Absence Of Spontaneous Symmetry Breaking Research Articles

Lattice Homotopy Constraints on Phases of Quantum Magnets.

The Lieb-Schultz-Mattis (LSM) theorem and its extensions forbid trivial phases from arising in certain quantum magnets. Constraining infrared behavior with the ultraviolet data encoded in the microscopic lattice of spins, these theorems tie the absence of spontaneous symmetry breaking to the emergence of exotic phases like quantum spin liquids. In this work, we take a new topological perspective on these theorems, by arguing they originate from an obstruction to "trivializing" the lattice under smooth, symmetric deformations, which we call the "lattice homotopy problem." We conjecture that all LSM-like theorems for quantum magnets (many previously unknown) can be understood from lattice homotopy, which automatically incorporates the full spatial symmetry group of the lattice, including all its point-group symmetries. One consequence is that any spin-symmetric magnet with a half-integer moment on a site with even-order rotational symmetry must be a spin liquid. To substantiate the claim, we prove the conjecture in two dimensions for some physically relevant settings.

Anatomy of quantum critical wave functions in dissipative impurity problems

Quantum phase transitions reflect singular changes taking place in a many-body ground state, however, computing and analyzing large-scale critical wave functions constitutes a formidable challenge. New physical insights into the sub-Ohmic spin-boson model are provided by the coherent state expansion (CSE), which represents the wave function by a linear combination of classically displaced configurations. We find that the distribution of low-energy displacements displays an emergent symmetry in the absence of spontaneous symmetry breaking, while experiencing strong fluctuations of the order parameter near the quantum critical point. Quantum criticality provides two strong fingerprints in critical low-energy modes: an algebraic decay of the average displacement and a constant universal average squeezing amplitude. These observations, confirmed by extensive variational matrix product states (VMPS) simulations and field theory arguments, offer precious clues into the microscopics of critical many-body states in quantum impurity models.

Spontaneous symmetry breaking and optimization of functional renormalization group

The requirement for the absence of spontaneous symmetry breaking in the d=1 dimension has been used to optimize the regulator dependence of functional renormalization group equations in the framework of the sine-Gordon scalar field theory. Results obtained by the optimization of this kind were compared to those of the Litim-Pawlowski and the principle of minimal sensitivity optimization scenarios. The optimal parameters of the compactly supported smooth (CSS) regulator, which recovers all major types of regulators in appropriate limits, have been determined beyond the local potential approximation, and the Litim limit of the CSS was found to be the optimal choice.

A quantum Mermin–Wagner theorem for quantum rotators on two-dimensional graphs

This is the first of a series of papers considering symmetry properties of quantum systems over 2D graphs or manifolds, with continuous spins, in the spirit of the Mermin–Wagner theorem [N. D. Mermin and H. Wagner, “Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models,” Phys. Rev. Lett. 17, 1133–1136 (1966)]10.1103/PhysRevLett.17.1133. In the model considered here (quantum rotators), the phase space of a single spin is a d-dimensional torus M, and spins (or particles) are attached to sites of a graph \documentclass[12pt]{minimal}\begin{document}$(\Gamma ,\mathcal {E})$\end{document}(Γ,E) satisfying a special bi-dimensionality property. The kinetic energy part of the Hamiltonian is minus a half of the Laplace operator −Δ/2 on M. We assume that the interaction potential is C2-smooth and invariant under the action of a connected Lie group \documentclass[12pt]{minimal}\begin{document}${\tt G}$\end{document}G (i.e., a Euclidean space \documentclass[12pt]{minimal}\begin{document}${\mathbb {R}}^{d^{\prime }}$\end{document}Rd′ or a torus M′ of dimension d′ ⩽ d) on M preserving the flat Riemannian metric. A part of our approach is to give a definition (and a construction) of a class of infinite-volume Gibbs states for the systems under consideration (the class \documentclass[12pt]{minimal}\begin{document}$\mathfrak {G}$\end{document}G). This class contains the so-called limit Gibbs states, with or without boundary conditions. We use ideas and techniques originated from papers [R. L. Dobrushin and S. B. Shlosman, “Absence of breakdown of continuous symmetry in two-dimensional models of statistical physics,” Commun. Math. Phys. 42, 31–40 (1975)10.1007/BF01609432; C.-E. Pfister, “On the symmetry of the Gibbs states in two-dimensional lattice systems,” Commun. Math. Phys. 79, 181–188 (1981)10.1007/BF01942060; J. Fröhlich and C. Pfister, “On the absence of spontaneous symmetry breaking and of crystalline ordering in two-dimensional systems,” Commun. Math. Phys. 81, 277–298 (1981)10.1007/BF01208901; B. Simon and A. Sokal, “Rigorous entropy-energy arguments,” J. Stat. Phys. 25, 679–694 (1981)10.1007/BF01022362; D. Ioffe, S. Shlosman and Y. Velenik, “2D models of statistical physics with continuous symmetry: The case of singular interactions,” Commun. Math. Phys. 226, 433–454 (2002)]10.1007/s002200200627 in combination with the Feynman–Kac representation, to prove that any state lying in the class \documentclass[12pt]{minimal}\begin{document}$\mathfrak {G}$\end{document}G (defined in the text) is \documentclass[12pt]{minimal}\begin{document}${\tt G}$\end{document}G-invariant. An example is given where the interaction potential is singular and there exists a Gibbs state which is not \documentclass[12pt]{minimal}\begin{document}${\tt G}$\end{document}G-invariant. In the next paper, under the same title we establish a similar result for a bosonic model where particles can jump from a vertex i ∈ Γ to one of its neighbors (a generalized Hubbard model).

Local recoil of extended solitons: a string theory example

It is well-known that localized topological defects (solitons) experience recoil when they suffer an impact by incident particles. Higher-dimensional topological defects develop distinctive wave patterns propagating along their worldvolume under similar circumstances. For 1-dimensional topological defects (vortex lines), these wave patterns fail to decay in the asymptotic future: the propagating wave eventually displaces the vortex line a finite distance away from its original position (the distance is proportional to the transferred momentum). The quantum version of this phenomenon, which we call ``local recoil'', can be seen as a simple geometric manifestation of the absence of spontaneous symmetry breaking in 1+1 dimensions. Analogously to soliton recoil, local recoil of vortex lines is associated with infrared divergences in perturbative expansions. In perturbative string theory, such divergences appear in amplitudes for closed strings scattering off a static D1-brane. Through a Dirac-Born-Infeld analysis, it is possible to resum these divergences in a way that yields finite, momentum-conserving amplitudes.

Absence of spontaneous symmetry breaking in the ground state of one-dimensional spin-orbital model

Combining the Shastry inequality and infrared bound based on reflection positivity, we prove the absence of spontaneous symmetry breaking associated with antiferro-spin, antiferro-orbital, and ferro-spin-orbital ordering in the ground state of the one-dimensional spin-orbital model in the region satisfying reflection positivity.

Absence of spontaneous symmetry breaking in the ground state of one-dimensional spin-orbital model

Combining the Shastry inequality and infrared bound based on reflection positivity, we prove the absence of spontaneous symmetry breaking associated with antiferro-spin, antiferro-orbital, and ferro-spin-orbital ordering in the ground state of the one-dimensional spin-orbital model in the region satisfying reflection positivity.

Quantum dynamics ofΦ4field theory beyond leading order in1+1dimensions

We consider the out-of-equilibrium evolution of a classical condensate field $\ensuremath{\varphi}=〈\ensuremath{\Phi}〉$ and its quantum fluctuations for a ${\ensuremath{\Phi}}^{4}$ model in $1+1$ dimensions with a double well potential. We use the two-particle point-irreducible (2PPI) formalism in the two-loop approximation. We compare our results to those obtained in the Hartree approximation, in the bare vertex approximation, and in the two-particle irreducible next-to-leading order large-$N$ approach, with thermal initial conditions. In the 2PPI scheme, we find that the system tends to the symmetric configuration at late times, as expected in the absence of spontaneous symmetry breaking.

One-point functions in integrable quantum field theory at finite temperature

We determine the form factor expansion of the one-point functions in integrable quantum field theory at finite temperature and find that it is simpler than previously conjectured. We show that no singularities are left in the final expression provided that the operator is local with respect to the particles and argue that the divergences arising in the non-local case are related to the absence of spontaneous symmetry breaking on the cylinder. As a specific application, we give the first terms of the low temperature expansion of the one-point functions for the Ising model in a magnetic field.

Absence of spontaneous symmetry breaking on fractal lattices with d̃⩽2

We generalize the Mermin-Wagner theorem proving that classical O( n) and quantum Heisenberg ferromagnetic models with nearest neighbors interactions cannot have spontaneous magnetization at any finite temperature on fractal and disordered lattices whose spectral dimension is less or equal to 2, due to the infrared divergencies of the Gaussian models defined on them.

Grivov copies and absence of spontaneous symmetry breaking in compact U(1) lattice Higgs models

We consider abelian Higgs models with compact gauge group U(1) and gauge fixing term (1/α) σx cos ϖμAμ. As already pointed out by one of us (F.N. in ref. [8]), the gauge fixed action still allows for Gribov copies due to its invariance under a certain subgroup Grib of gauge transformations, containing in particular the global ones. We show that for α large enough Grib is unbroken and hence as well as 〈φxφy〉 (for x ≠ y) are zero. In addition in four dimensions a spin-wave arguemet implies that 〈φ> = 0〉 for any α>0.

On the absence of spin-flip symmetry breaking in long-range random systems

Using a correlation inequality we prove the absence of spontaneous symmetry breaking in one-dimensional long range random ssstems with potentials which are not necessarily square integrable.

Does the quenched reduced U(∞) chiral model break spontaneously in weak coupling?

The U( N) chiral model, when quenched using Parisi's rule, has a [U(1) × U(1)] N /U(1) global invariance. To determine whether this symmetry breaks spontaneously in weak coupling for N=∞, a one-loop calculation of the distribution of eigenvalues of the single U( N) matrix of the model is performed. This distribution is shown to be uniform on the unit circle and hence, no symmetry breaking occurs. Further, the order parameter | tr U| 2/ N 2, which should be zero at N=∞ in the absence of spontaneous symmetry breaking, is evaluated in the weak coupling phase for one, two and three dimensions for N varying from 2 to 50 by Monte Carlo simulation of the quenched model. The data indicate that this parameter indeed goes to zero as N→∞ implying that the symmetry does not break.

On the absence of spontaneous symmetry breaking and of crystalline ordering in two-dimensional systems

We develop a unified approach, based on Araki's relative entropy concept, to proving absence of spontaneous breaking of continuous, internal symmetries and translation invariance in two-dimensional statistical-mechanical systems. More precisely, we show that, under rather general assumptions on the interactions, all equilibrium states of a two-dimensional system have all the symmetries, compact internal and spatial, of the dynamics, except possibly rotation invariance. (Rotation invariance remains unbroken if connected correlations decay more rapidly than the inverse square distance.) We also prove that two-dimensional systems with a non-compact internal symmetry group, like anharmonic crystals, typically do not have Gibbs states.

Comments on the absence of spontaneous symmetry breaking in low dimensions

We present some comments intended to bring out simply and more explicitly the physics behind the rigorously proven results on the absence of continuous symmetry breaking in (1 + 1) dimensions at $T=0$ and (2 + 1) dimensions for $T>0$. It is also pointed out within a model that a massless boson is present, in a sense defined in the text, even though symmetry breaking disappears.