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arXiv:2106.01381 · Physical review. B./Physical review. B

Many-body energy invariant for T -linear resistivity

Aavishkar A. Patel, Hitesh J. Changlani

Published May 9, 2022• 10 citations
arXivHTMLPDF

Abstract

The description of the dynamics of strongly correlated quantum matter is a challenge, particularly in physical situations where a quasiparticle description is absent. In such situations, however, the many-body Kubo formula from linear response theory, involving matrix elements of the current operator computed with many-body wave functions, remains valid. Working directly in the many-body Hilbert space and not making any reference to quasiparticles (or lack thereof), we address the puzzle of linear in temperature ($T$-linear) resistivity seen in non-Fermi-liquid phases that occur in several strongly correlated condensed matter systems. We derive a simple criterion for the occurrence of $T$-linear resistivity based on an analysis of the contributions to the many-body Kubo formula, determined by an energy invariant ``$f$ function'' involving current matrix elements and energy eigenvalues that describes the dc conductivity of the system in the microcanonical ensemble. Using full diagonalization, we test this criterion for the $f$ function in the spinless nearest-neighbor Hubbard model and in a system of Sachdev-Ye-Kitaev dots coupled by weak single-particle hopping. We also study the $f$ function for the spin conductivity in the two-dimensional Heisenberg model and arrive at similar conclusions. Our work suggests that a general principle, formulated in terms of many-body Hilbert space concepts, is at the core of the occurrence of $T$-linear resistivity in a wide range of systems, and precisely translates $T$-linear resistivity into a notion of energy scale invariance far beyond what is typically associated with quantum critical points.

DOI
10.1103/physrevb.105.l201108

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The description of the dynamics of strongly correlated quantum matter is a challenge, particularly in physical situations where a quasiparticle description is absent. In such situations, however, the many-body Kubo formula from linear response theory, involving matrix elements of the current operator computed with many-body wave functions, remains valid. Working directly in the many-body Hilbert space and not making any reference to quasiparticles (or lack thereof), we address the puzzle of linear in temperature (TTT-linear) resistivity seen in non-Fermi-liquid phases that occur in several strongly correlated condensed matter systems. We derive a simple criterion for the occurrence of TTT-linear resistivity based on an analysis of the contributions to the many-body Kubo formula, determined by an energy invariant ``fff function'' involving current matrix elements and energy eigenvalues that describes the dc conductivity of the system in the microcanonical ensemble. Using full diagonalization, we test this criterion for the fff function in the spinless nearest-neighbor Hubbard model and in a system of Sachdev-Ye-Kitaev dots coupled by weak single-particle hopping. We also study the fff function for the spin conductivity in the two-dimensional Heisenberg model and arrive at similar conclusions. Our work suggests that a general principle, formulated in terms of many-body Hilbert space concepts, is at the core of the occurrence of TTT-linear resistivity in a wide range of systems, and precisely translates TTT-linear resistivity into a notion of energy scale invariance far beyond what is typically associated with quantum critical points.
We present a functional-based approach to compute thermal expectation values for actions expressed in the $G\text{\ensuremath{-}}\mathrm{\ensuremath{\Sigma}}$ formalism, applicable to any time sequence ordering. Utilizing this framework, we analyze the linear response to an electric field in various Sachdev-Ye-Kitaev (SYK) chains. We consider the SYK chain where each dot is a complex q/2q/2q/2-body interacting SYK model, and we allow for r/2r/2r/2-body nearest-neighbor hopping where $r=\ensuremath{\kappa}q$. We find exact analytical expressions in the large-qqq limit for conductivities across all temperatures at leading order in 1/q1/q1/q for three cases, namely, $\ensuremath{\kappa}={1/2,1,2}$. When $\ensuremath{\kappa}={1/2,1}$, we observe linear-in-temperature TTT resistivities at low temperatures, indicative of strange metal behavior. Conversely, when $\ensuremath{\kappa}=2$, the resistivity diverges as a power law at low temperatures, namely, as 1/T21/{T}^{2}1/T2, resembling insulating behavior. As TTT increases, there is a crossover to Fermi liquid behavior ($\ensuremath{\sim}{T}^{2}$) at the minimum resistivity. Beyond this, another crossover occurs to strange metal behavior ($\ensuremath{\sim}T$). In comparison to previous linear-in-TTT results in the literature, we also show that the resistivity behavior exists even below the MIR bound, indicating a true strange metal instead of a bad metal. In particular, we find for the $\ensuremath{\kappa}=2$ case a smooth crossover from an insulating phase to a Fermi liquid behavior to a true strange metal and eventually becoming a bad metal as temperature increases. We extend and generalize previously known results on resistivities to all temperatures, do a comparative analysis across the three models where we highlight the universal features and invoke scaling arguments to create a physical picture out of our analyses. Remarkably, we find a universal maximum DC conductivity across all three models when the hopping coupling strength becomes large.
"Strange metals" with resistivity depending linearly on temperature T down to low T have been a long-standing puzzle in condensed matter physics. Here, we consider a lattice model of itinerant spin-[Formula: see text] fermions interacting via onsite Hubbard interaction and random infinite-ranged spin-spin interaction. We show that the quantum critical point associated with the melting of the spin-glass phase by charge fluctuations displays non-Fermi liquid behavior, with local spin dynamics identical to that of the Sachdev-Ye-Kitaev family of models. This extends the quantum spin liquid dynamics previously established in the large-M limit of [Formula: see text] symmetric models to models with physical [Formula: see text] spin-[Formula: see text] electrons. Remarkably, the quantum critical regime also features a Planckian linear-T resistivity associated with a T-linear scattering rate and a frequency dependence of the electronic self-energy consistent with the marginal Fermi liquid phenomenology.
Despite considerable theoretical effort, there has been no consensus on the description of the strange metal state seen in high temperature superconductors and its linear in T resistivity. Here, we construct a simple solvable model of itinerant electrons interacting through random couplings on a honeycomb lattice composed of Sachdev-Ye-Kitaev quantum dots which describes electrons without quasiparticle excitations. This model is solvable in the large-N limit and displays a temperature dependant crossover between two regimes. By tuning the strength of this interaction relative to the bandwidth of the itinerant electrons, we can additionally obtain a finite T crossover to a fully incoherent regime that also has a linear in T resistivity. In the non-Fermi liquid regime, the dc resistivityincreases linearly with temperature over a broad range of temperatures. The self energy has a singular frequency dependence but lacks momentum dependence which is reminiscent of a dynamical mean-field-theory-like behavior but in dimensions d < ∞ . In the low-temperature and strong-coupling limit, the behaviour parallels that of Dirac fermions. We discuss the implications of theseresults and further possibilities in such studies.
A theoretical understanding of the enigmatic linear-in-temperature (T ) resistivity, ubiquitous in strongly correlated metallic systems, has been a long sought-after goal.Furthermore, the slope of this robust T -linear resistivity is also observed to stay constant through crossovers between different temperature regimes: a phenomenon we dub "slope invariance."Recently, several solvable models with T -linear resistivity have been proposed, putting us in an opportune moment to compare their inner workings in various explicit calculations.We consider two strongly correlated models with local self-energies that demonstrate T linearity: a lattice of coupled Sachdev-Ye-Kitaev models and the Hubbard model in single-site dynamical mean-field theory.We find that the two models achieve T linearity through distinct mechanisms at intermediate temperatures.However, we also find that these mechanisms converge to an identical form at high temperatures.Surprisingly, both models exhibit "slope invariance" across the two temperature regimes.Thus not only do we reveal some of the diversity in the theoretical inner workings that can lead to T -linear resistivity, but we also establish that different mechanisms can result in "slope invarance."
The discovery of high-temperature superconductors has raises questions that extend beyond the scope of quasiparticles, a foundational concept in condensed matter theory. This dissertation explores new theoretical approaches to quantum matter without quasiparticle excitations, with a particular focus on quantum criticality and the associated superconductivity in correlated electron compounds. The soluble Sachdev-Ye-Kitaev (SYK) model has recently emerged as a fascinating platform to address the unusual metallic states with T-linear resistivity. We first consider differ- ent variants of random Hubbard models, which share many similarities to key aspects of the original SYK model. Using a large M analysis and a renormalization group method, we propose the existence of quantum critical points with fractionalized excitations separating two distinct phases, which provides insights into the quantum phase transitions and non-Fermi liquid behaviors observed in cuprate superconductors. In the second part of this dissertation, we examine the interplay between non-Fermi liquids and superconductivity with two different pairing mechanisms. Under the framework of random Hubbard model, we demonstrate that superconductivity emerging from non-Fermi liquids strongly deviates from the BCS theory by tuning the relative strength of hopping and exchange interactions. Furthermore, we investigate a two-dimensional Yukawa-SYK model with spatially randomn interactions coupling a Fermi surface to a scalar field associated with a quantum phase transition. Our theory agrees well with transport properties analysed in cuprates, such as a T-linear resistivity and Planckian dissipation.
Abstract The theory of quantum critical phenomena is introduced to study some current many-body problems in condensed matter physics. Renormalization group concepts are applied to strongly correlated electronic materials which are close to a zero-temperature instability. These systems have enhanced effective masses and susceptibility. Scaling arguments yield the exponents which govern the critical behavior of these quantities in terms of the usual critical exponents associated with a zero-temperature phase transition. We show the existence of a new energy scale, related to the quantum nature of the many-body instability, which can be generally associated with the setting of Fermi-liquid behavior with decreasing temperature in three-dimensional strongly interacting electronic systems. The theory of quantum critical phenomena is used to investigate the Kondo lattice problem, which provides a model to describe heavy-fermion systems and to introduce a scaling theory of the Mott transition with special emphasis on charge fluctuation effects. However, this report is not a review on heavy fermions and Mott insulators. The microscopic theories of these systems are still controversial and present some of the most challenging and instigating problems in condensed matter physics. This state of affairs stimulated the author to review and extend the scaling approach. The scaling theory we develop provides a powerful tool, based on the notion of universality, to understand the physical properties of correlated systems beyond the mean-field level. This is illustrated by our treatment of the one-dimensional Hubbard model, where, although the Fermi-liquid fixed point does not survive the fluctuations, the scaling approach is still useful. Finally, we discuss briefly how disorder affect our results.