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arXiv:1508.00763 · Physical review. B./Physical review. B

Dimensionless ratios: Characteristics of quantum liquids and their phase transitions

Yi-Cong Yu, Yang-Yang Chen, Hai‐Qing Lin, Rudolf A. Römer, Xi‐Wen Guan

Published Nov 15, 2016• 16 citations

Abstract

Dimensionless ratios of physical properties can characterize low-temperature phases in a wide variety of materials. As such, the Wilson ratio (WR), the Kadowaki-Woods ratio, and the Wiedemann-Franz law capture essential features of Fermi liquids in metals, heavy fermions, etc. Here we prove that the phases of many-body interacting multicomponent quantum liquids in one dimension (1D) can be described by WRs based on the compressibility, susceptibility, and specific heat associated with each component. These WRs arise due to additivity rules within subsystems reminiscent of the rules for multiresistor networks in series and parallel---a novel and useful characteristic of multicomponent Tomonaga-Luttinger liquids (TLL) independent of microscopic details of the systems. Using experimentally realized multispecies cold atomic gases as examples, we prove that the Wilson ratios uniquely identify phases of TLL, while providing universal scaling relations at the boundaries between phases. Their values within a phase are solely determined by the stiffnesses and sound velocities of subsystems and identify the internal degrees of freedom of said phase such as its spin degeneracy. This finding can be directly applied to a wide range of 1D many-body systems and reveals deep physical insights into recent experimental measurements of the universal thermodynamics in ultracold atoms and spins.

DOI: 10.1103/physrevb.94.195129

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Dimensionless ratios of physical properties can characterize low-temperature phases in a wide variety of materials. As such, the Wilson ratio (WR), the Kadowaki-Woods ratio, and the Wiedemann-Franz law capture essential features of Fermi liquids in metals, heavy fermions, etc. Here we prove that the phases of many-body interacting multicomponent quantum liquids in one dimension (1D) can be described by WRs based on the compressibility, susceptibility, and specific heat associated with each component. These WRs arise due to additivity rules within subsystems reminiscent of the rules for multiresistor networks in series and parallel---a novel and useful characteristic of multicomponent Tomonaga-Luttinger liquids (TLL) independent of microscopic details of the systems. Using experimentally realized multispecies cold atomic gases as examples, we prove that the Wilson ratios uniquely identify phases of TLL, while providing universal scaling relations at the boundaries between phases. Their values within a phase are solely determined by the stiffnesses and sound velocities of subsystems and identify the internal degrees of freedom of said phase such as its spin degeneracy. This finding can be directly applied to a wide range of 1D many-body systems and reveals deep physical insights into recent experimental measurements of the universal thermodynamics in ultracold atoms and spins.

We calculate the Wilson ratio of the one-dimensional Fermi gas with spin imbalance. The Wilson ratio of attractively interacting fermions is solely determined by the density stiffness and sound velocity of pairs and of excess fermions for the two-component Tomonaga-Luttinger liquid phase. The ratio exhibits anomalous enhancement at the two critical points due to the sudden change in the density of states. Despite a breakdown of the quasiparticle description in one dimension, two important features of the Fermi liquid are retained; namely, the specific heat is linearly proportional to temperature, whereas the susceptibility is independent of temperature. In contrast to the phenomenological Tomonaga-Luttinger liquid parameter, the Wilson ratio provides a powerful parameter for testing universal quantum liquids of interacting fermions in one, two, and three dimensions.

A simple set of algebraic equations is derived for the exact low-temperature thermodynamics of one-dimensional multicomponent strongly attractive fermionic atoms with enlarged SU

(N)

spin symmetry and Zeeman splitting. Universal multicomponent Tomonaga-Luttinger liquid (TLL) phases are thus determined. For linear Zeeman splitting, the physics of the gapless phase at low temperatures belongs to the universality class of a two-component asymmetric TLL corresponding to spin-neutral

N

-atom composites and spin-$(N\ensuremath{-}1)/2$ single atoms. The equation of state which we obtained provides a precise description of multicomponent composite fermions and opens up the study of quantum criticality in one-dimensional systems of

N

-component Fermi gases with population imbalance.

The one-dimensional (1D) Hubbard model, describing electrons on a lattice with an on-site repulsive interaction, provides a paradigm for the physics of quantum many-body phenomena. Here, by solving the thermodynamic Bethe ansatz equations, we study the universal thermodynamics, quantum criticality, and magnetism of the 1D attractive Hubbard model. We show that the compressibility and the susceptibility of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-like state obey simple additivity rules at low temperatures, indicating an existence of two free quantum fluids. The magnetic properties, such as magnetization and susceptibility, reveal three physical regions: quantum fluids at low temperatures, a non-Fermi liquid at high temperatures, and the quantum fluid to non-Fermi liquid crossover in between. The lattice interaction is seen to significantly influence the nature of the FFLO-like state in 1D. Furthermore, we show that the dimensionless Wilson ratio provides an ideal parameter to map out the various phase boundaries and to characterize the two free fluids of the FLLO-like state. The quantum scaling functions for the thermal and magnetic properties yield the same dynamic critical exponent

z=2

and correlation critical exponent $\ensuremath{\nu}=1/2$ in the quantum critical region whenever a phase transition occurs. Our results provide a rigorous understanding of quantum criticality and free fluids of many-body systems on a 1D lattice.

Abstract It has been shown (Yang and You 2011 Chin. Phys. Lett. 28 020503 ) that at zero temperature the ground state of the one-dimensional (1D) w -component Fermi gas coincides with that of the spinless Bose gas in the limit . This behavior was experimentally evidenced through quasi-1D tightly trapping ultracold 173 Yb atoms in a recent paper (Pagano et al 2014 Nat. Phys. 10 198 ). However, understanding of low-temperature behavior of Fermi gases with a repulsive interaction requires spin-charge separated conformal field theories of an effective Tomonaga–Luttinger liquid and an antiferromagnetic SU ( w ) Heisenberg spin chain. Here we analytically derive universal thermodynamics of 1D strongly repulsive fermionic gases with SU ( w ) symmetry via the Yang–Yang thermodynamic Bethe ansatz method. The analytical free energy and magnetic properties of the systems at low temperature in a weak magnetic field are obtained through the Wiener–Hopf method. In particular, the free energy essentially manifests the spin-charge separated conformal field theories for high-symmetry systems with arbitrary repulsive interaction strength. We also find that the sound velocity of the Fermi gases in the large w limit coincides with that for the spinless Bose gas, whereas the spin velocity vanishes quickly as w becomes large. This indicates strong suppression of the Fermi exclusion statistics by the commutativity feature among the w -component fermions with different spin states in the Tomonaga–Luttinger liquid phase. Moreover, the equations of state and critical behavior of physical quantities at finite temperature are analytically derived in terms of the polylogarithm functions in the quantum critical region.

This article briefly reviews recent theoretical developments in quantum critical phenomena in one-dimensional (1D) integrable quantum gases of cold atoms. We present a discussion on quantum phase transitions, universal thermodynamics, scaling functions and correlations for a few prototypical exactly solved models, such as the Lieb–Liniger Bose gas, the spin-1 Bose gas with antiferromagnetic spin-spin interaction, the two-component interacting Fermi gas as well as spin-3/2 Fermi gases. We demonstrate that their corresponding Bethe ansatz solutions provide a precise way to understand quantum many-body physics, such as quantum criticality, Luttinger liquids (LLs), the Wilson ratio, Tan's Contact, etc. These theoretical developments give rise to a physical perspective using integrability for uncovering experimentally testable phenomena in systems of interacting bosonic and fermonic ultracold atoms confined to 1D.

The Tomonaga–Luttinger–Liquid (TLL) has been the cornerstone of our understanding of the properties of one dimensional systems. This universal set of properties plays in one dimension, the same role than Fermi liquid plays for the higher dimensional metals. I will give in these notes an overview of some of the experimental tests that were made to probe such TLL physics. In particular I will detail some of the recent experiments that were made in spin systems and which provided remarkable quantitative tests of the TLL physics.