Quantum Multicritical Point Research Articles

Quantum multicritical point in the two- and three-dimensional random transverse-field Ising model

Quantum multicritical points (QMCPs) emerge at the junction of two or more quantum phase transitions due to the interplay of disparate fluctuations, leading to novel universality classes. While quantum critical points have been well characterized, our understanding of QMCPs is much more limited, even though they might be less elusive to study experimentally than quantum critical points. Here, we characterize the QMCP of an interacting heterogeneous quantum system in two and three dimensions, the ferromagnetic random transverse-field Ising model (RTIM). The QMCP of the RTIM emerges due to both geometric and quantum fluctuations, studied here numerically by the strong disorder renormalization group method. The QMCP of the RTIM is found to exhibit ultraslow, activated dynamic scaling, governed by an infinite disorder fixed point. This ensures that the obtained multicritical exponents tend to the exact values at large scales, while also being universal---i.e., independent of the form of disorder---providing a solid theoretical basis for future experiments.

Quantification of local Ising magnetism in rare-earth pyrogermanates Er2Ge2O7 and Yb2Ge2O7

Recently, a local-Ising-type magnetic order was inferred from neutron diffraction of the antiferromagnetic ${\mathrm{Er}}_{2}{\mathrm{Ge}}_{2}{\mathrm{O}}_{7}$ (pg-ErGO) with an applied magnetic field. Here, we use neutron spectroscopy to investigate the energetics of pg-ErGO and the isostructural ${\mathrm{Yb}}_{2}{\mathrm{Ge}}_{2}{\mathrm{O}}_{7}$ (pg-YbGO) to evaluate the adequacy of the local-Ising description. To begin, we generate a model of the magnetic structure of pg-YbGO using neutron diffraction and find a net ferromagnetic moment. While pg-ErGO and pg-YbGO have highly symmetric crystal structures ($P{4}_{1}{2}_{1}2$ tetragonal space group 92) with only one trivalent rare-earth magnetic site, the point symmetry of the rare-earth site is low with only a single symmetry element (point group ${\mathrm{C}}_{1}$). For both compounds, the energy scale of the first excited state is large compared to the magnetic ordering temperature, suggesting Ising character. The ground-state Kramer's doublet of both compounds is dominated by the maximal ${m}_{\mathrm{J}}$ component. However, the low point group symmetry of the rare-earth site leads to finite mixing of all other ${m}_{\mathrm{J}}$'s, which suggests potential deviations from Ising behavior. Moreover, quasielastic scattering is observed deep in the ordered state of pg-ErGO and pg-YbGO that may be due to non-Ising behavior. The dominant magnetic interaction in both compounds is found to be magnetostatic by considering the magnetic excitations in the ordered state. From consideration of these data, the pg-YbGO is more Ising-like than pg-ErGO. Also, quantum multicritical points are anticipated with applied magnetic field in both compounds.

Simulation of a topological phase transition in a Kitaev chain with long-range coupling using a superconducting circuit

We report an experimental work on simulating the Kitaev chain of one-dimensional $p$-wave superconductivity with long-range coupling using superconducting quantum circuits. The effective Hamiltonian of this model in the momentum space is mapped onto that of a qubit driven by a tunable microwave control field. By monitoring the dynamics of the qubit, we generate intuitive yet precise visualization of topological characteristics of different quantum phases of the system. Topological invariants can be directly deduced from the experimental results, without relying on indirect extraction from measurements such as microwave spectroscopy and temporal averaging. Therefore a much enhanced efficiency of measurement can be realized compared to other methods that are often used to simulate topological phase transitions. As a consequence, a comprehensive phase diagram covering a wide range of the parameter space is obtained. Topological phase transitions and quantum multicritical points are clearly demonstrated. In particular, new topological phases emerge as a consequence of long-range coupling. The method used here can be readily generalized to simulate more complex systems on different experimental platforms.

Fermionic multicriticality near Kekulé valence-bond ordering on a honeycomb lattice

We analyze the possibility of emergent quantum multicritical points (MCPs) with enlarged chiral symmetry, when strongly interacting gapless Dirac fermions acquire comparable propensity toward the nucleation of Kekul\'{e} valence-bond solid (KVBS) and charge-density-wave ($N_b=1$) or $s$-wave pairing ($N_b=2$) or anti-ferromagnet ($N_b=3$) in honeycomb lattice, where $N_b$ counts the number of bosonic order parameter components. Besides the cubic terms present in the order parameter description of KVBS due to the breaking of a discrete $Z_3$ symmetry, quantum fluctuations generate new cubic vertices near the high symmetry MCPs. All cubic terms are strongly relevant at the bare level near three spatial dimensions, about which we perform a leading order renormalization group analysis of coupled Gross-Neveu-Yukawa field theory. We show that due to non-trivial Yukawa interactions among gapless bosonic and fermionic degrees of freedom, all cubic terms ultimately become irrelevant at an $O(2+N_b)$ symmetric MCP, near two spatial dimensions, where $N_b=1,2,3$. Therefore, MCPs with an enlarged $O(2+N_b)$ symmetry near KVBS ordering are stable.

Universal Phase Diagram and Scaling Functions of Imbalanced Fermi Gases

We discuss the phase diagram and the universal scaling functions of attractive Fermi gases at finite imbalance. The existence of a quantum multicritical point for the unitary gas at vanishing chemical potential $\mu$ and effective magnetic field $h$, first discussed by Nikoli\'{c} and Sachdev, gives rise to three different phase diagrams, depending on whether the inverse scattering length $1/a$ is negative, positive or zero. Within a Luttinger-Ward formalism, the phase diagram and pressure of the unitary gas is calculated as a function of the dimensionless scaling variables $T/\mu$ and $h/\mu$. The results indicate that beyond the Clogston-Chandrasekhar limit at $(h/\mu)_c\simeq 1.09$, the unitary gas exhibits an inhomogeneous superfluid phase with FFLO order that can reach critical temperatures near unitarity of $\simeq 0.03\, T_F$.

Itinerant quantum multicriticality of two-dimensional Dirac fermions

We analyze emergent quantum multi-criticality for strongly interacting, massless Dirac fermions in two spatial dimensions ($d=2$) within the framework of Gross-Neveu-Yukawa models, by considering the competing order parameters that give rise to fully gapped (insulating or superconducting) ground states. We focus only on those competing orders, which can be rotated into each other by generators of an exact or emergent chiral symmetry of massless Dirac fermions, and break $O(S_1)$ and $O(S_2)$ symmetries in the ordered phase. Performing a renormalization group analysis by using the $\epsilon=(3-d)$ expansion scheme, we show that all the coupling constants in the critical hyperplane flow toward a new attractive fixed point, supporting an \emph{enlarged} $O(S_1+S_2)$ chiral symmetry. Such a fixed point acts as an exotic quantum multi-critical point (MCP), governing the \emph{continuous} semimetal-insulator as well as insulator-insulator (for example antiferromagnet to valence bond solid) quantum phase transitions. In comparison with the lower symmetric semimetal-insulator quantum critical points, possessing either $O(S_1)$ or $O(S_2)$ chiral symmetry, the MCP displays enhanced correlation length exponents, and anomalous scaling dimensions for both fermionic and bosonic fields. We discuss the scaling properties of the ratio of bosonic and fermionic masses, and the increased dc resistivity at the MCP. By computing the scaling dimensions of different local fermion bilinears in the particle-hole channel, we establish that most of the four fermion operators or generalized density-density correlation functions display faster power law decays at the MCP compared to the free fermion and lower symmetric itinerant quantum critical points. Possible generalization of this scenario to higher dimensional Dirac fermions is also outlined.

Compatible orders and fermion-induced emergent symmetry in Dirac systems

We study the quantum multicritical point in a (2+1)-dimensional Dirac system between the semimetallic phase and two ordered phases that are characterized by anticommuting mass terms with $O(N_1)$ and $O(N_2)$ symmetry, respectively. Using $\epsilon$ expansion around the upper critical space-time dimension of four, we demonstrate the existence of a stable renormalization-group fixed point, enabling a direct and continuous transition between the two ordered phases directly at the multicritical point. This point is found to be characterized by an emergent $O(N_1+N_2)$ symmetry for arbitrary values of $N_1$ and $N_2$ and fermion flavor numbers $N_f$, as long as the corresponding representation of the Clifford algebra exists. Small $O(N)$-breaking perturbations near the chiral $O(N)$ fixed point are therefore irrelevant. This result can be traced back to the presence of gapless Dirac degrees of freedom at criticality, and it is in clear contrast to the purely bosonic $O(N)$ fixed point, which is stable only when $N < 3$. As a by-product, we obtain predictions for the critical behavior of the chiral $O(N)$ universality classes for arbitrary $N$ and fermion flavor number $N_f$. Implications for critical Weyl and Dirac systems in 3+1 dimensions are also briefly discussed.

Quantum multicriticality in disordered Weyl semimetals

In electronic band structure of solid state material, two band touching points with linear dispersion appear in pair in the momentum space. When they annihilate with each other, the system undergoes a quantum phase transition from three-dimensional Weyl semimetal (WSM) phase to a band insulator phase such as Chern band insulator (CI) phase. The phase transition is described by a new critical theory with a `magnetic dipole' like object in the momentum space. In this paper, we reveal that the critical theory hosts a novel disorder-driven quantum multicritical point, which is encompassed by three quantum phases, renormalized WSM phase, CI phase, and diffusive metal (DM) phase. Based on the renormalization group argument, we first clarify scaling properties around the band touching points at the quantum multicritical point as well as all phase boundaries among these three phases. Based on numerical calculations of localization length, density of states and critical conductance distribution, we next prove that a localization-delocalization transition between the CI phase with a finite zero-energy density of states (zDOS) and DM phase belongs to an ordinary 3D unitary class. Meanwhile, a localization-delocalization transition between the Chern insulator phase with zero zDOS and a renormalized Weyl semimetal (WSM) phase turns out to be a direct phase transition whose critical exponent $\nu=0.80\pm 0.01$. We interpret these numerical results by a renormalization group analysis on the critical theory.

Strong-randomness infinite-coupling phase in a random quantum spin chain

We study the ground-state phase diagram of the Ashkin-Teller random quantum spin chain by means of a generalization of the strong-disorder renormalization group. In addition to the conventional paramagnetic and ferromagnetic (Baxter) phases, we find a partially ordered phase characterized by strong randomness and infinite coupling between the colors. This unusual phase acts, at the same time, as a Griffiths phase for two distinct quantum phase transitions, both of which are of infinite-randomness type. We also investigate the quantum multicritical point that separates the two-phase and three-phase regions, and we discuss generalizations of our results to higher dimensions and other systems.

Quantum XY criticality in a two-dimensional Bose gas near the Mott transition

We derive the equation of state of a two-dimensional Bose gas in an optical lattice in the framework of the Bose-Hubbard model. We focus on the vicinity of the multicritical points where the quantum phase transition between the Mott insulator and the superfluid phase occurs at fixed density and belongs to the three-dimensional XY model universality class. Using a nonperturbative renormalization-group approach, we compute the pressure P(μ, T) as a function of chemical potential and temperature. Our results compare favorably with a calculation based on the quantum O(2) model —we find the same universal scaling function— and allow us to determine the region of the phase diagram in the vicinity of a quantum multicritical point where the equation of state is universal. We also discuss the possible experimental observation of quantum XY criticality in an ultracold gas in an optical lattice.

Non-equilibrium dynamics near a quantum multicritical point

We study the non-equilibrium dynamics of a quantum system close to a quantum multi-critical point (MCP) using the example of a one-dimensional spin-1/2 transverse XY spin chain. We summarize earlier results of defect generenation and fidelity susceptibility for quenching through MCP and close to the MCP, respectively. For a quenching scheme which enables the system to hit the MCP along different paths, we emphasize the role of path on exponents associated with quasicritical points which appear in the scaling relations. Finally, we explicitly derive the scaling of concurrence and negativity for two spin entanglement generated following a slow quenching across the MCP and enlist the results for different quenching schemes. We explicity show the dependence of the scaling on the quenching path and dicuss the limiting situations.

Path-dependent scaling of geometric phase near a quantum multi-critical point

We study the geometric phase of the ground state in a one-dimensional transverseXY spin chain in the vicinity of a quantum multi-critical point. We approach the multi-critical pointalong different paths and estimate the geometric phase by applying a rotation in all spins about thez axis by anangle η. Although the geometric phase itself vanishes at the multi-critical point, the derivative withrespect to the anisotropy parameter of the model shows peaks at different points on theferromagnetic side close to it where the energy gap is a local minimum; we call these points‘quasi-critical’. The value of the derivative at any quasi-critical point scales with the systemsize in a power-law fashion with the exponent varying continuously with the parameterα that defines a path,up to a critical value α = αc = 2. For α > αc, or on the paramagnetic side, no such peak is observed. Numerically obtained results are inperfect agreement with analytical predictions.

Oscillating fidelity susceptibility near a quantum multicritical point

We study scaling behavior of the geometric tensor ${\ensuremath{\chi}}_{\ensuremath{\alpha},\ensuremath{\beta}}({\ensuremath{\lambda}}_{1},{\ensuremath{\lambda}}_{2})$ and the fidelity susceptibility ${\ensuremath{\chi}}_{F}$ in the vicinity of a quantum multicritical point (MCP) using the example of a transverse $\mathit{XY}$ model. We show that the behavior of the geometric tensor (and thus of ${\ensuremath{\chi}}_{F}$) is drastically different from that seen near a critical point. In particular, we find that it is a highly nonmonotonic function of $\ensuremath{\lambda}$ along the generic direction ${\ensuremath{\lambda}}_{1}~{\ensuremath{\lambda}}_{2}=\ensuremath{\lambda}$ when the system size $L$ is bounded by the shorter and longer correlation lengths characterizing the MCP: $1/|\ensuremath{\lambda}|{}^{{\ensuremath{\nu}}_{1}}\ensuremath{\ll}L\ensuremath{\ll}1/|\ensuremath{\lambda}|{}^{{\ensuremath{\nu}}_{2}}$, where ${\ensuremath{\nu}}_{1}&lt;{\ensuremath{\nu}}_{2}$ are the two correlation-length exponents characterizing the system. We find that the scaling of the maxima of the components of ${\ensuremath{\chi}}_{\ensuremath{\alpha}\ensuremath{\beta}}$ is associated with the emergence of quasicritical points at $\ensuremath{\lambda}~1/{L}^{1/{\ensuremath{\nu}}_{1}}$, related to the proximity to the critical line of the finite-momentum anisotropic transition. This scaling is different from that in the thermodynamic limit $L\ensuremath{\gg}1/|\ensuremath{\lambda}|{}^{{\ensuremath{\nu}}_{2}}$, which is determined by the conventional critical exponents. We use our results to calculate the defect density following a rapid quench starting from the MCP and show that it exerts a steplike behavior for small quench amplitudes. A study of heat density and diagonal entropy density also shows signatures of quasicritical points.

Anomalous nonergodic scaling in adiabatic multicritical quantum quenches

We investigate nonequilibrium dynamical scaling in adiabatic quench processes across quantum multicritical points. Our analysis shows that the resulting power-law scaling depends sensitively on the control path and that anomalous critical exponents may emerge depending on the universality class. We argue that the observed anomalous behavior originates in the fact that the dynamical excitation process takes place asymmetrically with respect to the static multicritical point and that noncritical energy modes may play a dominant role. As a consequence, dynamical scaling requires introducing genuinely nonstatic exponents.

Competing orders and hidden duality symmetries in two-leg spin ladder systems

A unifying approach to competing quantum orders in generalized two-leg spin ladders is presented. Hidden relationship and quantum phase transitions among the competing orders are thoroughly discussed by means of a low-energy field theory starting from an SU(4) quantum multicritical point. Our approach reveals that the system has a relatively simple phase structure in spite of its complicated interactions. On top of the U(1) symmetry which is known from previous studies to mix up antiferromagnetic order parameter with that of the $p$-type nematic, we find an emergent U(1) symmetry which mixes order parameters dual to the above. On the basis of the field-theoretical and variational analysis, we give a qualitative picture for the global structure of the phase diagram. Interesting connection to other models (e.g., the bosonic $t\text{\ensuremath{-}}J$ model) is also discussed.

Quantum critical point associated with the electronic topological transition in a two-dimensional electron system as a driving force for anomalies in underdoped high-Tccuprates

We study an electronic topological transition (ETT) (the transition due to change in topology of Fermi surface) that occurs in a two-dimensional electron system on a square lattice with hopping beyond nearest neighbors under a change of electronic concentration n (or of hole doping $\ensuremath{\delta}=1\ensuremath{-}n).$ We show that the ETT point $(\ensuremath{\delta}={\ensuremath{\delta}}_{c}, T=0)$ is an exotic quantum critical point (QCP) with several aspects of criticality. The first trivial one is related to singularities in thermodynamic properties. A nontrivial and never considered aspect concerns a behavior of the electron-hole response function: the ETT point is a quantum multicritical point, the end point of the critical line $(T=0, \ensuremath{\delta}&gt;{\ensuremath{\delta}}_{c})$ of static Kohn singularities. We show that the existence of this QCP results in global anomalies of the system in the presence of interaction. (We consider the interaction corresponding to the ${t\ensuremath{-}t}^{\ensuremath{'}}\ensuremath{-}J$ model.) The anomalies take place on one side of the ETT $(\ensuremath{\delta}&lt;{\ensuremath{\delta}}_{c}$ in the case of ${t}^{\ensuremath{'}}/t&lt;0$ corresponding to the high-${T}_{c}$ cuprates) and have a striking similarity with the anomalies observed in the high-${T}_{c}$ cuprates in the underdoped regime. The most important consequence of the ETT (besides the appearance of superconducting instability of the d-wave symmetry) is the existence of the pseudocritical zone, ${T}^{*}(\ensuremath{\delta})\ensuremath{\propto}{\ensuremath{\delta}}_{c}\ensuremath{-}\ensuremath{\delta},$ which grows from the point $\ensuremath{\delta}={\ensuremath{\delta}}_{c}$ on the side $\ensuremath{\delta}&lt;{\ensuremath{\delta}}_{c}.$ Below and around this zone the metal state is anomalous. Some anomalies are considered in the present paper and compared with experiment (NMR, neutron scattering).

Modified finite-size scaling for anharmonic crystals with quantum fluctuations

The modified finite-size scaling (for dimensions d ⩾ d >, where d > is the upper critical dimensionality) is verified in the closed vicinity of both the classical and the quantum multicritical points by the example of a quantum model for an anharmonic cyrstal, confined to a fully finite (block) geometry under periodic boundary conditions. Unified scaling equations corresponding to the two kinds of correlation lengths related with the fixed dimensionless temperature t when the quantum parameter λ is varied, and with fixed λ when t is varied, are obtained.

Universal amplitudes in finite-size scaling for an anharmonic crystal

We consider an exactly solvable d-dimensional lattice model of an anharmonic crystal confined to a geometry L d− d × ∞ d′ and subject to periodic boundary conditions. The general idea of finite-size scaling on the whole phase diagram is tested and the universal amplitudes of the correlation length near the upper and lower critical dimensions in both the classifical and the quantum multicritical points are computed. A detailed analysis is given in terms of the dimensions d and d′ of the lattice and parameter σ of the harmonic force decreasing at long distances as 1/ r d + σ (0 ⩽ σ ⩽ 2).

Quantum Para-, Ferro-, and Random Electric Behaviors in Oxide Perovskites

An introduction to and a survey of quantum para- and ferro-electric behavior in oxide perovskites are given. In the paraelectric case, quantum fluctuations stabilize the disordered state with a high, temperature-independent dielectric constant. Application of uniaxial stress or ion substitution can induce a quantum ferroelectric state at a special quantum multicritical point (QMCP). The occurrence of or crossover to a random-field induced domain state in partially substituted crystals is reviewed. The emphasis is on SrTiO3 studies and on Sr1-xCaxTiO3 mixed crystals, but contact is made to stressed KTaO3 and substitution of K+ or Ta5+ ions