.
- Tinkham M

- ↵.
- Schmidt VV

- ↵
- ↵.
- Chirikov BV

- ↵
- ↵
- ↵.
- Yao NY,
- Laumann CR,
- Cirac JI,
- Lukin MD,
- Moore JE

- ↵
- ↵.
- Papic Z,
- Stoudenmire EM,
- Abanin DA

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵

## Citation Manager Formats

## Sign up for Article Alerts

## Jump to section

- Article
- Abstract
- Qualitative Arguments for MBL Transition
- Qualitative Interpretation
- Quantum Behavior
- Possible Experimental Realization
- Methods
- 1. Berezinski–Kosterlitz–Thouless Critical Point at Zero Temperature
- 2. Charge Propagation in the Quantum Insulating Phase
- 3. Free Energy of the Chain in Thermodynamic Equilibrium
- 4. Noise Distribution
- 5. Charge and Entropy Relaxation in the Quantum Regime
- 6. KL Divergence
- Acknowledgments
- Footnotes
- References

- Figures & SI
- Info & Metrics

## You May Also be Interested in

### More Articles of This Classification

### Physical Sciences

### Related Content

- No related articles found.

### Cited by...

- No citing articles found.