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Simple Fermionic Model of Deconfined Phases and Phase Transitions

Fakher F. Assaad, Tarun Grover

Published Jan 1, 2016• 77 citations
OpenAlex

Abstract

Using Quantum Monte Carlo simulations, we study a series of models of fermions coupled to quantum Ising spins on a square lattice with $N$ flavors of fermions per site for $N=1,2$ and $3$. The models have an extensive number of conserved quantities but are not integrable, and have rather rich phase diagrams consisting of several exotic phases and phase transitions that lie beyond Landau-Ginzburg paradigm. In particular, one of the prominent phase for $N>1$ corresponds to $2N$ gapless Dirac fermions coupled to an emergent $\mathbb{Z}_2$ gauge field in its deconfined phase. However, unlike a conventional $\mathbb{Z}_2$ gauge theory, we do not impose the `Gauss's Law' by hand and instead, it emerges due to spontaneous symmetry breaking. Correspondingly, unlike a conventional $\mathbb{Z}_2$ gauge theory in two spatial dimensions, our models have a finite temperature phase transition associated with the melting of the order parameter that dynamically imposes the Gauss's law constraint at zero temperature. By tuning a parameter, the deconfined phase undergoes a transition into a short range entangled phase, which corresponds to N\'eel/Superconductor for $N=2$ and a Valence Bond Solid for $N=3$. Furthermore, for $N=3$, the Valence Bond Solid further undergoes a transition to a N\'eel phase consistent with the deconfined quantum critical phenomenon studied earlier in the context of quantum magnets.

Related papers

Simple Fermionic Model of Deconfined Phases and Phase Transitions

Fakher F. Assaad, Tarun Grover

Using quantum Monte Carlo simulations, we study a series of models of fermions coupled to quantum Ising spins on a square lattice with N flavors of fermions per site for N ¼ 1, 2, and 3.The models have an extensive number of conserved quantities but are not integrable, and they have rather rich phase diagrams consisting of several exotic phases and phase transitions that lie beyond the Landau-Ginzburg paradigm.In particular, one of the prominent phases for N > 1 corresponds to 2N gapless Dirac fermions coupled to an emergent Z 2 gauge field in its deconfined phase.However, unlike a conventional Z 2 gauge theory, we do not impose "Gauss's Law" by hand; instead, it emerges because of spontaneous symmetry breaking.Correspondingly, unlike a conventional Z 2 gauge theory in two spatial dimensions, our models have a finite-temperature phase transition associated with the melting of the order parameter that dynamically imposes the Gauss's law constraint at zero temperature.By tuning a parameter, the deconfined phase undergoes a transition into a short-range entangled phase, which corresponds to Néel antiferromagnet or superconductor for N ¼ 2 and a valence-bond solid for N ¼ 3. Furthermore, for N ¼ 3, the valence-bond solid further undergoes a transition to a Néel phase consistent with the deconfined quantum critical phenomenon studied earlier in the context of quantum magnets.

arXiv:1607.03912•Physical Review X•Dec 12, 2016•76 citations

Monte Carlo Study of Lattice Compact Quantum Electrodynamics with Fermionic Matter: The Parent State of Quantum Phases

Xiao Yan Xu, Yang Qi, Long Zhang +3 more

The interplay between lattice gauge theories and fermionic matter accounts for fundamental physical phenomena ranging from the deconfinement of quarks in particle physics to quantum spin liquid with fractionalized anyons and emergent gauge structures in condensed matter physics.However, except for certain limits (for instance, a large number of flavors of matter fields), analytical methods can provide few concrete results.Here we show that the problem of compact Uð1Þ lattice gauge theory coupled to fermionic matter in ð2 þ 1ÞD is possible to access via sign-problem-free quantum Monte Carlo simulations.One can hence map out the phase diagram as a function of fermion flavors and the strength of gauge fluctuations.By increasing the coupling constant of the gauge field, gauge confinement in the form of various spontaneous-symmetrybreaking phases such as the valence-bond solid (VBS) and Néel antiferromagnet emerge.Deconfined phases with algebraic spin and VBS correlation functions are also observed.Such deconfined phases are incarnations of exotic states of matter, i.e., the algebraic spin liquid, which is generally viewed as the parent state of various quantum phases.The phase transitions between the deconfined and confined phases, as well as that between the different confined phases provide various manifestations of deconfined quantum criticality.In particular, for four flavors N f ¼ 4, our data suggest a continuous quantum phase transition between the VBS and Néel order.We also provide preliminary theoretical analysis for these quantum phase transitions.

arXiv:1807.07574•Physical Review X•May 2, 2019•98 citations

Deconfined criticality and a gapless Z 2 spin liquid in the square-lattice antiferromagnet

Henry Shackleton, Alex Thomson, Subir Sachdev A

The theory for the vanishing of N\'eel order in the spin $S=1/2$ square lattice antiferromagnet has been the focus of attention for many decades. A consensus appears to have emerged in recent numerical studies on the antiferromagnet with first and second neighbor exchange interactions (the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model): A gapless spin liquid is present for a narrow window of parameters between the vanishing of the N\'eel order and the onset of a gapped valence bond solid state. We propose a deconfined critical SU(2) gauge theory for a transition into a stable ${\mathbb{Z}}_{2}$ spin liquid with massless Dirac spinon excitations; on the other side of the critical point, the SU(2) spin liquid (the `$\ensuremath{\pi}$-flux' phase) is presumed to be unstable to confinement to the N\'eel phase. We identify a dangerously irrelevant coupling in the critical SU(2) gauge theory, which contributes a logarithm-squared renormalization. This critical theory is also not Lorentz invariant and weakly breaks the SO(5) symmetry which rotates between the N\'eel and valence bond solid order parameters. We also propose a distinct deconfined critical U(1) gauge theory for a transition into the same gapless ${\mathbb{Z}}_{2}$ spin liquid; on the other side of the critical point, the U(1) spin liquid (the `staggered flux' phase) is presumed to be unstable to confinement to the valence bond solid. This critical theory has no dangerously irrelevant coupling, dynamic critical exponent $z\ensuremath{\ne}1$, and no SO(5) symmetry. All of these phases and critical points are unified in a SU(2) gauge theory with Higgs fields and fermionic spinons which can naturally realize the observed sequence of phases with increasing ${J}_{2}/{J}_{1}$: N\'eel, gapless ${\mathbb{Z}}_{2}$ spin liquid, and valence bond solid.

arXiv:2104.09537•Physical review. B./Physical review. B•Jul 6, 2021•27 citations

Deconfined Thermal Phase Transitions with $Z_2$ Gauge Structures

Eun-Gook Moon

Fathoming deconfined phases is one of the key issues in modern condensed matter. Striking many-body effects including massive quantum entanglement and coherence may be realized as manifested in quantum spin liquids and topological orders. Here, we demonstrate that deconfined phases even host exotic thermal phase transitions, dubbed deconfined thermal transitions. Constructing a $Z_2$ lattice gauge model with strong interactions between $Z_2$ gauge fluxes, we prove the existence of a thermal phase transition between deconfined and confined phases in two spatial dimensions in sharp contrast to its absence in the Wegner model. Incorporating deconfined fermions, it is shown that gapless excitations from Fermi surfaces endow line-tension to $Z_2$ gauge fluxes at zero temperature, and we argue that a deconfined thermal transition with deconfined fermions may be interpreted as a hidden order transition with thermal gap-opening in Fermi surfaces. Moreover, it is shown that symmetry breaking transitions in deconfined phases may be unconventional. Global $Z_2$ and $U(1)$ symmetry breaking transitions in deconfined phases may be in the same universality class, which is impossible under the conventional Landau-Ginzburg-Wilson paradigm. Characteristic signatures of the transitions in experiments and candidate strongly correlated systems such as Kitaev materials are also discussed.

arXiv:1812.05621•Dec 13, 2018•2 citations

Deconfined quantum critical point in one dimension

Brenden Roberts, Shenghan Jiang, Olexei I. Motrunich

We perform a numerical study of a spin-1/2 model with ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry in one dimension which demonstrates an interesting similarity to the physics of two-dimensional deconfined quantum critical points (DQCP). Specifically, we investigate the quantum phase transition between Ising ferromagnetic and valence bond solid (VBS) symmetry-breaking phases. Working directly in the thermodynamic limit using uniform matrix product states, we find evidence for a direct continuous phase transition that lies outside of the Landau-Ginzburg-Wilson paradigm. In our model, the continuous transition is found everywhere on the phase boundary. We find that the magnetic and VBS correlations show very close power-law exponents, which is expected from the self-duality of the parton description of this DQCP. Critical exponents vary continuously along the phase boundary in a manner consistent with the predictions of the field theory for this transition. We also find a regime where the phase boundary splits, as suggested by the theory, introducing an intermediate phase of coexisting ferromagnetic and VBS order parameters. Interestingly, we discover a transition involving this coexistence phase which is similar to the DQCP, being also disallowed by the Landau-Ginzburg-Wilson symmetry-breaking theory.

arXiv:1904.00010•Physical review. B./Physical review. B•Apr 29, 2019•78 citations

Dynamics of compact quantum electrodynamics at large fermion flavor

Wei Wang, Da-Chuan Lu, Xiao Yan Xu +2 more

Thanks to the development in quantum Monte Carlo technique, the compact U(1) lattice gauge theory coupled to fermionic matter at ($2+1$)D is now accessible with large-scale numerical simulations, and the ground state phase diagram as a function of fermion flavor (${N}_{f}$) and the strength of gauge fluctuations is mapped out. Here we focus on the large fermion flavor case (${N}_{f}=8$) to investigate the dynamic properties across the deconfinement-to-confinement phase transition. In the deconfined phase, fermions coupled to the fluctuating gauge field to form U(1) spin liquid with continua in both spin and dimer spectral functions, and in the confined phase fermions are gapped out into valence bond solid phase with translational symmetry breaking and gapped spectra. The dynamical behaviors provide supporting evidence for the existence of the U(1) deconfined phase and could shine light on the nature of the U(1)-to-VBS phase transition which is of the ${\mathrm{QED}}_{3}$-Gross-Neveu chiral O(2) universality whose properties are still largely unknown.

arXiv:1906.06929•Physical review. B./Physical review. B•Aug 13, 2019•28 citations
Using Quantum Monte Carlo simulations, we study a series of models of fermions coupled to quantum Ising spins on a square lattice with NNN flavors of fermions per site for N=1,2N=1,2N=1,2 and 333. The models have an extensive number of conserved quantities but are not integrable, and have rather rich phase diagrams consisting of several exotic phases and phase transitions that lie beyond Landau-Ginzburg paradigm. In particular, one of the prominent phase for N>1N>1N>1 corresponds to 2N2N2N gapless Dirac fermions coupled to an emergent Z2\mathbb{Z}_2Z2​ gauge field in its deconfined phase. However, unlike a conventional Z2\mathbb{Z}_2Z2​ gauge theory, we do not impose the `Gauss's Law' by hand and instead, it emerges due to spontaneous symmetry breaking. Correspondingly, unlike a conventional Z2\mathbb{Z}_2Z2​ gauge theory in two spatial dimensions, our models have a finite temperature phase transition associated with the melting of the order parameter that dynamically imposes the Gauss's law constraint at zero temperature. By tuning a parameter, the deconfined phase undergoes a transition into a short range entangled phase, which corresponds to N'eel/Superconductor for N=2N=2N=2 and a Valence Bond Solid for N=3N=3N=3. Furthermore, for N=3N=3N=3, the Valence Bond Solid further undergoes a transition to a N'eel phase consistent with the deconfined quantum critical phenomenon studied earlier in the context of quantum magnets.
Using quantum Monte Carlo simulations, we study a series of models of fermions coupled to quantum Ising spins on a square lattice with N flavors of fermions per site for N ¼ 1, 2, and 3.The models have an extensive number of conserved quantities but are not integrable, and they have rather rich phase diagrams consisting of several exotic phases and phase transitions that lie beyond the Landau-Ginzburg paradigm.In particular, one of the prominent phases for N > 1 corresponds to 2N gapless Dirac fermions coupled to an emergent Z 2 gauge field in its deconfined phase.However, unlike a conventional Z 2 gauge theory, we do not impose "Gauss's Law" by hand; instead, it emerges because of spontaneous symmetry breaking.Correspondingly, unlike a conventional Z 2 gauge theory in two spatial dimensions, our models have a finite-temperature phase transition associated with the melting of the order parameter that dynamically imposes the Gauss's law constraint at zero temperature.By tuning a parameter, the deconfined phase undergoes a transition into a short-range entangled phase, which corresponds to Néel antiferromagnet or superconductor for N ¼ 2 and a valence-bond solid for N ¼ 3. Furthermore, for N ¼ 3, the valence-bond solid further undergoes a transition to a Néel phase consistent with the deconfined quantum critical phenomenon studied earlier in the context of quantum magnets.
The interplay between lattice gauge theories and fermionic matter accounts for fundamental physical phenomena ranging from the deconfinement of quarks in particle physics to quantum spin liquid with fractionalized anyons and emergent gauge structures in condensed matter physics.However, except for certain limits (for instance, a large number of flavors of matter fields), analytical methods can provide few concrete results.Here we show that the problem of compact Uð1Þ lattice gauge theory coupled to fermionic matter in ð2 þ 1ÞD is possible to access via sign-problem-free quantum Monte Carlo simulations.One can hence map out the phase diagram as a function of fermion flavors and the strength of gauge fluctuations.By increasing the coupling constant of the gauge field, gauge confinement in the form of various spontaneous-symmetrybreaking phases such as the valence-bond solid (VBS) and Néel antiferromagnet emerge.Deconfined phases with algebraic spin and VBS correlation functions are also observed.Such deconfined phases are incarnations of exotic states of matter, i.e., the algebraic spin liquid, which is generally viewed as the parent state of various quantum phases.The phase transitions between the deconfined and confined phases, as well as that between the different confined phases provide various manifestations of deconfined quantum criticality.In particular, for four flavors N f ¼ 4, our data suggest a continuous quantum phase transition between the VBS and Néel order.We also provide preliminary theoretical analysis for these quantum phase transitions.
The theory for the vanishing of N\'eel order in the spin S=1/2S=1/2S=1/2 square lattice antiferromagnet has been the focus of attention for many decades. A consensus appears to have emerged in recent numerical studies on the antiferromagnet with first and second neighbor exchange interactions (the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model): A gapless spin liquid is present for a narrow window of parameters between the vanishing of the N\'eel order and the onset of a gapped valence bond solid state. We propose a deconfined critical SU(2) gauge theory for a transition into a stable Z2{\mathbb{Z}}_{2}Z2​ spin liquid with massless Dirac spinon excitations; on the other side of the critical point, the SU(2) spin liquid (the `$\ensuremath{\pi}$-flux' phase) is presumed to be unstable to confinement to the N\'eel phase. We identify a dangerously irrelevant coupling in the critical SU(2) gauge theory, which contributes a logarithm-squared renormalization. This critical theory is also not Lorentz invariant and weakly breaks the SO(5) symmetry which rotates between the N\'eel and valence bond solid order parameters. We also propose a distinct deconfined critical U(1) gauge theory for a transition into the same gapless Z2{\mathbb{Z}}_{2}Z2​ spin liquid; on the other side of the critical point, the U(1) spin liquid (the `staggered flux' phase) is presumed to be unstable to confinement to the valence bond solid. This critical theory has no dangerously irrelevant coupling, dynamic critical exponent $z\ensuremath{\ne}1$, and no SO(5) symmetry. All of these phases and critical points are unified in a SU(2) gauge theory with Higgs fields and fermionic spinons which can naturally realize the observed sequence of phases with increasing J2/J1{J}_{2}/{J}_{1}J2​/J1​: N\'eel, gapless Z2{\mathbb{Z}}_{2}Z2​ spin liquid, and valence bond solid.
Deconfined Thermal Phase Transitions with Z2Z_2Z2​ Gauge Structures
Fathoming deconfined phases is one of the key issues in modern condensed matter. Striking many-body effects including massive quantum entanglement and coherence may be realized as manifested in quantum spin liquids and topological orders. Here, we demonstrate that deconfined phases even host exotic thermal phase transitions, dubbed deconfined thermal transitions. Constructing a Z2Z_2Z2​ lattice gauge model with strong interactions between Z2Z_2Z2​ gauge fluxes, we prove the existence of a thermal phase transition between deconfined and confined phases in two spatial dimensions in sharp contrast to its absence in the Wegner model. Incorporating deconfined fermions, it is shown that gapless excitations from Fermi surfaces endow line-tension to Z2Z_2Z2​ gauge fluxes at zero temperature, and we argue that a deconfined thermal transition with deconfined fermions may be interpreted as a hidden order transition with thermal gap-opening in Fermi surfaces. Moreover, it is shown that symmetry breaking transitions in deconfined phases may be unconventional. Global Z2Z_2Z2​ and U(1)U(1)U(1) symmetry breaking transitions in deconfined phases may be in the same universality class, which is impossible under the conventional Landau-Ginzburg-Wilson paradigm. Characteristic signatures of the transitions in experiments and candidate strongly correlated systems such as Kitaev materials are also discussed.
We perform a numerical study of a spin-1/2 model with ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry in one dimension which demonstrates an interesting similarity to the physics of two-dimensional deconfined quantum critical points (DQCP). Specifically, we investigate the quantum phase transition between Ising ferromagnetic and valence bond solid (VBS) symmetry-breaking phases. Working directly in the thermodynamic limit using uniform matrix product states, we find evidence for a direct continuous phase transition that lies outside of the Landau-Ginzburg-Wilson paradigm. In our model, the continuous transition is found everywhere on the phase boundary. We find that the magnetic and VBS correlations show very close power-law exponents, which is expected from the self-duality of the parton description of this DQCP. Critical exponents vary continuously along the phase boundary in a manner consistent with the predictions of the field theory for this transition. We also find a regime where the phase boundary splits, as suggested by the theory, introducing an intermediate phase of coexisting ferromagnetic and VBS order parameters. Interestingly, we discover a transition involving this coexistence phase which is similar to the DQCP, being also disallowed by the Landau-Ginzburg-Wilson symmetry-breaking theory.
Thanks to the development in quantum Monte Carlo technique, the compact U(1) lattice gauge theory coupled to fermionic matter at (2+12+12+1)D is now accessible with large-scale numerical simulations, and the ground state phase diagram as a function of fermion flavor (Nf{N}_{f}Nf​) and the strength of gauge fluctuations is mapped out. Here we focus on the large fermion flavor case (Nf=8{N}_{f}=8Nf​=8) to investigate the dynamic properties across the deconfinement-to-confinement phase transition. In the deconfined phase, fermions coupled to the fluctuating gauge field to form U(1) spin liquid with continua in both spin and dimer spectral functions, and in the confined phase fermions are gapped out into valence bond solid phase with translational symmetry breaking and gapped spectra. The dynamical behaviors provide supporting evidence for the existence of the U(1) deconfined phase and could shine light on the nature of the U(1)-to-VBS phase transition which is of the QED3{\mathrm{QED}}_{3}QED3​-Gross-Neveu chiral O(2) universality whose properties are still largely unknown.