Title: Heuristic bounds on superconductivity and how to exceed them

Abstract What limits the value of the superconducting transition temperature ( T c ) is a question of great fundamental and practical importance. Various heuristic upper bounds on T c have been proposed, expressed as fractions of the Fermi temperature, T F , the zero-temperature superfluid stiffness, ρ s (0), or a characteristic Debye frequency, ω 0 . We show that while these bounds are physically motivated and are certainly useful in many relevant situations, none of them serve as a fundamental bound on T c . To demonstrate this, we provide explicit models where T c / T F (with an appropriately defined T F ), T c / ρ s (0), and T c / ω 0 are unbounded. more »« less

Vyas, Nikhil; Williams, R. Ryan(
, Theory of Computing Systems)

Abstract

We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in$\mathsf {Quasi}\text {-}\mathsf {NP} = \mathsf {NTIME}[n^{(\log n)^{O(1)}}]$$\mathrm{Quasi}-\mathrm{NP}=\mathrm{NTIME}\left[{n}^{{\left(\mathrm{log}n\right)}^{O\left(1\right)}}\right]$and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes$\mathcal { C}$$C$, by showing that$\mathcal { C}$$C$admits non-trivial satisfiability and/or#SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a non-trivial#SAT algorithm for a circuit class${\mathcal C}$$C$. Say that a symmetric Boolean functionf(x_{1},…,x_{n}) issparseif it outputs 1 onO(1) values of${\sum }_{i} x_{i}$${\sum}_{i}{x}_{i}$. We show that for every sparsef, and for all “typical”$\mathcal { C}$$C$, faster#SAT algorithms for$\mathcal { C}$$C$circuits imply lower bounds against the circuit class$f \circ \mathcal { C}$$f\circ C$, which may bestrongerthan$\mathcal { C}$$C$itself. In particular:

#SAT algorithms forn^{k}-size$\mathcal { C}$$C$-circuits running in 2^{n}/n^{k}time (for allk) implyNEXPdoes not have$(f \circ \mathcal { C})$$(f\circ C)$-circuits of polynomial size.

Applying#SAT algorithms from the literature, one immediate corollary of our results is thatQuasi-NPdoes not haveEMAJ∘ACC^{0}∘THRcircuits of polynomial size, whereEMAJis the “exact majority” function, improving previous lower bounds againstACC^{0}[Williams JACM’14] andACC^{0}∘THR[Williams STOC’14], [Murray-Williams STOC’18]. This is the first nontrivial lower bound against such a circuit class.

Ajtai, M.; Braverman, V.; Jayram, T.S.; Silwal, S.; Sun, A.; Woodruff, D.P.; Zhou, S.(
, Proceedings of the 41st ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems (PODS 2022))

There has been a flurry of recent literature studying streaming algorithms for which the input stream is chosen adaptively by a black-box adversary who observes the output of the streaming algorithm at each time step. However, these algorithms fail when the adversary has access to the internal state of the algorithm, rather than just the output of the algorithm. We study streaming algorithms in the white-box adversarial model, where the stream is
chosen adaptively by an adversary who observes the entire internal state of the algorithm at each time step. We show that nontrivial algorithms are still possible. We first give a randomized algorithm for the L1-heavy hitters problem that outperforms the optimal deterministic Misra-Gries algorithm on long streams. If the white-box adversary is computationally bounded, we use cryptographic techniques to reduce the memory of our L1-heavy hitters algorithm even further
and to design a number of additional algorithms for graph, string, and linear algebra problems. The existence of such algorithms is surprising, as the streaming algorithm does not even have a secret key in this model, i.e., its state is entirely known to the adversary. One algorithm we design is for estimating the number of distinct elements in a stream with insertions and deletions achieving a multiplicative approximation and sublinear space; such an algorithm is impossible for deterministic algorithms. We also give a general technique that translates any two-player deterministic communication lower bound to a lower bound for randomized algorithms robust to a white-box adversary. In particular, our results show that for all p ≥ 0, there exists a constant Cp > 1 such that any
Cp-approximation algorithm for Fp moment estimation in insertion-only streams with a white-box adversary requires Ω(n) space for a universe of size n. Similarly, there is a constant C > 1 such that any C-approximation algorithm in an insertion-only stream for matrix rank requires Ω(n) space with a white-box adversary. These results do not contradict our upper bounds since they assume the adversary has unbounded computational power. Our algorithmic results based on cryptography thus show a separation between computationally bounded and unbounded adversaries. Finally, we prove a lower bound of Ω(log n) bits for the fundamental problem of deterministic approximate counting in a stream of 0’s and 1’s, which holds even if we know how many total stream updates we have seen so far at each point in the stream. Such a lower bound for approximate counting with additional information was previously unknown, and in our context, it shows a separation between multiplayer deterministic maximum communication and the white-box space complexity of a streaming algorithm

The proximity of many strongly correlated superconductors to density-wave or nematic order has led to an extensive search for fingerprints of pairing mediated by dynamical quantum-critical (QC) fluctuations of the corresponding order parameter. Here we study anisotropics-wave superconductivity induced by anisotropic QC dynamical nematic fluctuations. We solve the non-linear gap equation for the pairing gap$$\Delta (\theta ,{\omega }_{m})$$$\Delta \left(\theta ,{\omega}_{m}\right)$and show that its angular dependence strongly varies below$${T}_{{\rm{c}}}$$${T}_{c}$. We show that this variation is a signature of QC pairing and comes about because there are multiples-wave pairing instabilities with closely spaced transition temperatures$${T}_{{\rm{c}},n}$$${T}_{c,n}$. Taken alone, each instability would produce a gap$$\Delta (\theta ,{\omega }_{m})$$$\Delta \left(\theta ,{\omega}_{m}\right)$that changes sign$$8n$$$8n$times along the Fermi surface. We show that the equilibrium gap$$\Delta (\theta ,{\omega }_{m})$$$\Delta (\theta ,{\omega}_{m})$is a superposition of multiple components that are nonlinearly induced below the actual$${T}_{{\rm{c}}}={T}_{{\rm{c}},0}$$${T}_{c}={T}_{c,0}$, and get resonantly enhanced at$$T={T}_{{\rm{c}},n}\ <\ {T}_{{\rm{c}}}$$$T={T}_{c,n}\phantom{\rule{0ex}{0ex}}<\phantom{\rule{0ex}{0ex}}{T}_{c}$. This gives rise to strong temperature variation of the angular dependence of$$\Delta (\theta ,{\omega }_{m})$$$\Delta \left(\theta ,{\omega}_{m}\right)$. This variation progressively disappears away from a QC point.

Shi, Zhenzhong; Baity, P. G.; Terzic, J.; Pokharel, Bal K.; Sasagawa, T.; Popović, Dragana(
, Nature Communications)

null
(Ed.)

Abstract The origin of the weak insulating behavior of the resistivity, i.e. $${\rho }_{xx}\propto {\mathrm{ln}}\,(1/T)$$ ρ x x ∝ ln ( 1 / T ) , revealed when magnetic fields ( H ) suppress superconductivity in underdoped cuprates has been a longtime mystery. Surprisingly, the high-field behavior of the resistivity observed recently in charge- and spin-stripe-ordered La-214 cuprates suggests a metallic, as opposed to insulating, high-field normal state. Here we report the vanishing of the Hall coefficient in this field-revealed normal state for all $$T\ <\ (2-6){T}_{{\rm{c}}}^{0}$$ T < ( 2 − 6 ) T c 0 , where $${T}_{{\rm{c}}}^{0}$$ T c 0 is the zero-field superconducting transition temperature. Our measurements demonstrate that this is a robust fundamental property of the normal state of cuprates with intertwined orders, exhibited in the previously unexplored regime of T and H . The behavior of the high-field Hall coefficient is fundamentally different from that in other cuprates such as YBa 2 Cu 3 O 6+ x and YBa 2 Cu 4 O 8 , and may imply an approximate particle-hole symmetry that is unique to stripe-ordered cuprates. Our results highlight the important role of the competing orders in determining the normal state of cuprates.

We perform path-integral molecular dynamics (PIMD), ring-polymer MD (RPMD), and classical MD simulations of H$$_2$$${}_{2}$O and D$$_2$$${}_{2}$O using the q-TIP4P/F water model over a wide range of temperatures and pressures. The density$$\rho (T)$$$\rho \left(T\right)$, isothermal compressibility$$\kappa _T(T)$$${\kappa}_{T}\left(T\right)$, and self-diffusion coefficientsD(T) of H$$_2$$${}_{2}$O and D$$_2$$${}_{2}$O are in excellent agreement with available experimental data; the isobaric heat capacity$$C_P(T)$$${C}_{P}\left(T\right)$obtained from PIMD and MD simulations agree qualitatively well with the experiments. Some of these thermodynamic properties exhibit anomalous maxima upon isobaric cooling, consistent with recent experiments and with the possibility that H$$_2$$${}_{2}$O and D$$_2$$${}_{2}$O exhibit a liquid-liquid critical point (LLCP) at low temperatures and positive pressures. The data from PIMD/MD for H$$_2$$${}_{2}$O and D$$_2$$${}_{2}$O can be fitted remarkably well using the Two-State-Equation-of-State (TSEOS). Using the TSEOS, we estimate that the LLCP for q-TIP4P/F H$$_2$$${}_{2}$O, from PIMD simulations, is located at$$P_c = 167 \pm 9$$${P}_{c}=167\pm 9$ MPa,$$T_c = 159 \pm 6$$${T}_{c}=159\pm 6$ K, and$$\rho _c = 1.02 \pm 0.01$$${\rho}_{c}=1.02\pm 0.01$ g/cm$$^3$$${}^{3}$. Isotope substitution effects are important; the LLCP location in q-TIP4P/F D$$_2$$${}_{2}$O is estimated to be$$P_c = 176 \pm 4$$${P}_{c}=176\pm 4$ MPa,$$T_c = 177 \pm 2$$${T}_{c}=177\pm 2$ K, and$$\rho _c = 1.13 \pm 0.01$$${\rho}_{c}=1.13\pm 0.01$ g/cm$$^3$$${}^{3}$. Interestingly, for the water model studied, differences in the LLCP location from PIMD and MD simulations suggest that nuclear quantum effects (i.e., atoms delocalization) play an important role in the thermodynamics of water around the LLCP (from the MD simulations of q-TIP4P/F water,$$P_c = 203 \pm 4$$${P}_{c}=203\pm 4$ MPa,$$T_c = 175 \pm 2$$${T}_{c}=175\pm 2$ K, and$$\rho _c = 1.03 \pm 0.01$$${\rho}_{c}=1.03\pm 0.01$ g/cm$$^3$$${}^{3}$). Overall, our results strongly support the LLPT scenario to explain water anomalous behavior, independently of the fundamental differences between classical MD and PIMD techniques. The reported values of$$T_c$$${T}_{c}$for D$$_2$$${}_{2}$O and, particularly, H$$_2$$${}_{2}$O suggest that improved water models are needed for the study of supercooled water.

Hofmann, Johannes S., Chowdhury, Debanjan, Kivelson, Steven A., and Berg, Erez. Heuristic bounds on superconductivity and how to exceed them. Retrieved from https://par.nsf.gov/biblio/10412396. npj Quantum Materials 7.1 Web. doi:10.1038/s41535-022-00491-1.

Hofmann, Johannes S., Chowdhury, Debanjan, Kivelson, Steven A., & Berg, Erez. Heuristic bounds on superconductivity and how to exceed them. npj Quantum Materials, 7 (1). Retrieved from https://par.nsf.gov/biblio/10412396. https://doi.org/10.1038/s41535-022-00491-1

@article{osti_10412396,
place = {Country unknown/Code not available},
title = {Heuristic bounds on superconductivity and how to exceed them},
url = {https://par.nsf.gov/biblio/10412396},
DOI = {10.1038/s41535-022-00491-1},
abstractNote = {Abstract What limits the value of the superconducting transition temperature ( T c ) is a question of great fundamental and practical importance. Various heuristic upper bounds on T c have been proposed, expressed as fractions of the Fermi temperature, T F , the zero-temperature superfluid stiffness, ρ s (0), or a characteristic Debye frequency, ω 0 . We show that while these bounds are physically motivated and are certainly useful in many relevant situations, none of them serve as a fundamental bound on T c . To demonstrate this, we provide explicit models where T c / T F (with an appropriately defined T F ), T c / ρ s (0), and T c / ω 0 are unbounded.},
journal = {npj Quantum Materials},
volume = {7},
number = {1},
author = {Hofmann, Johannes S. and Chowdhury, Debanjan and Kivelson, Steven A. and Berg, Erez},
}

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