## 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

# Quantum chaos on a critical Fermi surface

Contributed by Subir Sachdev, December 31, 2016 (sent for review November 3, 2016; reviewed by Sung-Sik Lee and John McGreevy)

## Significance

All high-temperature superconductors exhibit a remarkable “strange metal” state above their critical temperatures. A theory of the strange metal is a prerequisite for a deeper understanding of high-temperature superconductivity, but the ubiquitous quasiparticle theory of normal metals cannot be extended to the strange metal. Instead, strange metals exhibit many-body chaos over the shortest possible time allowed by quantum theory. We characterize the quantum chaos in a model of fermions at nonzero density coupled to an emergent gauge field. We find a universal relationship between the chaos parameters and the experimentally measurable thermal diffusivity. Our results establish a connection between quantum dephasing and energy transport in states of quantum matter without quasiparticles.

## Abstract

We compute parameters characterizing many-body quantum chaos for a critical Fermi surface without quasiparticle excitations. We examine a theory of

States of quantum matter without quasiparticle excitations are expected (1) to have a shortest-possible local thermalization or phase coherence time of order **1** but still of order **1** as *U*(1) gauge field in two spatial dimensions. Such a theory has a Fermi surface in momentum space that survives in the presence of the gauge field, even though the fermionic quasiparticles do not. (The Fermi surface is defined by the locus of points where the inverse fermion Green’s function vanishes and is typically computed in the gauge

It has been recognized for some time (13) that the naive vector

Here, we use an extended RPA theory to compute the Lyapunov time, and the associated butterfly velocity **1**. Notably the value of

This depends on both

Blake (23, 24) recently suggested, using holographic examples, that there is a universal relation between transport properties, characterized by the energy and charge diffusivities (28) and the parameters characterizing quantum chaos

Notably, the factors of **3** cancel precisely in the relationship Eq. **4**. This result supports the universality of the relationship between thermal transport and quantum chaos in strongly coupled states without quasiparticles.

A simple intuitive picture of this connection between chaos and transport follows from recognizing that quantum chaos is intimately linked to the loss of phase coherence from electron–electron interactions. As the time derivative of the local phase is determined by the local energy, phase fluctuations and chaos are linked to interaction-induced energy fluctuations and hence to thermal transport. On the other hand, other physical ingredients enter into the transport of other conserved charges, and so we see no reason for a universal connection between chaos and charge transport.

## Model

We consider a single patch of a Fermi surface with *A*). The (Euclidean) action is given by

This is derived from the action of a Fermi surface coupled to a *U*(1) gauge field with gauge coupling constant *A*) in

The bare frequency-dependent term in the fermion propagator is irrelevant in the scaling limit and is hence multiplied by the positive infinitesimal

The coupling

Because we need to perform all computations at finite temperature, it is imperative that we understand what the finite-temperature Green’s functions are. In the above patch theory, the gauge boson does not acquire a thermal mass due to gauge invariance (34). However, we nevertheless add a very small “mass” by hand to use as a regulator. The boson Green’s function then is

This boson Green’s function may then be used to derive the thermally corrected fermion Green’s function via the one-loop self-energy starting from free fermions (35) (Fig. S1*B*).

This thermally corrected Green’s function is not exact, owing to the uncontrolled nature of the large-

## Lyapunov Exponent

To study out-of-time order correlation functions, we define the path integral on a contour *B*). The generating functional is given by

To measure scrambling, we use fermionic operators, and hence we replace the commutators (2) by anticommutators. We evaluate the index-averaged squared anticommutator (6, 8)

This function is real and invariant under local **[****12****]** to have a small prefactor. This can be provided here by examining spatially separated correlators (which we do in the next section), although not by the large-

The approach described in ref. 8 involves summing a series of diagrams to obtain

(*SI Appendix B*. There are two types of rungs at leading order in

The first diagram in the ladder series that has no rungs is given by

This bare term remarkably ends up being

We have the Bethe–Salpeter equation of the ladder series

The

Here, we use the bare fermion Wightman functions, as the self-energy corrections will come in at higher orders in

(*A* is the fermion spectral function.) There is a *SI Appendix B*). We then have

Because there are no IR divergences, we drop the

Due to the sliding symmetry along the Fermi surface (14), we expect the eigenfunction that we are interested in to obey

Interestingly, both pieces of the kernel no longer depend on **14**, we can see that the IR divergent piece

As a matrix equation, this is of the form *SI Appendix D*. We find that

At high temperatures, when **23**. Counting powers, we then might expect *SI Appendix C* we consider a few higher order (in

## Butterfly Velocity

The out-of-time order correlation function evaluated at spatially separated points characterizes the divergence of phase space trajectories in both space and time. This process is described by the function **11**, which is the same as the function

Repeating the same steps that led to the derivation of Eq. **23**, we simply obtain

For small

The structure of the above equation indicates that chaos propagates as a wave pulse that travels at the butterfly velocity. The wave pulse is not a soliton and broadens as it moves (25); this is encoded in the function *SI Appendix D*. Note that this shows **3** once the factors of Fermi velocity *SI Appendix D*).

This is again strictly valid only at the lowest temperatures, where

With the scalings

## Energy Diffusion

It has been conjectured, and shown in holographic models (23, 28), that the butterfly effect controls diffusive transport. The thermal diffusivity

This computation is carried out in *SI Appendix E*. We obtain

Because the theory of a single Fermi surface patch is chiral, currents are nonzero even in equilibrium. We must thus define conductivities with respect to the additional change in these currents when electric fields and temperature gradients are applied. The thermal conductivity

We compute the conductivities using the one-loop dressed fermion propagators in *SI Appendix E* (the boson again does not contribute directly due to the absence of an

Using Eqs. **2****,** **3****,** **26****,** and **29** we then see that

The factors of powers of

## Outlook

We have computed the Lyapunov exponent

## SI Appendix A: Self-Energies

The one-loop self-energy graphs are shown in Fig. S1. The derivation of the one-loop boson self-energy is standard (14)

The formally infinite piece

The one-loop fermion self-energy is given by

## SI Appendix B: Wightman Functions

The Wightman function for two operators

## SI Appendix C: Higher Order Corrections

We consider the corrections to the ladder series of the main text coming from diagrams with crossed rungs. We show that certain diagrams with crossed boson rungs vanish and that diagrams with crossed fermion rungs contribute to

There are two simple types of crossed ladder insertions in the Bethe–Salpeter equation. The first one is shown in Fig. S2*A* and is given by

The integral over *C*) also vanish for exactly the same reason.

The insertion of Fig. S2*B* does not vanish. However, unlike the third diagram on the right-hand side of the Bethe–Salpeter equation in the main text, the flavor indexes on the two sides of the insertion are not decoupled. Thus, there is no factor of

Finally, we must mention that, due to the uncontrolledness of the large-

## SI Appendix D: Numerical Methods

Numerically, it is easier to solve the integral equation in the main text by keeping the IR divergent term explicit:

We keep *A*. We see that there is one zero for *B*. A plot of *C*.

For the butterfly velocity, we solve the appropriate integral equation using the same technique as above. Now *D*, leading to the result in the main text. To determine the function **25** of the main text, we need to numerically find

## SI Appendix E: Specific Heat and Thermal Conductivity

The expression for the free energy may be rewritten as a contour integral, keeping in mind the branch cuts in the fermion propagators along the real frequency axis

Evaluating this integral and differentiating with respect to

We now turn to the computation of the energy current correlator required to determine

These integrals must be done numerically, and it is then easily seen that they reproduce the sum correctly at bosonic Matsubara

For the conductivity

The momentum integrals in the simple two-loop vertex correction to these contributions were considered in ref. 14 for the higher loop renormalizations of the boson propagator. The authors found that the momentum integrals in the vertex correction vanish, owing to the obtainment of terms with denominators possessing poles on the same side of the real axis.

## Acknowledgments

We thank M. Blake, D. Chowdhury, R. Davison, A. Eberlein, D. Stanford, and B. Swingle for valuable discussions. This research was supported by the National Science Foundation under Grant DMR-1360789 and Multidisciplinary University Research Initiative Grant W911NF-14-1-0003 from Army Research Office. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. S.S. also acknowledges support from Cenovus Energy at Perimeter Institute.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: sachdev{at}g.harvard.edu or aavishkarpatel{at}g.harvard.edu.

Author contributions: A.A.P. and S.S. designed research, performed research, and wrote the paper.

Reviewers: S.-S.L., McMaster University, Hamilton, ON, Canada; and J.M., University of California, San Diego.

The authors declare no conflict of interest.

↵*Kitaev AY, Entanglement in strongly-correlated quantum matter, Talks at KITP, April 7, 2015, Univ of California, Santa Barbara, CA.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1618185114/-/DCSupplemental.

## References

- ↵.
- Sachdev S

- ↵.
- Larkin AI,
- Ovchinnikov YN

- ↵.
- Maldacena J,
- Shenker SH,
- Stanford D

- ↵.
- Shenker SH,
- Stanford D

- ↵
- ↵.
- Maldacena J,
- Stanford D

- ↵.
- Gu Y,
- Qi XL,
- Stanford D

- ↵.
- Stanford D

- ↵.
- Damle K,
- Sachdev S

- ↵.
- Sachdev S

- ↵.
- Aleiner IL,
- Faoro L,
- Ioffe LB

- ↵.
- Banerjee S,
- Altman E

- ↵.
- Lee SS

- ↵.
- Metlitski MA,
- Sachdev S

- ↵.
- Mross DF,
- McGreevy J,
- Liu H,
- Senthil T

- ↵.
- Dalidovich D,
- Lee SS

- ↵.
- Holder T,
- Metzner W

- ↵.
- Halperin BI,
- Lee PA,
- Read N

- ↵
- ↵.
- Kim YB,
- Furusaki A,
- Wen XG,
- Lee PA

- ↵.
- Roberts DA,
- Stanford D,
- Susskind L

- ↵.
- Shenker SH,
- Stanford D

- ↵.
- Blake M

- ↵.
- Blake M

- ↵.
- Roberts DA,
- Swingle B

- ↵.
- Alishahiha M,
- Astaneh AF,
- Mohammadi Mozaffar MR

- ↵.
- Lucas A,
- Steinberg J

- ↵
- ↵.
- Hartnoll SA,
- Mahajan R,
- Punk M,
- Sachdev S

- ↵.
- Lucas A,
- Sachdev S

- ↵.
- Eberlein A,
- Mandal I,
- Sachdev S

- ↵.
- Nave CP,
- Lee PA

- ↵.
- Sur S,
- Lee SS

- ↵.
- Hartnoll SA,
- Mahajan R,
- Punk M,
- Sachdev S

- ↵.
- Dell’Anna L,
- Metzner W

- ↵.
- Swingle B,
- Chowdhury D

- ↵.
- Moreno J,
- Coleman P

- ↵.
- Zhang JC, et al.

_{2}Cu_{3}O_{6+x}. arXiv, 1610.05845.

## Citation Manager Formats

## Sign up for Article Alerts

## Article Classifications

- Physical Sciences
- Physics

## Jump to section

## You May Also be Interested in

_{2}into liquid methanol using artificial marine floating islands.